Fast signal transforms with lifting steps

Abstract

This invention introduces a class of multi-band linear phase lapped biorthogonal transforms with fast, VLSI-friendly implementations via lifting steps called the LiftLT. The transform is based on a lattice structure which robustly enforces both linear phase and perfect reconstruction properties. The lattice coefficients are parameterized as a series of lifting steps, providing fast, efficient in-place computation of the transform coefficients as well as the ability to map integers to integers. Our main motivation of the new transform is its application in image and video coding. Comparing to the popular 8×8 DCT, the 8×16 LiftLT only requires 1 more multiplication, 22 more additions, and 6 more shifting operations. However, image coding examples show that the LiftLT is far superior to the DCT in both objective and subjective coding performance. Thanks to properly designed overlapping basis functions, the LiftLT can completely eliminate annoying blocking artifacts. In fact, the novel LiftLT's coding performance consistently surpasses that of the much more complex 9/7-tap biorthogonal wavelet with floating-point coefficients. More importantly, our transform's block-based nature facilitates one-pass sequential block coding, region-of-interest coding/decoding as well as parallel processing.

Description

FIELD OF THE INVENTION

where U_i0, U_i1, V_i0, V_i1 are diagonalizing orthogonal matrices and Γ_i, Δ_i are diagonal matrices with positive elements.

It is well known that any M/2×M/2 orthogonal matrix can be factorized into M(M−2)/8 plane rotations θ_i and that the diagonal matrices represent simply scaling factors α_i. Accordingly, the most general LT lattice consists of KM(M−2)/2 two dimensional rotations and 2M diagonal scaling factors α_i. Any invertible matrix can be expressed as a sequence of pairwise plane rotations θ and scaling factors α_i as shown in FIG. 3.

It is also well known that a plane rotation can be performed by 3 “shears”: $\left[\begin{array}{cc}\cos {\theta }_i& -\sin {\theta }_i\\ \sin {\theta }_i& \cos {\theta }_i\end{array}\right]=\left[\begin{array}{cc}1& \frac{\cos {\theta }_i-1}{\sin {\theta }_i}\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ \sin {\theta }_i& 1\end{array}\right]\left[\begin{array}{cc}1& \frac{\cos {\theta }_i-1}{\sin {\theta }_i}\\ 0& 1\end{array}\right].$
This can be easily verified by computation

Each of the factors above is capable of a “lifting” step in signal processing terminology. The product of two which effects a linear transform of pairs of coefficients: $\left[\begin{array}{c}a\\ b\end{array}\right]\to \left[\begin{array}{cc}1+\mathrm{km}& k\\ m& 1\end{array}\right]\times \left[\begin{array}{c}a\\ b\end{array}\right].$
The signal processing flow diagram of this operation is shown in FIG. 4. The crossing arrangement of these flow paths is also referred to as a butterfly configuration. Each of the above “shears” can be written as a lifting step.

Combining the foregoing, the shears referred to can be expressed as computationally equivalent “lifting steps” in signal processing. In other words, we can replace each “rotation” by 3 closely-related lifting steps with butterfly structure. It is possible therefore to implement the complete LT lattice shown in FIG. 2 by 3KM(−2)/2 lifting steps and 2M scaling multipliers.

In the simplest but currently preferred embodiment, to minimize the complexity of the transform we choose a small overlapping factor K=2 and set the initial stage E₀ to be the DCT itself Many other coding transforms can serve for the base stage instead of the DCT, and it should be recognized that many other embodiments are possible and can be implemented by one skilled in the art of signal processing.

Following the observation in H. S. Malvar, “Lapped biorthogonal transforms for transform coding with reduced blocking and ringing artifacts,” ICASSP97, Munich, April 1997, we apply a scaling factor to the first DCT's antisymmetric basis to generate synthesis LT basis functions whose end values decay smoothly to exact zero—a crucial advantage in blocking artifacts elimination. However, instead of scaling the analysis by √{square root over (2)} and the synthesis by 1/√{square root over (2)}, we opt for 25/16 and its inverse 16/25 since they allow the implementation of both analysis and synthesis banks in integer arithmetic. Another value that works almost as well as 25/16 is 5/4. To summarize, the following choices are made in the first stage: the combination of U₀₀ and V₀₀ with the previous butterfly form the DCT; ${\Delta }_0=\mathrm{diag}\left[\frac{25}{16},1,\dots ,1\right],$
and Γ_o=U₀₀=V₀₀=I_M/2. See FIG. 2.

