Odd-length armlets with flipping property and its application in image compression

doi:10.1016/j.eswa.2009.01.026

Expert Systems with Applications

Volume 36, Issue 6, August 2009, Pages 10025-10029

https://doi.org/10.1016/j.eswa.2009.01.026 Get rights and content

Abstract

Armlet introduced by Lian is a novel property of multiwavelet designed for signal processing. In this paper, we first recall some concepts of armlet. Then, we construct a new class of analysis ready multiwavelets (armlets) with odd-length filters based on Lian’s method. To test the performance of armlet in application, we study the statistical analysis of their discrete multiwavelet transforms (DMWTs) and use them to compress images. In comparison with other wavelets, armlets show potential advantages.

Introduction

Though multiwavelet can possess orthogonal, smooth, compact support and symmetry in theory at the same time, its computational drawbacks in application somehow shadow its theoretical advantages. It is not easy to deal with scalar-valued signal effectively by such a multi-channel filter. To tackle this problem, some notions are introduced: prefiltering (Hardin and Roach, 1998, Xia, 1998, Xia et al., 1996), balancing (Chui and Jiang, 2005, Lebrun et al., 1997, Lebrun and Vetterli, 1998, Lebrun and Vetterli, 2002, Lebrun, 2000, Lian and Chui, 2004, Weidmann et al., 1998), and armlet (Lian and Chui, 2004, Lian, 2005). By applying certain appropriate prefilter/postfilter processing to input/output data, prefiltering can translate the data to fit for the multi-channels. However, this approach sometimes destroys the good properties of multiwavelets such as orthogonal and symmetrical. It also makes the signal processing complicated, and costs more computing time. Balancing introduced by Lebrun can avoid prefiltering effectively by designing multiwavelets, but the nonlinearities of n-balancing conditions make it difficult to construct multiwavelets with balanced order for a relatively large n.

More recently, Lian and Chui (2004) introduced the notion of armlet. It is another way to avoid prefiltering, but is easier to construct multiwavelets with high armlet order than to construct balanced multiwavelets. Though it cannot preserve/cancel discrete-time polynomial signals in the lowpass/highpass subbands at the same time as balancing, it can provide precise tool to guarantee wavelet decomposition with highpass output not being effected by polynomial perturbation of the input. In other words, it can ensure that the matrix-valued wavelet decomposition process guarantees the same highpass output for the two output data sequence ${x_{l}}$ and ${x_{l} + p (l)}$ , where $p (x)$ is any polynomial of degree n (or degree $n - 1$ ) for a desirable integer $n ⩾ 1$ , which is called nth order wavelet analysis consistency ( $n - WAC$ ) requirement. The inner relationship between balancing and armlet is that n-balanced multiwavelets are always armlet of order at least n. Since annihilation of high-order polynomials is more crucial for exacting high-frequency contents than low-frequency ones and since the construction of armlets is easier than that of balancing multiwavelets, it is interesting to construct armlets and study its application.

The paper is organized as follows: first, we introduce some preliminaries about armlet and the method of constructing armlets. Then by modifying the parameters in constructing the algorithm, we construct a class of odd-length armlets. Finally, we make a statistical analysis of its DMWT and apply them (both of even-length and of odd-length) to image compression. In comparison with other wavelets, we show the statistical results.

Section snippets

Preliminaries

To be more simple, we only discuss the multiplicity of $r = 2$ . Let $Φ = [ϕ_{1}, ϕ_{2}]^{T}$ be an orthogonal scaling function vector, and $Ψ = [ψ_{1}, ψ_{2}]^{T}$ be the orthogonal multiwavelet corresponding to $Φ$ . They satisfy the following two scaling equation: $Φ (t) = \sum_{k = 0}^{M} P_{k} Φ (2 t - k), Ψ (t) = \sum_{k = 0}^{N} Q_{k} Φ (2 t - k),$ where $P_{k} and Q_{k}$ are square matrices of order 2 with $P_{0}, P_{M}, Q_{0}, Q_{N} \neq 0$ . As usual, define the two-scale(matrix) symbols of $Φ$ and $Ψ$ are defined as $P (z) = \frac{1}{2} \sum_{k = 0}^{M} P_{k} z^{k}, Q (z) = \frac{1}{2} \sum_{k = 0}^{N} Q_{k} z^{k},$ respectively, or in matrix style $P (z) = [\begin{matrix} P_{11} (z) & P_{12} (z) \\ P_{21} (z) & P \end{matrix}]$

Construction of odd-length armlets

In Lian’s paper (2005), he established an explicit formulation to express $Q_{k}$ in terms of $P_{k}$ for the setting of orthogonal multiwavelet.

