GB2302245A

GB2302245A - Wavelet transform filter

Info

Publication number: GB2302245A
Application number: GB9618683A
Authority: GB
Inventors: Ahmad Zandi; James D Allen; Edward L Schwartz; Martin Boliek
Original assignee: Ricoh Co Ltd
Current assignee: Ricoh Co Ltd
Priority date: 1994-09-20
Filing date: 1995-09-07
Publication date: 1997-01-08
Anticipated expiration: 2015-09-07
Also published as: GB9622826D0; GB2305829B; GB2302244B; GB2305829A; GB9618630D0; GB2302244A; GB2302245B; GB9618683D0

Description

0 # t WAVELET TRANSFORM FILTER 2302245 The present invention relates to

the field of data compression and decompression systems; particularly, the present invention relates to wavelet transform filters for use in lossless and lossy encoding and decoding of data in compression/decompression systems.

a 1 Data compression is an extremely useful tool for storing and 0 transmitting large amounts of data. For example, the time required to transmit an image, such as a facsimile transmission of a document, is reduced drastically when compression is used to decrease the number of bits required to recreate the image.

Many different data compression techniques exist in the Frior art. Compression techniques can be divided into two broad categories, loss-sy coding and lossless coding. Lossy coding involves coding that results in, the loss of information, such that there is no guarantee of perfect reconstruction of the original data. The goal of lossy compression is that changes to the original data are done in such a way that they are not objectionable or detectable. In lossless compression, all the Information Js retaIned and the data is compressed in a manner which allows for perfect reconstruction.

In lossless compression, input symbols or intensity data are converted to output codewords. The input may include image, audio, 0 one-di.mensional (e.g., data changing spatially or ternporally), two- dimensional (e.g., data changing in two spatial directions (or one spatial e- and one temporal dimension)), or multi-dimensional/multi-spectraI data. If the compression is successful, the codewords are represented in fewer bits than the number of bits required for the uncoded input symbols (or intensity data). Lossless coding methods include dictionary methods of coding (e.g., Lempel-Ziv), run length encoding, enumerative coding and entropy coding. In lossless image compression, compression is based on predictions or contexts, plus coding. The JBIG standard for facsimile compression and DPCM (differential pulse. code modulation - an option in the JPEG standard) for continuous-tone images are -examples of lossless compression for images. In lossy compression, input symbols or intensity data are quantized prior to conversion to output codewords. Quantization is intended to preserve relevant characteristics of the data while eliminating unimportant characteristics. Prior to quantization, lossy compression system often use a- transform to provide energy compaction.

JPEG is an example of a lossy coding method for image data.

Recent developments in image signal processing continue to focus attention on a need for efficient and accurate forms of data compression coding. Various forms of transform or pyramidal signal processing have been proposed, including multiresolution pyramidal processing and wavelet pyramidal processing. These forms are also referred to as subband processing and hierarchical processing. Wavelet pyramidal processing of image data is a specific type of multi-resolution pyramidal processing that may use'quadrature mirror filters (QN11s) to produce subband decomposition of an original image. Note that other types of non-QN1F wavelets exist. For more information on wavelet processing, see Antonind, M., et al., "Image Coding Using Wavelet Transform", JEU Transactions on Image Processing Vol. 1, No. 2, April 1992; Shapiro, "An Embedded Hierard-dcal Image Coder Using Zerotrees of Wavelet Coefficients", Proc. IEEE Data Compression Conference, pgs. 214-223,1993.

One problem associated with much of prior art wavelet processing is that a large memory is required to store all of the data ivbdle it is being processed. In other words, in performing wavelet processing, all of the data must be examined before encoding is performed on the data. In such a case, there is no data output until at least one full pass has been made through all of the data. In fact, wavelet processing typically involves multiple passes through the data. Because of this, a large memory is often required. It is desirable to utilize wavelet processing, while avoiding the requirement of a large memory. Fur-thermore, it is desirable to perform wavelet processing using only a single pass through the data.

Many wavelet or subband transform implementations require filters in a particular canonical form. For example, low and high-pass filters must be the same length, the surn of the squares of the coefficients must be one, the ldgh-pass filter must be the time and frequency reverse of the low-pass filter, etc. (See U.S. Patent No. 5,014, 134 issued May 1991 to Lawton et al.). It is desirable to allow a wider class of filters. That is, it is desirable to provide wavelet or subband transform implementations that use low and high-pass filters that are not the sarrie- length, the sum of the squares of the coefficients need not be one, the high-pass filter need not be the time'and frequency reverse of the low-pass filter, etc.

1 The present invention provides a wavelet transform filter for filtering an input data signal comprising: a first adder for adding a first pair of samples of the input data signal to generate a first result; a first output logic for outputting a first low-pass coefficient based on the first result; a first subtracter having inputs of a low-passed coefficient and the first low-pass coefficient, such that the low-passed coefficient is subtracted from the first low-pass coefficient to produce a second result, wherein the lowpassed coefficient is received by the first subtracter as a feedback; a second subtracter for subtracting a second pair of samples of the input data signal from each other to produce a third result; a second adder coupled to receive and add the second result and an input based on the third result to produce a fourth result; a second output logic coupled to generate a second low- pass coefficient based on the fourth result, such that the filter outputs the first and second low-pass coefficients.

The present invention is illustrated by way of example, and not by way of limitation, in thefigures of the accompanying drawings in which like reference numerals refer to similar elements and in which:

Fig-ure 1A is a block diagram of the encoding portion of a coding system which employs the wavelet filter of the 6 present invention; Figure IB is a block diagram of the bit significance embedding of the system of Figure 1A; Figure 2A is a block diagram of a wavelet analysis/synthesis system; Figure 2B illustrates forward and reverse representations of transform systems for filtering with nonoverlapped minimal length reversible filters; Figure 3A-D illustrate results of performing a four level decomposition; Figure 4A is a block diagram of a three-level pyramidal transform; Figure 4B is a block diagram of a two-dimensional, two level transform; is Figure 4C is a block diagram illustrating one dimensional filters performing a multi-resolution decompression; Figure 4D is a block diagram of a system using reversible wavelets; Figure 4E are block diagrams of an enhancement and analysis system; Figure 5 illustrates a tree structure on wavelets coefficients; Figure 6A and 6A (continued) is a flow chart of a single list zerotree modelling for encoding; Figure 6B and 6B (continued) is a flow chart of a single list zerotree modelling for encoding using reduced 1 7 flag memory; Figure 6C is a flow chart of a single list zerotree modelling for decoding; Figure 6D is a flow charge of a single list zerotree 5 modelling for decoding using reduced flag memory; Figure 7A is a flow chart of a horizon modelling for encoding; Figure 7B is a flow charge of a horizon modelling for encoding using reduced flag memory; Figure 7C is a flow charge of a horizon modelling for decoding; Figure 7D is a flow chart of a horizon modelling for decoding using reduced flag memory; Figure BA is a flow chart of a B-pass for encoding; Figure 8B is a flow chart of a B-pass for encoding using reduced flag memory; Figure 9A is a flow charge of a B-pass for decoding; Figure 9B is a flow chart of a B-pass for decoding using reduced flag memory; Figure 10 is one embodiment of the forward wavelet filter of the present invention; Figure 11 is a block diagram of one embodiment of a reverse wavelet filter; Figure 12 illustrates an image and coefficients in a line buffer for a four level pyramidal decomposition; Figure 13 is a block diagram of one example of wavelet filtering using a filter control unit, and the filter of the is 8 present invention; Figure 14 is a block diagram of another example of wavelet filtering using a filter control unit and the filter of the present invention; Figure 15 illustrates the assignment of memory banks to support horizontal and vertical accesses; Figure 16 illustrates the filter operation for a two level decomposition; Figure 17 is a block diagram of one example of a context model; Figure 18 is a block diagram of one example of a sign/magnitude unit; Figure 19 is a block diagram of one example of a magnitude memory unit; is Figure 20 is a block diagram of one example of a significance unit; Figure 21 is a block diagram of one example of a tree memory unit; Figure 22 is a block diagram of one example of coefficient shifting; Figure 23 is a block diagram of an alternative example of the significance unit using an alignment by 1.5; Figure 24 illustrates the dynamic allocation of coded data memory for one pass operation; Figures 25A and B are a flow chart of one example of an encoding process; Figures 26A and B are a flow chart of the decoding of 9 one example of the decoding process; Figures 27A and B are a flow chart of one example of the process for modelling each coefficient for both the encoding and decoding processes; Figures 28A and B are a flow chart of an alternate example of the encoding process; Figures 29A and B are a flow chart of an alternative example of a decoding process; Figures 30A and B are a flow chart of an alternate example of the process for modelling each coefficient in the encoding and decoding processes; and Figure 31 is one example of the multipliers for the frequency band used for coefficient alignment.

1 D A method and apparatus for compression and decompression is described. In the following detailed descriptiog numerous specific details are set forth, such as types of coders, numbers of bits, signal names, etc., in order to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that the present invention may be practiced without these specific details. In other instances, well-known structures and devices are shown in block diagram form, rather than in detail, in order to avoid obscur, _ng 10 the present invention.

Some portions of the detailed descriptions which follow are presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algoriturnilc

0 descriptions and representations are the means used by those skiLled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or mag-netic signals capable of being stored, transfer-red, combined, compared, and otherwise manipulated. It has proven conver-dent at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.

It should be bome in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely conver-dent labels applied to these quantities. Unless specifically - 11 stated otherwise as apparent from the following discussions, it is appreciated that throughout this description, discussions utilizing terms such as "processing" or "computing" or calculating" or "determiningn or ndisplaying11 or the like, refer to the action and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system's registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices.

The apparatus for performing the operations herein may be specially constructed for the required purposes, or it may comprise a general purpose computer selectively activated or reconfigured by a computer program stored in the computer. The algorithms and displays presented herein are not inherently related to any particular computer or other apparatus. Various general purpose machines may be used with programs in accordance with the teachings herein, or it may prove convenient to construct more specialized apparatus to perform the required method steps. The required structure for a variety of these machines willappear from the description below. In addition, the description is not made with reference to any particular programming language. it will be appreciated that a variety of programming languages may be used to implement the teachings described herein.

Figure 1A is a block diagram of one example of an encoding portion of the system. Note the decoding portion of the system operates in reverse order, along with the data flow. Referring to Figure 1A, an input image data 101 is received by wavelet transform block 102. The output of wavelet transform block 102 is coupled to a bit-significance embedding block 103. In response to the output from wavelet transform block 102, the bitsignificance embedding block 103 produces at least one bit stream that is received by a n entropy coder 104. In response to the input from bitsignificance embedding 103, entropy coder 104 produces a code stream 107.

In one example, the bit significance embedding block 103 comprises a sign magnitude formatting unit 109, a frequency-based context 1 model 105 and a joint space/frequency context model 106, such as shown in Figure 1B. In one example, the joint space/frequency context model 106 comprises a horizon context model. In some examples, frequency-based context model block 105 comprises a zerot-ree model. In another embodiment, frequency-based context model 105 comprises significance tree model. The sign-magnitude unit 109, f-,equency-based context model 105 and the joint space/frequency USF) con.ext model 106 perform bit-sig:,nificance embedding The input 0 of the sign magnitude unit 109 is coupled to the output of tle N-,-avelet transform coding block 102. The output of sign map. itaide unit 109 is coupled to a switch 108. Switch 108 is coupled to provicie the output of the sign magnitude unit 109 to an input of either frequency-based modeling block 105 or joint space/frequency modeling block 106. The output of fre-uency-based coding block 105 and horizon order codi-ic, block 1,36 are coupled to the input of entropy coder 104. Entropy coder 104 produces the output code stream 107.

Referring back to Figure 1A the imaoe data 101 is received and transform coded using revensible wavelets in wavelet transform-block 102 as defiled later below, to produce a series of coefficients representin g a multi-resolution decomposition of the image. These coefficients are received by the bit-signifi-cance embedding 103.

The bit significant embedding 103 orders and converts the coefficients into sign-magnitude format and, based on their significall,' (as described below later), the formatted coefficients are subjected to a combination of different embedded modeling methods.

The formatted coefficients are subjected to either one of tivo embedded modelling methods (e.g. frequency-based modelling and JSF modelling).

In one example, the formatted coefficients are subjected to either frequency-based modelling or joint spatial/frequency modelling. When the input data comprises image data having multiple bitplanes, a number of bitplanes are encoded with frequency-based modelling, while the remaining bitplanes are encoded with JSF modelling. The decision as to which method to use on which bitplanes may be a user parameter. In one example, the high-order bitplanes of the coefficients are ordered and coded with the frequencybased modelling of the present invention. in the frequencybased context model method, the prediction of significance of the coefficient bits is related to the pyramidal structure of the wavelet. The low-order coefficient bitplanes are ordered and encoded with the joint space/frequency context model of the present invention. The JSF modelling, for example horizon modelling, provides advantages over the frequencybased encoding for bitplanes that are less correlated with respect to the frequency domain coefficient relations.

The results of bit-significance embedding is decisions (or symbols) to be coded by the entropy coder. In one example, all decisions are sent to a single coder. In another example, decisions are labelled by significance, and decisions for each significance level are processed by different (physical or virtual) multiple coders.

The bit streams resulting from the frequency-based context model block 105 and JSF context model block 106 are encoded in order of significance using entropy coder 104.

an example, entropy coder 104 comprises a binary entropy coder. In one example, entropy coder 104 comprises a Q-coder, a B-coder defined in U.S. Patent No. 5,272,478, or a coder such as described in U.S. patent application serial no. 08/016,035, entitled "Method and Apparatus for Parallel Decoding and Encoding of Data", filed February 10, 1993. more information on the Q-coder, see Pennebaker, W.B., et al., "An Overview of the Basic Principles of the Q-coder Adaptive Binary Arithmetic," IBM Journal of Research and Development, Vol. 32, pg. 717-26, 1988. In one example, a single coder produces a single output code stream. In another example, multiple (physical or virtual) coders produce multiple (physical or virtual) data streams. Wavelet Decomposition The described system initially performs decomposition of an image (in the form of image data) or another data signal using reversible wavelets. A reversible wavelet transform comprises an implementation of an exactreconstruction system in integer arithmetic, such that a signal with integer coefficients can be losslessly recovered. By using reversible wavelets, it is possible to provide lossless compression with finite precision arithmetic. The results generated by applying the reversible wavelet transform to the image data are a series of coefficients. In one example the reversible wavelet transform is implemented using a set of filters. In one example, the filters are a two-tap low-pass filter and a six-tap high-pass filter. The present invention provides such filters, which may be implemented using only addition and subtraction operations (plus hardwired bit shifting).

Also, the high-pass filter generates its output using the results; of the low-pass filter. The resulting high-pass coeffic"Lents are only a few bits g-reater than the pixel resolution and the low-pass coefficients are the same as the pixel resolution. Because only the low-pass coefficients are repeatedly filtered in a pyramidal decomposition, resolution is not increased in multi-level decompositions.

A wavelet transform system is defined by a pair of FIR analysis filters ho W, hl W, and a pair of FIR synthesis filters go(n), gl(W.

ho and go are the low-pass filters and hl and gj are the high-pass filter5. A block diagram of the wavelet system is shown in Figure 2.A. Referring to Figure 2.A, for an input signal, x(n), the analysis filters ho and hl are applied and the outputs are decL.,nated by 2 (critically subsarnpled) to generate the trarLsfo=ed signals yo(?i) and yj I(n), referred to herein as low-passed and high-passed coefficients respectively. The analysis filters a-nd their corresponding decimation, or subsarnpling, blockIs forrn the analysis portion of the wavelet transform system. The coder/decoder contain all the processing logic and routines performed Lri the transformed domain (e.g., prediction, quantization, coding, etc.). The -hesis portion in wavelet systern sho,.%,n in Figure 2-A also includes a synt which the tran-sformed signals are upsampled by 2 (e.g., a zero is inserted after every te=) and then passed through synthesis filters, go(n) and EI(n). The low-passed coefficients yo(n) are passed -through the low-pass synthesis filter go and the high-passed y1(n) are passed through the I-Lighpassed filter gi. The output of filters go(n) and gl(n) are combined to produce (n)- While downsampling and upsampling are performed in some embodiments, in other embodiments, filters are used such that computations which are unneeded due to downsampling and upsampling are not performed.