After 2 series of ±1 butterflies W and the delay chain Λ(z), the LT symmetric basis functions already have good attenuation, especially at DC (ω=0). Hence, we can comfortably set U₁=I_M/2.

As noted, V₁ is factorizable into a series of lifting steps and diagonal scalings. However, there are several problems: (i) the large number of lifting steps is costly in both speed and physical real-estate in VLSI implementation; (ii) the lifting steps are related; (iii) and it is not immediately obvious what choices of rotation angles will result in dyadic rational lifting multipliers. In the current invention, we approximate V₁ by (M/2)−1 combinations of block-diagonal predict-and-update lifting steps, i.e., $\left[\begin{array}{cc}1& u_i\\ 0& 1\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ -p_i& 1\end{array}\right].$

Here, the free parameters u_i and p_i can be chosen arbitrarily and independently without affecting perfect reconstruction. The inverses are trivially obtained by switching the order and the sign of the lifting steps. Unlike popular lifting implementations of various wavelets, all of our lifting steps are of zero-order, namely operating in the same time epoch. In other words, we simply use a series of 2×2 upper or lower diagonal matrices to parameterize the invertible matrix V₁.

Most importantly, fast-computable VLSI-friendly transforms are readily available when u_i and p_i are restricted to dyadic rational values, that is, rational fractions having (preferably small) powers of 2 denominators. With such coefficients, transform operations can for the most part be reduced to a small number of shifts and adds. In particular, setting all of the approximating lifting step coefficients to −1/2 yields a very fast and elegant lapped transform. With this choice, each lifting step can be implemented using only one simple bit shift and one addition.

The resulting LiftLT lattice structures are presented in FIGS. 5 and 6. The analysis filter shown in FIG. 5 comprises a DCT block 1, 25/16 normalization 2, a delay line 3 on four of the eight channels, a butterfly structured set of lifting steps 5, and a set of four fast dyadic lifting steps 6. The frequency and impulse responses of the 8×16 LiftLT's basis functions are depicted in FIG. 8.

The inverse or synthesis lattice is shown in FIG. 6. This system comprises a set of four fast dyadic lifting steps 11, a butterfly-structured set of lifting steps 12, a delay line 13 on four of the eight channels, 16/25 inverse normalization 14, and an inverse DCT block 15. FIG. 7 also shows the frequency and impulse responses of the synthesis lattice.

The LiftLT is sufficiently fast for many applications, especially in hardware, since most of the incrementally added computation comes from the 2 butterflies and the 6 shift-and-add lifting steps. It is faster than the type-I fast LOT described in H. S. Malvar, Signal Processing with Lapped Transforms, Artech House, 1992. Besides its low complexity, the LiftLT possesses many characteristics of a high-performance transform in image compression: (i) it has high energy compaction due to a high coding gain and a low attenuation near DC where most of the image energy is concentrated; (ii) its synthesis basis functions also decay smoothly to zero, resulting in blocking-free reconstructed images.

Comparisons of complexity and performance between the LiftLT and other popular transforms are tabulated in Table 1 and Table 2. The LiftLT's performance is already very close to that of the optimal generalized lapped biorthogonal transform, while its complexity is the lowest amongst the transforms except for the DCT.

