CN1399766A - Fast and efficient computation of cubic-spline interpolation for data compression - Google Patents

Abstract

A fast and efficient method and system for computation of cubic-spline interpolation for data compression is described (II). In one aspect, the present invention is a method and system for obtaining a transform of the compressed signal and computing an inverse transform of the transform of the compressed signal to obtain the compressed signal (I).

Description

Be used for the quick and effectively calculating of the cubic spline interpolation of data compression

Invention field

The present invention relates to data compression.More particularly, the present invention relates to a kind of 1-D of being used for and 2-D signal new cubic spline interpolation (CSI) to subsample signal and image compression data.

Background of invention

In most of multimedia systems, the amount of view data is so big, so that the use of Image Data Compression almost is enforceable.Image Data Compression allows image in transmission in real time on the Internet.And it reduces the requirement to the image storage.Current space and ephemeral data reduce technology and all are suitable for, and continue to improve the performance of Image Data Compression.The basic problem of Image Data Compression is increasing compression ratio and reduces computational complexity in the fidelity can receiving.

Interpolation be in the process of the intermediate value of estimating one group of discrete sampling point, can use than one of critical function.Interpolation is widely used in the Image Data Compression, to amplify or to reduce image and correction space distortion.For example, see " cubic convolution interpolation that is used for Digital Image Processing " IEEE Trans.on Acoustics of R.G.Keys, Speech, and Sighal Processing, vol.ASSP-29, no.6, pp.1153-1160, in Dec, 1981 [1], its content by the reference table expressivity be included in here.Usually, the process that reduces data rate is called the process that extracts and increase data sample and is called interpolation, as at " cubic spline that is used for image interpolation and digital filtering " IEEE Trans.on of H.S.Hou and H.C.Andrews Acoustics, Speech, andSignal Processing, vol.ASSP-26, no.6, pp.508-517, as described in Dec, 1978 [2], its content by the reference table expressivity be included in here.

As everyone knows, several interpolating functions (are seen W.K.Pratt, Digital Image Processing, second edition, John Wiley ﹠amp as linear interpolation; Sons, Inc., New York, 1991, [3], its content by the reference table expressivity be included in here.), cubic convolution interpolation (seeing [1] and [3]), cubic B-spline interpolation (at C.de Boor, A Practical Guide to Splines.New York:Springer-Verlag, 1978, [4]; M.Unser, A.Aldroubi and M.Eden, " B-Spline Signal Processing:Part II-Efficient Design andApplications; " IEEE Trans.on Signal Processing, vol.41, pp.834-848, in February, 1993, [5]; M.Unser, A.Aldroubi and M.Eden, " Enlargement orReduction of Digital Images with Minimum Loss of Information; " IEEE Trans.on Image Processing, vol.4, pp.247-258, March nineteen ninety-five, [6]; And describe in [2]) can be used in the Image Data Compression process.

The shortcoming of these interpolation schemes is that generally they are not designed so that the error minimum between original image and its reconstructed image.At Reed (I.S.Reed in 1981, Notes on ImageData Compression Using Linear Spline Interpolation, Department ofElectrical Engineering, University of Southern California, LosAngeles, Califirnia, 90089-2565, U.S.A., in November, 1981 [7], its content is by with reference to being included in here) and at Reed in 1998 and Yu (I.S.Reed and A.Yu, Optimal Spline Interpolation for Image Compression, in 5822456,1998 on October 13, [8] of U.S. Patent No., its content is by with reference to being included in here) developed a kind of linear spline interpolation scheme of the view data that is used for resampling.This linearity spline interpolation is based on having linear interpolation minimum of a function square law.

Use the expansion of the attention of Reed in [7,8], developed the linear spline interpolation algorithm of a kind of modification for the son sampling of view data in the present invention, be called cubic spline interpolation (CSI) algorithm.(in [8], explain and by America On Line ^TM(AOL) the linear spline interpolation of Shi Yonging will be from being called " AOL algorithm " here the document.)

Drawn by [1], the cubic convolution interpolation different with the B spline interpolation can much effectively carry out than cubic B-spline interpolation method.In the present invention, new CSI scheme least square method with by Keys[1] cubic spline function developed for extraction process is combined.And the cubic spline reconstruction is used in the interpolation process.Therefore, CSI constitutes a kind of and cubic B-spline interpolation [2,4-6] and the very different new departure of cubic convolution interpolation [1,3].

The notion that is used for the CSI of 1-D and 2-D signal is described and is shown at following joint.In addition, show that the CSI scheme obtains subjective quality preferably for reconstructed image than linear interpolation, cubic convolution interpolation, cubic B-spline interpolation and linear spline interpolation by computer simulation.A significant advantage of this new CSI scheme is that it can be calculated by the use of FFT technology.The complicacy of CSI computation schemes is basically less than other conventional method.

Its content is by with reference to being included in here W.B.Pennebaker and J.L. Mitchell, JPEG Still Image Data Compression Standard, Van Nostrand Reinhold, New York, 1993, [9] have described JPEG static image data compression standard.As everyone knows, JPEG (seeing [9]) algorithm is the international compression standard that is used for still image.The shortcoming of conventional jpeg algorithm is that when the high quantization parameter was used for obtaining high compression ratio, it caused visual interference blockage effect.One embodiment of the present of invention comprise one than jpeg coder-code translator simple and modification, to improve Joint Photographic Experts Group and a kind of good quality reconstructed image of static maintenance by means of high compression ratio.

Recently, its content by with reference to be included in here at T.K.Truong, L.J.Wang, I.S.Reed, W.S.Hsieh and T.C.Cheng " use the Image Data Compression of three convolution spline interpolations ", accepted for Publication in IEEE Transactions on ImageProcessing[10] the author, proposed to be used for the modification jpeg coder-code translator of τ=2, this jpeg coder-code translator is the pre-treatment level of the CSI scheme with 4 to 1 ratio of compression as jpeg coder, and the cubic spline with 1 to 4 ratio is rebuild as the aftertreatment level against the JPEG code translator, to realize a high compression ratio.

In a kind of like this jpeg coder of modification, the CSI scheme is the pre-treatment level of jpeg coder.It can be realized by the use of fft algorithm.In addition, the packed data that will transmit is represented in the output of revising jpeg coder.It can precomputation and storage.In a kind of like this modification JPEG code translator, cubic spline is rebuild the aftertreatment level that constitutes the JPEG code translator.This aftertreatment level is different with conventional post-processing algorithm, this routine post-processing algorithm propose to hold within it by with reference to be included in here B.Ramamurthi and the non-linear space variable aftertreatment of the understaffed sign indicating number image of A.Gersho " IEEE Trans.on Acoustics; Speech; SignalProcessing; vol.ASSP-34; pp.1258-1267; 1986, [11], its content is by with reference to being included in the Y.Yang here, " spatially adaptive based on projection of piece conversion compressed image is rebuild " IEEE Trans.on ImageProcessing of N.Galatsanos and A.Katsaggelos, vol.4, pp.896-908, July nineteen ninety-five is in [12], to reduce the blockage effect of block-based coding.

The aftertreatment level that proposes is to use a process of cubic convolution interpolation.In [10], computer simulation shows, the modification jpeg coder-code translator that is used for τ=2 obtains than holding within it by with reference to the T.Lane that is included in here, Independent JPEG Group ' s free JPEGsoftware, 1998, [13]; [9] subjective quality of good reconstructed image and objective PSNR.And, revise the computing time that contrary JPEG code translator requirement is lacked than conventional JPEG code translator.But the modification jpeg coder-code translator that is used for τ=2 is in shortcoming, revises jpeg coder desired computing time greater than conventional jpeg coder.

Thereby in one aspect, the present invention describes a kind of fast method of revising jpeg coder that calculates.Show that at this respect of the present invention the speed approximation ratio that is used for calculating the new method of revising jpeg coder still has the fast twice of speed of conventional jpeg coder of the good quality of reconstructed image.

The present invention's general introduction

The present invention describes a kind of 1-D of being used for and the 2-D signal new cubic spline interpolation (CSI) to subsample signal and image compression data.This new interpolation scheme based on the least square method that has a cubic spline function can be implemented by Fast Fourier Transform (FFT) (FFT) and/or by Winograd discrete Fourier transformation (WDFT).The result is the simple and interpolation design fast that a kind of ratio obtains by conventional method.Show that by computer simulation a kind of so new CSI produces and is used for level and smooth exact algorithm.Linear interpolation, linear spline interpolation, cubic convolution interpolation and cubic B-spline interpolation are often relatively poor on performance.In addition, show in the present invention that the CSI scheme can be undertaken by calculating fast and effectively.The proposition method is used better simply technology in sampling process.It requires the addition of significantly lacking than original CSI algorithm and multiplies each other.

In one aspect, the present invention is a kind of being used for: define a cubic spline wave filter; Make wave filter and signal correction to obtain a coherent signal; The autocorrelation filter device is to obtain autocorrelation filter device coefficient; Calculate the conversion and the autocorrelation filter device coefficient of coherent signal; The conversion of coherent signal divided by autocorrelation filter device coefficient to obtain the conversion of compressed signal; And the inverse transformation of conversion of calculating compressed signal to be obtaining compressed signal, method and system.Signal, wave filter, and conversion can be one dimension or bidimensional.And conversion can be Fast Fourier Transform (FFT) (FFT) or have the overlapping Winograd discrete Fourier transformation (WDFT) of avoiding scheme.And, can define a zonal filter, to simplify relevant and autocorrelative step.

And a kind of overlapping scheme of avoiding of newtype can be with solve the problem of boundary conditions that occurs between two adjacent sub-images of the real image that is used for higher compression ratios.Show also that in the present invention a kind of very effective 9 Winograd discrete Fourier transformations (WDFT) can be used for replacing the FFT of the CSI conceptual scheme picture needs of realizing being used for 9 to 1 higher compression ratios.At last, a kind of fast new CSI algorithm is used for designing a kind of modification jpeg coder-code translator that is used for Image Data Compression with Joint Photographic Experts Group (JPEG) standard.As a result of, for higher compression ratios, the modification jpeg coder-code translator of proposition obtain reconstructed image than good quality, and also require computing time of lacking than conventional JPEG method and America on Line (AOL) (America Online) algorithm.

Brief description of the drawings

By the consideration of following the detailed description and the accompanying drawings, it is more obvious that purpose of the present invention, advantage and feature will become, wherein:

Fig. 1 is a demonstration periodic function with cycle of τ=5;

Fig. 2 is a demonstration 1-D cubic spline function;

Fig. 3 is a mobile cubic spline function of demonstration that is used for n=6;

Fig. 4 is the demonstration side view of 2-D cubic spline function;

Fig. 5 is that function is rebuild in a demonstration between sampling period;

Fig. 6 (a)-(d) is used in (25) for the CSI scheme

With in (26) Calculating in the demonstration area mask of the 2-D cubic spline function that uses;

Fig. 7 is the demonstration overlay chart grid point that is used for calculating α for τ=3;

Fig. 8 is the demonstration overlay chart grid point that is used for calculating β for τ=3;

Fig. 9 is the demonstration overlay chart grid point that is used for calculating γ for τ=3;

Figure 10 is the demonstration overlay chart grid point that is used for calculating δ for τ=3;

Figure 11 is 19 * 19 subimages of size 9 * 9 in the exemplary image of size 171 * 171;

Figure 12 is the reconstructed image with serious artistic work, and this image is by using the FCSI method generation by the direct use realization of 9 * 9 Winograd DFT that are used to compress;

Figure 13 is by 5 * 5 Winograd DFT and the overlapping illustrative example of avoiding the FCSI algorithm of subimage method realization;

Figure 14 is the reconstructed image that does not have obvious artistic work, and this image produces by using the FCSI that is realized by the overlapping avoiding method that directly uses and be used to compress of 9 * 9Winograd DFT;

Figure 15 is used for τ=2, a kind of exemplary modification jpeg coder of 3;

Figure 16 is used for τ=2, a kind of exemplary modification jpeg coder of 3;

Figure 17 (a)-(d) shows some reconstructed images with 200: 1 compression ratios.

