Wavelet transform method and apparatus

[0002] The present invention relates to a method and apparatus for executing a wavelet transform on an image signal.

[0003] The wavelet transform is a mathematical tool for characterizing the local properties of a signal at a variety of resolutions. In recent years, this has been found to be an effective way to compress digitized image signals.

[0004] When the wavelet transform is applied to an image signal by the pyramid method of the prior art, which will be described in more detail later, the image signal passes in succession through a plurality of stages. In each stage, the signal is filtered horizontally and vertically, and the resolution of the signal is reduced by half in each dimension.

[0005] One problem with this method is the repeated need for temporary storage of the image signal. Signal storage is necessary between the horizontal filtering and vertical filtering operations in each stage of the transform, and is also necessary between the different stages. A filtering operation cannot begin until the signal has been stored, so besides consuming memory space, the repeated storage operations impair the speed of the transform.

[0006] Another problem is that in the final output signal of the transform, high and low spatial frequencies are mixed in a way that is not advantageous for compression. This problem will be described in more detail later.

[0007] A further problem is that the prior-art method works only with rectangular images. If an image is not rectangular, it must be embedded in a rectangle, and the non-image parts of the rectangle must be filled in with, for example, the average value of the image signal, or with signal values copied from the border of the image. Efficient compression of the resulting rectangular image tends to be impaired by high-frequency artifacts generated by the abrupt transitions between the image area and the filled-in area.

[0008] It is accordingly an object of the present invention to execute a wavelet transform on a two-dimensional image signal without requiring a repeated storing of the image signal.

[0009] Another object of the present invention is to produce a wavelet transform output signal that can be highly compressed by further encoding.

[0010] Yet another object is to execute a wavelet transform on a two-dimensional image signal representing an image with an arbitrary shape, without generating artifacts.

[0011] According to a first aspect of the invention, an N-level wavelet transform is executed on an image signal, where N is an integer greater than one. The image signal represents an image having a first dimension and a second dimension. First, an N-level one-dimensional wavelet transform is executed on the image signal in the first dimension, generating an intermediate signal which is temporarily stored in a memory device. Then an N-level one-dimensional thereby wavelet transform is executed on the intermediate signal in the second dimension. Each N-level one-dimensional wavelet transform is preferably executed by a single matrix operation.

[0012] According to a second aspect of the invention, an N-level wavelet transform is executed on an image signal, where N is an integer greater than zero. The image signal represents an image having a first dimension and a second dimension, and having an arbitrary shape. The image signal is accompanied by shape information describing the shape of the image. First, N single-level one-dimensional wavelet transforms are executed on the image signal in a cascaded series in the first dimension, thereby generating an intermediate signal which is temporarily stored in a memory device. When each single-level one-dimensional wavelet transform is executed, the shape information is altered to describe the shapes of the component signals resulting from the transform. Next, N single-level one-dimensional wavelet transforms are executed on the intermediate signal in a cascaded series in the second dimension, accompanied by further similar alterations of the shape information.

[0013] FIG. 1 is a block diagram illustrating a first embodiment of the invention;

[0014] FIG. 2 illustrates the decomposition of an image signal into high-frequency and low-frequency components by the first embodiment;

[0015] FIG. 3 mathematically illustrates the operation of the N-level vertical wavelet filter in the first embodiment;

[0016] FIG. 4 is a block diagram illustrating a second embodiment;

[0017] FIG. 5 mathematically illustrates the operation of the N-level vertical wavelet filter in the second embodiment;

[0018] FIG. 6 is a block diagram illustrating a third embodiment;

[0019] FIG. 7 is a block diagram illustrating a fourth embodiment;

[0020] FIG. 8 is a block diagram illustrating a single-level one-dimensional wavelet filter;

[0021] FIG. 9 is a block diagram illustrating the prior art; and

[0022] FIG. 10 illustrates the decomposition of an image signal into high-frequency and low-frequency components in the prior art.

[0023] Embodiments of the invention will be described with reference to the attached drawings, but first, it will be useful to give some general information and describe the prior art in more detail.