To assess the new method in image coding, we compared images coded and decoded with four different transforms:

• • DCT: 8-channel, 8-tap filters
  • Type-I Fast LOT: 8-channel, 16-tap filters
  • LiftLT: 8-channel, 16-tap filters
  • Wavelet: 9/7-tap biorthogonal.
    In this comparison, we use the same SPIHT's quantizer and entropy coder, A. Said and W. A. Pearlman, “A new fast and efficient image coder based on set partitioning in hierarchical trees,” IEEE Trans on Circuits Syst. Video Tech., vol. 6, pp. 243-250, June 1996, for every transform. In the block-transform cases, we use the modified zero-tree structure in T. D. Tran and T. Q. Nguyen, “A lapped transform embedded image coder,” ISCAS, Monterey, May 1998, where each block of transform coefficients is treated analogously to a full wavelet tree and three more levels of decomposition are employed to decorrelate the DC subband further.

Table 1 contains a comparison of the complexity of these four coding systems, comparing numbers of operations needed per 8 transform coefficients:

	No.
Transform	Multiplications	No. Additions	No. Shifts
8 × 8 DCT	13	29	0
8 × 16 Type-I Fast LOT	22	54	0
917 Wavelet, 1-level	36	56	0
8 × 6 Fast LiftLT	14	51	6

In such a comparison, the number of multiplication operations dominates the “cost” of the transform in terms of computing resources and time, and number of additions and number of shifts have negligible effect. In this table, it is clear that the fast LiftLT is almost as low as the DCT in complexity and more than twice as efficient as the wavelet transform.

Table 2 sets forth a number of different performance measures for each of the four methods:

		DC	Stopband
	Coding	Atten.	Atten.	Mir. Freq.
Transform	Gain (dB)	(−dB)	(−dB)	Atten. (−dB)
8 × 8 DCT	8.83	310.62	9.96	322.1
8 × 16 Type-I Fast LOT	9.2	309.04	17.32	314.7
8 × 16 Optional LT	9.62	327.4	13.5	55.54
8 × 16 Fast LiftLT	9.54	312.56	13.21	304.85

The fast LiftLT is comparable to the optional 8×16 LT transform in coding gain and stopband attenuation an significantly better than the DCT.

Reconstructed images for a standard 512×512 “Barbara” test image at 1:32 compression ratio are shown in FIG. 9 for aesthetic and heuristic evaluation. Top left 21 is the reconstructed image for the 8×8 DCT (27.28 dB PSNR); top right shows the result for the 8×16 LOT(28.71 dB PSNR); bottom left is the 9/7 tap wavelet reconstruction (27.58 dB PSNR); and bottom right, 8×16 LiftLT (28.93 dB PSNR). The objective coding results for standard 512×512 “Lena,” “Goldhill,” and “Barbara” test image (PSNR in dB's) are tabulated in Table 3:

	Lena	Goldhill	Barbara
Comp.	9/7 WL	8 × 8	8 × 16	8 × 16	9/7 WL	8 × 8	8 × 16	8 × 16	9/7 WL	8 × 8	8 × 16	8 × 16
Ratio	SPIHT	DCT	LOT	LiftLT	SPIHT	DCT	LOT	LiftLT	SPIHT	DCT	LOT	LiftLT
8	40.41	39.91	40.02	40.21	36.55	36.25	36.56	36.56	36.41	36.31	37.22	37.57
16	37.21	36.38	36.69	37.11	33.13	32.76	33.12	33.22	31.4	31.11	32.52	32.82
32	34.11	32.9	33.49	34	30.56	30.07	30.52	30.63	27.58	27.28	28.71	28.93
64	31.1	29.67	30.43	30.9	28.48	27.93	28.34	28.54	24.86	24.58	25.66	25.93
100	29.35	27.8	28.59	29.03	27.38	26.65	27.08	27.28	23.76	23.42	24.32	24.5
128	28.38	26.91	27.6	28.12	26.73	26.01	26.46	26.7	23.35	22.68	23.36	23.47

PSNR is an acronym for power signal to noise r-atio and represents the logarithm of the ratio of maximum amplitude squared to the mean square error of the reconstructed signal expressed in decibels (dB).

The LiftLT outperforms its block transform relatives for all test images at all bit rates. Comparing to the wavelet transform, the LiftLT is quite competitive on smooth images—about 0.2 dB below on Lena. However, for more complex images such as Goldhill or Barbara, the LiftLT consistently surpasses the 9/7-tap wavelet. The PSNR improvement can reach as high as 1.5 dB.

