An Adaptive Total Generalized Variation Model with Augmented Lagrangian Method for Image Denoising

Abstract

We propose an adaptive total generalized variation (TGV) based model, aiming at achieving a balance between edge preservation and region smoothness for image denoising. The variable splitting (VS) and the classical augmented Lagrangian method (ALM) are used to solve the proposed model. With the proposed adaptive model and ALM, the regularization parameter, which balances the data fidelity and the regularizer, is refreshed with a closed form in each iterate, and the image denoising can be accomplished without manual interference. Numerical results indicate that our method is effective in staircasing effect suppression and holds superiority over some other state-of-the-art methods both in quantitative and in qualitative assessment.

1. Introduction

In the past few decades, many variation or partial differential equation (PDE) based restoration models [1–7] have been proposed to recover images from degraded observations, due to the ability of preserving significant image features such as edges or textures. Among these models, the total variation (TV) model, also named the Rudin-Osher-Fatemi (ROF) model [1], is distinguished for excellent edge preserving ability and becomes one of the most widely used regularizers in image restoration [1, 2, 8–10]. In particular, the TV denoising problem is in the following form: argmin_𝑢Ω||𝜆∇𝑢d𝐱+2_Ω𝑢−𝑢₀²d𝐱, (1) where Ω is an open bounded domain in two dimensions, 𝑢 is the image to be restored, 𝑢₀ is the observation containing Gaussian white noise, and 𝜆 is the regularization parameter which balances the regularization term and the data fidelity term. ∫_Ω|∇𝑢|d𝐱 is the TV seminorm of the bounded variation (BV) space BV(Ω). The TV model is highly effective in preserving edges and corners, compared with the quadratic Tikhonov model. However, only when the original image is piecewise constant, the TV model is proved to be optimal. In fact, staircasing effect usually appears because most of natural images are not piecewise constant. Staircasing effect cannot meet the demands of human vision, due to the new artificial edges which do not exist in original images.

To overcome the drawback of the TV model, researchers suggest introducing the higher-order derivatives of image functions [3–7, 12–16]. In order to eliminate the staircasing effect of TV model, Chambolle and Lions [14] proposed the following infimal-convolution minimization functional: argmin_𝑢,𝑣Ω|∇𝑢−∇𝑣|d𝐱+𝛼_Ω𝜕²𝑣d𝐱+𝜆_Ω𝑢−𝑢₀²d𝐱, (2) where discontinuous components of the image are allotted to 𝑢−𝑣 while regions of moderate slopes are assigned to 𝑣. The above model was proved to be practically efficient. Later, a modified form of (2) was proposed in [5] and its regularizer is of the following form: argmin_𝑢,𝑣Ω|∇𝑢−∇𝑣|d𝐱+𝛼_Ω|Δ𝑣|d𝐱. (3) That is, the second-order derivative in (2) is substituted by the Laplacian in (3). The similar use of Laplacian operator can also be seen in some PDE-based methods [3].

Since the classical TV model could not distinguish jumps from smooth transitions, Chan et al. [12] considered an additional penalization of the discontinuities in images. Precisely, they adopt _Ω|∇𝑢|d𝐱+𝛼_Ω𝜓(|∇𝑢|)ℎ(Δ𝑢)d𝐱 (4) as the regularization term, where 𝜓 is a real-valued function whose value approaches 0 while |∇𝑢| approaches infinity. The absence of the staircasing effect for this choice was verified in [15].

Bredies et al. [17] proposed the concept of total generalized variation (TGV), which is considered to be the generalization of TV. The TGV model is defined as TGV^𝑘_𝜶(𝑢)=sup_Ω𝑢div^𝑘𝑣d𝐱∣𝑣∈𝐶^𝑘_𝑐Ω,Sym^𝑘ℝ^𝑑,div^𝑙𝑣_∞≤𝛼_𝑙,𝑙=0,…,𝑘−1withSym^𝑘ℝ^𝑑=𝜁∶ℝ^𝑑×ℝ^𝑑⋯×ℝ^𝑑_𝑘times,⟶ℝ𝜁is𝑘-linearandsymmetric (5) where 𝑑≥1 denotes the image dimension, and, throughout this paper, we assume 𝑑=2; Sym^𝑘(ℝ^𝑑) is the space of symmetric 𝑘-tensors on ℝ^𝑑; 𝐶^𝑘_𝑐(Ω,Sym^𝑘(ℝ^𝑑)) is the space of compactly supported symmetric tensor field; 𝛼_𝑙 is fixed positive parameter. From the definition of TGV^𝑘_𝜶, we learn that it involves the derivatives of 𝑢 of order one to 𝑘. When 𝑘=1 and 𝛼₀=1, TGV^𝑘_𝜶 degenerates to the classical TV. Thus TGV can be seen as a generalization of TV.