[0024] Referring to FIG. 8, a single-level one-dimensional wavelet filter comprises a low-pass filter 2 and a high-pass filter 4, both of which receive the same input signal S. If S_Lis the output of the low-pass filter 2 and S_His the output of the high-pass filter 4, the operation of the wavelet filter can be described by the equations shown below, in which m is a positive integer, n is a non-negative integer, h(k) are the filter coefficients of the low-pass filter 2, and g(k) are the filter coefficients of the high-pass filter 4 (k=−n, . . . , m). $\begin{array}{c}{S}_L\left(i\right)=\sum _{k=-n}^{k=m}S\left(2i+k\right)h\left(k\right)\\ S_H\left(i\right)=\sum _{k=-n}^{k=m}S\left(2i+k\right)g\left(k\right)\end{array}$

[0025] The two output signals S_Land S_Heach have half the resolution of the input signal S. Resolution can be expressed as the number of signal samples. For integers N greater than unity, an N-level one-dimensional wavelet transform is carried out by iterating this process on the low-frequency component S_L.

[0026] When the input signal represents a two-dimensional image, wavelet filtering is performed separately in each dimension. FIG. 9 shows an apparatus for performing a three-level wavelet transform on a two-dimensional image signal by the pyramid method of the prior art.

[0027] In the first stage 6 of the apparatus, a horizontal wavelet filter 8 executes a single-level one-dimensional wavelet transform on each horizontal row of picture elements, thereby obtaining a low-frequency component signal S_L3which is temporarily stored in a memory device 10, and a high-frequency component signal S_H3which is temporarily stored in a memory device 12. Then a first vertical wavelet filter 14 performs a single-level one-dimensional wavelet transform on each vertical column of picture elements in S_L3, thereby obtaining a low-frequency component signal S_LL3and a high-frequency component signal S_LH3, and a second vertical wavelet filter 14 performs a single-level one-dimensional wavelet transform on each vertical column of picture elements in S_H3, thereby obtaining a low-frequency component signal S_HL3and a high-frequency component signal S_HH3. Of these four signals, the purely low-frequency component signal S_LL3is temporarily stored in a memory device 18, while S_LH3, S_HL3, and S_HH3, which contain high-frequency components, are output without further processing.

[0028] The internal memory devices 10 and 12 are necessary because the horizontal wavelet filter 8 generates the picture elements of the S_L3and S_H3signals in one order, working horizontally one row at a time, but the vertical wavelet filters 14 and 16 process the same S_L3and S_H3signals in a different order, working vertically one column at a time.

[0029] The second stage 20 has the same internal configuration as the first stage 6, and operates in similar fashion on the signal S_LL3stored in memory device 18 to obtain four more signals, denoted S_LL2, S_LH2, S_HL2, and S_HH2. Of these signals, S_LL2is stored in a memory device 22, and S_LH2, S_HL2, and S_HH2are output without further processing.

[0030] The third stage 24 has the same internal configuration, and operates in a similar fashion on the signal S_LL2stored in memory device 22, obtaining four more output signals S_LL1, S_LH1, S_HL1, and S_HH1. It can be seen that five temporary storage operations are needed to obtain each of these signals, including one storage operation in each of the three stages 6, 22, and 24, and two more storage operations between the stages, in memory devices 18 and 22.

[0031] FIG. 10 locates the ten resulting output signals in the two-dimensional plane. The plane is iteratively decomposed by dividing lines, each separating a low spatial frequency component (above or to the left of the dividing line) from a high spatial frequency component (below or to the right of the dividing line). The signal S_HL3, for example, represents a high horizontal spatial frequency component and a low vertical spatial frequency component.

[0032] The presence of this low vertical spatial frequency component means that the S_HL3signal possesses a strong residual correlation in the vertical dimension, which limits the amount by which this signal can be compressed by further encoding. Compression of S_LH3, S_HL2, and S_LH2is similarly limited by residual correlation.

[0033] First Embodiment

[0034] Referring to FIG. 1, a first novel wavelet transform apparatus comprises an N-level horizontal wavelet filter 26, a memory device 28, and an N-level vertical wavelet filter 30. In this embodiment N is equal to three. Filter 26 comprises three cascaded single-level horizontal wavelet filters 32, 34, and 36, while filter 30 comprises three cascaded single-level vertical wavelet filters 38, 40, and 42. The memory device 28 stores the output of the N-level horizontal wavelet filter 26 as an intermediate signal X, which is provided as input to the N-level vertical wavelet filter 30.

