A biorthogonal wavelet design technique using Karhunen–Loéve transform approximation

doi:10.1016/j.dsp.2015.06.002

Digital Signal Processing

Volume 51, April 2016, Pages 202-222

https://doi.org/10.1016/j.dsp.2015.06.002 Get rights and content

Abstract

The lifting style biorthogonal wavelet implementation has a nice property of enabling flexible design; it is immediately reversible and has a simple relation to subband filters. In this work, we present a method to design prediction (P) and update (U) filters of two-channel lifting structures by minimising the difference between Block Wavelet Transform (BWT) matrix of the wavelet and the Karhunen–Loéve Transform (KLT) of a stochastic process with certain autocorrelation. Here, BWTs are transform matrices that are generated by constructing columns through balanced wavelet trees fed by shifted impulse trains. Although wavelets already have fast implementations through subband filtering or lifting, parametric optimisation of the filter coefficients is still possible through indirectly mimicking the corresponding wavelet and a class of signals with certain KLT. This paper describes the above optimisation by putting constraints on the P and U filters for regularity and constructing the filter coefficients in a least-squares sense. In this part of the paper, the iterated orthogonality constraint over the BWT is resolved with a generalised $12 k + 1$ -tap/ $6 k + 1$ -tap P / U construction with numerical results for the $4 \times 4$ and $8 \times 8$ BWT cases. Experimental approximation to several KLT cases with numerical results in terms of filter coefficients, their spectral behaviour, compaction gain, etc. are provided.

Section snippets

Nomenclature

•
Matrices are notated by bold-face upper-case letters (i.e. X).
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Vectors are notated by a bold-face lower-case letters (i.e. x).
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Sizes of some matrices are indicated on the top-right of the matrix name (i.e. $X^{2 \times 2}$ ).
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Coefficients of the matrix (e.g. X) are shown with a lower-case letter having subscripts that stand for the matrix index coordinates (e.g. $X = {[x_{i, j}]}_{i, j = 1, \dots, N}$ ).
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${(ζ)}_{ξ}$ stands for $ζ (\mod ξ)$ . Such modulo operations may even exist in index subscripts or superscripts.
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For a vector $x [i] = {[x_{0}, x_{1}, x_{2}, \dots x_{n}]}^{T}$

DAUB5/3 wavelet

The prediction and update filters of daub5/3 wavelet are $P (z) = 0.5 (1 + z)$ and $U (z) = 0.25 (1 + z^{- 1})$ , respectively [10]. This particular wavelet produces orthogonal $2 \times 2$ and $4 \times 4$ BWTs. However, the condition in Equation (B.21) is violated for the $8 \times 8$ case. These BWT matrices of daub5/3 are $A_{daub 5 / 3}^{2 \times 2} = (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ - 1 & 1 \end{matrix})$ $A_{daub 5 / 3}^{4 \times 4} = (\begin{matrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ - 1 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 1 \\ - \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \end{matrix})$ $A_{daub 5 / 3}^{8 \times 8} = (\begin{matrix} \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} \\ - \frac{3}{4} & - \frac{1}{4} & 0 & \frac{1}{4} & \frac{3}{4} & \frac{1}{4} & 0 & - \frac{1}{4} \\ 0 & - \frac{1}{4} & - \frac{3}{4} & - \frac{1}{4} & 0 & \frac{1}{4} & \frac{3}{4} & \frac{1}{4} \\ - \frac{1}{2} & 0 & \frac{1}{2} & 0 & - \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & - \frac{1}{2} & 0 & \frac{1}{2} & 0 & - \frac{1}{2} & 0 & \frac{1}{2} \\ - \frac{1}{2} & 0 & \frac{1}{2} & - 1 & \frac{1}{2} & 0 & - \frac{1}{2} & 1 \\ \frac{1}{2} & - 1 & \frac{1}{2} & 0 & - \frac{1}{2} & 1 & - \frac{1}{2} & 0 \\ - \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} \end{matrix})$

CRF13/7 wavelet

The crf13/7

Wavelets which develop circulant and orthogonal submatrices: $X_{00}$ , $X_{01}$ and $X_{10}$