TGV involves and balances higher-order derivatives of 𝑢. Image reconstruction with TGV regularization usually leads to result with piecewise polynomial intensities and sharp edges. Therefore, TGV can effectively suppress the staircasing effect. In [17], an accelerated first-order method of Nesterov [18] was proposed to solve the TGV-regularized denoising problem.

In this paper, we propose an adaptive second-order TGV-regularized model for denoising and derive an augmented Lagrangian approach to handle the suggested model. Our denoising model is as follows: argmin_𝑢TGV²_𝜶(𝑢)s.t._Ω𝑢−𝑢₀²d𝐱≤𝑐. (6) According to the standard Lagrange duality, for a given 𝑐, there exists a nonnegative 𝜆 such that argmin_𝑢TGV²_𝜶(𝑢)+𝜆2_Ω𝑢−𝑢₀²d𝐱 (7) is equivalent to (6). However, with (6), we can automatically estimate the regularization parameter 𝜆. We first utilize an indicator function of the feasible set to transform problem (6) into an unconstrained one; then the variable splitting technique is applied to transform the resulting unconstrained problem into a problem with linear penalizing constraints; finally, the obtained constrained problem is solved by the alternating direction method of multipliers (ADMM) [19–22], which is an instance of the classical ALM. The resulting image denoising algorithm is effective in staircasing effect suppression compared with some TV-based denoising methods, due to the second-order TGV regularizer. Besides, it achieves the adaptive estimation of the regularization parameter without inner iterative scheme. It is worth noting that the idea of this paper can be extended to TGV models with higher order than two. However, for simplicity, we only treat the second-order model and this is adequate for a large class of natural images.

Our method differs from the previous works on at least two aspects. On one hand, compared with [16], which adopted the accelerated first-order method of Nesterov [18] to handle the unconstrained TGV-based denoising problem (7), we apply ALM to the constrained TGV-based denoising problem (6) and achieve the automatic estimation of the regularization parameter 𝜆. Our strategy avoids the extra cost on the manual selection of 𝜆 by try-and-error. On the other hand, compared with the existing TV-based adaptive methods [10, 23, 24], we propose a more complicated adaptive method based on TGV, and it is apt to achieve more attractive results than the TV-based methods.

The outline of the rest of the paper is organized as follows. Section 2 provides the description of the adaptive second-order TGV-based model for image denoising. Based on the Lagrange duality, an equivalent form of TGV²_𝜶 is suggested. The derivation of the proposed method is presented in Section 3. Section 4 gives the numerical results that demonstrate the effectiveness of the proposed method. At last, Section 5 ends this paper with a brief conclusion.

2. Adaptive Second-Order TGV-Based Model for Image Denoising

The space of bounded generalized variation (BGV) functions of order 𝑘 with weight 𝜶 is defined as BGV^𝑘_𝛼(Ω)=𝑢∈𝐿¹(Ω)TGV^𝑘_𝜶(𝑢)<∞. (8) Correspondingly, the BGV norm is defined as ‖𝑢‖_BGV^𝑘_𝛼=‖𝑢‖₁+TGV^𝑘_𝜶(𝑢). (9) The TGV seminorm rather than the BGV norm is usually used as a regularizer.

In this paper, we just take 𝑘=2 into consideration for simplicity. The second-order TGV can be written as TGV²_𝜶(𝑢)=sup_Ω𝑢div²𝑣d𝐱𝑣∈𝐶²_𝑐Ω,Sym²ℝ^𝑑,‖𝑣‖_∞≤𝛼₀,‖div𝑣‖_∞≤𝛼₁, (10) where the divergences are defined as (div𝑣)_𝑖=^𝑑_𝑗=1𝜕𝑣_𝑖𝑗𝜕𝑥_𝑗,1≤𝑖≤𝑑;div²𝑣=^𝑑_𝑖,𝑗=1𝜕²𝑣_𝑖𝑗𝜕𝑥_𝑖𝜕𝑥_𝑗. (11) In fact, Sym²(ℝ^𝑑) is equivalent to the space of all symmetric 𝑑×𝑑 matrices. The infinite norms in (10) are given by ‖𝑣‖_∞=sup_𝐱∈Ω^𝑑_𝑖,𝑗=1𝑣_𝑖𝑗(𝐱)^21/2,‖div𝑣‖_∞=sup_𝐱∈Ω^𝑑_𝑖=1(div𝑣)_𝑖^21/2. (12)

