US20070276894A1 - Process And Device For Determining A Transforming Element For A Given Transformation Function, Method And Device For Transforming A Digital Signal From The Time Domain Into The Frequency Domain And Vice Versa And Computer Readable Medium - Google Patents

Process And Device For Determining A Transforming Element For A Given Transformation Function, Method And Device For Transforming A Digital Signal From The Time Domain Into The Frequency Domain And Vice Versa And Computer Readable Medium Download PDF

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US20070276894A1
US20070276894A1 US10/573,955 US57395504A US2007276894A1 US 20070276894 A1 US20070276894 A1 US 20070276894A1 US 57395504 A US57395504 A US 57395504A US 2007276894 A1 US2007276894 A1 US 2007276894A1
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matrix
transformation
lifting
matrices
transforming element
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Haibin Huang
Xiao Lin
Susanto Rahardja
Rongshan Yu
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Agency for Science Technology and Research Singapore
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/147—Discrete orthonormal transforms, e.g. discrete cosine transform, discrete sine transform, and variations therefrom, e.g. modified discrete cosine transform, integer transforms approximating the discrete cosine transform
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/141—Discrete Fourier transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/148—Wavelet transforms
    • G—PHYSICS
    • G10—MUSICAL INSTRUMENTS; ACOUSTICS
    • G10L—SPEECH ANALYSIS TECHNIQUES OR SPEECH SYNTHESIS; SPEECH RECOGNITION; SPEECH OR VOICE PROCESSING TECHNIQUES; SPEECH OR AUDIO CODING OR DECODING
    • G10L19/00—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis
    • G10L19/02—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis using spectral analysis, e.g. transform vocoders or subband vocoders
    • G10L19/0212—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis using spectral analysis, e.g. transform vocoders or subband vocoders using orthogonal transformation
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/60—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/60—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding
    • H04N19/63—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding using sub-band based transform, e.g. wavelets
    • G—PHYSICS
    • G10—MUSICAL INSTRUMENTS; ACOUSTICS
    • G10L—SPEECH ANALYSIS TECHNIQUES OR SPEECH SYNTHESIS; SPEECH RECOGNITION; SPEECH OR VOICE PROCESSING TECHNIQUES; SPEECH OR AUDIO CODING OR DECODING
    • G10L19/00—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis
    • G10L19/0017—Lossless audio signal coding; Perfect reconstruction of coded audio signal by transmission of coding error
    • G—PHYSICS
    • G10—MUSICAL INSTRUMENTS; ACOUSTICS
    • G10L—SPEECH ANALYSIS TECHNIQUES OR SPEECH SYNTHESIS; SPEECH RECOGNITION; SPEECH OR VOICE PROCESSING TECHNIQUES; SPEECH OR AUDIO CODING OR DECODING
    • G10L19/00—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis
    • G10L19/04—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis using predictive techniques
    • G10L19/16—Vocoder architecture
    • G10L19/18—Vocoders using multiple modes
    • G10L19/24—Variable rate codecs, e.g. for generating different qualities using a scalable representation such as hierarchical encoding or layered encoding

Definitions

  • This invention relates to a process and a device for determining a transforming element for a given transformation function, a method and a device for transforming a digital signal from the time domain into the frequency domain and make versa and a computer readable medium.
  • DCT discrete cosine transform
  • the DCT is a real-valued block transform. Even if the input block consists only of integers, the output block of the DCT can comprise non-integer components. For convenience, the input block is referred to as input vector and the output block as output vector. If a vector comprises only integer components, it is called an integer vector. In contrast to the DCT, the integer DCT generates an integer output vector from an integer input vector. For the same integer input vector, the integer output vector of integer DCT closely approximates the real output vector of DCT. Thus the integer DCT keeps all the good properties of the DCT in spectrum analysis.
  • An important property of the integer DCT is reversibility. Reversibility means that there exists an integer inverse DCT (IDCT) so that if the integer DCT generates an output vector y from an input vector x, the integer IDCT can recover the vector x from the vector y.
  • IDCT integer inverse DCT
  • the integer DCT is also referred to as the forward transform, and the integer IDCT as the backward or inverse transform.