Describe in detail

The present invention describes a kind of 1-D of being used for and the 2-D signal new cubic spline interpolation (CSI) to subsample signal and image compression data.This new interpolation scheme based on the least square method with cubic spline function can be implemented by Fast Fourier Transform (FFT) (FFT).And in one embodiment of the invention, a kind of overlapping scheme of avoiding of newtype is with solve the problem of boundary conditions that occurs between two adjacent sub-images of the real image that is used for higher compression ratios.In one embodiment of the invention, a kind of effective 9 Winograd discrete Fourier transformations (WDFT) can be used for replacing the FFT of the CSI conceptual scheme picture needs of realizing being used for 9 to 1 higher compression ratios.At last, a kind of fast new CSI algorithm is used for designing a kind of modification jpeg coder-code translator that is used for Image Data Compression with Joint Photographic Experts Group (JPEG) standard.

In order to quicken to be used for the modification jpeg coder of τ=2 with excellent picture quality, the ratio of compression of CSI scheme is expanded to 9 to 1 (τ=3).Yet,, the needs for the other calculating that comprises quite a lot of addition and multiplication are arranged if increase the ratio of compression that is used for the CSI scheme.Therefore, in the present invention, develop a kind of fast new and efficient algorithm that is used for CSI.Be called the basic thought of this new algorithm of quick cubic spline interpolation (FCSI) scheme, be based on the CSI scheme, but than being used for the simpler form of having of original CSI scheme.The FCSI scheme significantly reduces for the needed other complexity of calculation of increasing compression ratio.Moreover, be used for calculating the item that is used for new FCSI scheme Constant alpha, β, γ and the δ that uses at length accurately calculates in the present invention.

For τ=3, the real image that will compress, the overlapping subimage method of avoiding of a kind of novelty is used for obtaining the boundary condition that needs.And implement simplification and the efficient algorithm that a kind of 9 WinogradDFT of use (WDFT) replace FFT, with the compression real image.Computer run shows, size 512 is taken advantage of some gray images of 512, on the 400-MHz Intel Pentium II personal computer that uses the C code, novelly overlappingly avoid sharply to reduce the computing time of the FCSI scrambler that subimage method and 9 Winograd DFT implement by a kind of.

When comparing for about 0.57 second of the original CSI scrambler under the ratio of compression of 9 to 1 (τ=3), the FCSI scrambler only requires about 0.15 second.And, new FCSI scheme obtain one with the similar PSNR of other more complicated zonal filter of consideration in the present invention.Finally, it is combined quickening the being used for modification jpeg coder of color image encoding to have this FCSI schemes of τ=3 and a Joint Photographic Experts Group, and still obtains than the measured reconstructed image of jpeg algorithm matter that is used for higher compression ratios.In other words, the modification jpeg coder that is used for τ=3 requires 0.71 second, 0.38 second and 0.67 second, respectively than being used for τ=2[10] time of modification jpeg coder, conventional jpeg coder [13] and AOL algorithm [8] lack.

The document is pressed as undertissue: in the II joint, at length derive the encryption algorithm that is used for this new interpolation method.In addition, show now that the character of FFT and convolution theorem can be used for calculating the CSI scheme.Decoding algorithm is explained in the III joint.In the IV joint, describe FCSI and calculate.In the V joint, calculate the needed constant of FCSI.In the VI joint, develop a kind of novel overlapping novel FCSI algorithm of avoiding subimage technology and Winograd DFT algorithm that uses.In the VII joint, present the jpeg coder-code translator of modification.Finally, test findings is presented in the VIII joint.

II. the encryption algorithm that is used for new interpolation method

By means of the coding of CSI scheme, utilize and carry out the extraction process that Image Data Compression needs.The ultimate principle of CSI scheme is, uses cubic spline function to recomputate the sample value of signal or view data by means of least square method.Show that in this joint the method for this new proposition is by following 1-D and the 2-D signal of being applied to:

A. the CSI that is used for the 1-D signal

Allow the τ be a fixing positive integer.And allow the data function X (t) be have cycle n τ periodic, wherein n is an integer.An example for the X (t) of n=5 shows in Fig. 1.By Fig. 1, the 1-D cubic spline function R (t) that is illustrated among Fig. 2 is defined by following formula:

Secondly need by as the move function of cubic spline function R (t) that give a definition:

Ψ _k(t)＝R(t-kτ)for?0≤k≤n-1.????????????????????????(2)

Move function Ψ for n=6 _k(t) a example is illustrated among Fig. 3.Target is by the n point and is similar to X (t), provided by following formula $S\left(t\right)=\underset{k=0}{\overset{n-1}{\mathrm{\Σ}}}{X}_k{\mathrm{\ψ}}_k\left(t\right)=\underset{k=0}{\overset{n-1}{\mathrm{\Σ}}}{X}_{k}R(t-\mathrm{k\τ})---\left(3\right)$

In the least square pattern, X wherein ₀..., X _N-1It is the value of reconstructing at the sample point place that represents the packed data that will transmit or store.Function S in (3) (t) is to use weight X ₀..., X _N-1The cubic spline of function X (t) rebuild.Drawn by [7], S (t) is defined by following formula to the least square approximation of X (t) $L({\mathrm{<figure-callout\; id="0"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_0,{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_1,\&CenterDot;\&CenterDot;\&CenterDot;X_{n-1})=\underset{t=-2\mathrm{\&tau;}}{\overset{\mathrm{n\&tau;}}{\mathrm{\&Sigma;}}}{(X\left(t\right)-S\left(t\right))}^2=\underset{t=-2\mathrm{\&tau;}}{\overset{\mathrm{n\&tau;}}{\mathrm{\&Sigma;}}}{(X\left(t\right)-\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_k{\mathrm{\&psi;}}_k\left(t\right))}^2,---\left(4\right)$

Wherein add on 2 τ and sue for peace at the one-period n of data τ.Fig. 1 represents to have the demonstration periodic function of cycles 5 τ.Fig. 2 is a demonstration 1-D cubic spline function, and Fig. 3 is for the mobile cubic spline function of the demonstration of n=6.

The identical process that use is described in [7,8] can be obtained and make function L (X in (4) ₀, X ₁..., X _N-1) minimum weight X ₀, X ₁..., X _N-1In order to make (4) minimum, L (X ₀, X ₁..., X _N-1) about X _jPartial differential produce following formulary for 0≤j≤n-1: $\frac{\&PartialD;L({\mathrm{<figure-callout\; id="0"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_0,{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_1,\&CenterDot;\&CenterDot;\&CenterDot;X_{n-1})}{\&PartialD;X_j}=\underset{t=-2\mathrm{\&tau;}}{\overset{\mathrm{n\&tau;}}{\mathrm{\&Sigma;}}}2(X\left(t\right)-\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_k{\mathrm{\&Psi;}}_k\left(t\right)),{\mathrm{\&psi;}}_j\left(t\right)=0,$

Or $\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{A}_{j,k}{X}_k=Y_j\mathrm{for}0\&le;j\&le;n-1,---\left(5\right)$

Wherein $A_{j,k}=\underset{t=-2\mathrm{\&tau;}}{\overset{\mathrm{n\&tau;}}{\mathrm{\&Sigma;}}}{\mathrm{\&psi;}}_k\left(t\right){\mathrm{\&psi;}}_j\left(t\right)\mathrm{for}0\&le;j,k\&le;n-1,---\left(6\right)$

With $Y_j=\underset{t=-2\mathrm{\&tau;}}{\overset{\mathrm{n\&tau;}}{\mathrm{\&Sigma;}}}X\left(t\right){\mathrm{\&psi;}}_j\left(t\right)\mathrm{for}0\&le;j\&le;n-1.---\left(7\right)$

Item Y in (7) _jCan deduct by following: $Y_j=\underset{t=-2\mathrm{\&tau;}}{\overset{\mathrm{n\&tau;}}{\mathrm{\&Sigma;}}}X\left(t\right)R(t-\mathrm{j\&tau;})=\underset{t=(j-2)\mathrm{\&tau;}}{\overset{(j+2)\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}X\left(t\right)R(t-\mathrm{j\&tau;}).---\left(8\right)$

Allow t-j τ=m, so $Y_j=\underset{m=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m+\mathrm{j\&tau;})R\left(m\right).---\left(9\right)$

Note, in (9), calculate Y _jRelate to n related coefficient having only 3 τ-1 points.

Allow the cubic spline function periodically be R (t)=R (t+n τ) now, promptly R (t) has the cycle of n τ.A in (6) then _{J, k}Matrix form can be simplified to: $A_{j,k}=\underset{m=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m+(j-k)\mathrm{\&tau;})R\left(m\right)$

Draw by [7], by allowing the A of following formula establishment in (10) _{J, k}Can represent with circulation form $A_{j,k}=B_{{(k-j)}_n},---\left(11\right)$ Wherein (k-j) _nIndication residue (k-j) mould n, and

B ₀＝α，B ₁＝β，B ₂＝γ，B ₃＝δ，B ₄＝0，…，B _n-4＝0，B _n-3＝δ，B _n-2＝γ，(12)

B _n-1＝β。Therefore, the A in (11) and (12) _{J, k}Have following symmetry, cyclic representation: (13) are updated to generation Matrix Formula in (5),

A·X＝Y，????????????????????????????(14)

Wherein matrix A provides in (13), X=(X ₀, X ₁..., X _N-1) ^TAnd Y=(Y ₀, Y ₁..., Y _N-1) ^T

Because the n * n matrix in (14) left side is a circular matrix, so (14) are reduced to immediately $Y_j=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_k{B}_{{(k-j)}_n}\mathrm{for}0\&le;j\&le;n-1.---\left(15\right)$

But for i=1,2 ...,

, wherein Indication is less than or equal to the maximum integer of x.Thereby (15) become $Y_j=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_k{B}_{{(j-k)}_n}\mathrm{for}0\&le;j\&le;n-1.---\left(16\right)$

FFT can be used for obtaining X in (16) _kIn order to see this point, allow Y _j, X _kAnd B _jFor 0≤j, k, the FFT of m≤n-1 respectively by

With

Definition.By use hold within it by the reference table expressivity be included in the The FastFourier Transform and its Application of the E.O.Brigham here, Prentice-Hall International, Inc., Englewood Cliffs, New Jersey, 1988[14]; With the Digital Signal Processing of A.V.Oppenheim and R.W.Schafer, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1975[15] the middle convolution theorem of describing, see that easily separating of (16) can be expressed as in frequency domain ${\tilde{Y}}_m={\tilde{X}}_m\&CenterDot;{\tilde{B}}_m$ Or ${\tilde{X}}_m={\tilde{Y}}_m/{\tilde{B}}_m$ , wherein ${\tilde{B}}_m\&NotEqual;0.$ Thereby, use Contrary FFT, can obtain X for 0≤k≤n-1 _k

The coding method that is used for the 1-D signal is summarized as follows:

Select the appropriate value of integer τ.Compression ratio is τ roughly.