For the convenience of the derivation of our algorithm, we apply the discrete form in the following and the tensors and vectors are denoted in bold type font. In order to make use of ADMM, we apply an equivalent definition of TGV²_𝜶 [17, 22] based on the Lagrange duality. With this definition, we have TGV²_𝜶(𝐮)=min_𝐩𝛼₀‖𝜀(𝐩)‖₁+𝛼₁‖∇𝐮−𝐩‖₁, (13) where 𝐮∈ℝ^𝑚𝑛 denotes an 𝑚×𝑛 image, 𝐩∈ℝ^𝑚𝑛×ℝ^𝑚𝑛 belongs to the two-dimensional 1-tensor field, and 𝜀 denotes the symmetrized derivative operator. Suppose that 𝑢_𝑖,𝑗 and 𝐩_𝑖,𝑗 denote the (𝑖,𝑗)th components of 𝐮 and 𝐩, respectively. Then we have 𝐩_𝑖,𝑗=[𝑝_𝑖,𝑗,1,𝑝_𝑖,𝑗,2]∈Sym¹(ℝ²) and the (𝑖,𝑗)th component of 𝜀(𝐩) is given by 𝜀(𝐩)_𝑖,𝑗=𝜀(𝐩)_𝑖,𝑗,1𝜀(𝐩)_𝑖,𝑗,3𝜀(𝐩)_𝑖,𝑗,3𝜀(𝐩)_𝑖,𝑗,2=∇_𝑥1𝑝_𝑖,𝑗,1∇_𝑥2𝑝_𝑖,𝑗,1+∇_𝑥1𝑝_𝑖,𝑗,22∇_𝑥2𝑝_𝑖,𝑗,1+∇_𝑥1𝑝_𝑖,𝑗,22∇_𝑥2𝑝_𝑖,𝑗,2∈Sym²ℝ²0≤𝑖≤𝑚,0≤𝑗≤𝑛, (14) where ∇_𝑥1 and ∇_𝑥2 denote the difference operators in directions 𝑥₁ and 𝑥₂. According to the definition of operators ∇ and 𝜀, (∇𝐮)_𝑖,𝑗 and 𝜀(𝐩)_𝑖,𝑗 are two-dimensional 1-tensor and symmetric 2-tensor, respectively. Besides, the ‖‖₁s of 𝐩 and 𝜀(𝐩) are defined as ‖𝐩‖₁=∑^𝑚,𝑛_𝑖,𝑗|𝐩_𝑖,𝑗|=∑^𝑚,𝑛_𝑖,𝑗𝑝²_𝑖,𝑗,1+𝑝²_𝑖,𝑗,2 and ‖𝜀(𝐩)‖₁=∑^𝑚,𝑛_𝑖,𝑗|𝜀(𝐩)_𝑖,𝑗|=∑^𝑚,𝑛_𝑖,𝑗𝜀(𝐩)²_𝑖,𝑗,1+𝜀(𝐩)²_𝑖,𝑗,2+2𝜀(𝐩)²_𝑖,𝑗,3, respectively. The deduction of (13) is given in the Appendix.

Then the constrained second-order TGV-regularized denoising problem (6) can be rewritten as argmin_𝐮,𝐩𝛼₀‖𝜀(𝐩)‖₁+𝛼₁‖∇𝐮−𝐩‖₁s.t.𝐮−𝐮₀²₂≤𝑐. (15)

3. Methodology

3.1. The Augmented Lagrangian Model of Adaptive TGV-Based Denoising

Problem (15) can be transformed into an unconstrained problem, with the following discontinuous objective functional: argmin_𝐮,𝐩𝛼₀‖𝜀(𝐩)‖₁+𝛼₁‖∇𝐮−𝐩‖₁+𝐼_Φ(𝐮), (16) where 𝐼_Φ(𝐮) is the indicator function of the feasible set defined by 𝐼_Φ(𝐮)=0,if𝐮∈Φ≜𝐮∶𝐮−𝐮₀²₂≤𝑐;+∞otherwise. (17) Note that Φ is a closed Euclidean ball centered at 𝐮₀ with radius √𝑐.

The solution of problem (16) suffers from its nonlinearity and nondifferentiability. Referring to the variable splitting, we introduce three auxiliary variables to simplify the solution process of (16): a variable 𝐰 for liberating 𝐮 out from the constraint of the feasible set; a variable 𝐲 and a variable 𝐳 for liberating 𝜀(𝐩) and ∇𝐮−𝐩 out from the nondifferentiable 1-norms, respectively. Then problem (16) can be transformed into the following equivalent constrained problem: argmin_𝐮,𝐩𝛼₀‖𝐲‖₁+𝛼₁‖𝐳‖₁+𝐼_Φ(𝐰)s.t.𝐮=𝐰,𝜀(𝐩)=𝐲,∇𝐮−𝐩=𝐳. (18)

In order to liberate 𝐮 out from the feasible set constraint, we introduce auxiliary variable 𝐰. Similar operation can also be found in [10]. Without this operation, we should resort to an inner iterative scheme to update the regularization parameter.