  • IntMDCT integer modified discrete cosine transform
  • MDCT modified discrete cosine transform
  • Malvar gave an efficient realization of MDCT by cascading a bank of Givens rotations with a DCT-IV block. It is well known that Givens rotation can be factorised into three lifting steps for mapping integers to integers, see for example [2].
  • IntMDCT relies on an efficient implementation of integer DCT-IV.
  • Integer transforms can be directly converted from their prototypes by replacing each Givens rotation with three lifting steps. Because in each lifting step there is one rounding operation, the total rounding number of an integer transform is three times the Givens rotation number of the prototype transform.
  • the number of Givens rotations involved is usually at N log 2 N level, where N is the size of the blocks, i.e. the amount of data symbols included in each block, the digital signal is divided into. Accordingly, the total rounding number is also at N log 2 N level for the family of directly converted integer transforms. Because of the roundings, an integer transform only approximates its floating-point prototype. The approximation error increases with the number of rounding operations.
  • the invention solves the problem of determining a transforming element for a given transformation function, which transformation function comprises a transformation matrix and corresponds to a transformation of a digital signal from the time domain into the frequency domain or vice versa, such that the number of roundings comprised by the transforming element is significantly reduced.
  • the invention further provides a method for transforming a digital signal from the time domain into the frequency domain or vice versa according to the determined transforming element.
  • the problem is solved by a process and a device for determining a transforming element for a given transformation function, a method and a device for transforming a digital signal from the time domain into the frequency domain or vice versa and a computer readable medium with the features according to the independent claims.
  • a process for determining a transforming element for a given transformation function which transformation function comprises a transformation matrix and corresponds to a transformation of a digital signal from the time domain into the frequency domain or vice versa, wherein the transformation matrix is decomposed into a rotation matrix and an auxiliary matrix which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix; the rotation matrix and the auxiliary matrix are each decomposed into a plurality of lifting matrices; and the transforming element is determined to comprise of a plurality of lifting stages which correspond to the lifting matrices.
  • a method for transforming a digital signal from the time domain into the frequency domain or vice versa using a transforming element wherein the transforming element corresponds to a given transformation function, which transformation function comprises a transformation matrix wherein the transforming element is determined by a process comprising decomposing the transformation matrix into a rotation matrix and an auxiliary matrix which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix; decomposing the rotation matrix and the auxiliary matrix each into a plurality of lifting matrices; and determining the transforming element to comprise of a plurality of lifting stages which correspond to the lifting matrices; and wherein further each lifting stage comprises the processing of sub-blocks of the digital signal by an auxiliary transformation and by a rounding unit.
  • a computer readable medium having a program recorded thereon, wherein the program is adapted to make a computer perform a process for determining a transforming element for a given transformation function, which transformation function comprises a transformation matrix and corresponds to a transformation of a digital signal from the time domain into the frequency domain or vice versa, wherein the transformation matrix is decomposed into a rotation matrix and an auxiliary matrix which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix; the rotation matrix and the auxiliary matrix are each decomposed into a plurality of lifting matrices; and the transforming element is determined to comprise of a plurality of lifting stages which correspond to the lifting matrices.
  • a computer readable medium having a program recorded thereon, wherein the program is adapted to make a computer perform a method for transforming a digital signal from the time domain into the frequency domain or vice versa using a transforming element, wherein the transforming element corresponds to a given transformation function, which transformation function comprises a transformation matrix wherein the transforming element is determined by a process comprising decomposing the transformation matrix into a rotation matrix and an auxiliary matrix which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix; decomposing the rotation matrix and the auxiliary matrix each into a plurality of lifting matrices; determining the transforming element to comprise of a plurality of lifting stages which correspond to the lifting matrices; and wherein further each lifting stage comprises the processing of sub-blocks of the digital signal by auxiliary transformations and by a rounding unit.
  • the invention provides a process and a method for realizing the integer type-IV DCT transformation.
  • the method according to the invention requires a significantly reduced number of roundings compared to prior art methods.
  • the approximation error is greatly reduced, in the case of DCT-IV it can be reduced from the usual N log 2 N level to be as low as 2.5N, where N denotes the block size of the digital signal.