Use (9) and obtain Y _jAnd use (10), (11) and (12) obtain B _j

Obtain Y _jAnd B _jFFT to obtain respectively With Also calculate ${\tilde{X}}_m={\tilde{Y}}_m/{\tilde{B}}_m.$

Get Contrary FFT be the X of the packed data that will transmit or store to obtain _k

B. the CSI that is used for the 2-D signal

Allow X (t ₁, t ₂) be about integer variable t ₁And t ₂Cycle n ₁τ and n ₂The double periodicity signal of τ (for example image), wherein n ₁And n ₂It also is integer.A 2-D cubic spline function R (t ₁, t ₂) define by following formula:

R(t ₁，t ₂)＝R(t ₁)·R(t ₂)，?????????????(17)

R (t wherein ₁) and R (t ₂) be respectively the 1-D cubic spline function.This cubic spline function is illustrated among Fig. 4.Well known fact is, the 2-D interpolation can be finished [1,3] by the use with respect to the 1-D interpolation of each coordinate.

Simulation by for the 1-D situation allows ${\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,k_2}({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2)=R(t_1-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{\&tau;},t_2-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2\mathrm{\&tau;})$

＝R(t ₁-k ₁τ)·R(t ₂-k ₂τ)for?0≤k _i≤n _i-1?and?i＝1，2.????(18)

By one with 1-D situation in (3) in the similar process used, 2-D CSI by as give a definition: $S({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2)=\underset{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,k_2}({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2),---\left(19\right)$

Wherein

It is the value of reconstructing at the sample point place that represents the compressed image that will transmit or store.Want to obtain best weight equally Thereby, $L\left(X_{k_1,k_2}\right)=\underset{t_1=-2\mathrm{\&tau;}}{\overset{n_1\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}\underset{t_2=-2\mathrm{\&tau;}}{\overset{n_2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}(X(t_1,t_2)-\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{\mathrm{\&psi;}}_{k_1,k_2}(t_1,t_2){)}^2---\left(20\right)$ It is a minimum value.Draw by [7], minimize (20) and produce $\frac{\&PartialD;L\left(X_{k_1,k_2}\right)}{\&PartialD;X_{j_1,j_2}}=\underset{t_1=-2\mathrm{\&tau;}}{\overset{n_1\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}\underset{t_2=-2\mathrm{\&tau;}}{\overset{n_2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}2(X(t_1,t_2)-\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{\mathrm{\&psi;}}_{k_1,k_2}(t_1,t_2)){\mathrm{\&psi;}}_{j_1,j_2}(t_1,t_2)=0,$ Or $\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{A}_{j_1{j}_2,k_1{k}_2}=Y_{j_1,j_2},0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2}.---\left(21\right)$ Wherein $A_{j_1{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k_1{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}=\underset{{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=-2\mathrm{\&tau;}}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_1\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=-2\mathrm{\&tau;}}{\overset{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}{\mathrm{\&Psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_1,j_2}({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2){\mathrm{\&Psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,k_2}({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2),0\&le;j_i,k_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2},---\left(22\right)$ With ${\mathrm{<figure-callout\; id="1"\; label="Y\; j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\underset{{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=-2\mathrm{\&tau;}}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_1\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-2\mathrm{\&tau;}}{\overset{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}X({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2){\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_1,j_2}({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2),0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2}.---\left(23\right)$

Item in (23) Can deduct by following: ${\mathrm{<figure-callout\; id="1"\; label="Y\; j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\underset{{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=-2\mathrm{\&tau;}}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_1\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-2\mathrm{\&tau;}}{\overset{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}X({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2)R(t_1-{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\mathrm{\&tau;},t_2-{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\mathrm{\&tau;})---\left(24\right)$ $=\underset{{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=(j_1-2)\mathrm{\&tau;}}{\overset{({\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_1+2)\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=(j_2-2)\mathrm{\&tau;}}{\overset{({\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2+2)\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}X({\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2)R(t_1-{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\mathrm{\&tau;},t_2-{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\mathrm{\&tau;})$

For k=1,2 allow t _k-j _kτ=m _k, so ${\mathrm{<figure-callout\; id="1"\; label="Y\; j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\underset{{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X({\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_1+{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\mathrm{\&tau;},{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_2+{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\mathrm{\&tau;})R({\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_1,m_2),---\left(25\right)$

R (m wherein ₁, m ₂) be the 2-D cubic spline function that is illustrated among Fig. 4.Item in (22) In a similar manner by following processing: $A_{j_1{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k_1{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}=\underset{{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R({\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_1+(j_1-k_1)\mathrm{\&tau;},{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_2+(j_2-k_2)\mathrm{\&tau;})R({\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_1,m_2)---\left(26\right)$

By the above formula that provides (26), array

Press following expression with the 2-D circulation form: $A_{j_1{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k_1{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}=B_{{(k_1-j_1)}_{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_1},{(k_2-j_2)}_{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_2}},---\left(27\right)$

Wherein Indication is for i=1,2 residue (k _i-j _i) mould n _i, and

Wherein for i=1,2,0≤s _i≤ n _i-1.Note, if array [

] represent that with matrix form then it is a block circulant matrix.Fig. 4 is the side view of 2-D cubic spline function.

Because the matrix in (28) [ ] be a block circulant matrix, so (21) can be expressed from the next $Y_{j_{}}$ ${{}_1,{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">j</figure-callout>}}_2}_{}=\underset{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{B}_{{(k_1-j_1)}_{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_1},{(k_2-j_2)}_{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">\; <figure-callout\; id="0"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>\; </figure-callout>}}_2}},0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2},---\left(29\right)$

Wherein Indication is for i=1,2 (k _i-j _i) mould n _iUse and (16) the middle similar process of using, (29) become then ${\mathrm{<figure-callout\; id="1"\; label="Y\; j"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\underset{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{B}_{{(j_1-k_1)}_{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>}}_1},{(j_2-k_2)}_{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">\; <figure-callout\; id="0"\; label="n"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">n</figure-callout>\; </figure-callout>}}_2}},0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2}.---\left(30\right)$

2-D FFT[14 in (30), 15] can be used for obtaining Allow With

For 0≤j _i, k _i≤ n _i-1 and 0≤s _i≤ n _i-1 2-D FFT is for 0≤m≤n _i-1,0≤n≤n ₂-1 respectively by

With Definition.Separating of (30) can be expressed as in frequency domain so

, wherein ${\tilde{B}}_{m,n}\&NotEqual;0$ 。At last, use The 2-D function of contrary FFT, for 0≤k _i≤ n _i-1, i=1,2 can obtain

The coding method that is used for the 2-D signal is summed up as follows:

■ selects the appropriate value of integer τ.Compression ratio is τ roughly ²

■ uses (25) and obtains

Secondly, application (26), (27) and (28) are obtained

■ gets With

2-DFFT to obtain respectively

With

Also calculate

■ gets

Contrary FFT be the packed data that will transmit or store to obtain

III. decoding algorithm

In decode procedure, use in the II joint, obtain at sampling spot (X for example _kWith The reconstructed value at place obtains reconstruction point between sampling spot by means of cubic spline function.This decoding algorithm is called cubic spline and rebuilds.

A. compress the decoding of 1-D signal

Because n reconstructed value X ₀..., X _N-1Be known, so can obtain reconstruction signal S (t) by the use of (3).In other words, recall signal is the convolution of the cubic spline function R (t) of definition in (1) and the sequence with n reconstructed value of sample interval τ.At two adjacent reconstructed value X _kWith X _K+1Between reconstruction function S (t _a) show in Fig. 5, and provide by summation,

S(t _a)＝X _k-1Ψ _k-1(ta)+X _kΨ _k(t _a)+X _k+1Ψ _k+1(t _a)+X _k+2Ψ _k+2(t _a)，(31)

K τ＜t wherein _a＜(k+1) τ and Ψ _k(t) be defined in (2), and given boundary condition is X in [1] _-1=3 (X ₀-X ₁)+X ₂And X _n=3 (X _N-1-X _N-2)+X _N-3Fig. 5 is the reconstruction function between sampling spot.

B. compress the decoding of 2-D signal

Because for 0≤k _i≤ n _i-1, i=1,2 reconstructed value Be known, so can obtain 2-D reconstructed image S (t by the use of (19) ₁, t ₂).In other words, retrieving images is the 2-D cubic spline function R (t that provides in (17) ₁, t ₂) 2-D convolution and 2-D sampling waveform As everyone knows, the calculating simpler method of describing in [3] is called bilinear interpolation, also can be used for carrying out the 2-D interpolation.The idea of bilinear interpolation can be used for realizing the reconstruction of 2-D cubic spline.In other words, the discrete data of every row can be from reconstructed value Interpolation has similar interpolation for the discrete data that provides of every row.Show the reconstructed image S (t between four adjacent reconstructed value easily _i, t _j) provide by following formula, $S(t_i,t_j)=\underset{m=-1}{\overset{2}{\mathrm{\&Sigma;}}}\underset{n=-1}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_1+m,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+n}(t_i,t_j),---\left(32\right)$

K wherein ₁τ＜t _i＜(k ₁+ 1) τ, k ₂τ＜t _j＜(k ₂+ 1) τ and the boundary condition that provides in [1] are $X_{-1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}=3({\mathrm{<figure-callout\; id="0"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{0,k_2}-{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{1,k_2})+{\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{2,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2},X_{n,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}=3(X_{n-1,k_2}-X_{n-2,k_2})+X_{n-3,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">k</figure-callout>}}_2},{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}=3\left({\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}\right)$ $-{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}})+{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">\; <figure-callout\; id="2"\; label="X\; k"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>\; </figure-callout>}}_{k_{}{}_{}},{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}=3({\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}-{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}})+{\mathrm{<figure-callout\; id="1"\; label="X\; k"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}},X_{-\mathrm{1,1}}=3({\mathrm{<figure-callout\; id="0"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{0,-1}-{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{1,-1})+{\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="A0081361300502.png,A0081361300521.png"\; state="\{\{state\}\}">X</figure-callout>}}_{2,-1,}$

X _n-1＝3(X _n-1，-1-X _n-2，-1)+X _n-3，-1，X _-1，n＝3(X _0，n-X _1，n)+X _2，n，X _n，n＝3(X _n-1，n-

X _n-2，n)+X _n-3，n。

The quick calculating of IV.CSI

In fact, the CSI scheme needs 2-D cubic spline function R (t ₁, t ₂) a large amount of pixels so that calculate in (25)

With in (26)

For example, for τ=2 (ratio of compression 4: 1) and τ=3 (ratio of compression 9: 1) calculating each in (25) With in (26) , relate separately to function R (t ₁, t ₂) 81 and 169 pixels of field of definition.As a result, (25) that need in the CSI scheme and the complexity of calculation of (26) expand to τ=3 at ratio of compression and o'clock enlarge markedly from τ=2.