The corresponding augmented Lagrangian functional of (18) is defined as ℒ_𝒜(𝐮,𝐩,𝐰,𝐲,𝐳;𝝁,𝝃,𝜼)≜𝐼_Φ(𝐰)−⟨⟩+𝛽𝝁,𝐰−𝐮₁2‖𝐰−𝐮‖²₂+𝛼₀‖𝐲‖₁−⟨(𝐩+𝛽𝝃,𝐲−𝜀)⟩₂2‖𝐲−𝜀(𝐩)‖²₂+𝛼₁‖𝐳‖₁−⟨⟩+𝛽𝜼,𝐳−∇𝐮+𝐩₃2‖𝐳−∇𝐮+𝐩‖²₂, (19) where 𝝁, 𝝃, and 𝜼 are Lagrange multipliers and 𝛽₁, 𝛽₂, and 𝛽₃ are penalty parameters which should be positive. According to the classical ADMM, we should solve the following iterative scheme: 𝐮^𝑘+1=argmin_𝐮ℒ_𝒜𝐮,𝐩^𝑘,𝐰^𝑘,𝐲^𝑘,𝐳^𝑘;𝝁^𝑘,𝝃^𝑘,𝜼^𝑘,𝐩^𝑘+1=argmin_𝐩ℒ_𝒜𝐮^𝑘+1,𝐩,𝐰^𝑘,𝐲^𝑘,𝐳^𝑘;𝝁^𝑘,𝝃^𝑘,𝜼^𝑘,𝐰^𝑘+1=argmin_𝐰ℒ_𝒜𝐮^𝑘+1,𝐩^𝑘+1,𝐰,𝐲^𝑘,𝐳^𝑘;𝝁^𝑘,𝝃^𝑘,𝜼^𝑘,𝐲^𝑘+1=argmin_𝐲ℒ_𝒜𝐮^𝑘+1,𝐩^𝑘+1,𝐰^𝑘+1,𝐲,𝐳^𝑘;𝝁^𝑘,𝝃^𝑘,𝜼^𝑘,𝐳^𝑘+1=argmin_𝐳ℒ_𝒜𝐮^𝑘+1,𝐩^𝑘+1,𝐰^𝑘+1,𝐲^𝑘+1,𝐳;𝝁^𝑘,𝝃^𝑘,𝜼^𝑘,𝝁^𝑘+1=𝝁^𝑘−𝛽₁𝐰^𝑘+1−𝐮^𝑘+1,𝝃^𝑘+1=𝝃^𝑘−𝛽₂𝐲^𝑘+1−𝜀𝐮^𝑘+1,𝜼^𝑘+1=𝜼^𝑘−𝛽₃𝐳^𝑘+1−∇𝐮^𝑘+1+𝐩^𝑘+1. (20)

3.2. Solution of the Subproblems

With the auxiliary 𝐰, the 𝐮 subproblem becomes quadratic and irrelevant to the constraint of the feasible set. It allows the following objective: 𝐮^𝑘+1=argmin_𝐮𝛽₁2𝐰^𝑘𝝁−𝐮−^𝑘𝛽₁²₂+𝛽₃2𝐳^𝑘−∇𝐮+𝐩^𝑘−𝜼^𝑘𝛽₃²₂. (21) The minimization problem (21) can be solved by the following equation: 𝐮^𝑘+1=𝛽₁𝛽₃𝐈+∇^𝑇∇⁻¹𝛽₁𝛽₃𝐰^𝑘−𝝁^𝑘𝛽₁+∇^𝑇𝐳^𝑘+𝐩^𝑘−𝜼^𝑘𝛽₃. (22) With the circulant boundary condition of images, we can solve (22) with several FFTs and IFFTs [8, 10].