The wavelet system may be described in terms of the Z-transform, %Yhere X(Z), (Z) are the input and output signals respectively, YO(Z), YI(Z) are the low-passed and high-passed transformed signals, Ho(Z), Hj(Z) t-he low-pass and the high-pass analysis filters and finally C-(Z), GI(Z) are the low-pass and the high-pass synthesis filters. If there is no alteration or uantizatlon in the transform domain, the output (Z) in Fic,,,--,re 2, is given by X (Z) H.(Z)G,(Z)+H,(Z)G,(Z)]X(Z)+ H,, (- Z) C (Z) +H, (- Z) G, (Z)] X (- Z).

2 The second term of X(Z), referred to as the aliasing" term, is canceled because the synthesis filters are defined to be the quadrature mirror of the analysis filters, i.e., GO (Z) = H, (-Z) { CI(Z) = -HO(-Z) In terms of the filter cocificients, (n) = (-1)n h 1 (n) 91 (n) = -(-1)n h 0 (n) Therefore, for a quadrature mirror filter pairs, after substit ution, the output is:

(z) = 1 [HO (Z)H 1 (-Z) -H, (Z)HO (-Z)]X(Z). 2 Thus, in the quadrature mirror system, the output is defined in terms of the analysis filters only. The wavelet transform is applied recursively to the transformed signals in that the outputs generated by the filters are used as inputs, directly or indirectly, into the filters. In the described example, only the low-passed transformed component yo(n) is recursively transformed such that the systern is pyramidal. An example of such a pyramidal system is sho,,vn in Figure 4A.

The Z transform is a convenient notation for expressing the operation of hardware and/or soffivare on data. Multiplication by Z-m models a rn clock cycle delay in hardware, and an array access to t.1-e -nth previous element in software. Such hardware implementations include memory, pipestages, shifters, registers, etc.

The signals, x(n) and 'R(n), are identical up to a multiplicative COnStant and a delay term, i.e. in terms of the Ztransform, X(Z) = cz--X(Z).

This is called an exact reconstruction system. Thus, the wavelet transform initially applied to the inpu.. data is exactly reconstructable.

One example using the Hadamard Transform is an exact reconstruction system, which in normalized form has the following representation in the Z-dornain:

1 19 HO(Z) -L(1+Z-1) V-2 H, (Z) (1-Z-1) 2 2 After substitution, the output (Z) Z-1X(z), which is clearly an exact-reconstruction. For more information on the Hadarnard Transform, see Anil K. Jain, Fundamentals of Image Processing, pg. 155.

A reversible version of the Hadamard Transform is referred to herein as the S-transform. For more information on S-transform, see Said, A. and Pearlman, W. "Reversible Image Compression via Multiresolution Representation and Predictive Coding," Dept. of Electrical, Computer and Systems Engineering, Renssealaer Polytechnic Institute, Troy, NY 1993. Since the Hadamard Transform is an exact reconstruction transform, the following unnormalized version (whic-h cliffers from the Hadamard Transform by constant factors) is also an exact reconstruction transform:

ho(Z) =!(I+Z-1) 2 hj(Z) = I-Z-1 Given the samples of the input signal as xo, -ri, the S-t-ransform is a reversible implementation of this system as, yo(O) = [(x(O)+x(1))/2j Yj(O) = X(O) - XM The notation L-1 means to round down or truncate and is sometimes referred to as the floor function. Similarly, the ceiling function F.1 means round up to the nearest integer.

-C> The proof that this irnplmentation is reversible follows from the fact that the only information lost in the approximation is the least significant bit of x(O)+x(l). But since the least significant bits of x(O)+x(l) and xffl)-x(l) are identical, this can be recovered from the high-pass output yj(0). In other words, x(O) = yo(O)+[(yl(O)+1)/2] x(l) = yo(O)-[(yl(O)-1)/21 The S-transform is a non-overlapping transform using minimal length reversible filters. Minimal length filters comprise a pair of filters, where both filters have two taps. Minimal length transforms do not provide good energy compaction. Minimal length filters implement a non-overlapped transform because the length of the filters is equal to the number of filters. Overlapped transforms use at least one filter which has length greater than the number of filters. Overlapped transforms using longer (non-mirdmal length) filters can provide better energy compaction. The present inventiorl provides non-minimal length reverEible filters which permits an overlapped transform.

Another example of an exact-reconstruction system comprises the Two/Six (TS)-Transform which has the Z-domain definition, HO(Z)= 1 (1+Z-1) _2 H 1 (-1-Z-1+8Z-2-8Z-3+Z-4+Z-5) 1(Z) = g72 After substitution, the output is X(Z) = 2Z-3 X(Z), which is an exact-reconstruction transform.

The rational unnormalized version of the TS-transform comprises:

h (Z) 1>Z-1) 0 2 hl(Z) -1(-1-Z-1+8Z-2-8Z-3+Z-4+Z-5) 8 If X(O), X(1),. x(5) are six samples of the signal, then the first duee low-passed coefficients yo(O), yo(l), yo(2) and the first hI;h-passed 0 coefficient yl(O) are given by:

yo(O) = 1(x(O)+x(l))/2j y,(1) = 1(x(2)+x(3))/2j 'yo(2) = LM4)+x(5))/21 y,(0)=[(-(x(O)+x(l)))+8(x(2)-X(3))+(X(4)+x(5))/8j.

However, the straight forward implementation of the rational unnormalized version of the TS-transform is not reversible. The following example shows that the implementation is non-reversible locally. A longer sequence can be constructed as an example for the global case. Since -(x(O) + x(l)) + (x(4) + x(S)) # - yo(O) + yc,(2) because of rounding to compute yo(O) and yo(2), this transform is not reversible usLng local information.

For example, if x(O) = 1, x(l) = 1, x(2) = 3, x(3) = 1, x(4) = 1, x(5) = 1, then 2-6)- yo(O) = L(I + 1) / 2J = 1 yo(l) = L(3+1)/2j = 2 yo(2) = L(1 + 1) / 2J = 1 y, (0) = L[-(1 + 1) + 8(3 - 1) + (1+ l& 8] = L(-2 +16 + 2) / 8J = 2 and if x(O) = 1, x(l) = 2, x(2)= 4, x(3)= 1, x(4)=1,x(5)= 1, then yo(O)= 1(1 + 2) / 2J = 1 Yo(l) = 1(4 + 1) / 2J = 2 yo(2)= LG + 1) / 2] = 1 Y(o)= U-G + 2) + 8(4 - 1) + (1 + 1)]j / 8 = L(-3 + 24 + 2) / 8J = L23 / 8J = 2 Since yo(O), yo(l), yOW and yl(O) are the same for two different sets of inputs x(O)... x(S), the transform is not reversible, since given yo(O),. yl(O) it cannot be determined from this local information which of the two sets were input. (Note that it can be proved that the transform is not reversible using global information from all coefficients.) Now consider a reversible TS-transform, which is referred to herein as an RTS-transform, which provides a different 1-dgh-pass filtering operation.

If x(O), x(l), x(2), x(3), x(4), x(5) are 6 samples of the signal, then the first three low-passed coefficients yo(O), yo(l), yo(2) and the first 1-dgh-passed coefficient yl (0) are given by, YO(O) = L(X(0)+x(l))/2j YO. (1) = LN2)+X(3))12j yo(2) = 1(x(4)+X(5))/2j 1 Y,(0) = L(-L(x(O)+x(l))/2j+4(x(2)-x(3))+L(x(4)+x(S))12J)14j = L(-yo(O) + 4(x(2) X(3)) +yc,(2)) / 4J.

Since x(2) - x(3) = yi(o) - L-(yo(O) +yo(2)) / 4J then x(2)-x(3) is completely known. With Yo(l)=LM2) + x(3)) / 2] and x(2) - x(3) and x(2)-x(3) defined above, x(2) and x (3) may be recovered becaiise the 'least significant bits of and -(0)-x(l) are identical.

Specifically, let d(O) = x(2)-x(3) = y,(0)-(-yc,(0)+yz(2),14)j x(2) = yo(1) + L(d(O) + 1) / 2 1 x(3) = yo (1) + [(d(O) - 1) / 21 One example of the forward filter for the RTS-transform is shown in Appendix A implemented in the programming lianpi.R,--- "C".

0 Note that mathematically the equation: 1 (-1 - Z-1 + 8Z-' - 8Z-' + Z + Z- 5) 8 and the equation:

is 1 1 (-1 - Z-) + 4 (Z-2 _ Z-3) + 1 (Z-4 + Z-1)) 4 2 17 are the same when perfo=ed with infinite precision arithmetic. The reason the second equation represents a reversible filter is apparent when physically implemented with integer arithinetic. Exemplary hardv.-are implementat-ons of 20 the low-p-ass filter and the high-pass filter are described in conjuncton with Figures 10 and 11.

Note that in both the S-transform and the RTS-transforrn, the lowpass filter is implemented so that the range of the input signal x(n) is the same as the output signal yo(n). For example, if the signal is an 8-bit image, the output of the low-pass filter is also 8 bits. This is an important property for a pyramidal system where the low-pass filter is successively applied because in prior art systems the range of the output signal is g- reater than that of the input signal, thereby making successive applications of the filter difficult. In addition, the low-pass filter has only two taps which makes it a non-overlapping filter. This property is important for the hardware implementation, as is described below later.

In one embodiment, with reference to the RTS-transform, the lowpass filter and high-pass filter are defined as:

ho(Z) =!(I+Z-1) 4 2 hj(Z) =!(-1(1+Z-')+4(Z-2_Z-3 +I(Z-4+Z-5)) 4 2) 2 Thus, the results from the low-pass filter may be used twice (in the first a.nd third te=s) in the dgh-pass filter. Therefore, only two other additions need to be performed to arrive at the results of the high-pass filter.

Many overlapped, non-minimal length reversible filters may be used in the present invention. Such forward and inverse representations of the transfo= system for filtering with non-overlapped mir-Limal length reversible filters is shown in Figure 2B. For instance, the following class of filters may be used. For an integer L z, d(O) = x(2(LL/2j + 1)) - x(2(LL/2j + 1) + 1) and and yo (0) = L(x(O) + x(l)) / 2] Y. (1) = LM2) + x(3)) / 2J yo (L - 1) = L(x(2L(I. - 1) / 2j) + x(2L(L - 1) / 2J + 1)) / 2] is 1Ljij L-I la,yo(i)+bd(O)+ lc,yo(j) yI(O) = io k j=[L2 j+2 The length of the high-pass filter is 2L. If L is odd, the filter may be closer to a symmetric filter. If ai, b, ci and k are integers and k-<b, then the filter is reversible. If ai, b, ci, and k are powers of two (or the negative or complement of a power of two), then the implementation of the filter may be simplified. If k = b (regardless of the values of ai and ci) then the rane of the output of the high-pass filter yj is minimized. For each ai, if there is exactly one cj where ai = -cj, then the high-pass filter will have no response to a constant input. If ai = -cj when j-(L-I)= i, then the filter may be closer to a symmetric filter.

Another useful property is L 1.12 j I [(a,)(2i)' + (a,)(2i + 1)'] + (b)(20/2 + 1))" i=G -(b)(2(LI/2] + 1) + 1)n + L-I I [c,(2,)m+c,(2,+1)m]=O j=LL/2j+2 This makes the high-pass filter have no response to a linearly changing input when m=1 and a quadratically changing input when m=2, etc., where m is the moment condition. This property is the principle 1161 reason that the RTS-transform has better energy compaction than the Stransform.

While filters must meet the minimum constraints for reversibility, for different applications, filters may be used that meet none, some or all of the other properties. In some examples, one of the following example high-pass filters is used. The filters are listed in a notation that just lists the integer coefficients of the rational version of the filter, to avoid obscurity.

1 1 -4 -4 16 -16 4 4 -1 -1 1 1-3-3 8-8 3 3-1-1 3 3 -22 -22 128 -128 77 22 -3 -3 The last filter is referred to as the (T,.,;o/Te.-i) 7I-filter, and it has --'-le property that it has no response to a cubically increasing function. Note that since 22=16+2.x." and 3=2+1, this filter can be implemented with a total of seven additions and subtractions.

The strict reversibility requirements for filters can be rela-xed by noting the following. High pass coefficients are encoded and decoded in the some order. Pixel values corresponding to previously decoded lraigh pass coefficients are known exactly, so they can be used in current high pass filtering. For example, the following filf-er can be used when a raster order is used.

HI(Z)=[1 [-!(11 Z-1)]+ [1 (8(Z-2 _ Z-3) + (z-4 + z-5), 4 2 2 1 C.1-7 The use of a single fixed Sigh-pass filter is not requiree. A.daptive filters may be used or multiple filters may be used. The ciata us-:,d to adapt or select among multiple filters must be restricted to data that is available in the decoder prior to a particular inverse filtering operation.

One way to use multiple filters is to process the high-pass coefficients proUessively. Altemate high-pass filtering operatiorLs (yj(0), yj(2), yj(4), -..) may be processed first with a reversible filter such as the TZTS high-pass filter. The remaining processing (3,1(1), yj(3), y (5), - .) nay use a non-reversible filter of up to six taps, because the exact vnlues of the inputs to the overlap portion of the filter are knovn. For example, any of the following filters may be used. -1 3 -3 1 -1 4 -4 1 -3 8 -8 3 1 -5 10 -10 5 -1 1 -4 8 -8 4 -1 In some examples, the high pass filter may be replaced with a prediction /interpolation operation. A predictor/interpolator may predict the difference between a pair of inputs using any data that is available in the decoder prior to a particular prediction /interpolation operation. The predicted difference is subtracted from the actual diffarenct of the inputs, and is output. In one example, prior art prediction methods used in

DPCM, prog-ressive coding or spatial domain coding areused. Using the lowpass and dgh-pass filters, multi-resolution decomposition is performed. The number of levels of a 29 composition is variable and ma be any number; however, currently the number of decomposition levels equals from two to five levels.

For example, if the reversible wavelet transform is r.cursively applied to an image, the first level of decomposition operates on the finest detail, or resolution. At a first decomposition level, the image is decomposed into four sub-images (e.g., subbands). Each subband represents a band of spatial frequencies. The first level subbands are designated LLo, LHo, HLO and HHO. The process of decomposing the original image involves subsampling by two in both horizontal and vertical dimensions, such that the first level subbands LLO, LHO, H and HHO each have one-fourth as many coefficients as the input has pixels (or coefficients) of the image, such as shown in Figure 3A.

Subba-nd LLO contains simultaneously low frequency horizontal and low frequency vertical information. Typically a large portion of the image 0 energy is concentrated in this subband. Subband LHO contains low frequency horizontal and high frequency vertical information (e.g., horizontal edge information). Subband H-Lo contains high frequency t horizontal information and low frequency vertical information (e.g., vertical edge information). Subband HHO contains 1-dgh frequency horizontal information and high frequency vertical information (e.g., texture or diagonal edge information).

