a block nonlocal tv method for image restorationmath0.bnu.edu.cn/~liujun/papers/siims_2017.pdf63...

A BLOCK NONLOCAL TV METHOD FOR IMAGE RESTORATION∗1

JUN LIU † AND XIAOJUN ZHENG†‡2

Abstract. In this paper, we propose a block nonlocal total variation (TV) regularization method3for image restoration. We extend the existing nonlocal TV method in two aspects: firstly, some block4nonlocal operators are introduced to extend the point based nonlocal diffusion as a block based non-5local diffusion process; secondly, the weighting function in the nonlocal method can be adaptively6determined by the cost functional itself. The proposed method is derived from a block based max-7imum a posteriori (MAP) estimation. By the assumption of the self similarity of small patches,8we formulate a regularization term as a log-likelihood functional of a mixture model. To optimize9this regularization term efficiently, we employ the idea of the expectation maximum algorithm and10give a variational framework to propose a block based nonlocal TV regularization. The weighting11function occurred in our model can be regarded as a probability of the similarity for image patches,12and it can be updated adaptively according to the newest estimation. Besides, we mathematically13prove the existence of a minimizer for one of the proposed models. Compared with the nonlocal14TV method, numerical results show that our method can greatly improve the quality of the restored15images, especially under heavy noise.16

Key words. Image Blocks, Nonlocal Total Variation, Image Restoration, Adaptive Weighting17Functions,Operator Splitting18

AMS subject classifications. 94A08, 68U1019

1. Introduction. Image denoising is a fundamental and active research field in20

image processing. The goal of image denoising is to recover a true image from some21

noisy observed data. The classical noise model is the additive Gaussian noise, that is,22

v = u+ n,23

where v is the observed image, u is the true image and n stands for noise.24

A vast variety of methods for image denoising are available after several decades25

of developments[4]. Gaussian (Gabor) filters[16] and anisotropic diffusion[21] are the26

early denoising approaches. Statistical methods (e.g.,[2]) considering the noise as a27

random variable is a popular denoising technique. They are mainly based on maxi-28

mum likelihood estimator (MLE) and Bayesian maximum a posteriori (MAP). Varia-29

tional approaches (e.g.,[22, 23]) is another popular and powerful technique for image30

denoising. The variational method obtains a clean image by minimizing a cost func-31

tional which typically consists of a fidelity term and a regularization term. Of course,32

all these methods are not separated from each other, such as many methods can be33

viewed within the variational framework. However, the variational method is more34

highly valued since the total variation (TV) based ROF model was proposed in [23].35

The ROF model is a classical and efficient model to remove Gaussian noise, and36

many variants [20, 17] based on TV had been designed for different denoising tasks37

due to its good edges-preserving property. However, the results obtained with TV38

∗This work was partially supported by National Natural Science Foundation of China (No.11201032) and the Fundamental Research Funds for the Central Universities. Jun Liu was alsosupported by the China Scholarship Council for a one year visiting at UCLA. This work was finishedduring his visit at UCLA. The authors would like to thank Professor Stanley Osher for his valuablecomments and suggestions.†School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing

Normal University, Beijing 100875, P.R. China ([email protected], ).‡College of Elementary Education, Hainan Normal University, Haikou 571158, P.R.

China([email protected], [email protected]).

1

This manuscript is for review purposes only.

mailto:[email protected]



2 JUN LIU, AND XIAOJUN ZHENG

would be piecewise constant and the image details such as textures would be removed39

together with noise. In order to better preserve the image textures and repetitive40

structures, the nonlocal denoising methods have received considerable attention in41

recent years. The concept of nonlocal averaging was introduced by Yaroslavsky via42

a nonlocal neighborhood filter [27]. The nonlocal neighborhood filter calculates the43

pixel value at point x according to the similar pixels at y which can be located far44

away from x. A well-known nonlocal method based on patch distances extending45

this idea is the nonlocal means method [3] proposed by Buades etc.. The novelty46

of the nonlocal means is to introduce the patch distances, i.e. the value at point y47

is used to denoise that at x if the local value pattern near y is similar to the local48

value pattern near x [24]. A variational interpretation of nonlocal means was given49

in [15, 10]. Motivated by the nonlocal means and the graph theory, the nonlocal TV50

variational framework based on nonlocal operators was proposed in [11]. The weight-51

ing functions occurred in these nonlocal methods play a very important role in the52

task of denoising. Although the above nonlocal methods show superior performance53

in textured or structure repeated images, the weighting functions in the nonlocal54

methods are given and fixed. It may give a bad estimation for the weight and lead to55

a undesirable restoration under the heavy noise. Furthermore, the estimation in these56

methods is point by point, it may not be robust for heavy noise. In [1], the authors57

proposed a block based cost functional by adding a entropy regularization and got an58