Following the same way, the subproblem with respect to 𝐩 is also quadratic and we have the objective functional as follows: 𝐩^𝑘+1=argmin_𝐩𝛽₂2𝐲^𝑘(𝐩)−𝝃−𝜀^𝑘𝛽₂²₂+𝛽₃2𝐳^𝑘−∇𝐮^𝑘+1𝜼+𝐩−^𝑘𝛽₃²₂. (23) Then, for 𝐩₁^𝑘+1, we have 𝛽₂∇^𝑇_𝑥1∇_𝑥1+𝛽₂2∇^𝑇_𝑥2∇_𝑥2+𝛽₃𝐈𝐩₁^𝑘+1=𝛽₂∇^𝑇_𝑥1𝐲^𝑘₁−𝝃^𝑘₁𝛽₂+∇^𝑇_𝑥2𝐲^𝑘₃−𝝃^𝑘₃𝛽₂−12∇_𝑥1𝐩^𝑘₂+𝛽₃∇_𝑥1𝐮^𝑘+1+𝜼^𝑘₁𝛽₃−𝐳^𝑘₁, (24) and for 𝐩₂^𝑘+1, we have 𝛽₂∇^𝑇_𝑥2∇_𝑥2+𝛽₂2∇^𝑇_𝑥1∇_𝑥1+𝛽₃𝐈𝐩₂^𝑘+1=𝛽₂∇^𝑇_𝑥2𝐲^𝑘₂−𝝃^𝑘₂𝛽₂+∇^𝑇_𝑥1𝐲^𝑘₃−𝝃^𝑘₃𝛽₂−12∇_𝑥2𝐩₁^𝑘+1+𝛽₃∇_𝑥2𝐮^𝑘+1+𝜼^𝑘₂𝛽₃−𝐳^𝑘₂, (25) where 𝐩₁ and 𝐩₂ are the combinations of 𝑝_𝑖,𝑗,1 and 𝑝_𝑖,𝑗,2 (0≤𝑖≤𝑚, 0≤𝑗≤𝑛), respectively. Similar to the solution of (22), problems (24) and (25) can also be solved conveniently through several FFTs and IFFTs under the assumption of the circulant boundary condition.

The subproblem for 𝐲 can be written as 𝐲^𝑘+1=argmin_𝐲‖𝐲‖₁+𝛽₂2𝛼₀𝐲−𝜀𝐩^𝑘+1−𝝃^𝑘𝛽₂²₂. (26) Problem (26) can be solved component-wisely through the following 4-dimensional shrinkage operation: 𝐲^𝑘+1_𝑖,𝑗=max𝜀𝐩^𝑘+1_𝑖,𝑗+𝝃^𝑘_𝑖,𝑗𝛽₂₂−𝛼₀𝛽₂×𝜀𝐩,0_{𝑘+1𝑖,𝑗}+𝝃^𝑘_𝑖,𝑗/𝛽₂𝜀𝐩^𝑘+1_𝑖,𝑗+𝝃^𝑘_𝑖,𝑗/𝛽₂₂0≤𝑖≤𝑚,0≤𝑗≤𝑛. (27)

The 𝐳 subproblem is given by 𝐳^𝑘+1=argmin_𝐳‖𝐳‖₁+𝛽₃2𝛼₁𝐳−∇𝐮^𝑘+1+𝐩^𝑘+1−𝜼^𝑘𝛽₃²₂, (28) and it can be solved component-wisely through the following 2-dimensional shrinkage operation: 𝐳^𝑘+1_𝑖,𝑗=max∇𝐮^𝑘+1_𝑖,𝑗+𝜼^𝑘_𝑖,𝑗𝛽₃−𝐩^𝑘+1_𝑖,𝑗2−𝛼₁𝛽₃×,0∇𝐮^𝑘+1_𝑖,𝑗+𝜼^𝑘_𝑖,𝑗/𝛽₃−𝐩^𝑘+1_𝑖,𝑗∇𝐮^𝑘+1_𝑖,𝑗+𝜼^𝑘_𝑖,𝑗/𝛽₃−𝐩^𝑘+1_𝑖,𝑗2,0≤𝑖≤𝑚,0≤𝑗≤𝑛. (29)

The subproblem with respect to 𝐰 can be written as 𝐰^𝑘+1=argmin_𝐰𝐼_Φ(𝐰)+𝛽₁2𝐰−𝐮^𝑘+1+𝝁^𝑘𝛽₁²₂=argmin_𝐰𝜆^𝑘+12𝐰−𝐮₀²₂+𝛽₁2𝐰−𝐮^𝑘+1+𝝁^𝑘𝛽₁²₂. (30) Consequently, the solution of problem (30) is 𝐰^𝑘+1=𝜆^𝑘+1𝐮₀+𝛽₁𝐮^𝑘+1+𝝁^𝑘/𝛽₁𝜆^𝑘+1+𝛽₁. (31) The solution of 𝜆^𝑘+1 falls into two cases according to the range of 𝐮^𝑘+1+(𝝁^𝑘/𝛽₁). On one hand, if 𝐮^𝑘+1+𝝁^𝑘𝛽₁−𝐮₀²₂≤𝑐, (32) we can set 𝜆^𝑘+1=0, and, obviously, 𝐰^𝑘+1=𝐮^𝑘+1+𝝁^𝑘/𝛽₁ satisfies the feasible set constraint. On the other hand, if (32) is not true, 𝐰^𝑘+1 should fulfill the following equation: 𝐰^𝑘+1−𝐮₀²₂=𝑐. (33) Substituting (31) into (33), we get 𝜆^𝑘+1=𝛽₁𝐮^𝑘+1+𝝁^𝑘/𝛽₁−𝐮₀₂√𝑐−𝛽₁. (34) The resulting image denoising algorithm is summarized in Algorithm 1 TGV²ID-ADMM.