Each of the succeeding second, third a-nd fourth lower decomposition levels is produced by decomposing the low frequency LL subband of the preceding level. This subband LLO of the first level is decomposed to produce subbands LLI, LHI, HL1 and HH1 of the moderate detail second level. Similarly, subband LLI is decomposed to produce I 12-9 coarse detail subbands LLI LH, HL2 and 11H2 of the third level. Also, subband LL2 is decomposed to produce coarser detail subbands LL3, LHI HL3 and HH3 of the third level, as shown in Figure 3D. Due to subsampling by two, each second level subband is one-sixteenth the size of the original image. Each sample (e.g., pel) at ffids level represents moderate detail in the original image at the same location. Similarly, each third level subband is 1/64 the size of the original image. Each pel at this level corresponds to relatively coarse detail in the original image at the 0 same location. Also, each fourth level subband is 1/256 the size of the original image.

Since the decomposed images are physically smaller than the original image due to subsampling, the same memory used to store the original image can be used to store all of the decomposed subbands. In other words, the original image and decomposed subbands LLo and LL1 are discarded and are not stored in a three level decomposition.

A parent-&dId relationship exists between a subba-nd component representative of coarse detail relative to a corresponding subband component at the next finer detail level.

Although only four subband decomposition levels are shown, additional levels could be developed in accordance with the requirements of a particular system. Also, with other transformations such as DCT or Enearly spaced subbands, different parent-cl-dld relationships may be defined.

The process of multi-resolution decomposition may be performed using a filtering system, such as that depicted in Figure 4A. An input signal representing a one-dimensional signal with length L is low-pass z> D and high-pass filtered by filter uruts 401 and 402 before being subsampled by two via ur-dts 403 and 405. A subsaznpled output signal from unit 403 is low-pass and high-pass filtered by units 405 and 406 before being subsampled by two via units 407 and 408, respectively. Subband components L and H appear at respective outputs of units 407 and 408. Similarly, the output signal from uni 405 is low-pass and high-pass filtered by units 409 and 410 before being subsampled by unIts 411 and 412, respectively. Subband components L and H appear at respective outputs of ur-dts 411 and 412. As described above, the filters 'used in subband decomposition are digital quadrature mirror filters for splitting the horizontal and vertical freuency bands into low frequency and high frequency bands. Figure 4B illustrates a two-dimensional, two-level tra-nsform. Fig7ure 4C also illustrates a two-dimensional, two-level t-ransfo= implernented using one-dimensional filters, such as those shown in Figure 10 and 11. The one-dimensional filters are applied at every otller position, to avoid computation rendered unnecessary by subsampling. In one embodiment, one-dimensional filters share computation between low- pass and high-pass computation.

Therefore, the system for compression and decompression uses non-minimal length, overlapped reversible filters. Figure 4D 3.9 a block diagram of one example of such a system. Referring to Figure 4D, 1-derarchical decompression is initially performed. The results of the 1-derarchical decomposition are sent to a compressor for compression. The compression performed may include vector quantization, scalar quantization, zero = length counting, 1 3 Huffin.an coding, etc. The output of the compressor compresses data representing a compressed version of the original input data. A decompressor may receive the data at sometime in the future and decompress the data. The system then performs an inverse decomposition using non-minimal length, overlapped reversible filters to generate a reconstructed version of the original data.

The reversible wavelet filters may also be used n exemplary analysis and enhancement systems, st,.cIi as sho,,,n in Figure '%E. Referring to Figure 4E, hierarchical decorp.pos,.fi',-nr. is perfornned on input data using non-minimal length, overlapped reversilDle wavelet filters. The analysis unit receives the coefAllc;ents generated by the filIters and classifies them into decisions, e.g., rather Chan encoding the coefficients completely, only relavent is 0 extracted. For example, in a document archiving systern, blank pages might be recogrL-ed using only the coarsest low-pass Anclher example ivould be to only use high pass information from a particular -ween image of text and!maces of natural scenes.

subband to distinguish bet 0 The hierarchical decomposition may be used for registering multiple images, such that coarse registration is done first with coarse subbands. In anollher embodirnent, the coefficients undergo -enhancement or filtering followed by inverse decomposition. Sharpening, edge enhancements, noise control, etc. may be performed using a derarchcal decompos,".tion.

Thus, the wavelet transform can be used in joint time/space and firequency domain analysis and filterLng/erihancernent systems.

- 32 Bit-Sicrnificance En.!)edded Coding The coefficients generated as a result of the wavelet decomposition are entropy coded. The coefficients initially undergo embedded coding in which the coefficients are ordered in a visually significant order or, more generally, ordered with respect to some error metric (e.g., distortion metric). Error or distortion metrics include peak error, and means squared error (MSE). Additionally, ordering can be p'erformed to. give preference to bit-significance spatial location, relevance for data base querring, and directionally (ver-tical, horizontal, diagonal, etc.). multiple embedded coding techniques are used, wherein a portion of the coefficients at one significance level are coded with one encoding technique, while the remaining coefficients are coded with another techniques. In the present invention, frequency-based modeling and joint spatial /frequency modeling are two different embedded coding systems used to encode the coefficients generated by the wavelet transform.i Frequency-based modeling involves predicting a number of coefficients at a higher frequency that when coding a coefficient at a lower frequency. The joint space/frequency modeling takes advantage of both the known frequency bands and the neighboring pixels (or data). one example of the joint space/frequency modeling is referred to herein as horizon modeling.

The data is initially formatted in sign magnitude format, which is followed by the data being sor-ted based on significance. After the data is sorted with respect to the given significance metric, the data is encoded.

\1 1 h 3:

Both the frequency-based coding and the horizon coding may based on bitsignificance ordering, but use different methods of encoding the e%, ents.

Assuming a digital signal, x(n), for each x(n) is represented with R bits of precision, then the embedded coding encodes the most significant bit (or bits) of every x(n) of the signal, then the next significant bit (or bits) and so on, For example, in the case of visually defined ordering, an image that requires better quality in the center than along the comers or near the edges (such as some medical images) may be subjected to encoding such that the low-order bits of the central pixels might be coded prior to the higher-order bits of the boundary pixels.

For an embedded system based on a bit significance distortion measure, binary values of the data are ordered by magnitude. In the claSe where the values are non-negative integers, such as occurs with respect to the intensity of pixels, the order that may be used is the bitplane order (e. g., from the most significant to the least significant bitplane). In embodiments where two's complement negative integers are also allowed, the embedded order of the sign bit is the same as the first nonzero bit of the absolute value of the integer. Therefore, the sign bit is not considered until a non-zero bit is coded. As a result, the possible values for an event in the bit significance embedded system of the present invention is ternary before the sign bit is coded. The ternary events are 1. not significant", "positive sigrdficant" and "negative significant". For example, using sig-n magnitude notation, the 16-bit numLer -7 is:

25,. 1000000000000111 - 34 on a bit-plane basis, the first twelve decisions will be "not significantn. The first 1-bit occurs at the thirteenth decision. The thirteenth decision will be "negative significantn. After the sign bit is coded, the possible events are reduced to binary, i.e. 0, 1. The fourteenth and fifteenth decisions are both nln.

In one example, a list is used to keep track of the coefficients. In one example, a one bit flag, referred to herein as the group flag, associated with each coefficient differentiates coefficients whose sign bit has not yet been coded from the coefficients with the sign bit already coded. In another example, two or more lists can be used instead of a flat bit. In another example, a single list is used without a flag.

is In another example lists are not used. All decisions for a coefficient are generated and labelled by significance, before any decisions for the next coefficient are generated. This eliminates the need for storing all coefficients in lists.

The Encoding and Decoding Process of the Patent Invention The following flow charts, Figures 25-30, depict examples of the encoding and decoding processes.

Figure 25 is a flow chart illustrating the encoder transform and modelling process. Referring to Figure 25, the encoder transform and modelling process begins by acquiring input data (processing block 2501). After acquiring input data, a reversible wavelet filter (processing block 2502) is applied.

-1 1-, Next, a test determines ifanother level of decomposition is desired (processing block 22J03). If another level of decomposition is desired, processing continues at processing 2504 where the reversible filter is applied to the LL coefficients that resulted from the immediately pre?ious decomposition, and processing continues back at processing 2503. In this manner, any number of levels of decomposition can be performed.

If a-nother level of decomposition is not desired, processing continues at processing block 2506 where the group flag for eaL',-, coefificient - ne 10 is initialized to the A-group. After initializing the group flag, the C for the A-pass, SA, is set to the most significant biltplane (max) (processing block 2507). Next, the bitplane for the B-pass, SE, is set to he next rno-st L::ant bif-plane (r.nax-1) (processing block 2508).

s igrul, Then, a test dete.=ines whether to code the bitplane for the Apass, 15 SA, with a frequency based model (processing block 2509). If the bit-plane 1.

SA is to be coded with the f-.equency-based model, processing cont,,-,es at 0 processing block 2510 where each coefficient is modeled with the frequency-based model and entropy code. On the other hand, if bitplane SA is not to be coded with the frequency-based model, processLng continues at processing block 2-511 where each coefficient is modeled with a joint space/frequency model and entropy code- In either case, processing thereafter continues at processing block 2512 %,,,here a test determines if the bitplane SA is greater than or equal to zero, thereby indicating whether it is the la-st bitplane. Lf bitplane S. _j, is g-reater than or equal to zero, processing loops back to processing block 2309. On the other hand, if bitplane SA is not greater than or equal to zero, ----1 36 processing continues at processing block 2513 where a test determines whether the bitplane SB is greater than or equal to zero, such that the process determines if the bitplane is the last bitplane to undergo a Bpass. If bitplane S13 is greater than or equal to zero, processing continues at processing block 2509. However, if bitplane SB is not greater than or equal to zero, processing continues at processing block 2514 where coded data is either transmitted onto a channel or stored in memory. After storing or transmitting the coded data, the encoder transform and modeling process of the present invention ends.

Figure 26 illustrates a decoder transform and modelling process. Referring to Figure 26, the decoder transform and modelling process begins by retrieving coded data (processLng block 2601). The coded data may be received from a channel or memory or other transmission system. After retrieving the coded data, a group flag for each coefficient is initialized to the A-g-roup (processing block 2602). Following this initialization, the bitplane for the A-pass, SA, is set to the most significant bitplane (max) (processing block 2603) and the bitplane for the B-pass, S13, is set to the next most significant bit-plane (max -1) (processing block 2604). Then, the value of each coefficient is set to an initial value of zero (processing block 2605).

After initializing the value of each coefficient to zero, a test determines whether the bitplane SA is to be decoded with a frequencybased model or not (processing 2606). If bitplane SA is to be decoded with a frequency-based model, processing continues to processing block 2607 where each coefficient is modeled with a frequency-based model and I'. entropy decode. If the bitplane SA is not to be decoded with a frequency- I k 3,7 based model, processing continues at processing 2608 where each coefficient is modeled with a joint space/frequency model and entropy decode.

After each coefficient is modeled, processing continues at proce-m-. ng block 2609 where the bitplane SA determines if it is the last bitplane by testing if it is g-reater than or equal to zero. If the bitpla-ne SA is greater than or equal to zero, processing continues at processing block 2606. On the other ha-nd, if bitplane SA is not greater than or equal to zero, then a test determines if the B-pass bitplane SB is greater than or e,-,al to zero (processing block 2610), thereby indicating that it is the last bll.plar. e for a B pass. If so, processing continues at processing block 2606 for further decoding. On the other hand, if the bitplane for the B-pass, SB, is not greater than or equal to zero, an inverse reversible filter is appLied on the coefficients from the coarsest level of decomposition (processing block 2611). A test then determines if all the levels have been inverse filtered (processing block 2612). If not, the inverse reversible filter is applied a-an on the coefficients on the coarsest remaining level of composition (processing block 2613). Thereafter, processing continues back at processing 2612 to test once again whether all of the levels have been inverse filtered.

Once all the levels have been inverse filtered, processing continues at processing block 2612 where a store or transmission of reconstructed data occurs.

Figure 27 illustrates one example of the process for modelling each coefficient. The process depicted represents the modelling process for either the frequency-based or JSF modelling and encode or decode. That in 1 --,(z each of the four blocks (2507, 2508, 2607, 2608) may be implemented with the modeling process of Figure 27. Referring to Figure 27, an initial process begins by initially testing whether the modeling is to be performed in one pass (processing block 2701). If the modeling is not to occur in one pass, a test determines whether the bitplane SA is greater than the bitplane SB (processing block 2702). If it is not, then the process transitions to processing block 2703 where a flag (do.A_flag) is cleared to indicate that an A-pass is not to be performed. If bitplane SA is greater than the bitplane SB then processing continues at processing block 2704 where the flag (do-A-flag) is set to indicate that an A-pass is to be performed.

After either of processing blocks 2703 or 2704, processing continues at processing block 2705 where a test determines if the bitplane SB is equal to bitplane SA. If the bitplanes are not equal, the present inVention clears a flag (do3-flag) to prevent a B-pass from occuring (processing block 2705) and processing thereafter continues at processing block 2707. If the bitplane SB is equal to the bitplane SA, the doBflag flag is set to indicate that a B-pass is to be performed (processing block 2706), and processing also thereafter continues at processing block 2707.

At processing block 2707, a test determines if the A-pass flag is set and the zerotree modeling is to be performed. If the flag indicates that an A-pass is to occur and zerotree modeling is to be performed, a "determined /undetermined" flag is initialized to the "undetermined" state for each coefficient (processing block 2708), and processing continues at processing block 2709. On the other hand, if either the A- pass indication flag or the zerotree modeling indication are not set, processing continues G directly to processing block 2709'.' At processing block 2709, the first coefficient is set to the variable C.

Once the first coefficient has been assigned to the variable C, a test deterrnines if the B-pass indication flag is set (processing block 2M9), If the B-pass indication flag (do-B_flag) is set, a B-pass is performed on coefficient C (processing block 2710), and processing continues at processing block 2711. On the other hand, if the B-pass flag is not set, then a B-pass is not perfo.rmed on C, and procesing continues directy to processing block 2711.

A test then determines whether the A-pass indicatinn flag (do-A-flag) is set (processing block 2711). If the A-pass indication laOc;s set, then an A-pass is performed on coefficient C (processing block 27177).

0 Thereafter, processir g continues at processing block 2713. If the A-pass indication flag is not set, processing continues at processing block 27/13 15 without pertorming an A-pass on coefficient C.

At processing block 2713, a test determines if ciefficient C 'is e last coefficient. If coefficient C is not the last coefficient, then processLn.

0 continues at processing block 2714 where the next coefficient is assigned to the variable C and processing continues at processing block 2719. Hove, ,er, if coefficient C is the last coefficient, processing continues at processing block 2715 where a test determines if the B-pass flag (do_B_flag) is set. Ii the B pass flag is set, the bitplane 5B is set equal to the bitplane SB -1 (processing block 2716), and processirig continues at processing block 2717. If the 13-pass r indication flag is not set, processing continues at processing bloc.k. 2717. At processing block 2717, a test determines if the A-pass flag is set. If it is set, then the bitpla-ne SA is set equal to the bitplane SA-1 (processing block 2718) 0 Z+_ 0 and processing ends. Also, if th A-pass flag is not set, then processing ends immediately.

in some examples, whether a coefficient at a particular bitplate is in the A-group or the B-group can be determined without using a flag bit. This saves one bit of memory per coefficient, which may be sigrtificant for large images. Instead a masIc is compared using AND logic to a coefficient. lf the result of the AND is zero, the bit is in the A-group; otherwise, it is in the B-g-roup. An example of these masks are shown in Table 7 for 8 bitpla.nes. Note that these masks are the hvo's complement of 2(-itpla-ne-1) 10 (without the sign bit).