The prediction and update filters of a particular biorthogonal wavelet example, where the corresponding BWT matrices become orthogonal, are given below: $P_{13 / 7} (z) = \frac{1}{2} (z^{- 1} - 1 + z + z^{2})$ $U_{13 / 7} (z) = \frac{1}{4} (z - 1 + z^{- 1} + z^{- 2})$ The corresponding $2 \times 2$ , $4 \times 4$ and $8 \times 8$ BWT matrices (which all happen to be orthogonal) are $A_{13 / 7}^{2 \times 2} = (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ - 1 & 1 \end{matrix})$ $A_{13 / 7}^{4 \times 4} = (\begin{matrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\ - \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} \\ - 1 & - 1 & 1 & 1 \\ - \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \end{matrix})$ $A_{13 / 7}^{8 \times 8} = (\begin{matrix} \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} \\ - \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} \\ - \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} \\ - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ - 1 & - 1 & - 1 & - 1 & 1 & 1 & 1 & 1 \\ - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & - \frac{1}{2} & - \frac{1}{2} \\ - \frac{1}{4} & \frac{1}{4} & - \frac{1}{4} & \frac{1}{4} & - 1 \end{matrix}$

Approximating a block wavelet transform matrix to a Karhunen–Loéve transform matrix

KLT is a transform matrix that minimises the reconstruction error between the signal in its original form and the one that is reconstructed from a “restricted” set of basis signals: $ε [n] = x [n] - \tilde{x} [n]$ , where the signal and its basis-restricted versions are: $\begin{matrix} x [n] & = & y_{1} e_{1} [n] + y_{2} e_{2} [n] + \dots + y_{m} e_{m} [n] \\ + y_{m + 1} e_{m + 1} [n] + \dots + y_{n} e_{n} [n] \\ \tilde{x} [n] & = & y_{1} e_{1} [n] + y_{2} e_{2} [n] + \dots + y_{m} e_{m} [n] \end{matrix}$

The minimisation of the squared-error yields a set of basis signals ( $e_{i} [n]$ ) as the eigenvectors of the autocorrelation matrix R corresponding to the signal, $x [n]$ .

Numerical results and discussion

In this research, 16 test images were used (out of which, 4 are deliberately constructed to contain different autocorrelation characteristics, 2 by using Canny edge detection over Lena and Mandrill images, and another 2 by taking frame differences of from Foreman and Bus video sequences). First, we consider the detailed case study of the Lena image and illustrate all the numerical steps during the $4 \times 4$ BWT approximation (in Section 4.1 and its subsections). Then we present results for the

Conclusions

In this research, a signal specific methodology to design lifting wavelets at certain sizes (5/3 for the $4 \times 4$ case and 13/7 for the $8 \times 8$ case) is presented. On the road, the dependence of $2^{k + 1} \times 2^{k + 1}$ BWT matrix to the $2^{k} \times 2^{k}$ BWT matrix using k-level lifting decomposition structure is discovered. With this dependency relationship, our method provided a recursion to achieve the BWT of a $k + 1$ -level lifting decomposition structure using only a k-level lifting scheme, while keeping the orthogonality

Mehmet Cemil Kale received his Ph.D. from the Ohio State University, Ohio, USA, in 2008. During his Ph.D., he was a Graduate Research Associate at the Department of Radiology and the Department of Electrical and Computer Engineering. Currently, he is an assistant professor at the Osmangazi University, Eskisehir, Turkey. His research interests include wavelet design.

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There are more references available in the full text version of this article.

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View full text

A biorthogonal wavelet design technique using Karhunen–Loéve transform approximation

Abstract

Section snippets

Nomenclature

DAUB5/3 wavelet

CRF13/7 wavelet

Wavelets which develop circulant and orthogonal submatrices: X00, X01 and X10

Approximating a block wavelet transform matrix to a Karhunen–Loéve transform matrix

Numerical results and discussion

Conclusions

Appl. Comput. Harmon. Anal.

Ten Lectures on Wavelets

Wavelets and Subband Coding

A block spin construction of ondelettes. Part I: Lemarie functions

Commun. Math. Phys.

Orthonormal bases for compactly supported wavelets

Commun. Pure Appl. Math.

A block spin construction of ondelettes. Part II: the QFT connection

Commun. Math. Phys.

Multiresolution approximations and wavelet orthonormal bases of L2(R)

Trans. Am. Math. Soc.

Biorthogonal bases of compactly supported wavelets

Commun. Pure Appl. Math.

Wavelets which develop circulant and orthogonal submatrices: $X_{00}$ , $X_{01}$ and $X_{10}$

Multiresolution approximations and wavelet orthonormal bases of $L_{2} (R)$