Convergence of fixed point iteration for deblurring and denoising problem

doi:10.1016/j.amc.2006.12.004

Applied Mathematics and Computation

Volume 189, Issue 2, 15 June 2007, Pages 1178-1185

https://doi.org/10.1016/j.amc.2006.12.004 Get rights and content

Abstract

To solve deblurring and denoising problem, we present a proof of the global and linear convergence of fixed point iteration, as well as an estimate for the rate of convergence is given. We show the equivalence among three different iterative methods: half-quadratic regularization, iteration based on Bergman distance, Weiszfeld’s method.

Introduction

Given an observed image f which is often blurry and noisy, we want to get the true image u. The deblurring and denoising problem can be modeled by f = Ku + r, here, K is a known linear blur operator, u is the true image, r is the noise. If K is identity, the corresponding problem is called a denoising problem.

The total variation (TV) deblurring and denoising models are based on a variational problem with constraints using the TV norm as a nonlinear nondifferentiable functional. The formulation of these models was first given by Rudin et al. in [7] for the denoising model and Rudin and Osher in [8] for the deblurring and denoising case. The unconstrained minimization problem: $\min_{u} (| \nabla u |_{β} + \frac{λ}{2} ‖ Ku - f ‖_{L^{2}}^{2})$ with $| \nabla u |_{β} = \int_{Ω} \sqrt{| \nabla u |^{2} + β} d x d y$ , Ω is the image area; β > 0 is a regularization parameter and is usually small. Furthermore, the smallest positive machine number [5] β = 10^-32 can be chosen. The limit of the solutions of the problems when β → 0 is the solution without the perturbed problems [1]; λ being the penalty parameter; $K : L^{2} (Ω) \to H$ being a bounded linear operator whose kernel does not include the space of continuous functions; and $H$ being some Hilbert space.

For the above problem, many methods have been shown to be effective, particularly for denoising problem. The Euler–Lagrange equation of the problem (1.1) in the more usual form is $0 = - \nabla \cdot (\frac{\nabla u}{| \nabla u |_{β}}) + λ K^{*} (Ku - f) .$ Eq. (1.2) is a nonlinear algebraic system, difficult to be solved. Vogel and Oman [10] proposed a fixed point iteration to linearize the corresponding Euler–Lagrange equation: $0 = - \nabla \cdot (\frac{\nabla u^{k + 1}}{| \nabla u^{k} |_{β}}) + λ K^{*} ({Ku}^{k + 1} - f) .$ Numerical experiments in [10] suggest that the convergence of this fixed point iteration is global and rapid, so its convergence theory has attracted some attention [2], [3], especially for the denoising problem. A proof of the global and linear convergence for the discrete version of this algorithm to solve the denoising problem is presented in [4]. For a particular (finite element) discretization of the denoising problem, global convergence in a finite-dimensional setting is established, linear convergence properties, including rates and their dependence on various parameters, are examined in [9].

The purpose of the paper is the analysis of the fixed point iteration to solve the deblurring and denoising problem. In Section 2, the generalized deblurring and denoising problem is introduced with the intent of establishing some basic notations and fixed point iteration is shown. In Section 3, how to get the fixed point iteration by half-quadratic regularization approach is shown. In Section 4, how to get the fixed point iteration based on Bergman distance is shown. In Section 5, how to get the fixed point iteration by generalized Weiszfeld’s method is shown. A proof of the global and linear convergence of the fixed point iteration, as well as an estimate for the rate of convergence is given. Some concluding remarks are presented in Section 6.

Section snippets

Deblurring and Denoising problem

Let u_ij be the approximation to the value $u (x_{i}, y_{j})$ , $i = 1, \dots, n, j = 1, \dots, n$ where $x_{i} = i Δ x$ , $y_{j} = j Δ y$ , where Δx and Δy are the spatial stepsizes, respectively. The discrete gradient operator $\nabla_{i, j} : R^{N} (N = n^{2}) \to R^{2}$ is defined as $\nabla_{i, j} u = (\frac{u_{i + 1, j} - u_{i, j}}{Δ x}, \frac{u_{i, j + 1} - u_{i, j}}{Δ y}) .$ Thus Eq. (1.1) can be simply written as $\min_{u} \sum_{i, j = 1}^{n} | \nabla_{i, j} u |_{β} + \frac{λ}{2} ‖ Ku - f ‖_{L^{2}}^{2}$ and the corresponding equation (1.2) can be written as $- \sum_{i, j = 1}^{n} \nabla_{i, j}^{T} (\frac{\nabla_{i, j} u}{| \nabla_{i, j} u |_{β}}) + λ K^{*} (Ku - f) = 0,$ where $\nabla_{i, j}^{T}$ is the transpose of the linear operator ∇_i,j. (1.3) can be written as $- \sum_{i, j = 1}^{n} \nabla_{i, j}^{T} (\nabla)$

Half-quadratic regularization

Now we will show how to get the fixed point iteration (2.8) by half-quadratic regularization approach.