Table 7 - Masks Bitplane Mask (binary) 7 00000000 6 10000000 11000000 4 11100000 3 11110000 2 11111000 1 11111100 111111110 Since independent masks can be assigned to the A-pass and B-pass respectively (called herein MA and MB), as many A-passes as desired can be performed before the corresponding B-pass. in one implementation with 17 bitplanes, 3 A-passes are performed, then 14 simultaneous A-passes a-nd B- passes are performed, and finally 2 B-passes are performed. Since A-pass Z 1 decisions can typically be coded"more efficiently than B-pass decisions, performing multiple A-passes initially can improve quality for lossy compression.

Figure 28 illustrates one example of an encoder -that uses a reduced flag memory (as is described later in the Detailed Description). Referring to Figure 28, the encoder transform and modeling process begins by acquiring input data (processing block 2801).

After acquiring input data, a reversible wavelet filter (processing block 2802) is applied.

Next, a test determines if another level of decomposition is desired (processing block 2903). If another level of decomposition is desired,processing continues at processing 2804 where the reversible filter is applied to the LL coefficients that resulted from the Lm.mediately previous decompression, and processing continues back at processin.-, 2803. In thii manner, any number of levels of decomposition can be performed.

If another level of decomposition is not desired, processing continues at processing block 2805 where the bitpla-ne for the A-pass, SA, is set to the most sig-nificant bitplane (max). Next, the bitplane for the B-pass, SB, is set to the next most significant bitplane (max-1) (processng block 2S06).

Next, mask MA is set to -2(SA+ 1) (processing block 2SO-7) and mask IMB is set to -2(SB+I) (processing block 2808). Then, a test determines whether to code the bitplane for the A-pass, SA, with a frequency based model (processing block 2808). If the bitplane SA is to be coded with the frequency-based model, processing continues at processing block 2809 & -2where one bit of each coefficienf is modeled with the frequency-based model and ent-ropy code. On the other hand, if bitplane SA is not to be coded with the frequency-based model, processing continues at processing block 2810 where one bit of each coefficient is modeled with a joint 5 space/frequency model and entropy code.

In either case, processing thereafter continues at processing block 2811 where a test determines if the bitplane SA is greater than or equal to zero, thereby indicating whether it is the last bitplane. If bitplane SA is g-reater tha-n or equal to zero, processing loops back to processng block 2608. On the other hand, if bitplane SA is not greater than or equal to zero, processLng contLnues at processing block 2812 where a test deterrnines whel-her the bit-plane SB is greater t-ha-n or equal to zero, such t2hat the process determ. L-ies if the bitplane is the last bitplane to undergo a B-pass. U bit-plane SB is greater than or equal to zero, processing continues at -al processing block 2808. However, if bitplane SB is not greater than or e-. to zero, processing continues at processing block 2813 where coded data is eitlier traiLsmitted onto a channel or stored in memory. After storing or 0 -he coded data, the encoder tran5form and modelling process transmitting of the present inyention ends.

Figure 29 illustrates an alternative arrangement of the decoder transform and modelling process when using a reduced flag memory. Referring to Figure 29, the decoder transform and modelling process begins by retrieving coded data (processing block 2901). The coded data may be received from a channel or memory or other trarLsmission system. Once the coded data is received, the bitplane for the A-pass, SA, is set to the most significant bitplane (max) 1 (processing block 2903) and the bitplane for the B-pass, SB, is set to the next most significant bitplane (max -1) (processing block 2904). After initializing the value of each coefficient to zero, the value of each coefficient is set to an initial value of zero (processing, block 2905). Then, mask MB is set to -2 (SI3+1) (processing block 2902) and mask MA is set to -2(SA+l) (processing block 2915).

Then, a test determines whether the bitplane SA is to be decoded with a frequency-based model or not (processing 2906). If bitplane SA is to be decoded with a frequency-based model, processing continues to processing block 2907 where one bit of each coefficient is modeled with a frequency-based model and entropy decode. If the bitplane SA is not to be decoded with a frequency-based model, processing continues at processing 2908 where one bit of each coefficient is modeled with a joint space/frequency model and entropy decode.

After each coefficient is modeled, processing continues at processing block 2909 where the bitplane SA determines if it is the last bitplane by testing if it is greater than or equal to zero. If the bitplane SA is greater than or equal to zero, processing continues at processing bluck 2906. On the other hand, if bitplane SA isnot g-reater than or equal to zero, then a test determines if the B-pass bitplane SB is greater than or equal to zero (processing block 2910), thereby indicating that it is the last bitplane for a B pass. If so, processing continues at processing block 2902 for further decoding.. On the other hand, if the bitplane for the B-pass, SB, is not greater than or equal to zero, an inverse reversible filter is applied on the f5 coefficients from the coarsest level of decomposition (processing block 2911). A test then determines if all the levels have been inverse filtered . Lt- (processing block 2912). If not, the inverse reversible filter is applied again on the coefficients on the coarsest rernairdng level of composidon (processing block 2913). Thereafter, processing continues back at processing 2912 where a test once again whether all of the levels have been inverse filtered.

Once all the levels have been inverse filtered, processing continues at processing block 2912 where a store or tran-smission of reconstructed data occurs.

Figure 30 illustrates one example of the process for modelling each coefficient. Note that like Figure 27, the process of Figure 30 may. be used to implement the modeling steps in Figures 28 and 29. Referring to Figure 30, an inital process begins by initially testing an A-pass is desized a.r.d vhel-I-Ler SA is greater than or equal to 0 (proces-zi7,c. block 3001). If so, 0 then the flag (doA_flag) indicating that an A-pass is to be performed is set (processLng block 3004), and processing continues at processing block 3002. Otherwise, the do-A-flag flag is cleared (processing block 3003).

lf bitplane SA is g-reater than the bitpla-ne 5B then processing continues at processing block 3004 where a flag is set to Lndirate Chat an Apass is to occur. If bitplane SA is not g-reater than the bitplane SB, then processing continues at processing block 3003 where the flag indicating an A-pass is supposed to occur is clear.

After either of processing blocks 3003 or 3004, processing continues at processing block 3002 where a test determines if the bitplane 5B Ls greater than or equal to bitplane SA and if a B-pass is desired. If the Ibitplanes are not equal, a flag (do_B_flag) is cleared to prevent a B-pass from occurring (processing block 3005), and processing -T- 1 L-- is e(;ual to thereafter continues at processing block 3007. If the bitplane SB the bitplane SA, the do-B-flag flag is set to indicate that a B-pass is to be performed (processing block 3006), and processing also thereafter continues at processing block 3007.

At processing block 3007, a test determines if the A-pass flag is set and the zerotree modeling is to be performed. If the flag indicates that an 0 A-pass is to occur and zerotree modeling is to be performed, a "determined /undete-mined" flag is initialized to the "undeter-mined" state for each coefficient which has children (processing block 30CS), and processing continues at processing block 33009. On the other hand, if eit'her 0 the A-pass indication flag or the zerotree modeling indication are not set, processing ccnt-inues directly to processlng block 3009. At processing block 3C-29, the first coefficient is set to the variable C.

Once the first coefficient has been assigned to the variable C, a test determines if the B-pass indication flag is set (processing block 3019). If the B-pass indication flag (do_B_flag) is set, a B-pass on coefficient I Cis performed (processing block 3010), and processing continues at I processing block 3011. On the other hand, if the B-pass flag is not set, then a B-pass is not performed on C, axid processing continues directly to processong block 3011.

The test then determines is the A-pass indication flag has been set (processing block 3011). If the A-pass indication flag has been 6ct, then an A-pass is performed on coefficient C (processing block 3017). ihereatter, processing continues at processing block 3013. If the A-pass indication flais set, processing continues at processing block 3013 without perfor-ning a_n. A-pass on coefficient C.

At processing block 3013,^a test detern-dnes if coefficient C is the last coefficient. lf coefficient C is not the last coefficient, then processing continues at processing block 3014 where the next coefficient is assigned to the variable C and processing continues at processing block 3019. However, if coefficient C is the last coefficient, processing continues at processing block 3015 where a test determines if the B-pass flag (doB_flag) is set. If the Bpass flag is set, the bitplane SB is set equal to the bitplane 513-1 (processing block 3016), and processing continues at processing block 3017. If the B-pass indication flag is not set, processing continues at processing block 3017. At processing block 3017, a test determines if the A-pass flag is set. If it is set, then the bitplane SA is set equal to the bitplane SA-1 (processing block 3018) and processing ends. Also, if the A-pass flag is not set, then processing ends immediately.

Coefficient Trees In a pyramidal system, the coefficients can be grouped into sets using a tree structure. The root of each tree is a purely low-pass coefficient. Figure 5 illustrates the tree structure of one purely low-pass coefficient of the transformed image. For a two-dimensional'signal such as an image, the root of the tree has three "children" and the rest of the nodes have four children each. The tree derarchically is not limited to two dimensional signals. For example, for a one dimensional signal, a root has one'child and non-root nodes have two children each. Higher dimensions follow from the one-dimensional and two-dimensional cases.

4- 7 The tree structure is also pparant from the operation of the filters shown in Figures 4A4C. The operation of the pairs of filters with subsampling causes the previously described coefficients to be related.

After the coefficients have been placed in sign magnitude format, a context model determines which of the multiple encoding methods is to be used to further encode the coefficients. A frequency-based coding scheme, such as zerotree coding, efficiently codes the significance data associated with a given subband decornpositrion for a specified threshold. In addition to using symbols indicating t1he signi-licance or in5ip. ificance of a single isolated coefficient in t-e associated subband decomposition, the entries of insi.p.ificant pare.-,ts with all children insignificant (those with rnagnitude5 less than or eclual to the given threshold) are grouped together and jointly ccded. These trees are sometimes referred to as zerotrees. These insignificant t- ,ees are _Sel 15 coded with a single dedicated symbol, sometimes called a zero tree root.

If, however, there is a significant descendant, the entT-y of an coefficient is encoded using the symbol for an "isolated zero". Thus, a tree is encoded with four symbols (positive significant, negative significant, 0 isolated zero or zerotree rodt) for decisions where the sign of the coefficient has not been coded yet.

Frequency-based coding is particularly useful in compression systems because the joint coding of insignfficant trees allows a small nurnber of parent coefficients to predict the insigT.Lifirance of a l.;i:. -e 0 n=ber of descendant coefficients. Since the entries in the tree associated with descendant coefficients can be predicted from the root, no additional symbols are needed to encode their insigrdficance. The insignificance of 1+9 the entire tree is encoded at a very low cost. Hence, the high order bil.pla.nes consist mostly of insig7lificant coefficients of which many are neither zerotree roots nor isolated zeros (i.e., they are children in in.sig-nificant trees which do not have to be coded).

Shapiro discloses a frequency-based model called Zerotree in U.S.

Patent No. 5,321,776. In Shapiro's method, two lists, a dominant list and a subordinate list are used to store all of the coefficients. For each significance level, two passes are made, a dominant pass and a subordinate i 1 C) pass. In one example, the frequency-based model is Zerotree.

In another example, a frequency-based model similar to Zerotree (as described by Shapiro) is used. Instead of using mmu'tiple 7151.5, only a single list is used, with each of the list elerrients being marked as being members of one Of 'the two groups. The separation of coefficients into an A-group and a B-group is equivalent to the separation Shapiro acldeves with dominant and subordinate lists (respectively). Shapiro's use of multiple lists allows greater fle)dbility in ordering of coefficients in the subordinate list at the expense of greater software/ hardware complexity. This Single List Zerotree method uses two passes, the A-pass and the B-pass that are equivalent to Shapiro's dominant pass and subordinate pass, respectively. The Single List Zerotree model will be described below.

The coding system maintains a list of the coefficients in sign-magnitude form in memory. Each element in the list has a one-bit label which indicates whether the element is a member of the "A-group or the "B-group". At the beginning of a stage, those Ls- 9 coefficients that have not yet been found to be significant are labeled as being in the A-gToup. Those coefficients that have been previously found to be significant with respect to previous, larger, thresholds are labe!ed as being in the B-group. The list contains the coefficients in the order in -vhich they will be processed for coding. At the beginning of the very first stage, all coefficients are labeled as members of the A- group, since no coefficient has been established as being significant. As coefficients are determined to be significant or insignificant, the labels for their entries are chanc-ed from the original A-cgroup designation to B-group desigra"lon.

0 0 Tliis list is subsequently refined at progressively finer threholosThat S, multiple passes through the list occur.

in one example, the binary events which correspond to the coefficients of the B-g-roup are binary arithmetically coded under a zeroth order Mark-ov context model. The 4-ary events corresponding to t.he coefficients of the A-group are also encoded under a zeroth order \12rkov context model.

The order of coefficients in the list preserves the tree structure such that no cl-dld can be modeled prior to its parent. Thus, an ordering which preserves the tree structure is fixed and used consistently. in one example, the coefficients are stored in memory in order from the first storage location used, alternatively, a linked list may be used.

In one example, the coefficients are coded in a bit significance, or bitplane embedded system. Since the coefficients are coded from most significant bitplane to least significant bitplane, the number of bitplanes in the data must be determined. This is To accomplished by finding an upper bound on the magnitudes of the coefficient values calculated from the data or derived from the depth of the image and the filter coefficients. For example, if the upper bound is. 149, then there are 8 bits of significance or 8 bitplanes.

Figure 6A illustrates one example of the single list zerotree encoding process. The process in Figure 6A may be used in the modeling process of Figure 27. Referring to Figure 6A, the process begins by testing whether the g-oup flag for the coefficient C is set to the "A- group" (processing 3221). If not, 0 then the process ends. On the other hand, if the group flag fDr coe-'. icie.-it C is set to the "A-group", then processing continues at processing block -'eterrnL-ied".a., 'or 3222 where a test determines if the "determined /un" W the coefficient C is set to "unde,Lerm.ined". If the "determ ined/undeterrnined" flag for the coefficient is not set to "undete.-n, ined", the process ends. If, however, the "determined /undetermined" flag for the coefficient C is set to undete--mined", processing continues at processing block 3203 wIrere a test determines if the bit SA of the coefficient C is one.

If the bit SA of the coefficient C is not one, processing continues at processing block 3207. On the other hand, if the bit SA of coefficient C is one, processing continues at processing block 3204 where a test determines is the sign of the coefficient C is positive. If the sign of the coefficient C is not positive, the decision is coded at "negallve significant" in "A-g-roup context(s) (processing block 3205), and the process continues at processing block 32-29. If the sign of the coefficient C is positive, the decision is coded as "posiltiive sigrdficant" in "A-g-roup" context(s) (processing block 3206), S-1 and the process continues at processing block 3229. At processing block 3229, the group flag for C is set to the "B-group".

At processing block 3207, a test deterrnines whether the bit SA iS zero for all the descendants (children) of the coefficient C. If bit SA is not zero, the decision is coded as 1nsignificant with significant children" (01) in "A-g-roup" context(s) (processing block 3208), and the process ends. On the other hand, if the bit SA is zero for all the descendants (chilldren) of t_he coefficient C, the decision is coded as "zerotree rooC (00) is context(s) (processing block,^)209). Thereafter, the s of t'he "dete.-mLned/undeter.,nined" flag for all of the descendan'. coeffic;ent C are sett, to "deter:-,iined" (processing block 32.21), and the process ends.

The termination test for the process may be i%hether or not a desired compression ratio is reached.

The binary events resulting from the B-pass may be entropy encoded under zeroth order Markov source context model. The 2bit alphabet (!,ize 4) resulting from the A-pass are encoded also under zeroth order Markov source, by a 4-ary (alphabet of size 4) aritlimetic coder.

