Correspondence between multiwavelet shrinkage and nonlinear diffusion☆
Introduction
Wavelets have been successfully used in signal and image processing [1], [2], [3], [4]. In particular, the undecimated wavelet transform (also called the shift-invariant wavelet transform) denoising [5] has been used widely for signal and image denoising. Let and be an undecimated finite impulse response (FIR) wavelet filter bank satisfying where for a scalar FIR filter , denotes Then undecimated (scalar) wavelet transform denoising of a 1-D noising signal consists of decomposition algorithm: and denoising algorithm: where is a shrinkage function. The hard- and soft-shrinkage functions [1], [2] are commonly used . The signal is called the denoised signal of the original signal .
Nonlinear diffusion filtering is a powerful method for signal and image denoising. Since the nonlinear diffusion was introduced by Perona and Malik in 1990 [6], a variety of nonlinear diffusion filters have been proposed, see e.g. [7], [8] and the references therein. High order nonlinear diffusion was also proposed and studied in [9], [10], [11].
The 1-D second-order nonlinear diffusion equation is given by: with initial condition , where denotes the partial derivative of the unknown function with respect to , and is the diffusivity. When is a signal with noise, then with an appropriate choice of , the solution of (1.2) (for some suitable ) is considered as the denoised signal of .
The nonlinear diffusion (1.2) is discretized in practice. The following is a simple discretization using the first order difference. More precisely, let and be the time and spatial step size respectively. Denote . Let be approximations to the solution at . Then and can be used to approximate respectively and at : Hence, (1.2) can be discretized as for .
The correspondence between Haar wavelet shrinkage and second-order nonlinear diffusion has been studied in [12], which shows that in (1.1) is in (1.3) if shrinkage function and the diffusivity have the relationship: [13] studied a relationship between multi-level wavelet shrinkage and nonlinear diffusion. In addition, [14] obtained the correspondence of coupled Haar wavelet shrinkage and 2-D second-order nonlinear diffusion.
Recently affine frames (also called wavelet frames) have been successfully used in noise removal and data recovery, see [15] and references therein. The correspondence between 1-D wavelet frame shrinkage and high order nonlinear diffusion was obtained in [16] and the correspondence in the multivariable case was studied in [17]. The study of the correspondence led to new types of diffusion equations and helped to design the frame-inspired diffusivity functions. In addition, a relationship between 1-D multi-level frame shrinkage and high order nonlinear diffusion was obtained in [18].
Multiwavelets have been studied by many researchers, see for example [19], [20], [21], [22], [23], [24], [25], [26], [27]. About applications and theoretical study, see [20], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47] and book [48]. A multiwavelet has two or more components. An orthogonal multiwavelet has important properties such as orthogonality, short support, and symmetry, etc. that scalar orthogonal wavelets fail to possess simultaneously. Furthermore, all the papers [42], [49], [50] showed that multiwavelets have a better performance in signal/image denoising than scalar wavelets. In addition, multiwavelet denoising has been used recently in applications including rolling bearing fault detection [51], [52], [53], [54] and study of load spectrum of computer numerical control lathe [55]. Therefore it is worth to further study multiwavelet denoising. In this paper we study the correspondence between multiwavelet denoising and nonlinear diffusion.
The remainder of the paper is organized as follows. In Section 2, we review multiwavelet denoising. In Section 3, we derive the correspondence between multiwavelet shrinkage and nonlinear diffusion. We also show that the multiwavelet shrinkages of the commonly used CL(2) and DGHM multiwavelets are associated with a second-order nonlinear diffusion equation. We provide some experimental results in Section 4. We draw the conclusion in Section 5.
Before moving to the next section, we provide some notations used in this paper. For a sequence of matrices, we define For a matrix , denotes its Hermitian, the complex conjugate and transpose of . For of 2 × 2 matrix, we use denote the -entry of :
Section snippets
Multiwavelet shrinkage
In this section we briefly review multiwavelet shrinkage. For simplicity of presentation, we consider in this paper multiwavelets with multiplicity . Two vector-valued functions and are called a pair of biorthogonal multiwavelets if the collections of and form biorthogonal bases of in the sense that they are Riesz bases of and they are orthogonal to each other:
Correspondence between multiwavelet shrinkage and nonlinear diffusion
We consider high-order nonlinear diffusion equation with initial condition: , where and are some natural numbers. (3.1) will be discretized. We use high-pass filters to discretize the partial derivatives and .