adaptive weighting function for image inpainting, however, they did not give a math-59

ematical interpretation for such a entropy regularization term. Many block based60

methods [8, 7, 13, 14] have shown that the block based estimation can improve the61

precision and be more robust to noise. Based on overlapping-patches technique and62

sparsity, block based nonlocal image denoising methods have been proposed in recent63

years, e.g. patch-based near-optimal image denoising (PLOW)[6], locally learned dic-64

tionaries (K-LLD) [5], clustering-based sparse representation (CSR) [9], a weighted65

dictionary learning model[18], the expected patch log likelihood (EPLL) [28] and so66

on.67

In this work, we formulate a block based nonlocal maximum a posterior estima-68

tion (MAP) framework, in which we construct a nonparametric mixture model to69

describe the prior probability density function for the small image patches. Instead of70

optimizing such a complicated regularization term derived from the mixture model,71

we propose a variant which comes from a well-known expectation maximum (EM)72

process. It leads to a block diffusion based BNLH1 model with an adaptive weighting73

function. The BNLH1 model can be regarded as an extension of the nonlocal means74

method. It also gives a statistical interpretation for the entropy regularization ap-75

peared in [1]. We also prove the existence of a minimizer for such a BNLH1 model76

with the minimizing sequence method. It is well-known that the L1 norm has a better77

performance on noncontinuity preservation than L2, so we define a block nonlocal TV78

operator, and propose a block nonlocal (BNLTV) TV model . The BNLTV can be79

efficiently solved by many recently popular splitting algorithms such as augmented80

Lagrangian [26], splitting Bregman [12] methods and ADMM[25]. Compared with the81

existing nonlocal TV, the experimental results in this paper show that our method82

can greatly improve the quality of the reconstruction.83

The rest of the paper is organized as follows: in section 2 we review some basic84

definitions and properties of the existing point based nonlocal TV operator. A block85

based maximum a posterior estimation (MAP) model and the two block nonlocal86

variational models BNLH1 and BNLTV are proposed in section 3; section 4 presents87

the algorithms and some details about the implementation; experimental results are88


A BLOCK NONLOCAL TV METHOD FOR IMAGE RESTORATION 3

shown in section 5; we summarize our method and conclude the paper in section 6.89

2. Nonlocal TV. In this section we will review some basic definitions of point90

based nonlocal derivative, nonlocal gradient, nonlocal divergence and Laplace operator91

as following [11].92

Let Ω ⊂ Rn, u : Ω → R, w : Ω × Ω → R be a symmetric and nonnegative93

weighting function.94

Nonlocal derivative of u at x with respect to y can be defined as

∂yu(x) := (u(y)− u(x))√w(x, y), x, y ∈ Ω,

then the nonlocal gradient operator ∇w : F1(Ω;R)→ F2(Ω×Ω;R), where F1(Ω;R).=

u : Ω→ R, F2(Ω× Ω;R).= v : Ω× Ω→ R, is defined as

(∇wu)(x, y) = (u(x)− u(y))√w(x, y),

the nonlocal divergence operator divw : F2(Ω × Ω;R) → F1(Ω;R) is given by theadjoint of the nonlocal gradient:

(divwv)(x) =

∫Ω

(v(x, y)− v(y, x))w(x, y)dy.

Obviously, the Laplace operator ∆w : F1(Ω;R)→ F1(Ω;R) will be

(∆wu)(x) =1

2divw((∇wu)(x, y)) =

∫Ω

(u(y)− u(x))w(x, y)dy.

Noting that w(x, y) = w(y, x), one can easily verify the adjoint equality

< ∇wu, v >=< u,−divwv >,

and obtain the following results:∫Ω×Ω

divwvdxdy = 0.

< ∆wu, v >=< u,∆wv >,

< ∆wu, u >= − < ∇wu,∇wu >≤ 0.

In the definitions of the nonlocal operators, the weighting function is always95

assumed as a known one or selected empirically. However, it is easy to see that the96

weighting function plays a key role in image denoising. If the empirically chosen97

weighting function is not good, the restoration would be very bad. Besides, the98

diffusion of ∆w is point based one and this leads to that the pixel reconstruction99

would be point by point, thus it may not be robust enough if the noise level is high.100

In order to solve these problems, we will extend the above definitions to some block101

based new ones.102

3. The Proposed Methods.103

3.1. Statistical Methods. In this section, we shall propose a block based max-104

imum a posteriori estimation (MAP) for nonlocal method, in which the weighting105

function can be determined by the cost functional itself and also can be updated in106

the iterations.107



3.1.1. Block Based Maximum a Posteriori (MAP) Estimation. Let Ω be108

a 2 dimensional discrete set. v : Ω → R is the observed noisy image, and u : Ω → R109

stands for the latent clear image. Denote B as a small symmetric neighborhood110

centered at 0, i.e. B = x : ||x|| < r. Moreover, we use uB(x) to represent a small111

image block of u, namely, uB(x) = u(x+ z) : z ∈ B. All the observed image blocks112

vB(x), x ∈ Ω can be regarded as some realizations of a random vector ξ. The MAP113

estimation problem for u can be written by114

u∗ = arg maxu

p(uB |vB) = arg maxu

p(vB |uB)p(uB)

p(vB).115

Since p(vB) is known for any given noisy image v, the MAP problem is equivalent to116