Theorem 3.1

Let $Φ$ be the orthogonal scaling function vectors and $Ψ$ be the orthogonal multiwavelets corresponding to $Φ$ , with their two-scale symbols $P (z) and Q (z)$ satisfying (1). Assume that $H_{1} (z), H_{2} (z), H_{3} (z), and H_{4} (z)$ satisfy Eqs. (7), (8). Then, all entries in $P (z) and Q (z)$ are determined by $P_{11} (z)$ , namely

\begin{matrix} P_{12} (z) = z^{M - 2 K + 1} P_{11} (- z) R (\frac{1}{z^{2}}), P_{21} (z) = z^{2 K - 1} P_{11} (- \frac{1}{z}) R (z^{2}), \\ P_{22} (z) = z^{M} P_{11} (\frac{1}{z}), \\ Q_{11} (z) = \pm \sqrt{2} \end{matrix}

Examples

By using the results from Section 3, we can construct a family of armlets with odd-length filters.

Example 4.1

Let $M = \deg (P_{11}) = 2$ , $Ψ$ has an armlet of order $n = 2$ . Assume $P_{11} (z) = a + bz + {cz}^{2}$ , then by applying Eq. (15), we get

P_{11} (z) = \frac{{cd}^{2} + c}{1 + d^{2}} z^{2} + \frac{- d + 1}{2 (1 + d^{2})} z + \frac{- 2 {cd}^{2} - 2 c + d + 1}{2 (1 + d^{2})} .

Using Eq. (14), we can calculate c in terms of d

c = \frac{d^{2} + 3 d - 1}{4 (1 + d^{2}) (d - 1)} .

Replacing c in Eq. (16) with the above-mentioned equation, together with Eq. (12), we find that the parameter d must satisfy

g (d) = d^{4} - 11 d^{2} + 1 = 0 .

To get smoother

Φ

and

Ψ

, we set

d =

Statistical analysis and application in image compression

As our filters have been balanced, we can directly apply them to image processing. To handle the boundary, we select the method that Tao Xia presented in his paper Xia and Jiang (1999). The images Lena and Baboon are tested to analyze the statistical quality of the armlets. We choose $bior 4.4 (db 9 / 7)$ , $CL 2$ (Chui & Lian, 1996), $Opt - recl (Opt)$ (Jiang, 1998), Armlets ( ${Arm}_{i}$ means that Armlets with $M = i$ ) for comparison. Here, $CL 2$ and $Opt - recl$ have been balanced. Table 4, Table 5 show the percent of

Conclusion

In this paper, we introduce a useful property to characterize multiwavelet. By computing odd-length armlets, we add a new class to armlet multiwavelet family. Then, we analyze the statistical feature of their DMWT. Results show that armlets can concentrate more energy in lowpass filter than in other ones. The simulating compress experiments also confirm that armlet is very useful to design multiwavelet for image compression.

Acknowledgements

The authors would like to thank the anonymous reviews for their comments that helped improve the presentation of the paper. The first author also thanks the support of the construction of disciplines on nonlinear analysis and computation of Beijing University of Chemical Technology.

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Odd-length armlets with flipping property and its application in image compression

Abstract

Introduction

Section snippets

Preliminaries

Construction of odd-length armlets

Examples

Statistical analysis and application in image compression

Conclusion

Acknowledgements

Applied Numerical Mathematics

Balanced multiwavelets in Rs

Mathematics of Computation

Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order p⩽2

IEEE Transactions on Circuits and Systems II – Analog and Digital Signal Processing

Orthogonal multiwavelets with optinum time–frequency resolution

IEEE Transactions on Signal Process

Balanced multiwavelets theory and design

IEEE Transactions on Signal Processing

Balanced multiwavelets in $R^{s}$

Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order $p ⩽ 2$