In order to show this point, for τ=3, some simplified solutions are used for overcoming the above problem of extra computation complicacy.At first, definition list is shown in the R (t among Fig. 6 (a) ₁, t ₂) the region mask of 169 pixels, be called zonal filter 1.This means, in order to calculate each in (25) , need to use R (t ₁, t ₂) 169 pixels region mask with one-period image X (t ₁, t ₂) relevant.And, in order to calculate each in (26)

, need to be used for function R (t ₁, t ₂) 169 pixels with R (t ₁, t ₂) other 169 pixel auto-correlations.

Fig. 6 (a)-6 (d) is for CSI scheme each in (25) With in (26)

Calculating in the 2-D cubic spline function that uses.Fig. 6 (a) is illustrated in 169 pixels in the zonal filter 1, and Fig. 6 (b) is illustrated in 133 pixels in the zonal filter 2, and Fig. 6 (c) is illustrated in 69 pixels in the zonal filter 3, and Fig. 6 (d) is illustrated in 25 pixels in the zonal filter 4.In order to reduce the computational complexity of this zonal filter 1, next proposes to use the zonal filter 2 that is illustrated in Fig. 6 (b), the zonal filter 4 that is illustrated in the zonal filter 3 of Fig. 6 (c) and is illustrated in Fig. 6 (d).These zonal filters 2,3 and 4 use R (t respectively _l, t ₂) 133,69 and 25 pixels or grid point, to calculate each in (25) With in (26) each

Secondly for τ=3, by the above-mentioned FCSI scheme of use exploitation of zonal filter 2, zonal filter 3 and zonal filter 4.According to the identical process in the former joint, obtain FCSI scheme easily for τ=3.At first, produce deviation [10] for these FCSI schemes of τ=3 by similar means a kind of and for the original CSI scheme of τ=2.For the FCSI scheme only in (25) the item With in (26)

Slightly different with original CSI scheme.And in the FCSI scheme, for the item of zonal filter 2, zonal filter 3 and zonal filter 4

With Complicacy be summarised among Table I and the II, and by following description:

1. by Fig. 6 (a), with (25) and (26) similarly mode obtain fully being used for zonal filter 1 With

It uses R (t ₁, t ₂) 169 pixels calculate each And each

This algorithm is very complicated, and original just CSI scheme.

2. by Fig. 6 (b), obtain item for zonal filter 2 , and except that σ=0, the result is similar with (26).For zonal filter 2 usefulness R (t ₁, t ₂) 133 pixels calculate in (25) each

This algorithm still relates to calculating widely.

3. by Fig. 6 (c), obtain item for zonal filter 3

, and result and (26) are similar, but for this situation μ=0 and σ=0.For zonal filter 3 usefulness R (t ₁, t ₂) 69 pixels calculate in (25) each This algorithm is also complicated.

4. by Fig. 6 (d), obtain item for zonal filter 4 , and result and (26) are similar, but η=λ=ρ=u=σ=0 in this case.For zonal filter 4 by only using R (t ₁, t ₂) 25 pixels calculate in (25) each This algorithm is than all compact at any of above other calculating that provides in situation 1,2 and 3.

Table I

Four zonal filters of use Complicacy

????λ	Non-zero	Non-zero	Non-zero	Zero
					????ρ	Non-zero	Non-zero	Non-zero	Zero
????μ	Non-zero	Non-zero	Zero	Zero
					????σ	Non-zero	Zero	Zero	Zero

Table II

Ratio of compression with 9 to 1 is taken advantage of four zonal filters calculating of image use of 512 by size 512

Complicacy

In the VIII joint, show that by computer run zonal filter 4 obtains and any one similar PSNR of other three zonal filters.Thereby these zonal filter 4 representatives are used for the most realistic and simple zonal filter of FCSI scheme.The major advantage that FCSI scheme with zonal filter 4 is better than original CSI scheme is that it significantly reduces computational complexity.

V. the calculating of constant

Reach with prosthomere according to (27), (28), for the item of zonal filter 4 needs Provide by following general formula:

Perhaps provide by following array,

Constant alpha, β, γ and δ are at 2-D splines R (m ₁, m ₂) between coefficient of autocorrelation.In following analysis, suppose that τ and m are integers, and R (t) is the 1-D cubic spline function of definition in (1).By the use of (33), by following constant alpha, β, γ and the δ of obtaining:

1) calculating of α: by the Fig. 7 for τ=3, the value of α is 2-D splines R (m ₁, m ₂) overlapping value sum, provide by following formula $\mathrm{\&alpha;}=1+4\&times;\underset{m=1}{\overset{2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}{R}^2(m/\mathrm{\&tau;}).---\left(34\right)$

2) calculating of β: by the Fig. 8 for τ=3, the value of β is 2-D splines R (m ₁, m ₂) and mobile 2-D splines R (m ₁+ τ, m ₂) overlapping value sum, provide by following formula $\mathrm{\&beta;}=\underset{m=1}{\overset{\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m/\mathrm{\&tau;})R((\mathrm{\&tau;}-m)/\mathrm{\&tau;})+2\&times;\underset{m=0}{\overset{\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}R((\mathrm{\&tau;}+m)/\mathrm{\&tau;})R(m/\mathrm{\&tau;}).---\left(35\right)$

3) calculating of γ: by the Fig. 9 for τ=3, the value of γ is 2-D splines R (m ₁, m ₂) and mobile 2-D splines R (m ₁+ 2 τ, m ₂) overlapping value sum, provide by following formula $\mathrm{\&gamma;}=\underset{m=0}{\overset{2\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}R(m/\mathrm{\&tau;})R((2\mathrm{\&tau;}-m)/\mathrm{\&tau;}).---\left(36\right)$

4) calculating of δ: by the Figure 10 for τ=3, the value of δ is 2-D splines R (m ₁, m ₂) and mobile 2-D splines R (m ₁+ 3 τ, m ₂) overlapping value sum, provide by following formula $\mathrm{\&delta;}=\underset{m=0}{\overset{\mathrm{\&tau;}}{\mathrm{\&Sigma;}}}R((\mathrm{\&tau;}+m)/\mathrm{\&tau;})R((2\mathrm{\&tau;}-m)/\mathrm{\&tau;}).---\left(37\right)$

For τ=3, above formula produces the following parameter for α, β, γ and δ specially: $\mathrm{\&alpha;}=1+4\&times;\underset{m=1}{\overset{2\&times;3}{\mathrm{\&Sigma;}}}{R}^2(m/3)=3.8916,$ $\mathrm{\&beta;}=\underset{m=1}{\overset{3-1}{\mathrm{\&Sigma;}}}R(m/3)R((3-m)/3)+2\&times;\underset{m=0}{\overset{3}{\mathrm{\&Sigma;}}}R((3+m)/3)R(m/3)=0.3786.$ $\mathrm{\&gamma;}=\underset{m=0}{\overset{2\&times;3}{\mathrm{\&Sigma;}}}R(m/3)R((2\&times;3-m)/3)=-0.1070,\mathrm{\&delta;}=\underset{m=0}{\overset{3}{\mathrm{\&Sigma;}}}R((3+m)/3)R((2\&times;3-m)/3)=0.0055.$

Fig. 7 represents to be used for calculating for τ=3 the overlap mapping grid point of α, Fig. 8 represents to be used for calculating for τ=3 the overlap mapping grid point of β, Fig. 9 represents to be used for calculating for τ=3 the overlap mapping grid point of γ, and Figure 10 represents to be used for calculating for τ=3 the overlap mapping grid point of δ.

VI. the novel FCSI algorithm of implementing by Winograd DFT and overlapping avoiding method

Hold within it by with reference to be included in here Dean P.Kolba and the prime factor fft algorithm of high speed convolution " use " IEEE Trans.on Acoustics.Speech of Thomas W.Parks, and Signal Processing, vol.ASSP-25, No.4, pp.281-294, in August, 1977 [16]; " about calculating discrete Fourier transformation " Mathematics ofComputation of S.Winograd, vol.32, No.141, pp.175-199, in January, 1978 [17]; And I.S.Reed,, T.K.Truong, R.L.Miller and B.Benjauthrit " about for n=4,5,6,8 at GF (2 ⁿ) on be used for deciphering the other result of the Fast transforms of Reed-Solomon sign indicating number " in the Deep Space Network Progress Report 42-50.Jet PropulsionLaboratory; Pasadena; CA; pp.132-155; the Winograd DFT algorithm of describing in January, 1979 and February [18]; be used in this joint by means of a kind of novel overlapping avoiding method, with for τ=3 realization FCSI schemes.

Consider the image of size N * N=512 * 512 pixels.If the FCSI scheme for τ=3 is used for compressing this original image, then the size of compressed image is reduced to

Pixel, wherein Indication is more than or equal to the smallest positive integral of x.Because 171 is not two power, so 2-D FFT can not be used for obtaining in (30)

In order to overcome this problem, a kind of possible method is by appending to the original edge of image that reduces zero, the data in (25)

With in (28)

, 0≤j wherein _i, s _i≤ n _i-1 for i=1, and 2, expand to 256 * 256 pixels from 171 * 171 pixels.Get then With

2-D FFT to obtain respectively With

This can produce Finally, can get

Contrary 2-D FFT with for

, 0≤k wherein ₁, k ₂≤ 255, obtain the packed data that will transmit or store.The shortcoming of using a kind of like this 2-D FFT method to calculate for the CSI scheme of τ=3 is, requires more computing time because data increase it from the size of 171 * 171 pixels to 256 * 256 pixels.

Because the size of the packed data that will transmit for τ=3 does not always equal two power, so in order to compress a kind of like this odd number sized image, a kind of 9 WinogradDFT (WDFT) that describe below are used for realizing the FCSI scheme.At first, because 171=9 * 19, so With , 0≤j wherein ₁, j ₂, s ₁, s ₂≤ 170, can be divided into 19 * 19 subimages, each has size 9 * 9 pixels, is illustrated among Figure 11.May think that the 2-D Fourier transform of each subimage can directly be passed through the use realization of 9 * 9WDFT algorithm, and size 171 * 171 With

The 2-D Fourier transform

With Can obtain respectively.In addition, by divided by

Can calculate

At last, may expect, Contrary 2-D Fourier transform can obtain by the use of 9 * 9WDFT algorithm.In other words, can obtain the reconstructed image of size 171 * 171.Yet test finds that this inverse transformation has serious artifact.See in this reconstructed image of in Figure 12, representing.Figure 12 shows the reconstructed image of the serious artifact that has the FCSI method that use realizes by the direct use of the 9 * 9Winograd DFT that is used to compress.Figure 11 is illustrated in 19 * 19 subimages of the size 9 * 9 in the exemplary image of size 171 * 171.

In order to remove the artifact of in Figure 12, finding by the direct use of above-mentioned 9 * 9WDFT compression, a kind of overlapping subimage technology of avoiding of novel type is applied to the FCSI method.An illustrative example of FCSI algorithm is at first realized by following simplification 5 * 5WDFT.This novel overlapping subimage method of avoiding shows that in this example, its calcspar is depicted among Figure 13.