  • the method according to the invention is low in computational complexity and modular in structure.
  • the method and the device according to the invention can be used for any types of digital signals, such as audio, image or video signals.
  • the digital signal which is a digitised signal, corresponds to a physical measured signal, may be generated by scanning at least one characteristic feature of a corresponding analog signal (for example, the luminance and chrominance values of a video signal, the amplitude of an analog sound signal, or the analog sensing signal from a sensor).
  • the digital signal comprises a plurality of data symbols.
  • the data symbols of the digital signal are grouped into blocks, with each block having the same predefined number of data symbols based on the sampling rate of the corresponding analog signal.
  • the method according to the invention can be used for transforming an input digital signal which represents integer values to an output signal which also represents integer values.
  • the transformation method according to the invention is reversible.
  • the output signal may be transformed back to the original input signal by performing the transformation method according to the invention.
  • Such a reversibility property of the transformation according to the method of the invention can be used in lossless coding in which the output signal should be identical to the original input signal.
  • Such an integer transformation of signals according to the invention can be used in many applications and systems such as MPEG audio, image and video compression, JPEG2000 or spectral analyzers (for analyzing Infrared, Ultra-violet or Nuclear Magnetic Radiation signals). It can also be easily implemented in hardware systems such as in a fixed-point Digital Signal Processor (DSP), without having to consider factors such as overflow in the case of a real-value signal transformation.
  • DSP Digital Signal Processor
  • the digital signal is transformed to the frequency domain by a transforming element, which is according to the process according to the invention determined for a given transformation function.
  • the transforming element comprises a plurality of lifting stages.
  • the transforming element can be illustrated based on the model of a lifting ladder.
  • the lifting ladder model has two side pieces, each for receiving one of two groups of data symbols.
  • Two or more cascading lifting stages are provided between the two side pieces.
  • Each lifting stage receives a signal at one end (input end), and outputs a signal at the other end (output end) via a summation unit.
  • a rounding unit is arranged at the output end.
  • the lifting stages are arranged between the side pieces in an alternating manner, such that the output (or input) ends of adjacent lifting stages are connected to the different side pieces.
  • the number of lifting stages of the transforming element is defined by the number of lifting matrices which is determined by the process according to the invention.
  • Discrete cosine transforms, discrete sine transforms, discrete Fourier transforms or discrete W transforms are examples of transformation functions that may be used as the transformation function according to the invention.
  • the number of lifting stages of the transforming element may be different, depending on the result of the process according to the invention for determining a transforming element for the respective transformation functions.
  • FIG. 1 shows the architecture of an audio encoder according to an embodiment of the invention.
  • FIG. 2 shows the architecture of an audio decoder according to an embodiment of the invention, which corresponds to the audio coder shown in FIG. 1 .
  • FIG. 3 illustrates an embodiment of the process according to the invention, wherein the transformation function is a DCT-IV transformation function.
  • FIG. 4 shows a flow chart of an embodiment of the method according to the invention.
  • FIG. 5 illustrates an embodiment of the method according to the invention using DCT-IV as the transformation function.
  • FIG. 6 illustrates the algorithm for the reverse transformation according to the embodiment of the method according to the invention illustrated in FIG. 5 .
  • FIG. 7 shows the architecture for an image archiving system according to an embodiment of the invention.
  • FIG. 8 illustrates an embodiment of the method according to the invention using DWT-IV as the transformation function.
  • FIG. 9 illustrates the algorithm for the reverse transformation according to the embodiment of the method according to the invention illustrated in FIG. 8 .
  • FIG. 1 shows the architecture of an audio encoder 100 according to an embodiment of the invention.
  • the audio encoder 100 comprises a conventional perceptual base layer coder based on the modified discrete cosine transform (MDCT) and a lossless enhancement coder based on the integer modified discrete cosine transform (IntMDCT).
  • MDCT modified discrete cosine transform
  • IntMDCT integer modified discrete cosine transform
  • An audio signal 109 which, for instance, is provided by a microphone 110 and which is digitalized by a analog-to-digital converter 111 is provided to the audio encoder 100 .
  • the audio signal 109 comprises a plurality of data symbols.