(1) u∗ = arg maxu

ln p(vB |uB) + ln p(uB).117

From the noisy model, one can easily find118

p(vB |uB) ∝ exp

(− (u− v)2

2σ2

),119

where σ is the standard deviation of the Gaussian noise.120

p(uB) is a prior term for u. As to natural images, there are usually many similar121

and repeated structures or textures in the sense of small blocks. We will construct122

a nonparametric mixture model to describe such a prior. Once u is estimated, the123

histogram of all the blocks uB(y), y ∈ Ω can be calculated with the formulation124 ∑y∈Ω

δ(z− uB(y)).125

Here δ is a vector-valued function which counts the number of blocks whose structures126

are the same as z, i.e.127

δ(x) =

1, x = 0,0, else.

128

For calculation convenience, usually δ can be approximated by a Gaussian func-129

tion130

δh(x) =1

(πh)|B|2

exp

(−||x||

2

h

),131

where h > 0 is a parameter which controls the precision of this approximation.132

Assume all the clear image blocks uB(x), x ∈ Ω are some realizations of a random133

vector with the probability density function134

p(z) =1

|Ω|∑y∈Ω

δh(z− uB(y)),135

then according to the independent and identically distributed assumption, one can136

obtain137

p(uB) =1

|Ω||Ω|∏x∈Ω

∑y∈Ω

δh(uB(x)− uB(y)).138



Therefore, ignoring any constant terms, the problem (1) becomes139

(2) u∗ = arg minu

E(u) =λ

2

∑x∈Ω

(u(x)− v(x))2 −∑x∈Ω

ln∑y∈Ω

δh(uB(x)− uB(y))

.140

The first term in the above cost functional is the fidelity term, which is used to141

measure the difference between the noisy image v and clear image u in terms of noisy142

model. The second term can be regarded as a regularization term, which describes the143

similarity among the small image blocks uB(x). λ > 0 is a regularization parameter144

which balances these two terms.145

To optimize E(u) directly is not easy since the regularization term, which is146

derived from a mixture model, is associated to a complex nonlinear equation if we147

employ the first order optimization conditions. However, this problem can be greatly148

simplified by the well-known expectation maximum (EM) algorithm. For convenience,149

we denote150

E1(u) = −∑x∈Ω

ln∑y∈Ω

δh(uB(x)− uB(y)).151

The key idea of EM algorithm to solve such a problem is as following: in order to152

minimize E1(u), we turn to construct another function H(u;un) which can be easily153

optimized and has a proposition:154

Proposition 3.1. If H(un+1;un) 6 H(un;un), then E1(un+1) 6 E1(un), where155

n is an iteration number.156

According to this proposition, the value of E1 would be descended during the iterations157

of minimizing H. Therefore, the original optimization problem of E1 can be replaced158

by finding the minimizer of H, which is easier to be minimized.159

In the next sections, we will give the formulation of H and give a variational160

interpretation for such a EM process.161

3.1.2. Expectation Maximum (EM) Process and Weighting Function.162

Please note minimizing E1 is a standard parameter estimation problem of mixture163

model. We assume that all the small image blocks uB(x), x ∈ Ω can be divided into164

|Ω| classes (some classes may be empty). Let we introduce a latent integer random165

variable, whose values are in the range of 1, 2, · · · , |Ω| and each η(x) indicates that166

each image block uB(x) centered at x should belong to the η(x)-th class. We use a167

function ω(x, y) to represent the probability of each block uB(x) belongs to the y-th168

class. According to the standard EM method, the expectation step can be written as169

E-step:170

171

(3) ωn(x, y) = Pη(x) = y|unB(x) =δh(unB(x)− unB(y))∑y∈Ω δh(unB(x)− unB(y))

.172

Then the related maximum step is173

M-step:174

175

(4)

un+1 = arg maxu

−∑x∈Ω

∑y∈Ω ln (δh(uB(x)− uB(y)))ωn(x, y)

= arg min

u

∑x∈Ω

∑y∈Ω ln (δh(uB(x)− uB(y)))ωn(x, y) := H(u;un)

.