For a (high-pass) scalar filter or with finitely many nonzero, we say that it has vanishing moment order if In the following, unless it is stated otherwise, the vanishing
Signals and image denoising using diffusion-inspired multiwavelet shrinkages
In this section we provide some experimental results on signal and image denoising using the diffusion-inspired multiwavelet shrinkages. We will focus on CL(2) multifilter bank with different shrinkage functions. In the following we give the corresponding shrinkage functions when diffusivity functions are the Perona–Malik diffusivity and Weickert diffusivity functions. We assume the spatial step size .
Let be the Perona–Malik diffusivity [6], where is a constant and
Conclusion and future work
In this paper, we present the correspondence between the diffusivity functions of high-order nonlinear diffusion equations and multiwavelet shrinkage functions. We show that the multiwavelet shrinkages of the commonly used CL(2) and DGHM multiwavelets are associated with a second-order nonlinear diffusion equation. CL(2) and DGHM multiwavelet signal/image denoising results with different shrinkage functions are also presented. Furthermore, we compare the denoising results of multiwavelets to a
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable suggestions and comments which have greatly improved the presentation of this paper.
References (62)
- et al.
From two-dimensional nonlinear difussion to coupled haar wavelet shrinkage
J. Vis. Commun. Image Represent.
(2007) Correspondence between frame shrinkage and high-order nonlinear diffusion
Appl. Numer. Math.
(2012)- et al.
A study of orthonormal multi-wavelets
Appl. Numer. Math.
(1996) - et al.
Fractal functions and wavelet expansions based on several scaling functions
J. Approx. Theory
(1994) - et al.
Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation
J. Comput. Appl. Math.
(2017) Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets
J. Approx. Theory
(2001)- et al.
Multiwavelet density estimation
Appl. Math. Comput.
(2013) - et al.
Symmetric-antisymmetric orthonormal multiwavelets and related scalar wavelets
Appl. Comput. Harmon. Anal.
(2000) - et al.
Biorthogonal balanced multiwavelets with high armlets order and their application in image denoising
Math. Comput. Simulation
(2015) - et al.
Weak fault detection and health degradation monitoring using customized standard multiwavelets
Mech. Syst. Signal Proc.
(2017)
Wind turbine fault detection using multiwavelet de-noising with the data-driven block threshold
Appl. Acoust.
Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review
Mech. Syst. Signal Proc.
Customized maximal-overlap multiwavelet denoising with data-driven group threshold for condition monitoring of rolling mill drivetrain
Mech. Syst. Signal Process.
Compound faults diagnosis based on customized balanced multiwavelets and adaptive maximum correlated kurtosis deconvolution
Measurement
De-noising by soft-thresholding
IEEE Trans. Inform. Theory
Ideal spatial adaptation via wavelet shrinkage
Biometrika
A Wavelet Tour of Signal Proc.
Wavelets and Filter Banks
Translation-invariant denoising
Scale space and edge detection using anisotropic diffusion
IEEE Trans. Pattern Anal. Mach. Intell.
Image selective smoothing and edge detection by nonlinear diffusion
SIAM J. Numer. Anal.
Denoising and Enhancement of Digital Images–Variational Methods, Integrodifferential Equations, and Wavelets
Properties of higher order nonlinear diffusion filtering
J. Math. Imaging Vision
Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time
IEEE Trans. Image Process.
Fourth-order partial differential equations for noise removal
IEEE Trans. Image Process.
Correspondences between wavelet shrinkage and nonlinear diffusion
A multiscale wavelet-inspired scheme for nonlinear diffusion
Int. J. Wavelets Multiresolut. Inf. Process.
Image restoration: Wavelet frame shrinkage, nonlinear evolution PDES, and beyond
Multiscale Model. Simul.
Correspondence between multiscale frame shrinkage and high-order nonlinear diffusion
Multiwavelet analysis and signal processing
IEEE Trans. Circuits Syst.
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