Figure 13 is by a 5 * 5Winograd DFT and an overlapping simple declaration example avoiding the FCSI algorithm of subimage method realization.The calcspar that is illustrated among Figure 13 is separated into the two parts of being indicated by dotted line.First by " I " mark, is a FCSI scrambler avoiding the subimage technology to use 5 * 5WDFT algorithm by means of overlapping.Second portion by " II " mark, is a FCSI code translator.In first, consideration is the source image data of size 24 * 24 pixels of expression in Figure 13 (a).By means of the use of zonal filter 4, produce the coefficient obtain size 8 * 8 pixels in Figure 13 (b), described for (25) of τ=3 What need reduces image.Secondly these coefficients Be divided into four overlapping 5 * 5 subimages of expression in Figure 13 (c).Notice that each subimage and each adjacent sub-images of size 5 * 5 pixels are overlapping, have the border of width 2.Test shows that when using Winograd DFT algorithm, this border can be used for obtaining the boundary condition between two adjacent sub-images.

As showing among Figure 13, use following three step handles Four overlapping 5 * 5 subimages transform to Four overlapping 5 * 5 subimages of correspondence.The first step is to get 5 * 5WDFT of four overlapping 5 * 5 subimages, to obtain Four conversion, 5 * 5 corresponding subimages.Second step was a handle This four 5 * 5 subimages divided by in (33) Or for τ=3

, to obtain

Four conversion of 5 * 5 subimages.The 3rd the step be for These four 5 * 5 subimages get contrary 5 * 5WDFT, finally to obtain shown in Figure 13 (d)

Four overlapping 5 * 5 subimages of correspondence.

Because

The superposition boundary of 5 * 5 subimages in some of pixel appear at Other adjacent 5 * 5 subimages in, so the deletion or remove

Four overlapping 5 * 5 subimages in duplicate pixel.By means of this means,

Four overlapping 5 * 5 subimages become Four non-overlapped 4 * 4 subimages.In order to show this point, in Figure 13 (d), each 5 * 5 subimage has the superposition boundary of width 2; Figure 13 (e) shows by using the residue sample of each subimage that this overlapping avoiding method obtains.In Figure 13 (d), at first consider to have all four number of sub images of superposition boundary at column direction.Because end effect, subimage 1 and 3 rank rear are the replicated columns in the superposition boundary that will remove.

Yet in the subimage 2 and 4 in Figure 13 (d), first row of this two number of sub images also are the replicated columns in the superposition boundary of needs deletion.At last, finish above overlapping avoiding method similarly with column direction at line direction.

The combination results of these four non-overlapped 4 * 4 subimages be illustrated among Figure 13 (f) Whole 8 * 8 subimages.These View data is the packed data that will transmit or store.In second portion, use whole 8 * 8 Packed data is rebuild function by means of the cubic spline that provides and is obtained being illustrated in 24 * 24 data reconstructions among Figure 13 (g) in (32).

Thereby the step that is included in image of compression is: define a cubic spline wave filter; Make wave filter and signal correction, to obtain a coherent signal; Make the wave filter auto-correlation, to obtain autocorrelation filter device coefficient; Calculate coherent signal and the transformation of coefficient of autocorrelation filter device; The conversion of coherent signal divided by the transformation of coefficient of autocorrelation filter device, to obtain the conversion of a compressed signal; And the inverse transformation of calculating compressed signal, to obtain compressed signal.

For by a kind of so overlapping real image of avoiding a size 512 * 512 of subimage method compression, 9 * 9WDFT rather than 5 * 5WDFT are used for the FCSI scheme.Then, use 9 * 9Winograd DFT and the overlapping FCSI encryption algorithm of subimage method of avoiding to be summarised in the following steps:

Select τ=3.Ratio of compression approximately is τ ²=9.

(25) that have zonal filter 4 are applied to the original image of size 512 * 512, obtaining 171 * 171 all coefficients,

Also use (33) to obtain Get then

9 * 9WDFT obtaining 9 * 9 coefficients,

All coefficients Be divided into border with width 2 Suitable overlapping 9 * 9 subimages.

Get 9 * 9WDFT of all overlapping 9 * 9 subimages is to obtain

The conversion of correspondence 9 * 9 subimages.

Calculate , to obtain

Each 9 * 9 subimage.

Get

The contrary 9 * 9WDFT of all 9 * 9 subimages, to obtain

9 * 9 subimages.

Remove

The superposition boundary of two adjacent 9 * 9 subimages in duplicate pixel.By this means, Overlapping subimage become Non-overlapped subimage.

Each non-overlapped subimage combined to obtain Entire image, 0≤k1 wherein, k2≤511.These view data It is the packed data that will transmit or store.

In the FCSI decoding algorithm, packed data Constitute encryption algorithm; Rebuild function by means of the cubic spline in (32) and obtain data reconstruction.Therefore,, obtain compression and reconstructed image, and test and find there is not tangible artifact, as shown in Figure 14 by means of the above-mentioned overlapping image subsection technology of avoiding.Figure 14 is to use the reconstructed image that does not have obvious artifact of the FCSI that is realized by 9 * 9Winograd DFT that is used to compress and overlapping avoiding method.

VII. revise jpeg coder-code translator for one kind

In this joint, present a kind of jpeg coder-code translator of modification for Image Data Compression.This algorithm is having τ ²The CSI of the ratio of compression to 1 or FCSI scheme are as the jpeg coder pre-treatment step for τ=2,3, as shown in Figure 15.As a result, have 1 to τ ²The cubic spline of ratio is rebuild this JPEG code translator post-processing step that is used for for τ=2,3, as shown in Figure 16.For this compression algorithm, prepare image at the another kind that before CSI or the FCSI pre-service original image in RGB (red, green and blue) color space is converted in the YUV color space.

This YUV image is followed has format 4: 1: 1 CCIR 601 color spaces.The supposition of original RGB size of images is 512 * 512 * 3=786,432 bytes, i.e. and 512 * 512=262, each collection of 144 bytes are used for redness, green and one of blue.After color space conversion, for a collection of Y use 512 * 512 bytes, and for U and V color component use 256 * 256=65, two collection of 536.By its content by the Practical Dogital Video with ProgrammingExamples in C with reference to the Phillip E.Mattison in being included in, John Wiley ﹠amp; Sons, Inc., 1994[19] draw, being used for from the formula of the conversion of RGB to YUV is Y=0.299R+0.587G+0.114B, U=0.493 (B-Y)=0.463B-0.147R-0.289G and V=0.877 (R-Y)=0.615R-0.515G-0.100B.

Two treatment steps are arranged in encoder stage.The first step is that each use for Y, U and V image has τ ²The pre-service of the CSI of ratio of compression or FCSI scheme to 1.In this process, input picture is the Y image of size 512 * 512 bytes, and output image is a size

The coded image of byte, wherein

Indication is more than or equal to the smallest positive integral of x.For U and V image, input picture has 256 * 256 bytes, thereby the output image that will encode is Byte.In other words, two kinds of situations using are arranged in this step.For first kind of situation τ=2, the CSI scheme that directly realizes by means of fft algorithm is used for original Y, U and V image, and output image is 256 * 256 bytes for the Y image, and is 128 * 128 bytes for U and V image.

For second kind of situation τ=3, be used for original Y, U and V image by means of overlapping avoiding method by the FCSI scheme that 9 * 9WDFT realizes, and output image is 171 * 171 bytes for the Y image, and is 85 * 85 bytes for U and V image.In the end of CSI or FCSI algorithm, the Y of three separation, U and the synthetic YUV image of V image sets.Second step was to use the encryption algorithm [9] based on JPEG DCT.Image after this step is called compressed image.This compressed image has the very pixel of smallest number now when comparing with original image.Generate image and still have the standard jpeg format.As a result, this compressed image can use standard JPEG code translator, also saves storage, and reduces transmission time of being used to communicate by letter.

Also be in the JPEG code translator of revising, the process in coding step two opposite uses in some is arranged.The first step is based on the encryption algorithm [9] of JPEG DCT.After this step, image file is separated into Y, U and the V image of three separation.Second step is to have 1 to τ for Y, U and the use of V image ²The post-processing step rebuild of the cubic spline of ratio of compression.This step is only used the cubic spline function reconstructed image data.After this interpolation, the Y size of images therefore from , promptly 256 * 256 for τ=2 or 171 * 171 for τ=3, be transformed into 512 * 512 bytes, and U and V image from

, promptly 128 * 128 for τ=2 or 85 * 85 for τ=3, be transformed into 256 * 256 bytes.Then three Y, U and V image are combined into a kind of yuv format once more.At last, this YUV image transitions is become to rebuild the RGB image.Equally, draw from [19], being used for from the formula of the conversion of YUV to RGB is R=Y+1.140V, G=Y-0.395U-0.58IV, and B=Y+2.032U.

VIII. test findings

Allow X (i, j) and S (i j) is original and reconstructed image respectively, and wherein 0≤i≤M-1 and 0≤j≤N-1 are the vertical indexes that separates with horizontal direction at image.The Mean Square Error (MSE) of the 2-D signal of image is provided by following formula $\mathrm{MSE}=\frac{1}{M\&times;N}\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}\underset{j=0}{\overset{N-1}{\mathrm{\&Sigma;}}}|X(i,j)-S(i,j){|}^2.---\left(38\right)$ Thereby the PSNR of 2-D signal is defined by following formula

Thus，the?PSNR?of?the?2-D?signal?are?defined?by $\mathrm{PSN}{R}_X\left(\mathrm{dB}\right)=10\mathrm{lo}{g}_{10}\left(\frac{{255}^2}{{\mathrm{MSE}}_X}\right),\mathrm{forX}\&Element;\{Y,U,V\},---\left(39\right)$

MSE wherein _Y, MSE _UAnd MSE _VBe respectively the MSE of color component Y, U and V.The PSNR of color component Y, U and V is defined by following formula $\mathrm{PSN}{R}_T\left(\mathrm{dB}\right)=10\mathrm{<figure-callout\; id="10"\; label="lo\; g"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">lo</figure-callout>}{g}_{}{}_{10}\frac{{255}^2}{({\mathrm{MSE}}_Y+{\mathrm{MSE}}_U+{\mathrm{MSE}}_V)/3}.---\left(40\right)$

The test findings that is used for the 2-D signal uses linear interpolation, linear spline interpolation, cubic convolution interpolation, cubic B-spline interpolation and CSI scheme to represent.The coding method that use is described in II joint and in the III joint for interpretation method about the 2-D signal description of τ=2, by (38) and (39), take advantage of some gray images of 512 to calculate the PSNR value of 2-D signal with 4: 1 (τ=2) ratio of compression for size 512.Use the test findings of the 2-D signal of above five kinds of interpolation schemes to be illustrated in the Table III.Result in this table shows that the CSI scheme obtains being better than all, and other compares the best PSNR of interpolation method.