  • the audio signal 109 is divided into a plurality of blocks, wherein each block comprises a plurality of data symbols of the digital signal and each block is transformed by a modified discrete cosine transform (MDCT) device 101 .
  • MDCT discrete cosine transform
  • the MDCT coefficients provided by the MDCT device 101 are quantized by a quantizer 103 with the help of a perceptual model 102 .
  • the perceptual model controls the quantizer 103 in such a way that the audible distortions resulting from the quantization error are low.
  • the output of the quantizer 103 is encoded by a bitstream encoder 104 which produces the lossy perceptually coded output bitstream 112 .
  • the bitstream encoder 104 losslessly compresses its input to produce an output which has a lower average bit-rate than its input by standard methods such as Huffman-Coding or Run-Length-Coding.
  • the input audio signal 109 is also fed into an IntMDCT device 105 which produces IntMDCT coefficients.
  • the quantized MDCT coefficients, which are the output of the quantizer 103 are used to predict the IntMDCT coefficients.
  • the quantized MDCT coefficients are fed into an inverse quantizer 106 and the output of the inverse quantizer 106 is fed into a rounding unit 107 by which they are rounded to integer values and the residual IntMDCT coefficients, which are the difference between the output of the rounding unit 107 and the IntMDCT coefficients, are entropy coded by an entropy coder 108 , which, analogous to the bitstream encoder 104 , losslessly reduces the average bit-rate of its input and produces a lossless enhancement bitstream 113 , which, together with the perceptually coded bitstream 112 , carries the necessary information to reconstruct the input audio signal 109 exactly.
  • FIG. 2 shows the architecture of an audio decoder 200 comprising an embodiment of the invention, which corresponds to the audio coder 100 shown in FIG. 1 .
  • the perceptually coded bitstream 207 is decoded by a bitstream decoder 201 , which performs the inverse operations to the operations of the bitstream encoder 104 of FIG. 1 and is fed to an inverse quantizer 202 . To its output, the inverse MDCT is applied by an inverse MDCT device 203 . Thus, the reconstructed perceptually coded audio signal 209 is obtained.
  • the lossless enhancement bitstream 208 is decoded by an entropy decoder 204 , which performs the inverse operations to the operations of the entropy encoder 108 of FIG. 1 and which produces the corresponding residual IntMDCT coefficients.
  • the output of the inverse quantizer 202 is rounded by a rounding device 205 and is added to the residual IntMDCT coefficients, thus producing the IntMDCT coefficients. Finally, the inverse IntMDCT is applied to the IntMDCT coefficients by an inverse IntMDCT device 206 to produce the reconstructed losslessly coded audio signal 210 .
  • the core of IntMDCT which plays an important role in lossless audio coding and which is used in the embodiment of the invention illustrated in FIGS. 1 and 2 , is an integer DCT-IV.
  • FIG. 3 illustrates a flowchart of an embodiment of the process according to the invention, wherein the transformation function is a DCT-IV transformation function.
  • the DCT-IV transformation function with a N-point real input sequence x(n) is defined as follows (see [2]).
  • the transformation matrix C N IV is decomposed into a rotation matrix and a matrix which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix.
  • N is assumed to be even.
  • step 300 the process is started.
  • the vector y 1 comprises the components of y corresponding to the indices from N/2 to N ⁇ 1 in reverse order.
  • y 0 and y 1 are each expressed by a DCT-IV matrix C N 2 IV _ and a DST-IV matrix S N 2 IV _ , each of dimension N/2.