Figure 6B and 6B (continued) illustrates an alternative implementation of the single list zerotree encoding process using reduced flag memory. The process of Figure 6B may be used as the A-pass in the process of Figure 30. Refe=g to FipLire ',B and 6B (continued), the process begirLs by testing whether the result of ANDing coefficient C with the mask MA is zero (processing 3201). If not, the process ends. On the other hand, if the result of AINTDing coefficient C with the mask MA is zero, processing continues at processing block 3202 where a test determines if the "determined /undetermined" flag for the parent of the coefficient C is set to "undetermined". If the flag for the parent of the coefficient is not set to "undetermined", the process ends. If, however, the "determined /undetermined" flag for the parent of the coefficient C is set to "undetermined", processing continues at processing block 3203 where a test determines if the bit SA of the coefficient C is one. If the bit SA of the coefficient C is not one, processing continues at 1 A processing block 3207. On the other hand, if the bit SA of coefficient C is one, processing continues at processing block 3204 where a test deter,-z- iL-ies is the sign of the coefficient C is positive. If the sign of the coefficient C is not positive, the decision is coded at "negative sim oup , ificant" in "A-g-. contex'L'k'S) (processing block.3205), and the process ends. Lf the -,ign of the coefficient C is positive, the decision is coded as "positive sig-nificant" in "A-&.-oup" context(s) (processing block 3206), and the process ends. In one ernbodime.nt, a 4-ary coder is used and 1-ary are coded in one context. A binary coder and three contexts may be used (e.g. the three contexts being the first bit of the decision, the second bit where the first bit is zero and the second bit when the first bit is a one).

At processing block 3207, a test determines whether the bit SA is zero for all the descendants (children) of the coefficient C. If bit SA is not zero, the decision is coded as "insignificant with significant children", "isolated zero", (01) in "A-group" context(s) (processing block 3208), and the process ends. On the other hand, if the bit SA is zero for all the descendants (cliildren) of the coefficient C, the decision is coded as 1 1 ) 5 1 #'zerotree root" (00) is "A-gro up context (s) (processing block 3209). Then, the "determined /undetermined" flag for the coefficient C is set to "determined" (processing block 3210). Thereafter, the "determined /undetermined" flag for all of the descendant> of the coefficient, which in turn have descendants, are set to "determined" (processing block 3211), and the process ends.

Decoding Lgps Decoding is performed in lockstep with the encoding.

Figure 6C illustrates one arrangement of the A-pass process for a zerotree horizon decoding process, and may be used in conjunction the process of Figu7e 27. Referring to Figure 6C, the process begins by testing svhetl,-ier the group flag for coefficient C is set to the "A-g- roup" (processing block 3521). If not, then the process ends. However, if so, processLng continues at processing block 3528 where a test dete-.mi:,- ,es ii the "determined /undetermined" flag for the coefficient C is set to undetermined". If not, the process ends. If it is, then the process continues to processing block 3502 where the temary decision is decoded 20 in A-group context(s).

Then a test determines if the decision is "positive significant" (processing block 3503). If the decision is "positive significant", the sign of the coefficient C is set to positive (processing blocl\ 3-505), the of 0 ' the coefficient is set to 25A (processing block 3507), the group flag for coefficient C is set to the "B-group" (processing block 3541), and the process ends.

1 S-J+ If the decision is not "positive significant" (processing block 3503), a test determines if the decision is "negative sigTdficanC (processing block 3504). If the decision is not "negative significant", the process continues at processing block 3509 where a test determines if the decision is a zerotree root. If the decision is not a zerotree root, the process ends. If the decision is zerotree root, the "determined /undetermined" flag for all decendants of coefficient C is set to "undetermined" (processing block 3531), and the process ends.

However, if the test of processing block 3504 dete.-...ines that the 10 decision is "negative significant", then the sign of the coefficient C is set to ficient C is set to negative (processing block 3506), the magnitude of coefl 25A (processing 'block 3507), the g-roup flag for coefficient C is set to the Bgroup (processing block,' 35411), and the process ends.

Figure 6D illustrates an alternative arrangement of the A-pass process for a zerotree horizon decoding process using reduced flag memory, and may be used in the process described in Figure 30. Referring 0 to Figure 61D, the process begins by testing whether the result of AN- Ding coefficient C with the mask MA is zero (processing block 3501). If not, then the process ends. However, if the result of AN1Ding coefficient C with the mask MA is zero, processing continues at processing block 3508 where a test determines if the "determined/ undetermineT flag for the parent of C is "undetermined". If not, the process ends. If it is, then the process continues to processing block 3502 where the ternary decision is decoded in A-group context(s).

Then a test determines if the decision is "positive sipificant" (processing block 3503). If the decision is "positive significant", the sip, of 1P, is J-5- the coefficient is set to positive (processing block 3505), the magnitude of the coefficient is set to 2SA (processing block 3507), and the process ends.

If the decision is not "positive significant", a test determines if it is "negative significant" (processing block 3504). If the decision is not "negative significant", the process continues at processing block 3509 0 where a test determines if the decision is a zerotree root. If the decision is not a zerotree root, the process ends. if the decision is zerotree root, the "determined /undetermined" flag for the coefficient C is set to "determined" (processing block 3510), the "determined /undetermined" flags for all decendants of coefficient C, which in tum have decendants, are set to "undetermined" (processing block 3511), and the process ends.

However, if the test of processing block 3504 deter-mines that the decision is "negative significant", then the sign of the coefficient C is set to negative (processing block 3506), the magnitude of coefficient C is set to 2SA (processing block 3507), and the process ends.

Alternatives exist to the choice made by Shapiro to use 4-ary decisions to describe trees. Larger alphabets may be used to further specify the characteristics of an entire tree when coding the root of the tree. In one embodiment, the following set of 6-ary decisions are used. insignificant with insignificant children (zerotree root) insignificant with at least one significant child significant, positive and aU children non-negative 0 sigUficant, positive and at least one c d1d is negative significant, negative and all children are non-positive significant, negative and at least one child is positive In this example, sign information in addition to insignificance is predicted for an entire tree. In other examples, trees with other sign constraints or with magnitude constraints can be predicted. Alternative. predictors might be especially useful in representing texture or in representing multi-resolution features. With larger alphabets, the use of higher order Markov contexts (as described later) may be useful.

Nlultil2ass List Based joint Space/Frequeney'Embedded Modeling Frequency embedded coding, such as Horizon order modeling disclosed herein, encodes the ternary events corresponding to the coefficients in the A-gToup. In horizon cod:,,-Lc,, all of 0 the Lnitializations preceding the coding steps are identical to the freauercy- based system. Binary entropy coding is performed is with three contexts, "A-group magnitude," "A-gro,.ip sign" and "B-Cc-oup. "

Figure 7A is a flow diagram of one example of the A-pass for a sinale list horizon encoding process. Tds process 0 1 may be used in the process of Figure 27. Referring to Figure 7A, the A- pass process begires by testing whether the group flag for coefficient C is set to the "A-g-roup" (processing block 3111). If not, the process ends. If the group flag for coefficient C is set to the "A-group", processing continues at processing block 3102 where a test determines if the bit SA of coefficient C is one. lf the bit SA of coefficient C is not one, the decision is coded as insignificant (0) in "A-group" context (processing block 3103) and the process ends. If the bit SA of coefficient C is one, then processing continues at processing block 3104 where a test determines whether the sign of coefficient C is positive. If the sign of coefficient C is positive, the decision SM 1 W.

;- 7 is coded as "positive significant'r (10) in "A-group" context(s) (processing block 3106), and the process continues at processing block 3117. On the other hand, if the sign of coefficient C is not positive, the decision is coeed as "negative significanC (11) in "A-group" context(s) (processing block 3105), and the process continues at processing block 3117. At processing block 3117, the group flag for coefficient C is set to the T-group".

Figure 7B is a flow diagram of an alternative arrangement of the A- pass for a single list horizon encoding process using reduced fiaThis process may be used in the process of Figure 30. Referring to Pi.=re the A-pass piocess be-ins by testing s,,,hether the result uf coeffic;ent C the mask NIA is Zero (processing nask. MA the process ends. If the result of Ai\rDing coefficient C %v,,h the n is zero, processing continues at processing block 3102,,.',ere a te- st dete--rr,,--ies if Che bit SA of coefficient C is one. If the bit SA of coefficient C 15 is not one, the decision is coded as Lnsignificant (0) in "A-c-oup" context C' (processing block 3103) and the process ends. If the bit SA of ece,.'lc; ent C s one, then processing continues at processing block 3104 where a test determines whether. the sign of coefficient C is positive. If the sign of coefficient C is positive, the decision is coded as "positive sig- .nificant" (K) in"A-group" context(s) (processing block 3106), and the process ends. On the other hand, if the sign of coefficient C is not positive, the decision is coded as "negative significant" (11) in "A-g-roup" context(s) (processing block 3105), and the process ends.

Decoding. Ste;2s Figure 7C illustrates one example of the A-pass process for a single list horizon decoding process and may be used in the process of Figure 27. Referring to Figure 7C, the process begins by testing whether the g-roup flag for coefficient C is set to the "A-g- roup" (processing block 3411). If not, then the process ends. However, if the g-roup flag for coefficient C is set to the "A-group", processing continues at processing block 3402 where the terna ry decision is decoded in A-gloup context(s).

Then a test determines if the decision is "positive significant" (processing block 3403). If the decision is "positive significant", the sign of the coefficient C is set to positive (processing block 3405), the inagnitude of the coefficient is set to 25A (processing block 3407), the group flag for coefficient C is set to the "B-g-r-,nup" (processing block 3418), and the process ends.

If the decision is not "positive significant", a test deter-nlines if it is negative significant" (processing block 3404). If the decLsion is not negative significanC, the process ends. However, if the decision is 1'negative significant", then the sign of the coefficient C is set to negative A (processing block (processing block 3406), the rnagnitude of C is set to 2S 3407), the g-roup flag for coefficient C is set to the T-group" (processing block 3418), and the process ends.

Figure 7D illustrates an alternative example of the A-pass process for a single list horizon decoding process using reduced flag mernory, and inay be used in the process of Figure 3 30. Referring to Figize 7 7 D, the process begins by testing whether the result of AND.ing coefficient C with the Mask MA is zero (processing block 3401). If not, then the :K - sq process ends. However, if the rsult of AN-Ding coefficient C -,,;1th the mask MA is zero, processing continues at processing block 3402 where the ternary decision is decoded in A-group context(s).

Then a test determines if the decision is "positive significant" (processing block 3403). If the decision is "positive significant", the si -n of the coefficient C is set to positive (processing block 3405), the magnitude of the coefficient is set to 2 A (processing block 3407), and the process ends.

If the decision is not "positive significant", a test deterrnines if it is negative signific int" (processing block 3404). If the decision is not "negati,,.,e significant", the process ends. However, if the decsion is 11 negative significant", then the sign of the coefficient C is set to negattive 0 0 (proce-ssLng block 3406), the magnitude of C is set to 2SA (processing block 1407), and the process ends.

B-Zas., fOr Both Zerotree and Horizon In one arrangement, the B-pass process for both zerotree and horizon of the present invention are the same. Examples of the B- pass algorythm for the encoding process and the decoding process are shown in Figures SA & 8B and 9A & 9B respechvely.

Figure BA illustrates one example of the B-pass process that in used partially for zerotree and single list horizon encoding process, and may be used in the process of Figure 27. Referring to Figure SA, the process initially tests whether the group flag for coefficient C is set (processing block 3311). If not, then the process ends. On the other hand, if the group flag is set, processing continues at processing block 3302 s,,here a test determines if the bit SB of coefficient C is 'T'. If the bit of SB of the 1 6c) coefficient C is not 'T', then the decision is coded as "0" in "B-group" context( s) (processing block 3303), and the process ends. lf the bit SB of Che coefficient C is 'T', then the decision is coded as 'T' in T- group" context(s) (processing block 3304), and the process ends.

Figure 8B illustrates an alternative example of the B-pass process that is used partially for zerotree and single list horizon encoding process and uses reduced flag memory, and may be used in the process of Figure 30. Referring to Figure 8B, the process initially tests whether the result of ANTDing the coefficient C with the mask MB is non-zero (processing block 33301). If not, then the process ends. On the other ha- nd, if the result of ANDing coefficient C with mask MB is non-zero, processing 0 continues at processing block 3302 where a test determines if Che bit 5B of coefficient C is "I". If the bit Of 5B of the coefficient C is not "I", then the decision is coded as "0" in "B-group" context(s) (processing block 330- 35), 2-nd the process ends. If the bit 513 of the coefficient C is 'T', then the dec.Sion is coded as 'T' in "B-group" context(s) (processing block 3-304), and the process ends.

Figure 9A illustrates one example of the B-pass decoding which may be used in the process of Figure 27. Referring to Figure 9A, a test initially determines if the group flag for coefficient C is set to the T-group" (processing block 3611). 11 not, then the process ends. However, if the group flag for coefficient C is set to the "B-group", then the decisions are decoded in the T-group" context(s) (processing block 3602). A test then decides if the decision is a 'T' (processing block 3603). If the decision is not a "I", the process ends. If the decision is a 'T', the bit 55B of coefficient C is set (processing block 3604) and the proces ends.

(, t Figure 9B illustrates an alternative arrangement of the B- pass decoding using reduced flag memory, and may be used in the process of Figure 30. Referring to Figure 9R, a test initially, deter-mines if the result of ANDing coefficient C with the mask MB is non- zero (processing block 3601). If the result of Ar\TDing coefficient C with the mask MB is zero, then the process ends. However, if the result of -k\,DLncy.

0 coefficient C with the mask MB is non-zero, then the decisions are decoled in the "B-g.,oup" context(s) (processing block 3602). A test then dec. des if the decision is a "I" (processing block 3603). If the decision is not a 'T', '11he 10 process ends. If the decision is a "I", the bit SB of coef':clen (processing block 3004) and the process ends.

0 Using the cornbinat.ion of zerotree order coding and hor.zon order coding, the arrangement provides bit-significance encoding of the coefficients generated by the reversible wavelets. Note that the use of both the A-group and the B-g-roup, and the ternary and the binary e-ents, corresponding to the 'W' and "B" passes, respectively, is especially important in view of the fact that a switch is made from using the zerotree ordering to the horizon ordering at the end of a-ny A pass. This compensates for the inefficiency in prediction that accompardes zerotree ordering at the low order bits. Therefore,. the system begins by zerotree coding the dgher order bits data, a-nd after a number of passes through the lists, i.e., after a n=ber of ID;t-il.nes are encoded, the encoder switches to encode the remainder of the data using horizon coding. The number of passes can be chosen statistically or can be chosen adaptively by monitoring the performance of zhe zerotree ordering coding block.

is Context Model Alternatives in one example, five binary context bins are used. This is small when compared with other systems such as JBIG which uses slightly more than 1024 contexts. Compression may be improved by using more context bins. Decisions may be conditioned on spatial location, level, and/or bit position. Decisions may also be conditioned on previously decoded data that is close to the current data in spatial location, level, and/or bit position. In general, zeroth order Markov contexts described earlier ma 10 be replaced with higher order Markov contexts.

Some exarnples are as follows. The most significant (a-nd therefore most easily predicted) bit of each mantissa (B-g-roup data in sorne embociLrnents) could use a differe:Lt context than the rest of the bits. The significant/ not significant decision could be conditioned on the same decision made for spatially close previous coefficients at the same tra.nsfo= level. Similarly, the sign bits for significant coefficients rnight be conditioned on the sign of spatially close previous coefficients at the same level or the sign of the coefficient of the parent.

Context model improvements might be especially important when compressing images that have spatial or multi-resolution structure. Grayscale images of line drawings or text are an example of images with both of these ty pes, of structure. Improvements are also impor-tant for compressing files that already have to be compressed and decompressed with a specified peak error.