Table III

PSNR (dB) with 2-D function of 4: 1 (τ=2) ratio of compression

Image	Linear	Linear batten (AOL)	Three convolution	Cubic B-spline	????CSI
						Pepper	????31.50	????32.50	????32.06	????30.38	????33.21
The lake	????29.11	????30.15	????29.56	????28.65	????30.87
						Two people	????29.09	????30.01	????29.16	????27.86	????30.51
The crowd	????32.10	????32.74	????32.93	????31.63	????33.86
						????Lena	????33.48	????34.13	????34.02	????32.11	????35.08

In Table IV,, take advantage of 512 identical gray image to present to have 9: 1 the test findings of (τ=3) ratio of compression for size 512 for the FCSI algorithm that uses zonal filter 1, zonal filter 2, zonal filter 3 and zonal filter 4.Observed by this table, the FCSI scheme with zonal filter 4 obtains and other three similar PSNR of zonal filter.In addition, have 9: 1 (τ=3) ratio of compression, use the test findings of linear interpolations, linear spline interpolation, cubic convolution interpolation and FCSI scheme by means of zonal filter 4, take advantage of 512 identical gray image to be illustrated in the Table V for size 512.Drawn by this table, the FCSI scheme with zonal filter 4 obtains the best PSNR of four kinds of interpolation methods.

Table IV

Have 9: 1 (τ=3) ratio of compression for FCSI scheme with different band mode filter

PSNR(dB)

Image	Zonal filter	1	Zonal filter 2	Zonal filter 3	Zonal filter 4
						Pepper	????30.44	????30.39	????29.84	????30.29
The lake	????27.74	????27.72	????27.38	????27.64

Two people	????27.33	????27.32	????27.06	????27.02
					The crowd	????29.87	????29.68	????29.46	????29.54
??Lena	????31.54	????31.50	????30.83	????31.43

Table V

The PSNR (dB) that has 9: 1 (τ=3) ratio of compression for four kinds of interpolation schemes

Image	Linear	Linear batten (AOL)	Three convolution	FCSI by means of zonal filter 4
					Pepper	????28.65	????29.80	????29.14	????30.59
The lake	????26.00	????27.11	????26.28	????27.64
					Two people	????25.75	????26.83	????25.60	????27.02
The crowd	????28.04	????28.98	????28.43	????29.54
					????Lena	????29.98	????30.96	????30.22	????31.43

Table VI for JPEG method [13], AOL algorithm [8], and, also see the jpeg coder-code translator of [10] for the τ that in Figure 15 and Figure 16, describes=2, be listed in the PSNR value of the colored test of reconstruction (Lena) image of size 512 * 512 under the different ratio of compression.For identical ratio of compression, the PSNR of the Lena image that is obtained by the modification jpeg coder-code translator for τ=2 is than those height of JPEG method and AOL algorithm.And, use (40), for JPEG method, AOL algorithm, and for the modification jpeg coder-code translator of τ=2 and τ=3, Table VII is listed in total PSNR value that the colour of size 512 * 512 under the higher compression ratios is rebuild the Lena image.For 250: 1 identical ratio of compression, total PSNR value of the Lena image that is obtained by the modification JPEG coder-decoder for τ=3 was than JPEG method, AOL algorithm, and for the modification jpeg coder-code translator height of τ=2.

Table VI

For JPEG, AOL, and for the jpeg coder-code translator of τ=2, in difference

The PSNR (dB) of the colored test of the reconstruction of size 512 * 512 Lena image under the ratio of compression

Ratio of compression	?????????????JPEG[13]			??????????????AOL[8]			Modification JPEG for τ=2
										????Y	????U	????V	????Y	????U	????V	????Y	????U	????V
	125∶1	??29.48	??32.67	??32.59	??30.1l	??34.80	??35.00	??30.27	??34.89										??35.00
100∶1										??30.76	??33.90	??33.90	??30.91	??35.60	??35.75	??31.20	??35.66	??35.85

????75∶1	??32.15	??35.37	??35.31	??31.80	???36.38	???36.41	???32.15	???36.45	???36.42
										????50∶1	??34.00	??37.08	??37.05	??32.86	???37.41	???37.31	???33.36	???37.42	???37.32

Table VII

For JPEG, AOL, and for the jpeg coder-code translator of τ=2 and τ=3,

The colored test of the reconstruction of size 512 * 512 Lena image is total under different ratio of compression

PSNR(dB)

Ratio of compression	??JPEG[13]	????AOL[8]	Modification JPEG[10 for τ=2]	Modification JPEG for τ=3
					????250∶1	????26.32	????30.25	????30.40	????30.53
????200∶1	????27.90	????31.06	????31.18	????31.19
					????150∶1	????30.13	????32.14	????32.20	????31.95
????100∶1	????32.59	????33.46	????33.66	????33.11

For the grey Lena image of size 512 * 512, on the 400-MHz Intel Pentium II personal computer that uses the C code, realizing the computing time of the CSI of τ=3 and FCSI scheme.In scrambler, by means of overlapping avoid FCSI scheme that subimage uses 9 WDFT with use for CSI when comparing in about 0.57 second of FFT, need about 0.15 second.Therefore, FCSI scheme fast than CSI scheme.

At last, JPEG method, AOL algorithm, and also on the identical 400-MHz Intel Pentium II personal computer that uses the C code, realize for the modification jpeg coder-code translator of τ=2 and τ=3.Be given in the Table VIII computing time for these four kinds of algorithms colored Lena image of size 512 * 512 under 200: 1 ratio of compression.When coding and decoding, for the modification jpeg coder-code translator of τ=3 with for 1.13 seconds of the modification jpeg coder-code translator of τ=2 with 0.34 second, for 1.09 seconds of the AOL algorithm with 0.30 second, and for 0.80 second of the JPEG method when 0.65 second compares respectively, only need 0.42 second and 0.27 second.Just by this means, for the modification jpeg coder that needs computing time of the modification jpeg coder of τ=3 time ratio for τ=2 respectively, AOL scrambler, and jpeg coder little 0.71 second, 0.67 second and 0.38 second.And, for the modification JPEG code translator that needs computing time of the modification JPEG code translator of τ=3 time ratio for τ=2 respectively, AOL code translator, and JPEG code translator little 0.07 second, 0.03 second and 0.38 second.

Table VIII

On 400-MHz Intel Pentium II personal computer, realize at 200: 1 ratio of compression

The computing time (second) of the colored Lena image of following size 512 * 512

	Scrambler	Code translator
			Modification JPEG for τ=3	????0.42	????0.27
Modification JPEG[10 for τ=2]	????1.13	????0.34
			AOL algorithm [8]	????1.09	????0.30
JPEG method [13]	????0.80	????0.65

Figure 17 represents to use JPEG method, AOL algorithm, reaches the reconstructed image for modification jpeg coder-code translator Lena under 100: 1 identical ratio of compression of τ=2.The reconstructed image of the subjective quality better than JPEG method and AOL algorithm is clearly indicated in use for the Lena image of the modification jpeg coder-code translator of τ=2.Figure 18 also represents to use JPEG method, AOL algorithm, reaches the reconstructed image for modification jpeg coder-code translator Lena under 200: 1 higher compression ratios of τ=2 and τ=3.In the figure, use reaches the reconstructed image for the good subjective quality of the modification jpeg coder-code translator of τ=2 for the Lena image indication of the modification jpeg coder-code translator of τ=3 than JPEG method, AOL algorithm.

Figure 17 (a) represents original Lena image, and Figure 17 (b) describes by for PSNR _Y=30.76dB, PSNR _U=33.90dB, and PSNR _VThe reconstructed image of the JPEG method of=33.90dB.Figure 17 (c) shows by for PSNR _Y=30.91dB, PSNR _U=35.60dB, and PSNR _VThe reconstructed image of the AOL algorithm of=35.75dB; And Figure 17 (d) representative is passed through for τ=2, PSNR _Y=31.20dB, PSNR _U=35.66dB, and PSNR _VThe reconstructed image of modification jpeg coder-code translator of=35.85dB.

Figure 18 (a) expression is by for scramble time=0.80 second, decoding time=0.65 second, and PSNR _TThe reconstructed image of the jpeg algorithm of=27.90dB, and Figure 18 (b) describes by for scramble time=1.09 second, decoding time=0.30 second, and PSNR _TThe reconstructed image of the AOL algorithm of=31.06dB.Figure 18 (c) shows by for τ=2, scramble time=1.13 second, decoding time=0.34 second, and PSNR _TThe reconstructed image of modification jpeg coder-code translator of=31.18dB; And Figure 18 (d) representative is by for τ=3, scramble time=0.42 second, decoding time=0.27 second, and PSNR _TThe reconstructed image of modification jpeg coder-code translator of=31.19dB.

9 Wingrad discrete Fourier transformations

Hold within it by with reference to be included in here Dean P.Kolba and the prime factor fft algorithm of high speed convolution " use " IEEE Trans.on Acoustfics of Thomas W.Parks, Speech, and Sigllal Processing, vol.ASSP-25, No.4, pp.281-294, the algorithm of exploitation is expressed from the next in order to calculate 9 Wingrad discrete Fourier transformations (DFT) in August, 1977 [16] $X\left(k\right)=\underset{n=0}{\overset{8}{\mathrm{\&Sigma;}}}x\left(n\right)W^{\mathrm{nk}},k=0,\&CenterDot;\&CenterDot;\&CenterDot;,8,---\left(41\right)$ W=e wherein ^{-j (2 π/9)}Be 9 roots of unit in complex field, and

The algorithm that is used for 9 Wingrad DFT:

a ₁＝x(1)+x(8)，a ₂＝x(1)-x(8)，a ₃＝x(2)+x(7)，a ₄＝x(2)-x(7)，a ₅=x(4)+x(5)，

a ₆＝x(4)-x(5)，a ₇＝x(3)+x(6)，a ₈＝x(3)-x(6)，a ₉＝-a ₁+a ₅，a ₁₀＝a ₁-a ₃，

a ₁₁＝-a ₃+a ₅，a ₁₂＝a ₂-a ₆，a ₁₃＝a ₂+a ₄，a ₁₄＝-a ₄-a ₆，a ₁₅＝a ₁+a ₃+a ₅，

a ₁₆＝a ₂-a ₄+a ₆，a ₁₇＝x(0)+a ₁₅+a ₇，

m ₁＝0.19740a ₉，m ₂＝0.56858a ₁₀，m ₃＝0.37111a ₁₁，m ₄＝0.54253a ₁₂，m ₅＝0.10026a ₁₃，

m ₆＝0.44228a ₁₄， $m_7=\frac{1}{2}{a}_7,$ m ₈＝0.86603a ₈， $m_9=\frac{1}{2}{a}_{15},$ m ₁₀＝0.86603a ₁₆，

c ₁＝x(0)-m ₇，c ₂＝m ₂-m ₃，c ₃＝m ₁+m ₃，c ₄＝m ₁+m ₂，c ₅＝c ₁+c ₂-c ₃，c ₆＝c ₁+c ₃+c ₄，

c ₇＝c ₁-c ₂-c ₄，c ₈＝m ₄-m ₆，c ₉＝m ₅-m ₆，c ₁₀＝m ₄-m ₅，c ₁₁＝c ₈+c ₉+m ₈，

c ₁₂＝c ₈+c ₁₀-m ₈，c ₁₃＝-C ₉+C ₁₀+m ₈，c ₁₄＝x(0)+a ₇-m ₉，

X(0)=a ₁₇，X(1)＝c ₅-jc ₁₁，X(2)＝c ₆-jc ₁₂，X(3)＝c ₁₄-jm ₁₀，X(4)＝c ₇-jc ₁₃，

X(5)＝c ₇+jc ₁₃，X(6)＝c ₁₄+jm ₁₀，X(7)＝c ₆+jc ₁₂，X(8)＝c ₅+jc ₁₁。

Thereby 9 Wingrad DFT only need to multiply each other for 8 times, 49 additions, and 2 displacements, and calculation times is significantly lacked than other algorithm known.