FIG. 9 also shows pictorially the reconstruction performance in Barbara images at 1:32 compression ratio for heuristic comparison. The visual quality of the LiftLT reconstructed image is noticeably superior. Blocking is completely avoided whereas ringing is reasonably contained. Top left: 8×8 DCT, 27.28 dB. Top right: 8×16 LOT, 28.71 dB. Bottom left: 9/7-tap wavelet, 27.58 dB. Bottom right: 8×16 LiftLT, 28.93 dB. Visual inspection indicates that the LiftLT coder gives at least as good performance as the wavelet coder. The appearance of blocking artifacts in the DCT reconstruction (upper left) is readily apparent. The LOT transform result (upper right) suffers visibly from the same artifacts even though it is lapped. In addition, it is substantially more complex and therefore slower than the DCT transform. The wavelet transform reconstruction (lower left) shows no blocking and is of generally high quality for this level of compression. It is faster than the LOT but significantly slower than the DCT. Finally, the results of the LiftLT transform are shown at lower right. Again, it shows no blocking artifacts, and the picture quality is in general comparable to that of the wavelet transform reconstruction, while its speed is very close to that of the bare DCT.

Claims

1. An apparatus for coding, storing or transmitting, and decoding M×M sized blocks of digitally represented images, where m is an even number, comprising

a. a forward transform comprising

i. a base transform having m channels numbered 0 through M−1, half of said channel numbers being odd and half being even;

ii. an equal normalization factor in each of the m channels selected to be dyadic-rational;

iii. a full-scale butterfly implemented as a series of lifting steps with a first set of dyadic rational coefficients;

iv. m/2 delay lines in the odd numbered channels;

v. a full-scale butterfly implemented as a series of lifting steps with said first set of dyadic rational coefficients; and

vi. a series of lifting steps in the odd numbered channels with a second specifically selected set of dyadic-rational coefficients;

b. means for transmission or storage of the transform output coefficients; and

c. an inverse transform comprising

i. m channels numbered 0 through M−1, half of said channel numbers being odd and half being even;

ii. a series of inverse lifting steps in the odd numbered channels with said second set of specifically selected dyadic-rational coefficients;

iii. a full-scale butterfly implemented as a series of lifting steps with said first set of specifically selected dyadic-rational coefficients;

iv. m/2 delay lines in the even numbered channels;

v. a full-scale butterfly implemented as a series of lifting steps with said first set of specifically selected dyadic-rational coefficients;

vi. an equal denormalization factor in each of the m channels specifically selected to be dyadic-rational; and

vii. a base inverse transform having m channels numbered 0 through M−1.

2. The apparatus of claim 1 in which the normalizing factor takes the value 25/16 and simultaneously the denormalizing factor takes the value 16/25.

3. The apparatus of claim 1 in which the normalizing factor takes the value 5/4 and simultaneously the denormalizing factor takes the value 4/5.

4. The apparatus of claim 1 in which the first set of dyadic rational coefficients are all equal to 1.

5. The apparatus of claim 1 in which the second set of dyadic rational coefficients are all equal to ½.

6. The apparatus of claim 1 in which the base transform is any M×M invertible matrix of the form of a linear phase filter and the inverse base transform is the inverse of said M×M invertible matrix.

7. The apparatus of claim 1 in which the base transform is the forward M×M discrete cosine transform and the inverse base transform is the inverse M×M discrete cosine transform.

8. An apparatus for coding, compressing, storing or transmitting, and decoding a block of M×M intensities from a digital image selected by an M×M window moving recursively over the image, comprising:

a. an M×M block transform comprising:

i. an initial stage

ii. a normalizing factor in each channel

b. a cascade comprising a plurality of dyadic rational lifting transforms, each of said plurality of dyadic rational lifting transforms comprising

i. a first bank of pairs of butterfly lifting steps with unitary coefficients between adjacent lines of said transform;

ii. a bank of delay lines in a first group of m/2 alternating lines;