1 The above system may be implemented in hardware and/or software. A hardware implementation of the above system requires implementation of the wavelet filters, memory/data flow manarrEment to 0 provide the data for the filters, a context model to control the embedded coding of the present invention, memory/data flow management to provide the data for the context model, and a binary ent-ropy coder.

Wa-elet Filters One embodiment of the forward wavelet filter of the present invention is shown in Figure 10. The wavelet filter shmvn in Fi,-,,-,re 10 acco=odates 4 16-bit two's complement input pixels, shown as x(2)-x(5).

Referring to Figure 10, the two tap "1 1" low-pass filter uses one 0 16-bit adder 1001. The OUtpUt5 aze called S and D, respectively. The output of the adder (S) is truncated to 16 bits using shift-by-l block 1003. The shift by-I block 1003 perforTns a divide-by-2 function by shifting its 17-bit input to the right one bit.

The six tap "-1 -18 -8 1 1" high-pass filter requires the computation of -So + 4DI + S2. The function S2 - SO is computed with 16-bit subtractor 1005 receiving the output of shift-by-I block 1003 and the YO(O). The 4D1 term is computed using subtTactor 1002 and shift-by-2 block 1004. The output produced by 16-bit subtractor 1002 is shifted to the left two places, thereby effectively multiplying its output by four. AddLng the A-DI output from shift-by-2 block 1004 to the output of subtractor 1005 is performed by 20-bit adder 1006. The output of the final adder is truncated to 18 bits using shift-by-2 block 1007. Shift-by-2 block 1007 performs a divide-by-4 function by shifting its 20 bit input to the right two bits.

- -,,Q 64- Thus, the total computational hardware required (not counting registers for storing temporary results) is:

16-bit adder, 2 @ 16-bit subtractors, I @ 19-bit adder. Note that shifting is performed by the wiring, such that no logic is needed.

In other embodiments, for inputs of size N, one N-bit adder, two N-bit subtractors and one (N+3) bit adder may be used.

Due to the extremely low hardware cost of these adders /sub tractors, 10 parallel implementations of the filters can be used if desired.

Note that alternatively, instead of subtracting X(3) and X(2), X(4)-X(5) can be computed and saved until needed later as X(2)-X(3) for the next shift or application of the filter. Both the for-ward filter (and the inverse filter described below) may be pipelined to achieve higher throughput.

The inverse wavelet filter is shown in Figure 11. The inputs of YO(O) and YO(2) are subtracted by subtractor 1101. The result of the subtraction is shifted to the right two bits by shift-by-2 block 1102. This effectively divides the output of the subtractor by 4. A subtraction is performed between the output of shift-by-2 block 1104 and the Yj(O) input. The input YO(I) is shifted one bit to the left by shift-by-I block 1103, thereby multiplying the input by two. After YO(I) is shifted by 1 (multiplied by two), the LSB of the shifted value is the LSB taken from the output of subtractor 1104 and combined with the 16 bits output from sl-iift-by-1 block 1103 to form an input for adder 1105 and subtractor 1106. The other input for adder 1105 and subtractor 1106 is the output of subtractor 1104. The outputs of adder 1105 and subtractor 1106 may subsequently undergo clipping.

A choice of two chp operations may be used. In both cases, the 20-bit value is shifted by, 1 (divided by 2), to a 19-bitvalue. For a systtin that only performs lossless compression, the least significant 16 bits can be output (the remaining 3 bits can be ignored). In a lossy system (or a lossy/lossless system), the 19-bit value is set to zero if it is negative or set to 216-1 if it is gTeater than 216-1; othenvise, the least significant 16 bits can be output.

For inputs of size N bits, one N-bit subtractor, one (N+2) bit subtractor, one (N+3) bit adder and one (N+3) bit subtraclor may be use.,. , and the clip unit outputs N bits.

kle-noiZ,, Usage With respect to memory -and data flow managernent for the s,;a,,-eet filters of the present invention,' for images where a full frame can fit in memory, mernory/data flow management is not a difficult issue. Even,-- r 1024 X 1024 16-bit medical images (e.g., 2 Mbytes in size), requirin. a Lill franne buffer is reasonable for many applications. For larger images (e.g., A4, 400 DPI 4-color images are about 50 N1bytes in size), performing the wavelet transform with a limited amount of line buffer memory is desirable.

Note that a full frame buffer is not necessary for the present 1 this, the P-e,-ncjr-y invention to implement a one-pass system. Because o. required may be reduced by about a factor of 100 (compared to usin. a full frame buffer for large images).

later.

i The one-pass system is described 6 6 The data stored in the filter memory is a series of coefficients that are to be subjected to the embedded coding and binary entropy coding. The embedded coding uses a context model to coordinate the use of frequency based coding or horizon coding, and to provide data in the proper order. The context model operates in conjunction with a memory management scheme. For systems with a full frame buffer, providing dat?_ in the proper order is not difficult. For systems without a full frame buffer, the transform data management scheme of the one-pass embodiment of the present invention (descr-ibed below) provides coefficients to the context model such that the context model only needs to buffer coefflicients for one tTee. A one pass frequency based context model and one pass joint space/frequency context model operate on one tree at a time.

The result of the embedding operation - is to produce bit streams from the frequency based modeling mechanism, and the joint space/frequency modeling mechanism These bit streams are then encoded using a binary entropy coder.

For systern with a full frame buffer, any binary entropy coder (or other appropriate coder) can be used. For systems without a full frame buffer, either multiple independent coders must be used, or the coder must be able to simulate multiple independent coders. Also, memory or channel management is needed to keep track of the outputs from the independent coders. An advantage is that the data to be managed is prioritized (embedded). If sufficient space or ba-ndividth is not available during compression or transmission less 1 1 6,7 important data can be discarded^on the fly, providing for reasonable lossy compression.

One Pass System The one-pass transform allows the input data in the system to be completely processed as it is received. In such a system, the processing of the data is not dependent on data that follows. The memory required to compess an image is independent of the length of the image. By removing the dependency, the one-pass ', system can output compressed data before all of the data has been processed.

A.

Dgita '..lpr.ac.,ement for One-Pass Transform Figure 12 illustrates a sechon of an image that is being cernpressed 0 in a raster order by band fashion' -z. Consider a four level decomposition. Each tree has 24 x 24 = 16 x 16 =256 coefficients. However, since the high-pass filter of the wavelet transform is overlapped, each tree depends on more than 256 input pixels. The two tap I I" low-pass filter (L) does not cause any overlap, all of the overlap comes from the six tap "-1 -1 8 -8 1 1" high-pass filter (H). The largest overlap occurs for the cascade of three applications of the low-pass filter followed by one application of the high pass filter (LLLH). Three applications of the low-pass filter (LI-L) require a support of 23 = 8 input pixels. Support regions 8x8 pixels in size are shown in Figure 12. When the high-pass filter is included in the cascade, the J 619 support regions is (6 x 23) x (6 x f3) = 48 x 48 pixels. A 48 x 48 pixel support region, consisting of thirty six 8 x 8 blocks as shown in Figure 12.

Assume that the coefficients in 48 x 48 pixel support region shown in Figure 12 are currently being processed. The lightly shaded portion of the support region depicts pixels that have already been used in previous support regions. The lightly shaded part that is outside the support region illustrates pixels that have already been used in previous support regions and will be needed in future support regions. The black 16 x 16 region is the part of the support region that contains pixels that have not been used previously. Sim, ilarly, the darkly shaded 16 x 16 region contain.5 pLxels that have not been used previously but which will be used in the next 48 x 48 support region. One three-level 16 x 16 transfo= is computed, the perrvious results for eight other three-level 16 x 16 t-ransforms are recalled from a buffer, and the fourth level of the tTansform, is applied to the nine 16 x 16 three-level transforms. The buffering required to accomplish &ds is enough to Store the three level trarisform coefficients for (2 x width_oLimage +32) x 16 pixels, plus enough to store a 16 line (one band) 0 buffer of pixels.

Figure 13 is a block diagram of one example of the one pass wavelet filtering urdt which includes a filter control urdt 1301, a memory 1302 and a filter 1303. Filter 1303 comprises the filter described in conjunction with Figure 10. Memory 1302 refers to the memory described above in conjunction with Figure 12 and stores either pixels or coefficients. Filter control unit 1301 determines the data flow betwe-en mernory 1302 and filter 1303. The operation of the filter control unit 1301 is explained below.

i 1 b11 Figure 14 illustrates an afternative wavelet filter unit. To achieve high speed operation, multiple filters can be used. In one arrangement since filter 1303 requires 4 or 5 inputs (e.g., inverse filter, forward filter) and produces two outputs, the memory bandwidth required could be substantial. The memory might have multiple pixels /coefficients per location, multiple banks and/or multiple ports. A memory interface unit 1401 reduces the memory bandwidth required by providing small buffers for local data used during processing. Memory interface unit 11-01 also provides multiplexing/demultiplexing between the input/ou'put (1/0) of mernory 1'302 and the 1/0 of filter 1303.

In addition to the memory bandwidth required for filtennaddiftnal bandwidt'n may be required for input of the pixels into rnemor" 1302 and output of the coefficients to the context model. If pixels a.-e Lnput in raster order, additional memory may be required for the band buffer.

If a memory stores multiple elements (pixels or coefficients) per location, instead of storing horizontally or vertically adjacent elements in a row or column, it may reduce the amount of memory accesses and buffering required if elements in a NxN block, when N is a power of 2, share the same location. This allows equal conver-ence for vertical and 20 horizontal accesses.

Multiple banks of memory may also be irnplemented so that both horLzontal and vertical access can take equal advantage of multiple banl,-s as shown in Figure 15. For the two bank case, a bank select bit for selecting either of the banks may be formed, in one embodLment, by exclusive ORing the LS13s of the horizontal and vertical coordinates. For the four bank case, the two bank select bits may be formed by adding _?0 (module 4 with a 2-bit adder) th7e two L5Bs of the horizontal and vertical coordinates.

Figure 16 illustrates the one-pass filter operation for a 2-level decomposition implementation by the filter control unit 1301 (Figure 13). Note that for illustration purposes, a 2-level description will be discussed first to illustrate the general technique. in other examples, 3level, 4-level, or high level decompositons are used. A two level decomposition has 16 coefficients, per tree and requires computation with 16 input pixels that have not been used previously.

The filtering for a tree of coefficients is performed in 16 or fewer time u-nits to correspond with the input and output rate. For this example, two filters operatLnc, in parallel are used to achieve the desired t2hroughput of two filtering operatio.-Ls per unit time. For each spatial location, -,%,here the leading edge of a filter is applied, Figure 16 shows a number that indicates the time that each filtering operation is performed.

Since the order of filtering is determined by the leading edge of the filter, filtering does not generate all the coefficients of one tree before generating any of the coefficients of the next tree. The filtering of the children of the tree occurs before the filtering of the parents, and low-pass filtering is done before the corresponding high-pass fUtering. The filtering operates on A-g-roup of coefficients which has the same number of coefficients of a type that a tree does.

The level I horizontal filtering is performed during time 0 through 7 and the results are stored in a temporary buffer. (Each spatial location results in two coefficients). During time 2 through 9, vertical filtering is performed (using the second filter) on data in the buffer and data from 1 -7 7 previous horizontal filtering frim memory (twice per spatial location). Vertical filtering can begin as soon as the second horizontal filtering operation is complete. The HH, HL and LH coefficients are ready for output to the context model (at the appropriate time). The LL coefficients 5 are used in the next level.

With only two filters, level 0 zero horizontal filtering cannot begin until time 8 when the level 1 horizontal filtering is complete, making a filter available. Level 0 horizontal filtering cannot finish until time 10 one cycle after the level 0 vertical filtering is complete, proving all the required data. Next during time 11 and 12, the level 1 vertical filtering can occur.

Table 1 below summarizes the operation of each filter during each time unit. The format of the entries is level number, horizontal or vertical ("H" or "V") and the spatial location of the leading edge. The inputs of vertical filtering operations are identified as either lowpass to highpass with a subscript I" or "H". Note that it is not necessary to assign 0 one filter to perform horizontal filtering and the other to perform vertical filtering, since both filters are identical.

-75L, Table 1

Time Filter 1 Filter 2 0 IH(OfO) 1 1H(0,1) 2 1H(2,0) 3 1H(2,1) 4 1H(0,2) 1H(0,3) 6 1H(2,2) 7 1H(2,3) 8 OH(0,0) 9 (idle) OH(0,1) 11 12 13 (idle) 14 (idle) 1VL(0,0) 1VH(OM 1VG0,2) 1VH(2,0) 1VG0,2) 1VH(0,2) 1VL(2,2) 1VH(2,2) (Idle) WL(OM WH(OM (idle) While level I horizontal filtering can begin again for the next g-roup of input pixels at time 11, this would cause the filter to operate faster than the input and output rate. Instead, the filters will be idle and the next gToup will be started at time 16. Idle filtering cycles may be used for memory transfers. Instead of occun-ing at the end ---5 of the filtering for each group, the idle cycles may be distributed among the filtering cycles if desired.

In view of the explanation of the 2-level case, the three level case shown in Table 2. Concatenations of two or four time units are used to transfer the information onto one page, thereby making it easier to read Table 2

Time Filter 1 10-3 4-7 8-11 12-15 16-19 20-23 24-27 28-31 -rFilter 2 (idle),(idle) 2VL(0,0), 2'H(0,0) 2VG2,0), 2VH(2,0), 2VG4,0), 2Vli(4,0) 2VL(6,0), 2VH(6,0), 2VI.(0,2), 2\11.1(0,2) 2VG2,2), 2VW2,2), 2YL(4,2), 2VW4,2) 2VG6,2), 2VW6,2), 2VL(0,4), 2VH(0,4) 2VG2,4), 2VH(2,4), 2VI,(4,4), 2VW4,4) 2VL(6,4), 2VH(6,4), 2VG0,6), 2VH(0,6) 2VG2,6), 2VH(2, 6), 2VL(4,6), 2VH(4,6) 2VL(6,6), 2VW6,6), IVI(0,0), IVW0,0) IVG2,0), IVH(2,0), IVI(0,2), WH(0,2) 2H(0,0), 2H(0,1), 2H(2,0). 2H(2,1) 2H(4,0), 2H(4,1), 2H(6,0) 2H(6,1) 2H(0, 2), 2H(0,3), 2H(2,2) 2H(2,3) 2H(4,2), 2H(4,3), 2H(6,2) 2.H(6,3) 2H(0,4), 2H(0,5), 2H(2,4) 2H(2,5) 2H(4,4), 2H(4,5), 2-H(6,4) 2HW5) 2H(0,6), 2H(0, 7), 21H(2,6) 2H(2,7) 2H(4,6), 2H(4,7), 2H(6,6) 2H(6,7) IH(0,0), IH(0,1), 1H(2,0) 1H(2,1) IH(0,2), IH(0,3), 1HM2) 1H(2,3) 32-35 36-39 40-43 44-47 48-51 52-55 56-59 60-63 OH(0,0, (idle), OH(0,1), (idle) WL(2,2), l'H(2,2), (idle), OVI_(0,0), _2VH(0,0), (idle), (idle), (idle) (idle) (idle) n1LI Table 3 illustrates the fou-r level case. Since there are now 2.56 time units per group of coefficients, for simplicity, only the level and direction of filtering is shown.

Table 3

Time Filter 1 Filter 2 0-1 2-127 128-129 9 13 0 - 15 160-161 162-167 168 169 171 172 173- 175 5 Level 3 Horizontal Level 3 Horizontal (idle) Level 3 Vertical Level 3 Vertical Level 2 Vertical Level 2 Vertical Level 1 Vertical Level 1 Vertical Level 1 Vertcal (idle) Level 0 Vertical Level 0 Vert-ical (idle) Level 2 Horizontal Level 2 Horizontal Level 1 Horizontal Level 1 Horizontal Level 0 Horizontal (idle) Level 0 Horizontal (idle) (idle) (idle) The output of the filtering and memory subsystem is a series of coefficients that undergo bit-significance embedded coding.