[0035] The wavelet transform performed by the first embodiment will be referred to as a separable wavelet transform, because it can be separated into a horizontal one-dimensional wavelet transform followed by a vertical one-dimensional wavelet transform.

[0036] The individual wavelet filters 32, 24, 36, 38, 40, and 42 comprise well-known electronic computing circuits, descriptions of which will be omitted to avoid obscuring the invention with unnecessary detail.

[0037] In the N-level horizontal wavelet filter 26, horizontal wavelet filter 32 decomposes the input two-dimensional image signal S into a low-frequency component S_L3and a high-frequency component S_H3by applying the equations given above to each horizontal row of picture elements in S. In the same way, horizontal wavelet filter 34 decomposes S_L3into a low-frequency component S_L2and a high-frequency component S_H2, and horizontal wavelet filter 36 decomposes S_L2into a low-frequency component S_L1and a high-frequency component S_H1.

[0038] The three single-level horizontal wavelet filters 32, 34, and 36 are thus cascaded by transmitting the low-frequency component from one filter to the next. In this process, it is not necessary to store the entire S_L3component signal between filters 32 and 34, because filter 34 processes the S_L3signal values in the same order as the order of output from horizontal wavelet filter 32, working horizontally one row at a time. Similarly, it is not necessary to store the entire S_L2signal between filters 34 and 36.

[0039] The memory device 28 stores the four signals S_L1, S_H1, S_H2, and S_H3. Vertical wavelet filter 38 decomposes each of these four signals into a low-frequency component and a high-frequency component. Vertical wavelet filter 40 then decomposes each of the four low-frequency components output by vertical wavelet filter 38 into a further low-frequency component and a high-frequency component, and vertical wavelet filter 42 decomposes each of the four low-frequency components output by vertical wavelet filter 38 into a still further low-frequency component and a high-frequency component. The output Y of the first embodiment comprises the high-frequency components output by vertical wavelet filters 38, 40, and 42, and the low-frequency component output by vertical wavelet filter 42.

[0040] FIG. 2 exhibits the output of the first embodiment in the two-dimensional plane, the dividing lines representing the separation of low-frequency from high-frequency components as in FIG. 10. First the N-level horizontal wavelet filter 26 divides the plane into four areas delimited by vertical lines in FIG. 2, representing the four signals S_L1, S_H1, S_H2, and S_H3. Then the N-level vertical wavelet filter 30 divides each of these four areas into four sub-areas, representing sixteen separate component signals in all.

[0041] Of these sixteen signals, S_LL1, S_LH1, S_HL1, S_HH1, S_HH2, and S_HH3are identical to the corresponding signals in FIG. 10. The area that produced one signal component S_HL3in FIG. 10, however, is divided into three components S_HL30, S_HL31, and S_HL32in FIG. 2. When S_HL32is compressed by further encoding, a higher compression ratio can be achieved than the ratio achievable for S_HL3in FIG. 10, because the lowest vertical spatial frequencies are eliminated and the residual correlation is less. Similar gains in compression are obtained in S_LH23and in several other components in FIG. 2, so the overall compression ratio is significantly improved.

[0042] The operation of the N-level vertical wavelet filter 30 will next be described mathematically as a triple matrix operation, for the case of an image with a height of eight picture elements. The operation is partly shown in FIG. 3. The column vector at the right represents one column in the image after filtering by the N-level horizontal wavelet filter 26. In the notation given above, x₀to x₇are elements of one of the four signals S_L1, S_H1, S_H2, and S_H3.

[0043] In this case, the high-pass and low-pass filters in each single-level wavelet filter have only two non-zero coefficients each. The coefficients of the low-pass filter are h₁and h₂. The coefficients of the high-pass filter are g₁, and g₂. The vertical wavelet filter 38 is mathematically represented by the matrix M₃₈. The vertical wavelet filters 40 and 42 are similarly represented by matrices M₄₀and M₄₂.