Input:  𝐮0,𝑐. (1) Initialize 𝐮0,𝐩0,𝐰0,𝐲0,𝐳0,𝝁0,𝝃0,𝜼0. Set 𝑘=0 and 𝛽1,𝛽2,𝛽3>0. (2)  while  stopping criterion is not satisfied,  do (3)  Compute  𝐮𝑘+1 according to (22); (4)  Compute  𝐩𝑘+1 according to (24) and (25); (5)  Compute  𝐲𝑘+1 according to (27); (6)  Compute  𝐳𝑘+1 according to (29); (5)  if  (32) holds,  then (6)  𝜆𝑘+1=0 and 𝐰𝑘+1=𝐮𝑘+1+(𝝁𝑘/𝛽1); (7)  else (8)  Update 𝜆𝑘+1 and 𝐰𝑘+1 according to (34) and (31); (9)  end  if (10)   Update  𝝁𝑘+1,𝝃𝑘+1, and  𝜼𝑘+1 according to (20); (11)  𝑘=𝑘+1; (12)  end  while (13)  return   𝐮𝑘+1.

Algorithm 1 

Algorithm TGV 𝟐 ID-ADMM:  Second-order TGV-regularized image denoising with ADMM.

The adoption of the variable 𝐰 is essential for the adaptive estimate of the regularization parameter 𝜆. With the assistance of 𝐰, 𝐮 is liberated out from the constraint of the feasible set. Thus, the update of 𝜆 is free from the disturbance of the update of 𝐮, and a closed form for updating 𝜆 is achieved in each step without inner iteration. From functionals (16) and (18) we learn that, by setting (𝛼₀,𝛼₁)=(0,1), (𝐲,𝐩)=(𝟎,𝟎), and 𝛽₂=0, Algorithm TGV²ID-ADMM will degenerate to a TV-based denoising algorithm, and we denote this case as TGV¹ID-ADMM.

The convergence of Algorithm TGV²ID-ADMM follows from the convergence analysis for the TV-based ADMM in [11, 25], due to the convex property of TGV²_𝜶. In this paper, we do not repeat the lengthy analysis procedure. However, we have the following essential convergence theorem for the proposed method.

Theorem 1. For fixed 𝛽₁,𝛽₂,𝛽₃>0, the sequence {𝐮^𝑘,𝐩^𝑘,𝐰^𝑘,𝐲^𝑘,𝐳^𝑘,𝝁^𝑘,𝝃^𝑘,𝜼^𝑘,𝜆^𝑘} generated by Algorithm TGV²ID-ADMM from any initial point (𝐮⁰,𝐩⁰,𝐰⁰,𝐲⁰,𝐳⁰,𝝁⁰,𝝃⁰,𝜼⁰) converges to (𝐮^∗,𝐩^∗,𝐰^∗,𝐲^∗,𝐳^∗,𝝁^∗,𝝃^∗,𝜼^∗,𝜆^∗), where 𝐮^∗ is the solution of functional (15) and 𝜆^∗ is the regularization parameter corresponding to the feasible set constraint 𝐮∈Φ.

4. Experiment Results

In this section, we illustrate the effectiveness of the proposed algorithm on suppressing staircasing effect and removing Gaussian noise in image. Besides, we also show the robustness of the results with respect to the penalty parameters. We performed our algorithm under MATLAB v7.8.0 and Windows 7 on a PC with an Intel Core (TM) i5 CPU at 3.20 GHz and 8 GB of RAM.

The root mean squared error (RMSE) and the peak signal-to-noise ratio (PSNR) used in comparison are defined as RMSE=𝐮−𝐮_clean2√,𝑚𝑛PSNR=10log₁₀255²⋅𝑚𝑛𝐮−𝐮_clean²₂, (35) where 𝐮_clean is the original image that contains no noise. Besides, in subsections 4.1 and 4.2, we set the penalty parameters as 𝛽₁=10^{(BSNR/10−1)}×𝛽 and 𝛽₂=𝛽₃=𝛽=0.3 for TGV²ID-ADMM (𝛽₂=0 for TGV¹ID-ADMM) to achieve consistently promising result with fast speed, where BSNR is the blurred signal-to-noise ratio defined by BSNR = 10 log₁₀ (var(𝐮₀)/𝜎²) (var(𝐮₀) denotes the variance of 𝐮₀).

