Correspondence between multiwavelet shrinkage and nonlinear diffusion

https://doi.org/10.1016/j.cam.2020.113074Get rights and content

Highlights

  • The correspondence between multiwavelet denoising and nonlinear diffusion is investigated.

  • The paper shows CL(2) and DGHM multiwavelets shrinkages are associated with a second-order diffusion equation.

  • The paper derives high-order nonlinear diffusion equations associated with general multiwavelet shrinkages.

  • Experimental results show diffusion-inspired multiwavelet shrinkage performs better than traditional hard- and soft-thresholding in signal and image denoising.

Abstract

Wavelet/frame shrinkage and nonlinear diffusion filtering are two popular methods for signal and image denoising. The relationship between these two methods has been studied recently. This relationship leads to new types of diffusion equations and helps to design the wavelet/frame-inspired diffusivity functions, and on the other hand it helps to design the diffusion-inspired shrinkage functions with a better performance in signal and image denoising.

Multiwavelets have important properties such as orthogonality, short support, and symmetry, etc. that scalar orthogonal wavelets cannot possess simultaneously. There is rich literature on the theoretical study, construction and applications of multiwavelets. In particular, it has been shown that multiwavelets perform better than the scalar wavelets in signal and image denoising. Recently multiwavelet denoising has been applied in different applications including rolling bearing fault detection and study of load spectrum of computer numerical control lathe. Therefore it is worth to further study multiwavelet denoising.

In this paper we investigate the correspondence between multiwavelet denoising and nonlinear diffusion. We show that the multiwavelet shrinkages of the commonly used CL(2) and DGHM multiwavelets are associated with a second-order nonlinear diffusion equation. We also derive high-order nonlinear diffusion equations associated with general multiwavelet shrinkages. The experimental results carried out in this paper show that the diffusion-inspired multiwavelet shrinkage performs better than the traditional multiwavelet hard- and soft-thresholding shrinkages.

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 {p,q} and {p˜,q˜} be an undecimated finite impulse response (FIR) wavelet filter bank satisfying p˜(ω)p(ω)+q˜(ω)q(ω)=1,ωR,where for a scalar FIR filter p={pk}kZ, p(ω) denotes p(ω)=kZpkeikω,ωR.Then undecimated (scalar) wavelet transform denoising of a 1-D noising signal {xk}kZ consists of decomposition algorithm: Ln=kZpkxk+n,Hn=kZqkxk+n,nZ,and denoising algorithm: uk=nZp˜nLkn+nZq˜nSθ(Hkn),kZ,where Sθ is a shrinkage function. The hard- and soft-shrinkage functions [1], [2] are commonly used Sθ. The signal {uk}kZ is called the denoised signal of the original signal {xk}kZ.

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: ut=x{g((ux)2)ux},with initial condition u(x,0)=f(x), where ut denotes the partial derivative ut of the unknown function u(t,x) with respect to t, and g is the diffusivity. When f(x) is a signal with noise, then with an appropriate choice of g(x), the solution u=u(x,t) of (1.2) (for some suitable t) is considered as the denoised signal of f(x).

The nonlinear diffusion (1.2) is discretized in practice. The following is a simple discretization using the first order difference. More precisely, let τ and h be the time and spatial step size respectively. Denote uk0=f(kh),kZ. Let ukj,j1 be approximations to the solution u(x,t) at (kh,jτ). Then uk+1jukjh and ukj+1ukjτ can be used to approximate respectively xu(x,t) and tu(x,t) at (kh,jτ): (xu)(kh,jτ)uk+1jukjh,ut(kh,jτ)ukj+1ukjτ.Hence, (1.2) can be discretized as ukj+1=ukj+τh2g((uk+1jukjh)2)(uk+1jukj)τh2g((ukjuk1jh)2)(ukjuk1j),for j=0,1,.

The correspondence between Haar wavelet shrinkage and second-order nonlinear diffusion has been studied in [12], which shows that uk in (1.1) is uk1 in (1.3) if shrinkage function Sθ and the diffusivity g have the relationship: Sθ(x)=x(14τh2g(2x2h2)). [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 P={Pk}kZ of matrices, we define P(ω)=kPkeikω.For a matrix M, M denotes its Hermitian, the complex conjugate and transpose of M. For Qk of 2 × 2 matrix, we use qij,k denote the (i,j)-entry of Qk: Qk=q11,kq12,kq21,kq22,k.

Section snippets

Multiwavelet shrinkage

In this section we briefly review multiwavelet shrinkage. For simplicity of presentation, we consider in this paper multiwavelets with multiplicity 2. Two vector-valued functions Ψ(x)=[ψ1(x),ψ2(x)]T and Ψ˜(x)=[ψ˜1(x),ψ˜2(x)]T,xR are called a pair of biorthogonal multiwavelets if the collections of 2j2ψ1(2jxk),2j2ψ2(2jxk),j,kZ and 2j2ψ˜1(2jxk),2j2ψ˜2(2jxk),j,kZ form biorthogonal bases of L2(R) in the sense that they are Riesz bases of L2(R) and they are orthogonal to each other: ψi(2jx

Correspondence between multiwavelet shrinkage and nonlinear diffusion

We consider high-order nonlinear diffusion equation ut=(1)1+ααxα{g((βuxβ)2)βuxβ}with initial condition: u(x,0)=f(x), where α and β are some natural numbers. (3.1) will be discretized. We use high-pass filters to discretize the partial derivatives αxα and βxβ.

For a (high-pass) scalar filter q(ω)=kZqkeikω or q={qk}kZ with finitely many qk nonzero, we say that it has vanishing moment order J if kZkjqk=0,jwith 0j<J.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 Sij when diffusivity functions g are the Perona–Malik diffusivity and Weickert diffusivity functions. We assume the spatial step size h=1.

Let g(x2)=c1+(xλ)2,be the Perona–Malik diffusivity [6], where c is a constant and λ>0

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.

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    This work was supported in part by Simons Foundation, USA (Grant No. 353185).

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