176



Please note H is easy to be optimized since it is a quadratic term.177

In our previous work [19], we reinterpreted the above EM process in a variational178

framework. The E-step and M-step can be regarded as two steps of alternating di-179

rection minimization method. For completeness, we list the main results of [19] here:180

181

Lemma 3.1 (Commutativity of log-sum operations, [19]). Given a function182

δh(x, y) > 0, we have183

−∑x∈Ω

ln∑y∈Ω

δh(x, y)184

= minω∈C

−∑x∈Ω

∑y∈Ω

ln δh(x, y)ω(x, y) +∑x∈Ω

∑y∈Ω

ω(x, y) lnω(x, y)

,185

186

where C =ω(x, y) : 0 6 ω(x, y) 6 1,

∑y∈Ω ω(x, y) = 1,∀x ∈ Ω

.187

Hence, we define188

H(u, ω) = −∑x∈Ω

∑y∈Ω

ln δh(uB(x)− uB(y))ω(x, y) +∑x∈Ω

∑y∈Ω

ω(x, y) lnω(x, y),189

and can get190

minu

E1(u) = minu

minω∈C

H(u, ω)

.191

One can choose the alternating direction minimization method to solve such a problem192

and get an iteration scheme193

(5)

ωn+1 = arg minω∈C

H(un, ω),

un+1 = arg minu

H(u, ωn+1).194

It is easy to check that the two problems in (5) are equal to the E-step (3) and M-step195

(4), respectively. Moreover, for the scheme (5), we have196

Lemma 3.2 (Energy Descent, [19]). The sequence un produced by iteration197

scheme (5) satisfies198

E1(un+1) 6 E1(un).199

Basing on the former analysis, we employ the functional H(u, ω) as the regular-200

ization term and propose the following image restoration functional201

(u∗, ω∗) = arg minu,ω∈C

λ2

∑x∈Ω

(u(x)− v(x))2 + h∑x∈Ω

∑y∈Ω

ω(x, y) lnω(x, y)202

+∑x∈Ω

∑y∈Ω

||uB(x)− uB(y)||2ω(x, y).203

204

By setting λ = 0, the above model can be regarded as an extension of nonlocal205

means method. To be different from the nonlocal means in which the weighting func-206

tions are manually chosen according to some experience, here the weighting functions207



are determined by the restoration cost functional itself. Besides, the estimation in208

nonlocal means is point by point but here is block based. Generally speaking, the209

block based estimation can improve the precision and be more robust to noise. It210

is well known that the nonlocal means is related to the nonlocal H1 regularization211

problem, hence we can extend the block based method to the nonlocal TV version.212

In the next section, from the view of the variational method, we will propose a block213

nonlocal TV model with adaptive weighting functions.214

3.2. Variational Framework.215

3.2.1. Some Definitions and Notations. Let Ω ⊂ R2 be a bounded and216

open set, Br ⊂ R2 be a small symmetric region centered at 0 with radius r, i.e.217

Br = z : ||z|| < r. ω : Ω × Ω → R+ is a nonnegative smooth weighting function,218

u : Ω→ R is an image function, and g : Br → R+, ρε : Bε → R+ are two other smooth219

weighting functions with g(z) = g(−z),∫Brg(z)dz = 1 and

∫Bερε(z)dz = 1. Denote220

Ω = Ω + Br + Bε = x|x = x1 + x2 + x3, x1 ∈ Ω, x2 ∈ Br, x3 ∈ Bε. We use the221

symbol “˜” to represent an expansion of a function. For example, u is an expansion222

of u with223

u(x) =

u(x), x ∈ Ω,

u0(x), x ∈ Ω\Ω.224

In which u0(x) is a given function associated to the so-called boundary condition.225

To simplify the following formulations, here we set the boundary condition u0(x) =226

0. Other boundary conditions such as Neumann boundary condition also can be227

discussed in the similar way. In the next discussion, without special statement, the228

boundary conditions are all set to 0 if we need the expansions of any functions.229

The symbol ”∗” is used to represent the convolution operator, namely, (ρε ∗ u)(x) =230 ∫Bερε(y)u(x− y)dy.231

Let F1(Ω;R), F2(Ω×Ω×Br;R) be two function spaces, we define a block nonlocal232

gradient operator233

∇gω : F1(Ω;R)→ F2(Ω× Ω×Br;R)234

which has the following representation:235

∇gωu(x) =√g(z)[(ρε ∗ u)(y + z)− (ρε ∗ u)(x+ z)]

√w(x, y), ∀x ∈ Ω, y ∈ Ω, z ∈ Br.236

Here we employ two smooth functions g and ρε in the definition of the block nonlocal237

gradient since we need ρε to keep the later constructed regularization term possessing238

some better mathematical properties, which can theoretically guarantee that we can239

proof the existence of solutions for our proposed model (please see in section 3.2.4).240

Hence, the related block nonlocal divergence operator divgω : F2(Ω × Ω × Br;R) →241