  • step 306 the N ⁇ N rotation matrix R po is calculated, which comprises the first three matrices of equation (44):
  • R po _ [ I N 2 _ J N 2 _ ] ⁇ [ C _ S _ - S _ C _ ] ⁇ [ I N 2 _ J N 2 _ ] ( 45 )
  • R po _ [ cos ⁇ ⁇ 4 ⁇ N sin ⁇ ⁇ 4 ⁇ N cos ⁇ 3 ⁇ ⁇ 4 ⁇ N sin ⁇ 3 ⁇ ⁇ 4 ⁇ N ⁇ ⁇ cos ⁇ ( N - 1 ) ⁇ ⁇ 4 ⁇ N sin ⁇ ( N - 1 ) ⁇ ⁇ 4 ⁇ N - sin ⁇ ( N - 1 ) ⁇ ⁇ 4 ⁇ N cos ⁇ ( N - 1 ) ⁇ ⁇ 4 ⁇ N ⁇ ⁇ - sin ⁇ 3 ⁇ ⁇ 4 ⁇ N cos ⁇ 3 ⁇ ⁇ 4 ⁇ N - sin ⁇ ⁇ 4 ⁇ N cos ⁇ ⁇ 4 ⁇ N ] ( 46 )
  • step 307 the auxiliary matrix, which, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix, is calculated.
  • the transformation matrix C N IV is decomposed into a rotation matrix R po , into an auxiliary matrix T that, when multiplied with itself, equals a permutation matrix multiplied with an integer diagonal matrix and into an even-odd matrix P eo .
  • R po and T are each factorised into a product of lifting matrices.
  • H 2 _ [ sin ⁇ ⁇ 4 ⁇ N sin ⁇ 3 ⁇ ⁇ 4 ⁇ N ⁇ sin ⁇ ( N - 1 ) ⁇ ⁇ 4 ⁇ N ] ( 56 )
  • step 309 the lifting matrices are aggregated as far as possible.
  • Equation (59) indicates that the transforming element for the integer DCT-IV transformation according to the invention comprises five lifting stages.
  • step 310 Since the final factorisation formula is determined, the process is stopped in step 310 .
  • FIG. 4 shows a flow chart 400 of an embodiment of the method according to the invention using five lifting stages, a first lifting stage 401 , a second lifting stage 402 , a third lifting stage 403 , a fourth lifting stage 404 and a fifth lifting stage 405 .
  • This method is preferably used in the IntMDCT device 105 of FIG. 1 and the inverse IntMDCT device 206 of FIG. 2 to implement IntMDCT and inverse IntMDCT, respectively.
  • x 1 and x 2 are first and second blocks of the digital signal, respectively.
  • z 1 , z 2 , z 3 are intermediate signals
  • y 1 and y 2 are output signals corresponding to the first and second block of the digital signal, respectively.
  • FIG. 5 shows a flow chart of an embodiment of the method according to the invention, wherein the transformation function is a DCT-IV transformation function.
  • the transforming element used in this embodiment corresponds to the equation (59), i.e., is the one which is determined by the embodiment of the process illustrated in FIG. 3 .
  • the transforming element comprises five lifting stages which correspond to the five lifting matrices of equation (59).
  • the transforming element comprises a data shuffling stage corresponding to the permutation matrix P eo .
  • the input of the first lifting stage are the two blocks of the digital signal x 1 and x 2 , z 1 , z 2 and z 3 are intermediate signals and y 1 and y 2 are output signals corresponding to the first and second block of the digital signal, respectively.
  • the first lifting stage 501 is explained, which is the lifting stage corresponding to the lifting matrix T 3 .
  • FIG. 6 illustrates the lifting stages of the transforming element for the reverse transformation of the transformation illustrated in FIG. 5 .
  • the input of the first lifting stage are the two blocks of the digital signal y 1 and y 2 , z 1 , z 2 and z 3 are intermediate signals and x 1 and x 2 are output signals corresponding to the first and second block of the digital signal, respectively.
  • the lifting stages 605 , 604 , 603 , 602 and 601 of FIG. 6 are inverse to the lifting stages 501 to 505 of FIG. 5 , respectively. Since also the permutation of the input signal corresponding to the matrix P eo can be reversed and an according data shuffling stage is comprised by the inverse transforming element, the provided method is reversible, thus, if used in the audio encoder 100 and the audio decoder 200 illustrated in FIGS. 1 and 2 , providing a method and an apparatus for lossless audio coding.
  • FIG. 7 shows the architecture for an image archiving system according to an embodiment of the invention.
  • an image source 701 for instance a camera, provides an analog image signal.
  • the image signal is processed by a analog-to-digital converter 702 to provide a corresponding digital image signal.