It is well known that a function ${vt}^{2} + \frac{1}{4 v}$ gets its minimum when $v = \frac{1}{2 t} .$ The minimization problem (2.5) can be written as $\min_{u} \sum_{i = 1}^{m} ϕ (| A_{i}^{T} u |_{β}) + \frac{λ}{2} ‖ Ku - f ‖_{L^{2}}^{2} = \min_{u} \sum_{i = 1}^{m} \min_{v_{i}} (v_{i} ϕ (| A_{i}^{T} u |_{β})^{2} + \frac{1}{4 v_{i}}) + \frac{λ}{2} ‖ Ku - f ‖_{L^{2}}^{2} = \min_{u} \min_{v} \sum_{i = 1}^{m} (v_{i} ϕ (| A_{i}^{T} u |_{β})^{2} + \frac{1}{4 v_{i}}) + \frac{λ}{2} ‖ Ku - f ‖_{L^{2}}^{2} = \min_{u} \min_{v} Θ (u, v),$ where $\frac{1}{2 ϕ (| A_{i}^{T} u |_{β})} = \arg \min_{v_{i}} (v_{i} ϕ (| A_{i}^{T} u |_{β})^{2} + \frac{1}{4 v_{i}}) \forall i$ . It can be seen that $Θ (u, v)$ is quadratic in u.

The algorithm ARTUR of Charbonnier consists of

Iteration based on Bregman distance

First we recall several definitions.

Definition 4.1

For convex functional $J (u) : = ∣ u ∣_{BV}$ , we shall denote the subdifferential of J at a point u by $\partial J (u) : = {p \in BV (Ω)^{*} | J (v) ⩾ J (u) + 〈 p, v - u 〉 \forall v \in BV (Ω)},$ where $〈 \cdot, \cdot 〉$ denotes the standard duality product.

Definition 4.2

A generalized Bregman distance associated with J(·) is $D^{p} (u, v) \equiv D_{J}^{p} (u, v) \equiv J (u) - J (v) - 〈 p, u - v 〉$ for $p \in \partial J (v)$ .

Osher et al. [6] proposed an iterative regularization procedure based on Bergman distance as follows: $\min_{u} J (u) + H (u, f),$ where J is a convex nonnegative regularization functional and

Generalized Weiszfeld’s method

In this section, we show that the fixed point iteration can be deduced from the generalized Weiszfeld’s method, establish the global and linear convergence of the generalized Weiszfeld’s method, and give an estimate for the rate of convergence.

The generalized Weiszfeld’s algorithm for (2.5) consists of the choice of a uniformly strictly convex quadratic functional G(v, u) which is an approximation to F(u), i.e., the following hypothesis:

Hypothesis 5.1

1.
$G (v, u) = F (u) + (v - u, F^{'} (u)) + \frac{1}{2} (v - u, C (u) (v - u)) .$
2.
C(u) is

Concluding remarks

This paper has considered the fixed point iteration for solving the deblurring and denoising problem. The global convergence and the linear convergence, as well as an estimate for the rate of convergence is given. Half-quadratic regularization, iteration based on Bergman distance and generalized Weiszfeld’s method are three different methods and proposed by different researchers. We construct the equivalence by the fixed point iteration among the three different methods.

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Convergence of fixed point iteration for deblurring and denoising problem

Abstract

Introduction

Section snippets

Deblurring and Denoising problem

Half-quadratic regularization

Iteration based on Bregman distance

Generalized Weiszfeld’s method

Concluding remarks

Phys. D

Analysis of total variation penalty methods for ill-posed problems

Inverse Prob.

Image recovery via total variation minimization and related problems

Numer. Math.

On the convergence of the lagged diffusivity fixed point method in total variation image restoration

SIAM J. Numer. Anal.