F1(Ω;R) can be given by242

divgωp(x, y, z) =

∫Ω

∫Br

∫Bε

√g(z)ρε(s)

[p(x− z + s, y, z)

√ω(x− z + s, y)243

− p(y, x− z + s, z)√ω(y, x− z + s)

]dsdzdy.244

245

It is easy to check246

< ∇gωu, p >= − < divgωp, u >247



y

x

ω(x, y)

(a) Nonlocal

ω(x, y)

x x + z1

y + z1

y − z1

x − z1

y

(b) Block Nonlocal

Fig. 1: Comparison of point based nonlocal operator and the proposed block basednonlocal operator.

with the 0 boundary conditions for p, ω and u. Here <,> is the standard inner product248

in L2.249

The related block nonlocal Laplacian operator 4gω : F1(Ω;R)→ F1(Ω;R) is given250

by251

4gωu(x) =

∫Ω

∫Br

∫Bε

g(z)ρε(s)[(ρε ∗ u)(y + z)− (ρε ∗ u)(x+ s)]·252

[ω(y, x− z + s) + ω(x− z + s, y)]dsdzdy,253254

which satisfies 4gω = divgω∇gω.255

The proposed block based nonlocal operators can be considered as the vector-256

valued extensions of the existing nonlocal operators. Please see the figure 1 for some257

comparisons.258

3.2.2. Block Nonlocal H1 with Adaptive Weighting Function. The block259

nonlocal H1 norm of u can be given as260

(H1)gω(u) =

∫Ω

∫Ω

∫Br

g(z)[(ρε ∗ u)(y + z)− (ρε ∗ u)(x+ z)]2ω(x, y)dzdydx.261

Thus, the previous model can be modified as BNLH1 model:262

(u∗, ω∗)263

= arg minu,ω∈C

λ2

∫Ω

(u(x)− v(x))2dx+ h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx+ (H1)gω(u),264

265

where C =ω(x, y) : 0 6 ω(x, y) 6 1,

∫Ωω(x, y)dy = 1,∀x ∈ Ω

266

3.2.3. Block Nonlocal TV. By defining267

TV gω (u) =

∫Ω

√∫Ω

∫Br

g(z)[(ρε ∗ u)(y + z)− (ρε ∗ u)(x+ z)]2ω(x, y)dzdydx.268



we get a BNLTV model:269

(u∗, ω∗)270

= arg minu,ω∈C

λ2

∫Ω

(u(x)− v(x))2dx+ h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx+ TV gω (u),271

272

where C =ω(x, y) : 0 6 ω(x, y) 6 1,

∫Ωω(x, y)dy = 1,∀x ∈ Ω

.273

In the proposed method, there are some convolutions ρε ∗ u in the regularization274

term. As mentioned earlier, the reason for such a design is that these convolution275

operators can help us to theoretically proof the existence of minimizers. From the276

viewpoint of statistics, it requires that there are some self-similarity blocks in the277

smooth version of image u. This requirement is reasonable. In the practical compu-278

tation, we can let ε be small enough, and then ρε becomes a delta function.279

3.2.4. Existence of a Minimizer for the Proposed Models. In this section,280

we will proof the existence of a minimizer for BNLH1. As to the BNLTV model, we281

have not yet investigated a theoretical result and let us leave it as a future work.282

Let us consider the energy functional for BNLH1 model283

(6) J(u, ω) =λ

2

∫Ω

(u(x)− v(x))2dx+ h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx+ (H1)gω(u).284

We will show the existence of a minimizer for (6) in the following space

X := (u, ω) : u ∈ L2(Ω), 0 ≤ ω ≤ 1, a.e., and

∫Ω

ω(x, y)dy = 1, a.e.x ∈ Ω.

Proposition There exists a solution for BNLH1 model285

(7) min(u,ω)∈X

J(u, ω).286

Proof: Let287

J1(u) =λ

2

∫Ω

(u(x)− v(x))2dx,

J2(ω) = h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx,

J3(u, ω) =

∫Ω

∫Ω

∫B

g(z)(ρε ∗ u(y + z)− ρε ∗ u(x+ z))2ω(x, y)dzdydx.

288

It is obviously that J1(u) ≥ 0 and J3(u, ω) ≥ 0. Futhermore, since x lnx ≥ − 1e

whenever x ≥ 0 and Ω is a bounded domain, J2(ω) ≥ −he |Ω|2. That is, J(u, ω) has

a lower bound and inf(u,ω)∈X

J(u, ω) exists. Let (un, ωn) be a minimizing sequence of

(7), i.e. a sequence such that

J(un, ωn)→ inf(u,ω)

J(u, ω).