iii. a second bank of butterfly lifting steps with unitary coefficients, and

iv. a bank of pairs of butterfly lifting steps with coefficients of 1/2 between m/2−1 pairs of said m/2 alternating lines;

c. means for transmission or storage of the output coefficients of said M×M block transform; and

d. an inverse transform comprising

i. a cascade comprising a plurality of dyadic rational lifting transforms, each of said plurality of dyadic rational lifting transforms comprising

a) a bank of pairs of butterfly lifting steps with coefficients of 1/2 between said m/2−1 pairs of said m/2 alternating lines;

b) a first bank of pairs of butterfly lifting steps with unitary coefficients between adjacent lines of said transform;

c) a bank of delay lines in a second group of m/2 alternating lines, ; and

d) a second bank of pairs of butterfly lifting steps with unitary coefficients between adjacent lines of said transform;

ii. a de-scaling bank, ; and

iii. an inverse initial stage.

9. A method of coding, storing or transmitting, and decoding M×M sized blocks of digitally represented images, where m is an even number a power of 2, comprising

a. transmitting the original picture signals to a coder, which effects the steps of

i. converting the signals with a base transform having m channels numbered 0 through M−1, half of said channel numbers being odd and half being even, ;

ii. normalizing the output of the preceding step with a dyadic rational normalization factor in each of said m channels;

iii. processing the output of the preceding step through two lifting steps with a first set of identical dyadic rational coefficients connecting each pair of adjacent numbered channels in a butterfly configuration;

iv. transmitting the resulting coefficients through m/2 delay lines in the odd numbered channels;

v. processing the output of the preceding step through two inverse lifting steps with the first set of dyadic rational coefficients connecting each pair of adjacent numbered channels in a butterfly configuration; and

vi. applying two lifting steps with a second set of identical dyadic rational coefficients connecting each pair of adjacent odd numbered channels to the output of the preceding step;

b. transmitting or storing the transform output coefficients;

c. receiving the transform output coefficients in a decoder; and

d. processing the output coefficients in a decoder, comprising the steps of

i. receiving the coefficients in m channels numbered 0 through M−1, half of said channel numbers being odd and half being even;

ii. applying two inverse lifting steps with dyadic rational coefficients connecting each pair of adjacent odd numbered channels;

iii. applying two lifting steps with dyadic rational coefficients connecting each pair of adjacent numbered channels in a butterfly configuration;

iv. transmitting the result of the preceding step through m/2 delay lines in the even numbered channels;

v. applying two inverse lifting steps with dyadic rational coefficients connecting each pair of adjacent numbered channels in a butterfly configuration;

vi. denormalizing the result of the preceding step with a dyadic rational inverse normalization factor in each of said m channels; and

vii. processing the result of the preceding step through a base inverse transform having m channels numbered 0 through M−1.

10. A method of coding, compressing, storing or transmitting, and decoding a block of M×M intensities from a digital image selected by an M×M window moving recursively over the image, comprising the steps of:

a. Processing the intensities in an M×M block coder comprising the steps of:

i. processing the intensities through an initial stage;

ii. scaling the result of the preceding step in each channel;

b. processing the result of the preceding step through a cascade comprising a plurality of dyadic rational lifting transforms, each of said plurality of dyadic rational lifting transforms comprising

i. a first bank of pairs of butterfly lifting steps with unitary coefficients between adjacent lines of said transform;

ii. a bank of delay lines in a first group of m/2 alternating lines;

iii. a second bank of butterfly lifting steps with unitary coefficients, and

iv. a bank of pairs of butterfly lifting steps with coefficients of 1/2 between m/2−1 pairs of said m/2 alternating lines;

c. transmitting or storing the output coefficients of said M×M block coder;

d. receiving the output coefficients in a decoder; and

e. processing the output coefficients in the decoder, comprising the steps of

i. processing the output coefficients through a cascade comprising a plurality of dyadic rational lifting transforms, each of said plurality of dyadic rational lifting transforms comprising

a) a bank of pairs of butterfly lifting steps with coefficients of 1/2 between said m/2−1 pairs of said m/2 alternating lines;

b) a first bank of pairs of butterfly lifting steps with unitary coefficients between adjacent lines of said transform;

c) a bank of delay lines in a second group of m/2 alternating lines;

d) a second bank of pairs of butterfly lifting steps with unitary coefficients between adjacent lines of said transform;

e) a de-scaling bank; and

f. processing the results of the preceding step in an inverse initial stage.