4.1. Staircasing Effect Reduction by the Proposed Method

We first compare Algorithm TGV²ID-ADMM with Algorithm TGV¹ID-ADMM to illustrate the effectiveness of TGV²_𝜶 model in staircasing effect reduction. We use ‖𝐮^𝑘+1−𝐮^𝑘‖²₂/‖𝐮^𝑘‖²₂≤10⁻⁶ as the stopping criteria for these two algorithms, where 𝐮^𝑘 denotes the restored result in the 𝑘th iteration. For the second-order case, we set (𝛼₀,𝛼₁)=(3,1), whereas for the one-order case, we set (𝛼₀,𝛼₁)=(0,1).

In this experiment, we use a synthetic piecewise affine image shown in Figure 1 as the test image. The original image is contaminated by Gaussian noise of standard variance 𝜎=15 at first. Then we imply TGV²ID-ADMM and TGV¹ID-ADMM to remove the noise. Table 1 shows the results in terms of RMSE, PSNR, total iterations, and CPU time. The ground truth, noised, and restored images by the two algorithms are displayed in Figure 1. Furthermore, for better visualization, we additionally provide the three-dimensional close-ups of the marked regions of the two restored images in Figure 1. From Table 1 we observe that TGV²ID-ADMM does better than TGV¹ID-ADMM in terms of both RMSE and PSNR. Figure 1 shows that the denoised image of TGV²ID-ADMM almost contains no artificial edges in affine regions. In contrast, the restored result of TGV¹ID-ADMM contains obvious staircasing effect in affine regions. The three-dimensional closed-ups vividly demonstrate this phenomenon. This illustrates that our TGV-based algorithm is effective in staircasing effect reduction.

Table 1 

Results of the staircasing effect reduction experiment.

(a)

(b)

(c)

Figure 1 

First row: ground truth and Gaussian noised (𝜎=15) piecewise affine images; second row: restored images by TGV2ID-ADMM and TGV1ID-ADMM; third row: three-dimensional close-ups of the marked regions in the two restored images.

Table 1 also shows that, to accomplish the denoising task, TGV²ID-ADMM usually costs more CPU time than TGV¹ID-ADMM, since TGV²_𝜶 model involves much more calculation. However, the cost is worthy due to the impressive improvement on both quantitative and qualitative restoration quality. Figure 2 displays the evolutions of 𝜆s and PSNRs achieved by the two algorithms. It is learnt that, the regularization parameters of both converge to the optimal points at last, which guarantees the automatic implementation of the two algorithms.

(a)

(b)

Figure 2 

Evolutions of 𝜆s (a) and PSNRs (b) achieved by TGV2ID-ADMM (TGV2) and TGV1ID-ADMM (TGV1).

4.2. Comparison in Accuracy

In this subsection, we compare TGV²ID-ADMM with the other two famous adaptive TV-based denoising algorithms: Chambolle’s projection algorithm [23] and Split Bregman algorithm [24], both possessing public online implementations at “http://www.ipol.im/”. Two natural images, Lena and Peppers both of size 512 × 512 shown in Figure 3, are used for comparison. The parameter setting for TGV²ID-ADMM is the same as that in the previous subsection. We obtain the test results of the two competitors through online experimental operation.

(a)

(b)

Figure 3 

Test images: Lena and Peppers.

We add Gaussian noise of standard variances of 20, 30, and 40 to Lena and Peppers to obtain the noised observations, respectively. Then we apply these three algorithms to restore the noisy images. Table 2 shows the comparison results in terms of RMSE and PSNR. The best result for each comparison item is highlighted in bold type font. Table 2 shows that TGV²ID-ADMM holds superiority on both RMSE and PSNR for all the tested cases. Figure 4 displays the noised Lena under Gaussian noise of 𝜎=30 and the restorations by the three algorithms, whereas Figure 5 exhibits the noised Peppers under Gaussian noise of 𝜎=40 and the corresponding restorations. Figures 4 and 5 demonstrate that TGV²ID-ADMM obtains results with better visual impression and efficiently suppresses the staircasing effect. In contrast, both TV-based Chambolle’s projection algorithm and TV-based Split Bregman algorithm achieve results with obvious staircasing effect. Since we apply test images with different levels of noise, the robustness of our algorithm towards the noise level is verified to a certain extent.

Table 2 

Comparison results in accuracy.

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(b)

(c)

(d)

Figure 4 

Noised Lena under Gaussian noise of 𝜎=30 and the restorations by TGV2ID-ADMM, Chambolle, and Split Bregman, respectively.