It is clear that ‖ωn‖L∞ ≤ 1 and ωn ∈ L∞(Ω×Ω) , as L∞ is the conjugate space ofL1 which is a separable linear normed space, by the Banach-Alaoglu Theorem, thereis a * weakly convergent subsequence(which we relabel by n), and a * weak limitω ∈ L∞(Ω× Ω) such that

ωn∗ ω in L∞(Ω× Ω),



That is, for any ϕ ∈ L1(Ω× Ω),∫Ω×Ω

ωn(x, y)ϕ(x, y)dxdy →∫

Ω×Ω

ω(x, y)ϕ(x, y)dxdy, n→ +∞.

Since ω lnω is continuous and convex with respect to ω, it is quasiconvex andJ2(ω) is * weakly lower semicontinuous, i.e.

lim infn→∞

∫Ω

∫Ω

ωn(x, y) lnωn(x, y)dydx ≥∫

Ω

∫Ω

ω(x, y) lnω(x, y)dydx.

We also need to verify the * weak limit ω is in X, that is, 0 ≤ ω ≤ 1 a.e., and289 ∫Ωω(x, y)dy = 1, a.e. x ∈ Ω.290

Owing to ωn∗ ω in L∞, for any ϕ ∈ L1,291

(8)

∫Ω

∫Ω

ωn(x, y)ϕ(x, y)dydx→∫

Ω

∫Ω

ω(x, y)ϕ(x, y)dydx.292

Set A = (x, y) ∈ Ω × Ω : ω(x, y) > 1, Br = (x, y) ∈ Ω × Ω : ω(x, y) < 0,293

ϕ1(x, y) = χA(x, y), ϕ2(x, y) = χBr(x, y), then ϕ1, ϕ2 ∈ L1(Ω × Ω). We will show294

that |A| = 0 and |Br| = 0.295

Substituting ϕ1 for ϕ in (8), we obtain296

(9)

∫A

ωn(x, y)dydx→∫A

ω(x, y)dydx.297

If |A| 6= 0, the right hand side of (9) is bigger than |A|, by sign-preserving theorem298

of limit, there is a N > 0, for any n > N ,∫Aωn(x, y)dydx > |A|. However, since299

ωn ≤ 1, the left hand side of (9) should be smaller than or equal to |A| for any n,300

which is a contradiction. Thus, we have |A| = 0. In the same way, we obtain |Br| = 0.301

The results show that 0 ≤ ω ≤ 1 a.e..302

Let f(x) =∫

Ωω(x, y)dy, ϕ(x, y) = sign(f(x)− 1), then ϕ ∈ L1, and∫

Ω

(∫Ω

ωn(x, y)dy

)ϕ(x)dx→

∫Ω

f(x)ϕ(x)dx,

i.e. ∫Ω

|f(x)− 1|dx = 0,

hence,∫

Ωω(x, y)dy = 1 for a.e. x ∈ Ω. Above all, we have ω ∈ X, and

lim infn→∞

J2(ωn) ≥ J2(ω).

Since J(u, ω) is coercive, un must be bounded, i.e. there is a constant M > 0such that

‖un‖L2(Ω) ≤M.

Boundedness of the sequence in a reflexive space implies the existence of a weaklyconvergent subsequence, which we still denote by un, then, there is a weak limitu ∈ L2(Ω) such that

un u.



Since J1 : u 7→ λ2

∫Ω

(u(x) − v(x))2dx is continuous and convex, and un u inL2(Ω), we have

lim infn→∞

λ

2

∫Ω

(un(x)− v(x))2dx ≥ λ

2

∫Ω

(u(x)− v(x))2dx,

i.e.

limn→∞

J1(un) = J1(u).

Now, we consider the convergence of J3(un, ωn).303

Here, we assume the support of g is a proper subset of B and choose ε equalling304

to one half of the distance between the support and B.305

Set bn(x, y) =∫Brg(z)(ρε ∗ un(x+ z)− ρε ∗ un(y+ z))2dz. By the smoothness of

convolution operation, integral bn depending on parameters x, y is in C1(Ω×Ω), and

∂bn∂x

(x, y) = 2

∫Br

g(z)(ρε ∗ un(x+ z)− ρε ∗ un(y + z))

(∂ρε∂x∗ un(x+ z)

)dz,

∂bn∂y

(x, y) = 2

∫Br

g(z)(ρε ∗ un(x+ z)− ρε ∗ un(y + z))

(∂ρε∂y∗ un(x+ z)

)dz.