11. The apparatus of claim 1 in which the constants coefficients are approximations chosen for rapid computing rather than exact constants coefficients.

12. A method of coding, storing or transmitting, and decoding a block of M×M intensities from a digital image selected by an M×M window moving recursively over the image, comprising:

a. processing the intensities in an M×M block coder comprising the steps of:

i. processing the intensities through an initial stage;

ii. scaling the result of the preceding step in each channel;

b. processing the result of the preceding step through a transform coder using a method of processing blocks of samples of digital signals of integer length m comprising processing the digital samples of length m with an invertible linear transform of dimension m, said transform being representable as a cascade, using the steps, in arbitrary order, of:

i) at least one +/−1 butterfly step,

ii) at least one lifting step with rational complex coefficients, and

iii) at least one scaling factor;

c. transmitting or storing the output coefficients of said M×M block coder;

d. receiving the output coefficients in a decoder; and

e. processing the output coefficients in the decoder into a reconstructed image using the inverse of the coder of steps a. and b.

13. The method of claim 12 wherein the method of processing blocks of samples of digital signals of integer length m additionally comprises the step of at least one time delay.

14. The method of claim 12, wherein the rational complex coefficients in the at least one lifting step are dyadic.

15. The method of claim 12, wherein

a) said invertible transform is an approximation of a biorthogonal transform;

b) said biorthogonal transformation comprises a representation as a cascade of at least one butterfly step, at least one orthogonal transform, and at least one scaling factor;

c) said at least one orthogonal transform comprises a cascade of

i) at least one ±1 butterfly step,

ii) at least one planar rotation, and

iii) at least one scaling factor;

b) said at least one planar rotation being represented by equivalent lifting steps and scale factors; and,

c) said approximation is obtained by replacing floating point coefficients in the lifting steps with rational coefficients.

16. The method of claim 15, wherein the coefficients of the lifting steps are chosen to be dyadic rational.

17. The method of claim 12, wherein the invertible transform is a unitary transform.

18. The method of claim 12, wherein

a) said invertible transform is an approximation of a unitary transform;

b) said approximation of the unitary transform comprises a representation of the unitary transform as a cascade of at least one butterfly step, at least one orthogonal transform, and at least one scale factor;

c) said at least one orthogonal transform being represented as a cascade of

(1) at least one ±1 butterfly steps,

(2) at least one planar rotation, and

(3) at least one scaling factor;

d) said at least one planar rotation being represented by equivalent lifting steps and scale factors; and,

e) said approximation being derived by using approximate rational values for the coefficients in the lifting steps.

19. The method of claim 18, wherein the invertible transform is an approximation of a transform selected from the group of special unitary transforms: discrete cosine transform (DCT); discrete Fourier transform (DFT); discrete sine transform (DST).

20. The method of claim 18, wherein the coefficients of the lifting steps are dyadic rational.

21. The method of claim 18, wherein at least one of the following lifting steps is used, whose matrix representations take on the form: $\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ b& 1\end{array}\right],$

22. The method of claim 21, wherein the invertible transform is an approximation of a transform selected from the group: discrete cosine transform (DCT); discrete Fourier transform (DFT); discrete sine transform (DST).

23. The method of claim 22, wherein the approximation of the 4 point DCT is selected from the group of matrices: $\left\{\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 1& -1& 1& -1\end{array}\right],\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 1& -2& 2& -1\end{array}\right],\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 5& -2& 2& -5\end{array}\right]\right\}.$

24. The method of claim 19 in which the invertible transform is an approximation of a transform selected from the group three point DCT, 4 point DCT, 8 point DCT, and 16 point DCT.

25. The method of claim 19 in which the invertible transform is an approximation of a transform selected from the group 512 point FFT, 1024 point FFT, 2048 point FFT, and 4096 point FFT.