B. The Context Model for the One-Pass System In one arrangement,the bit significance embedded context model for the one-pass system, each tree is processed in four parts. The root of the tree, the highest level LL coefficient, is coded by one-pass horizon order coding. The three subtrees starting xVith each of the roots three cdldren, the highest level HH, HL and LI-I coefficients, are processed with both one pass joint space/frequency modeling and one pass frequency-based modeling. Ihe coefficients are coded such that encoded data may be output prior to the bit signLficance embedding context model operating on all of the data.

* 0-e-Pass Si=,.ifcance Tree The zerotree context model carmot be used in the one-pass systern. Zerotree requires a list (or multiple lists) that contains every coefficient and zero-cee makes multiple passes throu h the lists(s). An alte.rnative freque-ncy-based model, one-pass significance tree, does not require any lists containing all the coefficients. Another difference between one pass signi.ficance tree and zerotree is that significance tree processes aLl children before processing its parents when generating decisions, in contrast to zerotree which process es parent first.

The context model. i is shown in block djag-ram form in Figure 17. Context model 1700 contains two processing units, the sign/magnitude unit 109 (Figure IA) and the significance unit 1702. Context model 1700 also uses two memories (with memory, control logic), a magnitude memory 1701 and a tree memory 1703. Each of these hvo memory units may be implemented with multiple storage areas tu -716 allow for alternating use during-l-dgh speed operation (i.e., %,;hile data is being written into one, the other is being read, or emptied).

The magnitude memory 1701 reorders coefficients in the tree into an order based on significance, such as an order based on their magnitude.

This is accomplished by maintaining a queue for each possible rnagrdtude. The significance unit 1702 receives coefficients in order of sigrdficance (e.g., magnitude) and generates decisions for a coder that handles the Apass algorythm. Tree memory 1703 is coupled to significance unit 1702 and eliminates zerotrees after all zeros.

The following discussion ass=es that the coefilicienLs are 18-bits and that the input data has undergone a four level decorriposition.

One example of the sign/magnitude unit log is shown in Fip.,.re 18 anconverts input coefficients into a sign /mac,-.j -L-de form at. Sign /ma gnitu de unit 109 is coupled to receive 18 bits of the coef,,cie.-its and includes an inverter 1801, a multiplexer (NfUTX) 1802, a priority encoder 1803 and a counter 1804. The sign /magnitude unit 109 oustputs a t sigrdficance indication (e.g., a 5-bit value), the mantissa Of 6he input coefficient, (e.g., 17 bits), the sign of the input coefficient I bit and an index from counter 1804, (e.g., 7 bits.) MTJX 1802 is coupled to receive 17 bits of the coefficient directly input into sign/magnitude ur-dt 109 and an inverted version of the 17 bits from two's complementer 1801. Based on the sign bit (coefficient bit 17) received on the select input of NfUX 1802, the positive of the two inputs is output as the mantissa.

The sign/mag7dtude unit 109 uses a priority encoder IS03 to deternnine the fixst significant bit of each coefficie-nt. Based on the first f is -) 7 sigrdfica.nt bit of each coefficient, a Significance level may be associated with the coefficient.

Counter 1804 is used to associate an index with the current tree element. For a four level decomposition, the index varies from 0 to 84 (because 1 + 4 + 16 + 64 = 85, the number of elements in a subtree). The input coefficients are in tree order, which is assumed to be parents first, children last in this example. The coefficients are from different levels of decomposition as shown in Table 4 for the coefficients in order.

Table 4

Level Coefficients index 0 0 1 1, 22, 43, 6,4 2, 7, 12, 17, 2.3, 28, 33, 38, 4-4, 49, 54, 59, 65, 70, 75, 80 3... 6,8... 11,13... 16,18... 21,24... 27, 29-32, 34-37, 39... 42, 4 5... 1IS, 5 0... 53, 55-58, 60... 63,66... 69,71... 74, 76-79, 81-84 2 3 1 Figure 19 is one example of the block diagram of magnitude memory 1701. A counter and a memory is associated with each possible significance level (except nothing is needed for zero coefficients wir-h do not need to be coded). For instance, counter 1916 and memory 1936 are associated with significance level 17. In one example, there are sixteen levels of significance. Therefore, there axe 17 counters and 17 associated memories.

t In one example, each memory must have 85 locations, for eAh possible coefficient in a subtree (since each subtree contaim 83 coefficients), but the memory size might be rounded up to a power of 2, such as 128, for converdence. Each memory entry can contain a sign bit, a 7-bit index and N magnitude bits, where N is the significant level. If the use of a fixed width memory is desired, entries for significant 16 and 0, 15 and 1, etc. can be combined, so that each word has two entries totaling 32 bits. Of course, with an odd number of significance levels, one word must contain only one entry, which is level 7 in this example.

Sign, index, and mantissa values received from sign /magnitude unit 109 are written to the appropriate memory at the address provided by t1he associated memory's counter. The associated counter is then incremen'ted so that Cne next coefficient at that sigT-dficance level may 'be stored in the next location.

Mernory is read out from each of the memories 1920-1926 in descending order of sigTLificance. The output of each coefficient includes its mantessa, sign, and index output. If counter for the highest level of sig-.uficance (e.g., level 16) is nonzero, it is decremented and the memory is read at that address. This is repeated until the counter value is zero. Then the next level of sigrdficance (e.g., level 15) is considered. Each level of significance is considered in t= until all the counters have been decremented to zero and aU the memories emptied.

In a real-time system, it may be desirable to use two banks of counters and memories, so that one bank can be used for input while the other is used for output.

t -7 The counters address their associated memory such that a LIFO (last in, first out) is implemented. A LIFO is the correct order for when subtrees are input in the order of parents first. Alternatively, if subtrees were input child first, the operation of the counters could be changed to implement a FIFO (first in, first out).

Figure 20 is a block diagram of one example of significance unit 1702. Referring to Figure 20, index counter 2001 is used to step through 0 each coefficient in a subtree, children first. in one example, index counter 2001 is iritialized to 84 and counts down to zero. Significance counter 2004 begins at the maximum significance level (e.g., 16 in the example) and counts down every time index counter 8.41 cornpletes a cycle (returns to 84), such that the significance counter 2004 keeps track of the bitplane. The level of a particular index is determined by lo,-,.c (Lndex to level 2003) that performs the function shown in Table 4 above.

Magnitude memory unit 1701 provides an index, magnitude and sign of the next coefficient in the memory enabled by the output of the significance counter 2004. If the index input from memory is the same as the index output of index counter 2001, the equivalence logic 2002 asserts the nonzero output indication. The non-zero output indication sigrdfies that the magnitude memory should provide the next index, etc. on the next cycle. If there is not a match, then a non match indication is sent to discussion generator 2008.

In one example, three flip-flops, shown as flag o (2005), flag 1 (2006), and flag 2 (2007), axe used to keep track of non-zero data and are assigned to decomposition levels 0, 1 and 2, respectively. Note that the number of flip-flops required is one less than the number of 6 So decomposition levels. Flip-flops- 2005-2007 are irLitially cleared. When the non-zero signal from equivalence logic 202 is asserted, all of the flip-flops in flip-flops 2005-2007 that assigned to a level less than the current level are set. The flip-flop assigned to the current level is cleared. The level is 5 provided by index-to-level logic 2003, which provides the level in respon.se to the index provided by index counter 2001.

"Coded" flags are stored (in some embodiments a register file), one bit for each index. When the non-zero signal is asserted, the bit associated with the current index counter value in the coded flags storage is set. Otherwise, if the significance counter value is the maximum value, the associated bit is cleared. Otherwise, the value of the bit remains unchanged. The already coded output signal from coded flags storage the same as the new value of the bit associated with the c=ent index.

i Note that in an alternative_example, the coded flags are not used and the already coded signal is never used.

In one example, decision generator 2008 determines when the c=ent level is 3 and the previous level was not. In response to this determination, decision generator 2008 asserts the start output and the start level output is the previous level. If the non-zero signz.1 is asserted, the decision generator 2008 outputs a decision as 1. slgrdfican'," and also outputs the sign (00,01) and the mantissa. Otherwise, if the already coded input is asserted, no decision is output. Otherwise, if the flag flipflop assigneci to the c=ent level is set, decision generator 2008 outputs the decision as 'Insignificant, with significant children" (10). Otherwise, decision generator 2008 outputs the decision as "insig-uficant, and children irLsig-nificant" (11) and asserts the all zero signal.

1 Note that to implement both frequency-based modeling and horizon one pass joint space/frequency modeling, the following change is made to significance unit 2000. Significance counter 2004 is compared with a threshold, and the all zero output is only asserted if the counter 5 value is gTe,-ter Oan the threshold.

In one example, the significance category input to tree memory 1703 (shown in Fgiure 21 and described below) is the output of the significance counter 2004. In this embodiment of the context model (e.0- 1 0-1 bit-significance embedding unit), the significance category is based on the number of bitplanes, and there are 17 different categories of significance.

This is an arbitrary choice. In another example, bitplanes may be combined to create fewer significance categories. Also, level ca.n be added to bitplane information to create more significance categories. More significance categories might provide better lossy compression while fewer might reduce hardware complexity.

Figure 21 is a block diagram of one example of a tree memory unit. Referring to Figure 21, memory 2101 has adecr.:ate space for storing a decision and a significance indication for each possible decision. In one example, for a four level decomposition with 17 sigrdficance levels, the number of locations in memory 2101 is equal to 85 x 17 = 1445.

To access memory 2101, addresses are generated. Counter 2102 is ir-Litially zero. When decision generator 2008 does not assert the all zero input, the value in counter 2102 is used to address memory. If dedsion generator 2008 asserts the start input, the current value of counter 2102 is 6 lg:i, stored in one of registers 2110-2112 according to the start level, which acts as a selection mechanism. Counter 2102 is then incremented.

When decision generator 2008 asserts the all zero input, the value in the register (e.g., 2110, 2111, 2112) selected by the level input is used to address memory 2101 and this value plus one is loaded into counter 2102.

This causes the memory locations used for insignificant children of an insignificant parent to be ignored.

During memory output. counter 2102 is decrernented to provide the address of the location to output. Output (and decrementing) stops when counter 2102 reaches zero. The output from tree memory unit 2100 is received by an entropy coder that properly codes the decision at the specified significance.

For real-time operation, two tree memory units can be used so '_'-at one is used for input while the other is used for output.

Coefficient Alignment In one arrangement, the zerotree context model uses an unnormalized I + Z-1 low-pass filter. However, the zerotree context model may be used with normalized filters, such as 1 + Z-1 4-2 In order to use normalized filters, an alignment unit, such as alignment unit 2200 in Figure 22 between the for-ward wavelet filter 1000 and the context inodel 105, can be used to compensate for the energy gained (or alternatively, lost) from the urmormalized filter, which improves compression. Because alignment allows non-unifo= quantization for lossy operation', alignment can enhance the visual quality of lossy image reconstructions. In the one-dimensional case, coefficients from each level of the tree would have different alignment (divisors = -2, 2, 2.2-, 4, multipliers = 2.-2-, 2, -2, 1). In the two-dimensional case, the divisors would be 2, 4, 8, 16 and the multipliers would be 8, 4, 2, 1.

Since the alignment is just for grouping similar binary iecisions for coding, using the exact normalization value is not critical. 71e alignment must be inverted during decoding,.so both multiplication and division are required. Using factors/ divisors that are powers of two would allow hardware efficient shifting to be performed instead. When coefficients are multiplied by a power of two, the lessor significant zero bits added do not have to be coded.

However, instead of limiting the alignment factors 11 divisors to power of two, an approximation like -2= 1.5 or -2= 2 -±1.5 can be used with the following method. Instead of multiplying/ dividing coefficients by the factor/divisor, only the "significant" coefficients would instead be scaled by the factor/divisor. The sign/magnitude urLit may be modified as shown. in Figure 23 to include a "1.5" priority encoder 2301 that returns the position of eie,er (1) the most significant "I" bit if the next most significant bit is "I" also, or otherwise (2) the one less than the position of the most significant "1" bit. A truth table for a "1.5" priority encoder for 3 input bits is shown in Table 5.

E-+ Table 5

Input (Binary) j output 001 010 011 100 101 110 ill 0 0 1 1 1 2 2 Depending on the level of the coefficient indicated by the current index value, a multiplexer 2302 selects the significa-nce. from either the standard 10 priority encoder or the 1.5" priority encoder. Whenever the "1-5" align.ment is used, the mantissa contains N+I bits, where N is the significance value. Otherwise, the mantissa contaii-Ls N bits.

An alignment unit 2200 comprising a two input multiplexer acting as a shifter can implement aligning by 1 or 2. Combining this with the 1.5 15 alignment provided by the sign/magrdtude unit allows ahgnments of 1, 1.5, 2, or 3 which is a good approximation of the desired multipUers for onedimensional signals since the nu.mbers are simpler (e.g., powers of 1 5, t-.%,o). (For two-dimensional signals like images, the numbers are simpler.) During decoding, the N+2th bit of the mantissa (which is not coded) is the complement of the N+Ith bit when the "1.5" priority encoder is used...

Coefficient alignment can be used for tuning the zerotree and for finer and non-uniform quantization. In case of images (two dimensional signals), one example of the RTS-transform aligns the coefficients by multiplying the frequency band by the numbers depicted in Figure 31. Multiplying these numbers results in the RTS-transfo.= bein. a very close approximation of the exact reconstruction wavelets of the TS10 transforms.

The er,,-.Aopy coder must take into account the alic,,=ent process to be ef.ficient.

is Frecuencv-based Context Moels by Fractional Bi21,-.nP5 An alternative method of frequency-based modeling uses fractional bitpla. nes or fractional bits of significance. One imp!e-mentation of this is to process each bitplane tvvice, such that the passes include an Al-pass, a BI-pass, an AO-pass, and a BO-pass. Note that the names of the passes were chosen because the Al-pass handles coefficients starting with "11" arvi the 20 AO-pass handles those starting with "10".

During the Al-pass for bitplane S, a coefficient in the A-g-roup is significant only if both bits S and S-I are non-zero. During the A-1-pass, a coefficient in the A-group is significant if bit S is non-zero. Since the two most significant bits are known, the BI-pass and the BO-pass only need to 25 process S-1 bits (assuming S=O is the least significant bitplane).

f 811.

Since alternate fractional Sitpla.-nes differ by a factor of 1.5 or 2/1.5, alignment for different levels can be achieved by gTouping the desired fractional bitplanes for each level.

Fractional bitplanes cause finer modeling of the data by the parent/cldld relationship used by frequency-based context model. More than two passes, for example, four or eight passes could be used for even finer modeling. For example, in the four pass case, the All-pass would handle coefficients that. started with "111". The other passes would handle "110, " "101" and "100." Less fine modeling could also be used. For example, a pass could be made only for every other bitplane. In the less fine -n.odeling case, more bits are coded by the B-group.

C. Coder and Memor] ? y/Chanrel Man Zement for One-Pass Svstern Memory management for coded data in the one pass system is presented for systems that store all of the data in mernory and for systems that transmit data in a channel. In the one-pass system, coded data must be stored such that it can be accessed in an embedded causal fashion, so that less sig-nificant data can be discarded without losing more significant data. Since coded data is variable length, dynamic memory allocation can be used.