As306 ∫Ω

∫Ω

|bn(x, y)|dxdy ≤ 2

∫Ω

∫Ω

∫Br

g(z)[(ρε ∗ un(x+ z))2 + (ρε ∗ un(y + z))2]dzdxdy307

≤ 4|Ω|‖g‖∞∫

Ω

∫Br

(ρε ∗ un(x+ z))2dzdx ≤ 4|Ω||B|‖g‖∞M2,308

309

∫Ω

∫Ω

|∇xbn(x, y)|dxdy310

≤ 2

∫Ω

∫Ω

∫Br

g(z)|ρε ∗ un(x+ z)− ρε ∗ un(y + z)|∣∣∣∣∂ρε∂x ∗ un(x+ z)

∣∣∣∣ dzdxdy311

≤ 4|Ω||B|‖g‖∞‖ρε‖1,∞M2,312313

then bn(x, y) is a bounded sequence in W 1,1(Ω×Ω). By Rellich-Kondrachov Com-314

pactness Theorem, W 1,1(Ω×Ω) is compactly embedded in L1(Ω×Ω), i.e. there exists315

a subsequence bn(x, y) which converges to b(x, y) in L1(Ω× Ω).316

Apparently, J3(u, ω) is continuous and convex in u, then it is weakly lower semi-317

continuous, that is,318

lim infn→+∞

∫Ω

∫Ω

bn(x, y)ω(x, y)dzdxdy319

≥∫

Ω

∫Ω

∫Br

g(z)(ρε ∗ u(x+ z)− ρε ∗ u(y + z))2ω(x, y)dzdxdy.320

321



Furthermore,322 ∫Ω

∫Ω

bn(x, y)(ωn(x, y)− ω(x, y))dydx323

=

∫Ω

∫Ω

(bn(x, y)ωn(x, y)− b(x, y)ω(x, y))dydx324

+

∫Ω

∫Ω

(b(x, y)ω(x, y)− bn(x, y)ω(x, y))dydx325

=

∫Ω

∫Ω

(bn(x, y)− b(x, y))ωn(x, y)dydx+

∫Ω

∫Ω

b(x, y)(ωn(x, y)− ω(x, y))dydx326

+

∫Ω

∫Ω

(b(x, y)− bn(x, y))ω(x, y)dydx.327328

Because of ωn∗ ω in L∞(Ω× Ω), bn → b in L1(Ω× Ω), we have

limn→+∞

∫Ω

∫Ω

bn(x, y)(ωn(x, y)− ω(x, y))dydx = 0.

As329

J3(un, ωn) =

∫Ω

∫Ω

bn(x, y)ωn(x, y)dxdy330

=

∫Ω

∫Ω

bn(x, y)(ωn(x, y)− ω(x, y))dxdy +

∫Ω

∫Ω

bn(x, y)ω(x, y)dxdy,331332

we havelim infn→+∞

J3(un, ωn) ≥ J3(u, ω).

In a word,

J(u, ω) = J1(u) + J2(ω) + J3(u, ω) ≤ lim infn→∞

J(un, ωn),

andJ(u, ω) = inf

(u,ω)∈XJ(u, ω),

which completes the proof.333

4. Algorithms. To simplify the computation, we can set ρε to the delta function334

δ, thus ρε ∗ u = δ ∗ u = u. In the following discussion, we set ρε = δ.335

The BNLH1 model can be directly minimized by an alternating direction mini-336

mization algorithm. We summary the steps in algorithm 4.1.337

Algorithm 4.1. Given an initial value u0 = v, for n = 1, 2, 3, · · · , do338

Step1, calculating the weighting function by339

ωn+1(x, y) =

exp

(−∫Brg(z)(un(y + z)− un(x+ z))2dz

h

)∫

Ω

exp

(−∫Brg(z)(un(y + z)− un(x+ z))2dz

h

)dy

.340

Step 2, restoring un+1 by solving the following equation341

−4gωn+1u+ λu = λv.342



Step 3, if the convergence condition ||un+1−un||2||un||2 is less than a tolerant error, then343

stop. Otherwise, go to the step 1.344

As to BNLTV, we need to employ some more efficient algorithms, such as aug-345

mented Lagrangian method an splitting Bregman method. Here, we write the iteration346

scheme as augmented Lagrangian method.347

Let us introduce a function q = (∇gω)u, the BNLTV model is equal to the con-348

straint minimization problem349

(u∗, ω∗, q∗) = arg minu,ω∈C,q=(∇g

ω)u

λ2

∫Ω

(u(x)− v(x))2dx+ h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx350

+

∫Ω

||q(x, ·, ·)||2dx.351

352

By applying standard augmented Lagrangian method, one can get a Lagrangian353

function354

L(u, ω, q, d) =λ

2

∫Ω

(u(x)− v(x))2dx+ h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx355

+

∫Ω

||q(x, ·, ·)||2dx+ < d, q − (∇gω)u > +η

2||q − (∇gω)u||22.356

357

The saddle of L can be split into 4 subproblems, each subproblem is easy to solve.358