[0044] The column vector at the right is multiplied in turn by matrices M₃₈, M₄₀, and M₄₂, producing the output column vector at the left. In the notation of FIG. 2, if x₀to x₇are elements of S_L1, for example, then y₀is an element of S_LL1, y₁is an element of S_LH1, y₂and y₃are elements of S_LH02, and y₄, y₅, Y₆, and Y₇are elements of S_LH03.

[0045] The operation of the N-level horizontal wavelet filter 26 can be similarly described as a triple matrix operation in which a row vector is successively multiplied by three matrices M₃₂, M₃₄, and M₃₆. If the input signal S is represented as a rectangular row-column matrix of picture elements, the entire operation of the first embodiment can be described in matrix form as follows:

[0046] X=SM₃₂M₃₄M₃₆

[0047] Y=M₄₂M₄₀M₃₈X

[0048] As noted above, the output signal Y is more compressible, by well-known techniques such as quantization, run-length encoding, and entropy encoding, than the output obtained in the prior art. In addition, during the transform process, the image signal has to be temporarily stored only once, instead of five times as in the prior art, so less memory space is required, and less time is taken up by memory access operations.

[0049] Second Embodiment

[0050] The second embodiment performs a separable wavelet transform by replacing the triple matrix operations described above with single matrix operations.

[0051] Referring to FIG. 4, the second embodiment comprises a matrix-type horizontal wavelet filter 44 that performs a single matrix operation on the input signal S, producing the intermediate signal X that is stored in the memory device 28. A matrix-type vertical wavelet filter 46 then completes the wavelet transform by performing a single matrix operation on the intermediate signal X.

[0052] The filters 44 and 46 are matrix processors. If the single matrix operation performed by the horizontal wavelet filter 44 is represented by a matrix M₄₄, and the single matrix performed by the vertical wavelet filter 46 by a single matrix M₄₆, then the operation of the second embodiment can be described mathematically as follows:

[0053] X=SM₄₄

[0054] Y=M₄₆X

[0055] The matrices M₄₄and M₄₆are the products of the corresponding matrices in the first embodiment:

[0056] M₄₄=M₃₂M₃₄M₃₆

[0057] M₄₆=M₄₂M₄₀M₃₈

[0058] FIG. 5 partially depicts the operation of the vertical wavelet filter 46, showing a column vector being multiplied by matrix M₄₆. In effect, the second embodiment performs the matrix multiplication operation M₄₂M₄₀M₃₈ahead of time, instead of performing this operation each time the wavelet transform is carried out. The second embodiment therefore operates faster than the first embodiment.

[0059] Third Embodiment

[0060] The third embodiment carries out a separable wavelet transform on images with arbitrary shapes.

[0061] Referring to FIG. 6, the third embodiment comprises an N-level horizontal wavelet filter 48, a pair of memory devices 50 and 52, and an N-level vertical wavelet filter 54. N is again equal to three.

[0062] The input to the third embodiment comprises two signals: an image signal S giving the values of picture elements in the image, and shape information R describing the shape of the image. The image is treated as being embedded in a rectangular area. The shape information R indicates which parts of the rectangular area belong to the image. The shape information R comprises, for example, a rectangular bit mask, or a run-length encoding of such a bit mask.

[0063] In the N-level horizontal wavelet filter 48, a first single-level horizontal wavelet filter 56 performs a wavelet filtering operation on the input signal S according to the shape information R. If, for example, the rectangular area has a width of m picture elements, but only the first n picture elements in a certain horizontal row belong to the image (where m and n are positive integers and m is greater than n), then if n is an even number, horizontal wavelet filter 56 operates on all of these n picture elements, producing component signals S_L3and S_H3with n/2 elements each. if n is an odd number, horizontal wavelet filter 56 operates on the first n−1 picture elements, and appends the n−th picture element to the resulting low-frequency component signal S_L3, so that S_L3has (n−1)/2+1 elements, and component signal S_H3has (n−1)/2 elements.

[0064] A first reshaper 58 then alters the shape information R to describe the shape of the component signals S_L3and S_H3. These component signals can be treated as being embedded in the left and right halves of a rectangular area with the same width m.