26. A method of coding, storing or transmitting, and decoding sequences of intensities of integer length m recursively selected from a time ordered string of intensities arising from electrical signals, the method comprising the steps of

a) recursively processing the sequences of intensities of integer length m with an invertible forward linear transform of dimension m, said transform being representable as a cascade using the steps, in a preselected arbitrary order, of:

ii) at least one ±1 butterfly step,

iii) at least one lifting step with rational complex coefficients, and

iv) applying at least one scaling factor;

b) compressing the resulting transform coefficients;

c) storing or transmitting the compressed transform coefficients;

d) receiving or recovering from storage the transmitted or stored compressed transform coefficients;

e) decompressing the received or recovered compressed transform coefficients; and

f) recursively processing the decompressed transform coefficients with the inverse of the forward linear transform of dimension m, said inverse transform being representable as a cascade using the steps, in the exact reverse order of the preselected arbitrary order, of:

ii) at least one inverse butterfly corresponding to each of the at least one ±1 butterfly step;

iii) at least one inverse lifting step corresponding to each of the at least one lifting step with rational complex coefficients; and,

iv) applying at least on inverse scaling factor corresponding to the at least one scaling factor.

27. The method of claim 26 wherein the method of processing blocks of samples of digital signals of integer length m additionally comprises the step of at least one time delay.

28. The method of claim 26, wherein the rational complex coefficients in the at least one lifting step are dyadic.

29. The method of claim 26, wherein

a) said invertible transform is an approximation of a biorthogonal transform;

b) said biorthogonal transformation comprises a representation as a cascade of at least one butterfly step, at least one orthogonal transform, and at least one scaling factor;

c) said at least one orthogonal transform comprising a cascade of

i) at least one ±1 butterfly step,

ii) at least one planar rotation, and

iii) at least one scaling factor;

b) said at least one planar rotation being represented by equivalent lifting steps and scale factors; and,

c) said approximation being obtained by replacing floating point coefficients in the lifting steps with rational coefficients.

30. The method of claim 29, wherein the coefficients of the lifting steps are chosen to be dyadic rational.

31. The method of claim 26, wherein the invertible transform is a unitary transform.

32. The method of claim 26, wherein

a) said invertible transform is an approximation of a unitary transform;

b) said approximation of the unitary transform comprises a representation of the unitary transform as a cascade of at least one butterfly step, at least one orthogonal transform, and at least one scale factor;

c) said at least one orthogonal transform being represented as a cascade of

(1) at least one ±1 butterfly steps,

(2) at least one planar rotation, and

(3) at last one scaling factor;

d) said at least one planar rotation being represented by equivalent lifting steps and scale factors; and,

e) said approximation being derived by using approximate rational values for the coefficients in the lifting steps.

33. The method of claim 32, wherein the invertible transform is an approximation of a transform selected from the group of special unitary transforms: discrete cosine transform (DCT); discrete Fourier transform (DFT); discrete sine transform (DST).

34. The method of claim 32, wherein the coefficients of the lifting steps are dyadic rational.

35. The method of claim 32, wherein at least one of the following lifting steps is used, whose matrix representations take on the form: $\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ b& 1\end{array}\right],$

36. The method of claim 35, wherein the invertible transform is an approximation of a transform selected from the group: discrete cosine transform (DCT); discrete Fourier transform (DFT); discrete sine transform (DST).

37. The method of claim 36, wherein the approximation of the 4 point DCT is selected from the group of matrices: $\left\{\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 1& -1& 1& -1\end{array}\right],\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 1& -2& 2& -1\end{array}\right],\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 1& -1& -2\\ 1& -1& -1& 1\\ 5& -2& 2& -5\end{array}\right]\right\}.$

38. The method of claim 33 in which the invertible transform is an approximation of a transform selected from the group three point DCT, 4 point DCT, 8 point DCT, and 16 point DCT.

39. The method of claim 33 in which the invertible transform is an approximation of a transform selected from the group 512 point FFT, 1024 point FFT, 2048 point FFT, and 4096 point FFT.