5 Wingrad discrete Fourier transformations

Provide by following formula for calculate the algorithm that 5 Wingrad discrete Fourier transformations (DFT) develop in [16]: $X\left(k\right)=\underset{n=0}{\overset{4}{\mathrm{\&Sigma;}}}x\left(n\right)W^{\mathrm{nk}},k=0,\&CenterDot;\&CenterDot;\&CenterDot;,4,---\left(42\right)$

W=e wherein ^{-j (2 π/5)}Be 5 roots of unit in complex field, and

The algorithm that is used for 5 Wingrad DFT is provided by following formula:

a ₁＝x(1)+x(4)，a ₂＝x(1)-x(4)，a ₃＝x(2)+x(3)，a ₄＝x(2)-x(3)，a ₅＝a ₂+a ₄，a ₆＝a ₁-a ₃，

a ₇=a ₁+a ₃，a ₈＝x(0)+a ₇，

m ₁＝0.95106a ₅，m ₂＝1.53884a ₂，m ₃＝0.36327a ₄，m ₄＝0.55902a ₆， ${\mathrm{<figure-callout\; id="5"\; label="m"\; filenames="A0081361300521.png"\; state="\{\{state\}\}">m</figure-callout>}}_5=\frac{1}{4}{a}_7,$

c ₁＝x(0)-m ₅，c ₂＝c ₁+m ₄，c ₃＝c ₁-m ₄，c ₄＝m ₁-m ₃，c ₅＝m ₂-m ₁，

X(0)＝a ₈，X(1)＝c ₂-jc ₄，X(2)＝c ₃-jc ₅，X(3)＝c ₃+jc ₅，X(4)＝c ₂+jc ₄。

Thereby 5 Wingrad DFT only need to multiply each other for 4 times, 17 additions, and 1 displacement, and calculation times is significantly lacked than other algorithm known.

In the present invention, proposed a kind of based on least square method by means of the new CSI scheme of cubic spline function with compressing image data.What show is that the CSI scheme that is realized by fft algorithm produces the PSNR performance better than other interpolation method that is used for reconstructed image.In addition, for a kind of quick CSI that is called FCSI of compression of images exploitation.A kind of like this FCSI scheme needs the addition of lacking than original CSI scheme and multiplies each other in extraction process.In FCSI, a kind of quick 9 Wingrad DFT algorithms are used for calculating the CSI scheme by means of zonal filter 4, and a kind of overlapping serious border artifact of avoiding the subimage technology to use and solve between any two adjacent sub-images of real image.Show that by computer run the FCSI scheme needs time ratio for little 0.42 second of the original CSI of τ=3.

At last, use with jpeg coder-code translator, with acceleration modification jpeg coder-code translator for τ=2 in color image encoding for this FCSI scheme of τ=3.Computer simulation shows, obtains the subjective quality and the PSNR of the reconstructed image better than JPEG method for high compression ratio for the modification jpeg coder-decoding of τ=3.And, in coding and decode procedure, all need to reach the computing time that the modification jpeg coder-code translator for τ=2 lacks than JPEG method, AOL algorithm.

Be appreciated that to describe here and only represent example embodiment of the present invention with the exemplary scheme and the various enforcement of expression in the accompanying drawings.Really, can carry out various modifications and interpolation to this embodiment, and not break away from the spirit and scope of the present invention.For example, the present invention's hardware of utilizing computer program, special electronic circuit or being used for the electronic image process chip can be realized.

And the professional who is familiar with present technique will recognize, method of the present invention can be applicable to the image that different value for τ has various different sizes and has different size for Wingrad DFT.Thereby it is 9 * 9 or 5 * 5 the description of Wingrad DFT only is illustrative here, rather than restrictive.And the professional who is familiar with present technique will recognize, imagine various different VLSI of the present invention and implement.Thereby, these and other modification and interpolation may be obvious for the professional who is familiar with present technique, and can implement so that the present invention is applicable to various different purposes, comprise video signal flow, combine, be applied to JPEG 2000 standards, reduce the size (as in digital camera) of digital photos etc. with the MPEG IV standard that is used for tableaux.

Claims (54)

1. one kind by the encode method of 1-D signal of computing machine, comprises step: by 1-D cubic spline wave filter of following formula definition Filter applies in an input signal Xm, is had $Y_j=\underset{m=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m+\mathrm{j\&tau;})R\left(m\right)---\left(9\right)$ To calculate Y _jUse $A_{j,k}=\underset{m=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m+(j-k)\mathrm{\&tau;})R\left(m\right)$

Wherein by allowing following formula set up A _{J, k}Can represent with circulation form $A_{j,k}={B_{(k-j)}}_n,---\left(11\right)$ And wherein (k-j) _nIndication residue (k-j) mould n, and B ₀=α, B ₁=β, B ₂=γ, B ₃=δ, B ₄=0 ..., B _N-4=0, B _N-3=δ, B _N-2=γ, (12) B _N-1=β is to calculate B _jCalculate Y respectively _iAnd B _iFFT to obtain With

Calculate ${\tilde{X}}_m={\tilde{Y}}_m/{\tilde{B}}_m$ And calculate Contrary FFT to obtain packed data X _k

2. method according to claim 1 further comprises step:

Existing for 0≤j≤n-1 $\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{A}_{j,k}{X}_k=Y_j$ , the X in (5) _kBe applied to carry out the X that provides by following formula _kAnd the convolution of R (t)

With the S (t) that obtains providing by following formula $S\left(t\right)=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_k{\mathrm{\&psi;}}_k\left(t\right)=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_{k}R(t-\mathrm{k\&tau;})----\left(3\right)$

3. method according to claim 1, wherein τ=2.

4. method according to claim 1, wherein τ=3.

5. scrambler that is used for data compression, comprising: a 1-D cubic spline wave filter is defined by following formula

Be used for input signal X of a filter applies _mDevice, have $Y_j=\underset{m=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m+\mathrm{j\&tau;})R\left(m\right)---\left(9\right)$ To calculate Y _jBe used for using the device of following formula $A_{j,k}=\underset{m=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m+(j-k)\mathrm{\&tau;})R\left(m\right)$

Wherein by allowing following formula set up A _{J, k}Can represent with circulation form $A_{j,k}={B_{(k-j)}}_n,---\left(11\right)$ And wherein (k-j) _nIndication residue (k-j) mould n, and B ₀=α, B ₁=β, B ₂=γ, B ₃=δ, B ₄=0 ..., B _N-4=0, B _N-3=δ, B _N-2=γ, (12)

B _N-1=β is to calculate B _j

Be used for calculating respectively Y _jAnd B _jFFT to obtain

With

Device;

Be used for calculating ${\tilde{X}}_m={\tilde{Y}}_m/{\tilde{B}}_m$ Device; And

Be used for calculating

Contrary FFT to obtain packed data X _kDevice.

6. scrambler according to claim 5 further comprises a scrambler, has to be used for existing for 0≤j≤n-1 $\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{A}_{j,k}{X}_k=Y_j$ , the X in (5) _kBe applied to carry out the X that provides by following formula _kAnd the device of the convolution of R (t)

With the S (t) that obtains providing by following formula $S\left(t\right)=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_k{\mathrm{\&psi;}}_k\left(t\right)=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{X}_{k}R(t-\mathrm{k\&tau;}).---\left(3\right)$

7. scrambler according to claim 5, wherein τ=2.

8. scrambler according to claim 5, wherein τ=3.

9. one kind by the encode method of 2-D signal of being used for of carrying out of computing machine, comprises step: by R (t ₁, t ₂)=R (t ₁) R (t ₂), (17) definition 2-D cubic spline wave filter, wherein a R (t ₁) and R (t ₂) be respectively the 1-D cubic spline function; Filter applies in having cycle n ₁τ and n ₂Input signal X (the t of τ ₁, t ₂), have $Y_{j_1,j_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m_1+j_1\mathrm{\&tau;},m_2+j_2\mathrm{\&tau;})R(m_1,m_2),---\left(25\right)$ To calculate $A_{j_1{j}_2,k_1{k}_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m_1+(j_1-k_1)\mathrm{\&tau;},m_2+(j_2-k_2)\mathrm{\&tau;})R(m_1,m_2)$ Use

Wherein, array Can be expressed as with the 2-D circulation form $A_{j_1{j}_2,k_1{k}_2}=B_{{(k_1-j_1)}_{n_1},{(k_2-j_2)}_{n_2}},---\left(27\right)$ And wherein Indication is for i=1,2 residue (k _i-j _i) mould n _i, and

Wherein for i=1,2,0≤s _i≤ n _i-1, to calculate

Calculate respectively

With 2-DFFT to obtain With Calculate ${\tilde{X}}_{m,n}={\tilde{Y}}_{m,n}/{\tilde{B}}_{m,n}$ And calculate

Contrary FFT to obtain a compressed image

10. method according to claim 9 further comprises step:

Be applied in the following formula $\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{A}_{j_1{j}_2,k_1{k}_2}=Y_{j_1,j_2},0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2}.---\left(21\right)$

To carry out (t by R ₁, t ₂)=R (t ₁) R (t ₂) provide And R (t ₁, t ₂) the 2-D convolution, to obtain a 2-D reconstructed image S (t ₁, t ₂).

11. method according to claim 9, wherein σ=0.

12. method according to claim 9, wherein σ=μ=0.

13. method according to claim 9, wherein η=λ=ρ=μ=σ=0.

14. method according to claim 9, wherein τ=2.

15. method according to claim 9, wherein τ=3.

16. a scrambler that is used for data compression comprises:

A 2-D cubic spline wave filter is by R (t ₁, t ₂)=R (t ₁) R (t ₂), (17) definition,

R (t wherein ₁) and R (t ₂) be respectively the 1-D cubic spline function;

Be used for a filter applies in having cycle n ₁τ and n ₂Input signal X (the t of τ ₁, t ₂) device, have ${n_2\mathrm{\&tau;},\mathrm{withY}}_{j_1,j_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m_1+j_1\mathrm{\&tau;},m_2+j_2\mathrm{\&tau;})R(m_1,m_2),---\left(25\right)$ To obtain Be used for using the device of following formula $A_{j_1{j}_2,k_1{k}_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m_1+(j_1-k_1)\mathrm{\&tau;},m_2+(j_2-k_2)\mathrm{\&tau;})R(m_1,m_2)$

Wherein, array

Can be expressed as with the 2-D circulation form $A_{j_1{j}_2,k_1{k}_2}=B_{{(k_1-j_1)}_{n_1},{(k_2-j_2)}_{n_2}},---\left(27\right)$

And (k wherein _i-j _i) _NiIndication is for i=1,2 residue (k _i-j _i) mould n _i, and

Wherein for i=1,2,0≤s _i≤ n _i-1, to obtain

Be used for calculating respectively With 2-D FFT to obtain

With Device; Be used for calculating ${\tilde{X}}_{m,n}={\tilde{Y}}_{m,n}/{\tilde{B}}_{m,n}$ Device; And be used for calculating

Contrary FFT to obtain a compressed image

Device.