[0065] The altered shape information R₃is supplied to a second single-level horizontal wavelet filter 60, which filters S_L3according to R₃, and to a second horizontal reshaper 62. Reshaper 62 further alters R₃to obtain shape information R₂describing the shape of the low-frequency and high-frequency component signals S_L2and S_H2output by filter 60. Shape signal R₂also retains information describing the shape of the high-frequency component signal S_H3output by the first single-level horizontal wavelet filter 56.

[0066] Continuing in the same way, a third single-level horizontal wavelet filter 64 filters S_L2according to R₂, and a third horizontal reshaper 66 alters R₂to obtain shape information R₁describing the shapes of the component signals S_L1and S_H1output by filter 64, as well as the shapes of S_H2and S_H3. Component signals S_L1, S_H1, S_H2, and S_H3are stored in memory device 50 as the intermediate signal X, while shape information R₁is stored in memory device 52.

[0067] The N-level vertical wavelet filter 64 comprises a first single-level vertical wavelet filter 68 and vertical reshaper 70, a second single-level vertical wavelet filter 72 and vertical reshaper 74, and a third single-level vertical wavelet filter 76 and vertical reshaper 78, which operate on the intermediate signal X and shape information R₁in the same way that the N-level horizontal wavelet filter 48 operated on S and R, although in the vertical dimension. Shape signal R₁is successively altered by reshapers 70, 74, and 78 to obtain shape informations R₁₃, R₁₂, and R₁₁. The final outputs of the third embodiment are a wavelet transform signal Y and shape information R₁₁describing the shapes of the various components of Y.

[0068] By altering the shape information as described above, the third embodiment is able to confine operations to the image area at every stage and level of the wavelet transform. No artifacts are introduced by transitions between image and non-image areas, and the output signal Y is free of spurious high-frequency components that would impede compression of the transformed image signal.

[0069] A further advantage of the third embodiment is that there is no need to generate dummy signal values to fill in non-image parts of the rectangular area.

[0070] Fourth Embodiment

[0071] The fourth embodiment applies the reshaping technique or the third embodiment to the pyramid method of the prior art.

[0072] FIG. 7 illustrates the fourth embodiment for a three-level wavelet transform. In the first stage 80, a single-level horizontal wavelet filter 82 filters the image signal S according to the shape information R, and a horizontal reshaper 84 alters R to obtain shape information R₃. The low-frequency and high-frequency components S_L3and S_H3generated by a filter 82 are stored in memory devices 86 and 88, and the altered shape information R₃is stored in a memory device 90. The two single-level vertical wavelet filters 92 and 94 then filter S_L3and S_H3according to shape information R₃, obtaining component signals S_LL3, S_LH3, S_HL3, and S_HH3, and a vertical reshaper 96 alters R₃to obtain shape information R₃₃describing the shapes of these component signals.

[0073] Of the first-stage output signals, the low-frequency component signal S_LL3and shape information R₃₃are stored in a pair of memory devices 98 and are 100 and furnished to a second stage 102, which operates in the same manner as the first stage. The low-frequency component S_LL2and shape information R₂₂output from the second stage are stored in a pair of memory devices 104 and 106 and are furnished to a third stage 108, which also operates in the same manner. The final output of the wavelet transform comprises ten component signals S_LL1, S_LH1, S_HL1, S_HH1, S_LH2, S_HL2, S_HH2, S_LH3, S_HL3, and S_HH3, and shape information R₁₁describing the shapes of these components.

[0074] The third and fourth embodiments both enable a wavelet transform of any number of levels to be performed on an image with an arbitrary shape, without generating high-frequency artifacts. The third embodiment is preferable to the fourth embodiment for the same reasons that the first embodiment is preferable to the prior art: less memory is required, operation is faster, and the output signal is more compressible because of being more finely divided in the two-dimensional plane.

[0075] Although horizontal filtering was performed before vertical filtering in the preceding embodiments, the order could be switched around. The second embodiment, for example, could be performed according to the following equations, without changing the final result Y.

[0076] X=M₄₆S

[0077] Y=XM₄₄

[0078] The individual single-level wavelet filters in the invention need not be separate electronic computing circuits. The invention can also be practiced by means of a suitably programmed general-purpose computing device such as a digital signal processor.

[0079] Those skilled in the art will recognize that other variations are possible within the scope claimed below.