  • the digital image signal is losslessly encoded by a lossless image encoder 703 which includes a transformation from the time domain to the frequency domain. In this embodiment, the time domain corresponds to the coordinate space of the image.
  • the lossless coded image signal is stored in a storage device 704 , for example a hard disk or a DVD. When the image is needed, the losslessly coded image signal is fetched from the storage device 704 and provided to a lossless image decoder 705 which decodes the losslessly coded image signal and reconstructs the original image signal without any data loss.
  • Such lossless archiving of image signals is important, for example, in the case that the images are error maps of semiconductor wafers and have to be stored for later analysis.
  • FIG. 8 illustrates an embodiment of the method according to the invention using DWT-IV as the transformation function.
  • I N/2 is the identity matrix of order N/2 (confer equation (29))
  • J N/2 is the counter identity matrix of order N/2 (confer equation (30)).
  • P N is a N ⁇ N permutation matrix
  • P N _ [ I N / 2 _ J N / 2 _ ] ( 69 )
  • the lifting steps are aggregated as far as possible.
  • the lifting matrices R 3 and T 1 can be aggregated to the lifting matrix S:
  • Equation (79) indicates that the transforming element for the integer DWT-IV transform according to the invention comprises five lifting stages.
  • the transforming element comprises a data shuffling stage corresponding to the permutation matrix P N .
  • the data shuffling stage rearranges the components order in each input data block.
  • the input of the first lifting stage are the two blocks of the digital signal x 1 and x 2 , z 1 , z 2 and z 3 are intermediate signals and y 1 and y 2 are output signals corresponding to the first and second block of the digital signal, respectively.
  • the first lifting stage 801 is explained, which is the lifting stage corresponding to the lifting matrix T 3 .
  • FIG. 9 illustrates the lifting stages of the transforming element for the reverse transformation of the transformation illustrated in FIG. 8 .
  • the input of the first lifting stage are the two blocks of the digital signal y 1 and y 2 , z 1 , z 2 and z 3 are intermediate signals and x 1 and x 2 are output signals corresponding to the first and second block of the digital signal, respectively.
  • the last lifting stage 905 illustrated in FIG. 9 is inverse to the first lifting stage 801 illustrated in FIG. 8 . So, in the first step 906 of the last lifting stage 905 , x 1 is multiplied by K 3 . The result of this multiplication is rounded to integer values in step 907 . The rounded values are then subtracted from z 1 in step 908 .
  • the signal x 2 fulfils the equation x 2 z 1 ⁇ K 3 ⁇ x 1 ⁇ (85) where ⁇ * ⁇ denotes rounding operation.
  • the lifting stages 905 , 904 , 903 , 902 and 901 of FIG. 9 are inverse to the lifting stages 801 to 805 of FIG. 8 , respectively. Since also the permutation of the input signal corresponding to the matrix P N can be reversed and an according data shuffling stage is comprised by the inverse transforming element, the whole provided method is reversible, thus, if used in the lossless image encoder 703 and the lossless image decoder 705 illustrated in FIG. 7 , providing a method and an apparatus for lossless image coding.
  • the method according to the invention for DCT-IV was used for audio coding and the method according to the invention for DWT-IV was used for image coding
  • the method according to the invention for DCT-IV can as well be used for image coding
  • the method according to the invention for DWT-IV can as well be used audio coding and both can be used as well for the coding of other digital signals, such as video signals.
  • F N [ I _ N / 2 / 2 I _ N / 2 / 2 I _ N / 2 / 2 - I _ N / 2 / 2 ] ⁇ [ I _ N / 2 W _ ] ⁇ [ F _ N / 2 F _ N / 2 ] ⁇ P _ eo ( 87 )
  • F N/2 be the transform matrix of the normalized FFT of order N/2.
  • I N/2 denotes, as above, the identity matrix of order N/2.
  • Equation (87) the first matrix from left is an even-odd matrix P eo , which only reorders the components in the input vector.
  • Equation (87) the third matrix from left is a counter-diagonal matrix, which merely multiplies half of the components in the input vector by a complex number residing on the unit circle.
  • decimation-in-frequency radix-2 FFT algorithm is merely a transposition of the decimation-in-time radix-2 algorithm in Equation (87).

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