17. scrambler according to claim 16 further comprises a code translator, has a device and is used for being applied in the following formula

$\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{A}_{j_1{j}_2,k_1{k}_2}=Y_{j_1,j_2},0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2}.---\left(21\right)$

To carry out (t by R ₁, t ₂)=R (t ₁) R (t ₂) provide

And R (t ₁, t ₂) the 2-D convolution, to obtain a 2-D reconstructed image S (t ₁, t ₂).

18. scrambler according to claim 16, wherein τ=2.

19. scrambler according to claim 16, wherein τ=3.

20. scrambler according to claim 16, wherein σ=0.

21. scrambler according to claim 16, wherein σ=μ=0.

22. scrambler according to claim 16, wherein η=λ=ρ=μ=σ=0.

23. one kind is used for coded image X (t by what computing machine carried out ₁, t ₂) method, comprise step:

A zonal filter R (m ₁, m ₂) be applied to image X (t ₁, t ₂), wherein $Y_{j_1,j_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m_1+j_1\mathrm{\&tau;},m_2+j_2\mathrm{\&tau;})R(m_1,m_2),---\left(25\right)$ To calculate Use $A_{j_1{j}_2,k_1{k}_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m_1+(j_1-k_1)\mathrm{\&tau;},m_2+(j_2-k_2)\mathrm{\&tau;})R(m_1,m_2)$

Wherein, array Can be expressed as with the 2-D circulation form $A_{j_1{j}_2,k_1{k}_2}=B_{{(k_1-j_1)}_{n_1},{(k_2-j_2)}_{n_2}},---\left(27\right)$

And wherein Indication is for i=1,2 residue (k _i-j _i) mould n _i, and

Wherein for i=1,2,0≤s _i≤ n _i-1, to calculate

Wherein constant alpha, β, γ, δ, η, λ, ρ, μ and σ are at zonal filter R (m ₁, m ₂) between coefficient of autocorrelation;

All coefficients

Be divided into border with width 2

Each overlapping subimage;

Calculate The Winograd discrete Fourier transformation (WDFT) of all overlapping subimages, to obtain coefficient

All coefficients

Be divided into border with width 2 Each overlapping subimage;

Calculate The WDFT of all overlapping subimages, to obtain The conversion of each overlapping subimage;

Calculate ${\tilde{X}}_{m,n}={\tilde{Y}}_{m,n}/{\tilde{B}}_{m,n}$ To obtain Each subimage;

Calculate The contrary WDFT of all subimages to obtain Subimage;

Remove The superposition boundary of two adjacent sub-images in duplicate pixel; And

Each non-overlapped subimage combination to obtain

Entire image.

24. method according to claim 23, wherein τ=2.

25. method according to claim 23, wherein τ=3.

26. method according to claim 23, wherein coefficient of autocorrelation σ=0 between zonal filter.

27. method according to claim 23, wherein coefficient of autocorrelation σ=μ=0 between zonal filter.

28. method according to claim 23, wherein coefficient of autocorrelation η=λ=ρ=μ=σ=0 between zonal filter.

29. method according to claim 23, wherein WDFT is the 9 * 9WDFT that is provided by following formula $X\left(k\right)=\underset{n=0}{\overset{8}{\mathrm{\&Sigma;}}}x\left(n\right)W^{\mathrm{nk}},k=0,\&CenterDot;\&CenterDot;\&CenterDot;,8,---\left(41\right)$ And Subimage be 9 * 9.

30. method according to claim 23, wherein WDFT is the 5 * 5WDFT that is provided by following formula $X\left(k\right)=\underset{n=0}{\overset{4}{\mathrm{\&Sigma;}}}x\left(n\right)W^{\mathrm{nk}},k=0,\&CenterDot;\&CenterDot;\&CenterDot;,4,---\left(42\right)$ And Subimage be 5 * 5.

31. method according to claim 23 further comprises step: be applied in the following formula $\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{A}_{j_1{j}_2,k_1{k}_2}=Y_{j_1,j_2},0\&le;j_i\&le;n_i-1\mathrm{andi}=\mathrm{1,2}.---\left(21\right)$

To carry out And R (t ₁, t ₂) the 2-D convolution, to obtain a 2-D reconstructed image S (t ₁, t ₂).

32. a modification jpeg coder that is used for compression of images comprises:

A zonal filter R (m ₁, m ₂) be applied to image X (t ₁, t ₂) device, wherein $Y_{j_1,j_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}X(m_1+j_1\mathrm{\&tau;},m_2+j_2\mathrm{\&tau;})R(m_1,m_2),---\left(25\right)$

To calculate

Be used for using the device of following formula $A_{j_1{j}_2,k_1{k}_2}=\underset{m_1=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}\underset{m_2=-2\mathrm{\&tau;}+1}{\overset{2\mathrm{\&tau;}-1}{\mathrm{\&Sigma;}}}R(m_1+(j_1-k_1)\mathrm{\&tau;},m_2+(j_2-k_2)\mathrm{\&tau;})R(m_1,m_2)$

Wherein, array

Can be expressed as with the 2-D circulation form $A_{j_1{j}_2,k_1{k}_2}=B_{{(k_1-j_1)}_{n_1},{(k_2-j_2)}_{n_2}},---\left(27\right)$

And wherein Indication is for i=1,2 residue (k _i-j _i) mould n _i, and

Wherein for i=1,2,0≤s _i≤ n _i-1, to calculate

Wherein constant alpha, β, γ, δ, η, λ, ρ, μ and σ are at zonal filter R (m ₁, m ₂) between coefficient of autocorrelation;

Be used for all coefficients

Be divided into border with width 2

The device of each overlapping subimage;

Be used for calculating The Winograd discrete Fourier transformation (WDFT) of all overlapping subimages, to obtain coefficient

Device;

Be used for all coefficients Be divided into border with width 2

The device of each overlapping subimage;

Be used for calculating

The WDFT of all overlapping subimages, to obtain

The device of conversion of each overlapping subimage;

Be used for calculating ${\tilde{X}}_{m,n}={\tilde{Y}}_{m,n}/{\tilde{B}}_{m,n}$ To obtain

The device of each subimage;

Be used for calculating

The contrary WDFT of all subimages to obtain The device of subimage;

Be used for removing

The superposition boundary of two adjacent sub-images in the device that duplicates pixel; And

Be used for handle Each non-overlapped subimage combination to obtain The device of entire image.

33. scrambler according to claim 32, wherein τ=2.

34. scrambler according to claim 32, wherein τ=3.

35. scrambler according to claim 32, wherein coefficient of autocorrelation σ=0 between zonal filter.

36. scrambler according to claim 32, wherein coefficient of autocorrelation σ=μ=0 between zonal filter.

37. scrambler according to claim 32, wherein coefficient of autocorrelation η=λ=ρ=μ=σ=0 between zonal filter.

38. scrambler according to claim 32, wherein WDFT is the 9 * 9WDFT that is provided by following formula $X\left(k\right)=\underset{n=0}{\overset{8}{\mathrm{\&Sigma;}}}x\left(n\right)W^{\mathrm{nk}},k=0,\&CenterDot;\&CenterDot;\&CenterDot;,8,---\left(41\right)$ And

Subimage be 9 * 9.

39. scrambler according to claim 32, wherein WDFT is the 5 * 5WDFT that is provided by following formula $X\left(k\right)=\underset{n=0}{\overset{4}{\mathrm{\&Sigma;}}}x\left(n\right)W^{\mathrm{nk}},k=0,\&CenterDot;\&CenterDot;\&CenterDot;,4,---\left(42\right)$ And Subimage be 5 * 5.

40. scrambler according to claim 32 comprises that is further translated a device, has to be used for being applied in the following formula

Device $\underset{k_1=0}{\overset{n_1-1}{\mathrm{\&Sigma;}}}\underset{k_2=0}{\overset{n_2-1}{\mathrm{\&Sigma;}}}{X}_{k_1,k_2}{A}_{j_1{j}_2,k_1{k}_2}=Y_{j_1,j_2},$ 0≤j _i≤n _i-1?and?i＝1，2。To carry out

And R (t ₁, t ₂) the 2-D convolution, to obtain a 2-D reconstructed image.

41. a method that is used for coded signal of being undertaken by computing machine comprises step:

Define a cubic spline wave filter;

Make wave filter and signal correction to obtain a coherent signal;

The autocorrelation filter device is to obtain autocorrelation filter device coefficient;

Calculate coherent signal and the transformation of coefficient of autocorrelation filter device;

The conversion of coherent signal divided by the transformation of coefficient of autocorrelation filter device to obtain the conversion of compressed signal; And

The inverse transformation of the conversion of calculating compressed signal is to obtain compressed signal.

42. according to the described method of claim 41, wherein signal, wave filter, and conversion be one dimension.

43. according to the described method of claim 41, wherein signal, wave filter, and conversion be bidimensional.

44., further comprise the step of the convolution of calculating compressed signal and wave filter, to obtain a reconstruction signal according to the described method of claim 41.

45. according to the described method of claim 41, wherein the step of computational transformation comprises calculating Fast Fourier Transform (FFT) (FFT), and the step of calculation reverse transformation comprises the contrary FFT of calculating.

46. according to the described method of claim 41, wherein the step of computational transformation comprises calculating to have the overlapping Winograd discrete Fourier transformation (WDFT) of avoiding scheme, and the step of calculation reverse transformation comprises the contrary WDFT of calculating.

47. according to the described method of claim 41, further comprise the step that defines a zonal filter, to simplify relevant and auto-correlation step.

48. a modification jpeg coder that is used for data compression comprises:

A cubic spline wave filter;

Be used for making wave filter relevant with input signal to obtain the device of a coherent signal;

Be used for the autocorrelation filter device to obtain the device of autocorrelation filter device coefficient;

Be used for calculating the device of coherent signal and the transformation of coefficient of autocorrelation filter device;

Be used for the conversion of coherent signal divided by the device of autocorrelation filter device transformation of coefficient with the conversion that obtains compressed signal; And

The inverse transformation of conversion that is used for calculating compressed signal is to obtain the device of compressed signal.

49. according to the described scrambler of claim 48, wherein signal, wave filter, and conversion be one dimension.

50. according to the described scrambler of claim 48, wherein signal, wave filter, and conversion be bidimensional.

51. according to the described scrambler of claim 48, further comprise a code translator, have the device of the convolution that is used for calculating compressed signal and wave filter, to obtain a reconstruction signal.

52. according to the described scrambler of claim 48, wherein be used for the device of computational transformation and comprise the device that is used for calculating Fast Fourier Transform (FFT) (FFT), comprise the device that calculates contrary FFT and be used for the device of calculation reverse transformation.

53. according to the described scrambler of claim 48, wherein being used for the device of computational transformation comprises being used for calculating to have the overlapping device of avoiding the Winograd discrete Fourier transformation (WDFT) of scheme, comprises the device that is used for calculating contrary WDFT and be used for the device of calculation reverse transformation.

54. according to the described scrambler of claim 48, further comprise the device that is used for defining a zonal filter, the device that is used for being correlated with simplification and be used for autocorrelative device.

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