(a)

(b)

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(d)

Figure 5 

Noised Peppers under Gaussian noise of 𝜎=40 and the restorations by TGV2ID-ADMM, Chambolle, and Split Bregman, respectively.

4.3. Solution Robustness with Respect to the Penalty Parameters

Although the positive assumption of penalty parameters is sufficient for the convergence of ADMM, the results of ADMM are commonly influenced by the choice of the penalty parameters to a certain extent in practice. As suggested by a referee, we add an experiment to show the robustness of the results of TGV²ID-ADMM with respect to the penalty parameters, under the two denoising background problems mentioned above, that is, the Lena denoising problem under Gaussian noise of 𝜎=30 and the Peppers denoising problem under Gaussian noise of 𝜎=40. We still set 𝛽₁=10^{(BSNR/10−1)}×𝛽 and 𝛽₂=𝛽₃=𝛽 but change 𝛽 from 0.01 to 1 with a step size of 0.01. In Figure 6, we plot PSNR versus 𝛽 for the denoised Lena and Peppers. Figure 6 demonstrates that the optimal 𝛽 should be focalized in [0.2,0.3] and its location is robust towards the variation of image and noise level. The results of our method possess sufficient robustness with respect to the variation of penalty parameters to a certain extent, since the absolute error between the maximum and the minimum of PSNR is less than 0.18 dB in the experiment, and this error could not introduce obvious distinction in visual quality. In the former two experiments, the setting of 𝛽=0.3 is approximately optimal for the proposed algorithm.

(a) Lena image under Gaussian noise of 𝜎=30

(b) Peppers image under Gaussian noise of 𝜎=40

Figure 6 

PSNR versus 𝛽 for (a) Lena image under Gaussian noise of 𝜎=30 and (b) Peppers image under Gaussian noise of 𝜎=40, respectively. The absolute error between the maximum and the minimum of PSNR for each image is less than 0.18 dB.

5. Concluding Remarks

We propose an adaptive TGV-based model for noise removal in this paper. The variable splitting (VS) and the classical augmented Lagrangian method are used to handle the proposed model. From the experimental results, we observe that the proposed algorithm is effective in suppressing staircasing effect and preserving edges in images, and it is superior to some other famed adaptive denoising methods both in quantitative and in qualitative assessment. Besides, our work can be smoothly generalized to image deblurring problems.

Appendix

The Equivalent Definition of TGV²_𝛼

In discrete version, we haveTGV²_𝜶(𝐮)=max_𝐯,𝐝⟩‖𝐯‖⟨𝐮,div𝐝∣div𝐯=𝐝,_∞≤𝛼₀,‖𝐝‖_∞≤𝛼₁, (A.1) where ‖𝐯‖_∞=max_𝑖,𝑗𝑣²_𝑖,𝑗,1+𝑣²_𝑖,𝑗,2+2𝑣²_𝑖,𝑗,3^1/2,‖𝐝‖_∞=max_𝑖,𝑗𝑑²_𝑖,𝑗,1+𝑑²_𝑖,𝑗,2^1/2𝐯_𝑖,𝑗=𝑣_𝑖,𝑗,1𝑣_𝑖,𝑗,3𝑣_𝑖,𝑗,3𝑣_𝑖,𝑗,2,𝐝_𝑖,𝑗=𝑑_𝑖,𝑗,1,𝑑_𝑖,𝑗,2. (A.2) Therefore, according to the Lagrange duality, we have TGV²_𝜶(𝐮)=min_𝐩max_{‖𝐯‖∞≤𝛼0,‖𝐝‖∞≤𝛼1}⟨𝐮,div𝐝⟩+⟨𝐩,𝐝−div𝐯⟩=min_𝐩max_{‖𝐯‖∞≤𝛼0,‖𝐝‖∞≤𝛼1}⟨−∇𝐮,𝐝⟩+⟨𝐩,𝐝⟩+⟨𝜀(𝐩),𝐯⟩=min_𝐩max_{‖𝐯‖∞≤𝛼0,‖𝐝‖∞≤𝛼1}⟨𝐩−∇𝐮,𝐝⟩+⟨𝜀(𝐩),𝐯⟩=min_𝐩𝛼₀‖𝜀(𝐩)‖₁+𝛼₁‖∇𝐮−𝐩‖₁. (A.3)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Editor Fatih Yaman and anonymous referees for their valuable comments. Their help has greatly enhanced the quality of this paper. This work was partially supported by the National Natural Science Foundation of China under Grant nos. 61203189, 61104223, and 61374120 and the National Science Fund for Distinguished Young Scholars of China under Grant no. 61025014.