In one arrangement, the embedded coding scheme uses 18 bitplanes and, thus, assigns 18 levels of significance to the data. The coder in a one-pass system must be "embedded causal." That is, the decoding events corresponding to a bitplane do not require information from lower order bitplanes. In the one-pass case, typically all of the bits from one tree will be coded before any of the bits in the next tree 1; 7 are coded, so bits of different siinificance are not separated. For coders that do not use internal state, like Huffman coders, this is not a problem.

However, many sophisticated compressors with better compression use internal state.

One way to solve this problem for these coders is to use 18 different coders, perhaps 18 Q-coder chips. A technique that would allow the use of 9 Q-coder chips is described in U.S. Patent No. 5,097,261 (T..ingcinn, Jr. ), entitled "Data Compression for Recording on a Record Medium," issued March 17, 1992. A better way uses a pipelined coder to implement different virtual codes with a single physical coder, such as that described in U.S. Patent Application Serial No. 08/016,035, entitled '"Niethod and Apparatus for Parallel Decoding and Encoding of Data", filed February 10, 1993. In such a coder, the multiple bit generator states for each probability are each assigned to a part of the data. For example, each of IS states could be assigned to a particular bitplane for 18 bit data. Registers in the shifter in the coder are also assigned to each part of the data. In the encoder, no interleaving is performed; each part of the data is simply bitpacked.

In arrangements either with multiple physical or virtual coders, memory is allocated to each part of the data. When compression is complete, a linked list describing the memory allocated plus the contents of the allocated memory is the result.

If the memory overflows, the memory allocation routing causes more important data to overwrite less important data. For example, the least significant bit of numeric data might be overwritten first. The information that describes how memory is allocated must be stored in addition to the coded data.

f 1 cc h Figure 24 shows an example dynamic memory allocation unit for three categories of significance. Only three categories are described to avoid obscurity; typically, a larger number of categories, such as 8, 16 or 18, would be used. A register file (or other storage) holds a pointer for each category of significance plus another pointer for indicating the next free memory location. The memory is divided into fixed size pages.

Initially, each pointer assigned to a significance category points to the start of a page of memory and the free pointer points to the next available page of memory. Coded data, identified with a significance category, is stored at the memory location addressed by the corresponding pointer. The pointer is then increme-nted to the next memory locati i on.

When the pointer reaches the maxi-n= for the current page, the address of the start of the next free page stored in the free pointer is stored with the current page an a link. In one example, the part of the coded data memory or a separate memory or register file could be used for this purpose. Then the current pointer is set to the next free page. The free pointer is incrernented. These steps cause a new page of memory to be allocated to a particular significance category and provide Links to pages of 0 memory containing data for a common significance category so that the order of allocation can be determined during decoding.

When all pages in the memory are in use and there is more data that is more signiEcant than the least significant data in memory, memory reassign.ment may be performed. Three such reassignment techniques are described. In all three cases, memory assigned to the least significant data f sq 1 is reassigned to more significant- data and no more least significant data is stored.

First, the page currently being used by the least significant data is.

simply assigned to the more significant data. Since most typical entropy coders use internal state information, all of the least significant data stored previously in that page is lost.

Second, the page currently being used by the least significant data is assigned to the more significant data. Unlike the previous case, the pointer is set to the end of the page and as more significant data is written to the page, the corresponding pointer is decremented. This has the advantage of preserving the least significant data at the start of the page if the more significant data does not require the entire page.

Third, instead of the current page of least significant data being 0 reassigned, any page of least sigrdficant data may be reassigned. This requires that the coded data for all pages be coded independently, which may reduce the compression achieved. It also requires that the uncoded data corresponding to the start of all pages be identified. Since any page of least significant data can be discarded, greater flexibility in quantization is available.

The third alternative might be especially attractive in a system that achieves a fixed rate of compression over regions of the image. A specified number of memory pages can be allocated to a region of the image. Whether lessor significant data is retained or not can depend on the compression achieved in a particular region. Note that the memory assigned to a region might not be fully utilized if lossless compression required less than the amount of memory assigned. Achieving a fixed r.-'te of compression on a region of tie image can support random access to the image regions.

When compression is complete, the data may be transferred, if desired, to a channel or storage device in order of significance. The various links and pointers would then no longer be needed and multipass decoding could be performed. Alternatively, for one-pass decoding, the pointers to the data for each significance can be kept.

In some applications, some significance categories might not be used. For example, a 16-bit compressor'might be used on a 12-bit medical image, so significance categories corresponding to bitplanes 15 12 would be unused. In implementations with large pages and many unused significance categories, this would waste memory (when the system does not know in advance that some categories are unused), since memory does not have to be allocated to them. Another solution to this memory waste would be to use a small memory (or register) to hold a count for each significance category. The count would keep track of the number of "insignificant, no significant children" decisions that occur before any other decision occurs. The memory required to store these counters must be "traded-off' against the memory used by unused significance categories.

The ability to write data into each page from both ends can be used to better utilize the total amount of memory available in the system.

When all pages are allocated, any page that has sufficient free space at the end can be allocated for use from the end. The ability to use both ends of a page must be balanced against the cost of keeping track of the location where the two types of data meet. Note that this is different from the case t 91 where one of the data types was^not significant and could simply be overwritten.

Using a Channel In a system where data is transmitted in a channel instead of being stored in a memory and fixed size pages of memory are used (but only one page per significance category is needed), when a page of memory is full, it 0 0 is transmitted in the channel, and memory location can be reused as soon as they are transmitted. In some applications, the page size of the rnernory can be the size of data packets used in the channel or a multiple of the packet size. (Note that in one application, two pages per significance level can be used so that data can be written into one while the ol,l,-, Le,- is read for output to the channel.

In some communications systems, for exarnple ATIM (Asynchronous Transfer Mode), priorities can be assigned to packets. AT.M has two priority levels, priority and secondary. Secondary packets are only t-,ansmitted if sufficient bandwidth is available. A threshold can be used to determine which significance categories are priority and which are secondary. Another method would be to use a threshold at the encoder to not transmit significance categories that were less significant than a threshold.

Lossy Corn12ression with Bounded PeakError In some applications, perfect (lossless) reconstruction is not needed.

It may be desirable to achieve compression with a specified maximum peak error. Let the peak error be E. This can be achieved by truncating f 1 is Cl 9 the compressed data, so that aU lessor significant data that is not needed to achieve the desired accuracy is discarded.

Another way for achieving compression with a specified maximum peak error is to divide (with integer division) by a value less than or equal to 2xE+ 1 each pixel of the image to be compressed. During reconstruction, each pixel in the image is processed with:

output pixel = (2xE+I) x input pixel +E.

(Altennatively, instead of adding-E during decompression, subtraction can occur during compression prior to dividing by 2xE+I.) Another way to achieve compression with a specified maximum peak error is to replace the division and multiplication by shifts. The 51-L"" amount is Llog2(2xE+l)j. Because shifting is convenient, a betLer error specification (replacing peak error) might be errors of the form 2n<error<-2n].

The preceding should not be confused with quantization of coefficients that is well known in the art of lossy image compression. In many lossy compression systems (e.g., JPEG), transforTn domain coefficients are assigned a maximum peak error, which only indirectly 0 controls the peak error of the image. A critical difference is 20 that the above performs the quantization on pixels and uses lossless compression of coefficients.

Transform domain quantization can also be used. Many coe fficients have an effect on peak error propagates through multiple levels of the transform. It is easier to determine the effect on peak error for the hig1h25 pass coefficients that have no children.

1 93 - Consider a one-dimension'al signal which is to be encoded with a maxim= peak error of E. This can be achieved by quantizing the finest detail high-pass coefficients to +7'E- For a two-dimensional signal, since. there are two applications of the high-pass filter, the finest detail HH coefficients can be quantized to 4E.

An alternative to using quantization of the input image is to control the decisions to the entropy coder. One example is the follwvLng 0, For each coefficient, if setting the coefficient to zero would not cause the er-ror in any pixel affected by that coefficient to exceed the maximum er-ror, the coefficient is s-!t to zero. In some implementations only particular coefficients will be tested, perhaps only the AC coefficients that have no children. Coefficients can be considered with a g-reedy strattogy where one is considered at a time. Other strategies can consider small g-roups of coefficients and choose to zero the largest possible subset of the gToup.

Attention is directed to Application No. 9518298.6 (Publication No. 2,293, 733) from which this case is divided and which claims the encoder described above.

Attention is also directed to co-divisional applications S KU. r (" X i c -'s 0 - cl (Publication No.) and (Publication No.) which respectively claim the wavelet filter of Figure 11 and other aspects of the coder described above.

Finally attention is directed to Application No. 9518299.4 (Publication No. 2,293,734) which claims related methods of coding data.

APPE1MIX void RTS (T,w,m,n) 5 int w, m, n; int T; int i, j, k; int X, Y; is for (i=o;i<n;i++) for (j=O;j<m;j+=2) Example Code For RTS Transform (Forward) - One Level Decomp X[j/21 = (T[iw+jl+T[iw+j+l])>>1; Y [j /21 = T Uw+j 1 -T Uw+j +11; for (j=o; j<m/2; j++) T [im+j 1 =X [j 1; TUm+m/2+j]= (-X[j]+(Y[(j+l)t(M/2)]<<2)+X[(j+2) % (m/2)])>>2; void InverseRTS (T,w^n) int w,m,n; int T; int i, j; int B, D; for (i=o;i<n;i++) for (j=o;j<m/2;j++) for (j=o;j<m/2;j++) Example Code For RTIS Transform (Inverse) - One Level Decomp B[j] = (-T[im+jl+T[im+((j+2)%(M/2))])&OX3; D[(j+IMM/2)l = ((T[im+jl-T[im+ ((j+2) % (m/2))]+ (T[iM+M/2+j]<<2)+B[j]>>2); T[iw+2j] = ((T[im+j]<<1)+D[jl+l)>>1; T[iw+2j+11= T[iw+2j] - D[j]; j for (j=O;j<m;j++) RTS Transform (Forward) void RTS (T, w, m, n) int w, m, n; int T; 5 int i, j, k; int temp; int X, Y; for (i=o; i<n;i++) Two Level Decomp for (j=O; j<m; j+=2) X[j/21 = (T[iw+jl+T[iw+j+l])>>1; Y[j/21 = TUw+j]-TUw+j+11; for (j=O;j<m/2;j++) temp [im+jl=X[j]; temp [im+m/2+jl= (-X[j]+(Y[(j+l)%.(m/2)]<<2)+X[(j+2)1r(m/2)])>>2; for (i=O;i<n;i+=2) for (i=o;i<n/2;i++) X[i/21 = (temp[im+jl+temp[(i+l)m+j])>>1; Y[i/21 = temp[im+j]-temp[(i+l)m+j]; void InverseRTS (T,w,m,n) int w,m,n; int T; T [iw+j]=X [i]; THi+n/2)w+j]= (-X[i]+(Y[(i+l)t(n/2)]<<2)+X[(i+2)%;(n/2)])>>2; Two Level Decomp - RTS Transform (Inverse) int i, j; int B, D; int temp; - 96 for (j=O;j<m;j++) for (i=o;i<n;i++) for (i=o;i<n/2;i++) B[i] = (-T[iw+jl+T[((i+2)t(n/2))w+j])&OX3; D[(i+1)t(n/2)l = ((T[iw+jlT[((i+2)t(n/2))w+jl+ (T[(n/2+i)w+j]<<2)+B[i])>>2); for (i=o;i<n/2;i++) temp[2im+j] = ((T[iw+j]<<1)+D[il+l)>>1; temp[(2i+l)m+j] = tempf2im+j] D[i]; for (j=O;j<m/2;j++) B[j] = (-temp[im+jl+temp[im+((j+2)t(m/2))))&OX3; D[(j+1)%(m/2)l = ((temp[im+jj- for (j=O;j<m/2;j++) temp[im+((j+2)%(M/2))]+ (temp[iM+M/2+j]<<2)+B[j])>>2); TEiw+2j] = ((temp[im+j]<<1)+D[jl+l)>>1; T[iw+2j+11 = T[iw+2j] - D[j]; 1 1

Claims

C L A I M S

A wavelet transform filter for filtering an input data signal comprising: a first adder for adding a first pair of samples of the input data signal to generate a first result; a first output logic for outputting a first low-pass coefficient based on the first result; a first subtractor having inputs of a low-passed coefficient and the first low-pass coefficient, such that the low-passed coefficient is subtracted from the first low-pass coefficient to produce a second result, wherein the lowpassed coefficient is received by the first subtractor as a feedback; a second samples of the third result; a second adder coupled to receive and add the second result and an input based on the third result to produce a fourth result;

20 a second output logic coupled to generate a second lowpass coefficient based on the fourth result, such that the filter outputs the first and second low-pass coefficients.

subtractor for subtracting a second pair of input data signal from each other to produce a
2. A filter according to claim 1 wherein the first output logic comprises a divider.
3. A filter according to claim 2 wherein the divider comprises a bit shifter.
4. A filter according to claim 1, 2 or 3 wherein the second output logic comprises a divider.
5. A filter according to claim 4 wherein the divider comprises a bit shifter.

t 3
6. A wavelet transform filter according to claim 1 wherein said first output logic comprises a first divider coupled to receive the first result to divide the first result by a first factor into a fifth result, wherein the fifth result is output as said first low-pass coefficient; further comprising a first multiplier coupled to receive the third result for multiplying the third result by a second factor to produce a sixth result, said sixth result being passed to said second adder as said input based on the third result; and wherein said second output logic comprises a second divider coupled to receive the fourth result and divide the fourth result by a third factor to generate a second low-pass coefficient.
7. A filter according to claim 6 wherein the multiplier comprises a shifter.
8. A filter according to claim 6 or 7 wherein at least one of the dividers comprises a shifter.
9. A wavelet transform filter constructed and arranged to eperate substantially as hereinbefore described with reference to and as illustrated in Figure 10 of the accompanying drawings.

9. A wavelet transform filter constructed and arranged to opeate substantially as hereinbefore described with reference to and as illustrated in Figure 10 of the accompanying drawings.

11 1 1 qc ---1..,.

Amendments to the claims have been filed as follows 1. A wavelet transform filter for filtering an input data signal comprising: a first adder for adding a first pair of samples of the input data signal to generate a first result; a first output logic for outputting a first low-pass coefficient based on the first result; a first subtractor having inputs of a low-passed coefficient and the first low-pass coefficient, such that the low-passed coefficient is subtracted from the first low-pass coefficient to produce a second result, wherein the lowpassed coefficient is received by the first subtractor as a feedback; a second subtractor for subtracting a second pair of samples of the input data signal f rom each other to produce a third result; a second adder coupled to receive and add the second result and an input based on the third result to produce a fourth result;

20 a second output logic coupled to generate a second lowpass coefficient based on the fourth result, such that the filter outputs the first and second low-pass coefficients.

2. A filter according to claim 1 wherein the first output logic comprises a divider.

3. A filter according to claim 2 wherein the divider comprises a bit shifter.

4. A filter according to claim 1, 2 or 3 wherein the second output logic comprises a divider.

5. A filter according to claim 4 wherein the divider comprises a bit shifter.

t _iw_ 6. A wavelet transform filter according to claim 1 wherein said first output logic comprises a first divider coupled to receive the first result to divide the first result by a first factor into a fifth result, wherein the fifth result in output as said first low-pass coefficient; further comprising a first multiplier coupled to receive t:he third result for multiplying the third result by a second factor to produce a sixth result, said sixth result being passed to said second adder as said input based on the third result; and wherein said second output logic comprises a second divider coupled to receive the fourth result and divide the fourth result by a third factor to generate a second low-pass coefficient.

7. A filter according to claim 6 wherein the multiplier comprises a shifter.

A filter according to claim 6 or 7 wherein at least one of the dividers comprises a shifter.