359

Subproblem 1: expectation maximum step

ωn+1 = arg minω∈C

L(un+1, w, qn, dn)360

= arg minω∈C

h

∫Ω

∫Ω

ω(x, y) lnω(x, y)dydx+η

2||qn − (∇gω)un+1 +

dn

η||22.361

362

Subproblem 2: restoration step

un+1 = arg minu

L(u,wn, qn, dn)363

= arg minu

λ

2

∫Ω

(u(x)− v(x))2dx+η

2||qn − (∇gωn)u+

dn

η||22.364

365

Subproblem 3: gradient updating

qn+1 = arg minq

L(un+1, wn+1, q, dn)366

= arg minq

∫Ω

||q(x, ·, ·)||2dx+η

2||q − (∇gωn+1)un+1 +

dn

η||22.367

368

Subproblem 4: Lagrangian multiplier updating

dn+1 = arg maxd

L(un+1, wn+1, qn+1, d)369

= arg maxd

< d, qn+1 − (∇gωn+1)un+1 >

.370

371



As to the subproblem 1, we can use the first order optimization condition to get372

an approximation solution. For simplifying the representation, let us denote373

An(x, y) = exp− η

2h

∫Br

[g(z)(un+1(y + z)− un+1(x+ z))2 +

1√ωn(x, y)

√g(z)·374

(un+1(y + z)− un+1(x+ z))(qn(x, y, z) +dn(x, y, z)

η)]dz.375

376

Then we obtain377

(10) ωn+1(x, y) =An(x, y)∫

ΩAn(x, y)dy

.378

To get the solution of subproblem 2, we need to solve equation379

(11) λu− η4gωu = λv − ηdivgω(qn +dn

η).380

This equation can be solved iteratively by many methods such as Gauss-Seidel itera-381

tion.382

qn+1 can be given by a shrinkage process383

(12) qn+1(x, y, z) = S((∇gωn+1)un+1(x, y, z)− dn(x, y, z)

η, η),384

where the shrinkage operator385

S(f(x, y, z), η) =

f(x,y,z)||f(x,·,·)||2 (||f(x, ·, ·)||2 − 1

η ), ||f(x, ·, ·)||2 > 1η ,

0, ||f(x, ·, ·)||2 < 1η ,

386

and ||f(x, ·, ·)||2 =√∫

Ω

∫Brf2(x, y, z)dzdy. The last subproblem can be easily maxi-387

mized by388

(13) dn+1(x, y, z) = dn(x, y, z) + τ(qn+1(x, y, z)− (∇gωn+1)un+1(x, y, z)),389

where 0 < τ < 2η is a parameter which controls the convergence of the augmented390

Lagrangian algorithm. Usually, it can be chosen as τ = η.391

We summarize the steps as algorithm 4.2.392

Algorithm 4.2. Given an initial value u0 = v, ω0 = 1|Ω| , d

0 = q0 = 0, for393

n = 1, 2, 3, · · · , do394

Step1, weighting function updating:395

Calculating the weighting function by (10).396

Step 2, image restoration:397

Restoring un+1 by solving the equation (11).398

Step 3, gradient updating:399

Computing qn+1 according to (12).400

Step 4, Lagrangian multiplier updating:401

Renewing dn+1 by formulation (13).402

Step 5, convergence checking:403

If the convergence condition ||un+1−un||2||un||2 is less than a tolerant error, then stop. Oth-404

erwise, go to the step 1.405



5. Experimental Results. In this section, we will show some numerical results406

for the proposed method and make some comparison with some related methods.407

Without special statements, in the proposed method, some parameters are chosen as408

following: h = 0.252, σ = 3.0, η = 10, Br is set to a 7× 7 rectangle path. λ is set as409

different values according to the noise levels, we will list the exact values later. For410

computational efficiency and saving memory storage, we do not let y go through the411

entire Ω and just set a 13× 13 search window for every x.412

Generally speaking, if the images have a lot of repeated structures (regular tex-413

tures), the size of the searching window should be enlarged because it can help us414

to find more nonlocal similar pixels which have the same textures. However, it will415

lead to some heavy computational burdens. On the other hand, if the images contain416

heavy noise, one should increase the size of the image blocks. The large image blocks417

can help us to produce some smooth results. It plays the role of a local smoother.418

Conversely, the image blocks size should be reduced if the noise level is low, because419

the improvements would be limited in this case and one can get some not too bad420

results with much less CPU time.421

According to the definition of the block gradient operator ∇gω, both local and422

nonlocal factors are considered by the variables y and z, respectively. Thus, the423

proposed block based ones are more robust for heavy noise than the point based424

nonlocal operators [11]. Let us give an intuitive interpretation for such a diffusion425

mechanism. Let ρε = δ and ω be symmetric, then the Gauss-Seidel iteration for426

−4gωu would be427

(14) un+1(x) =

∫Ω

∫Brun(y + z)ω(x− z, y)g(z)dzdy∫

Ω

∫Brg(z)ω(x− z, y)dzdy

.428

This equation means that all of the pixels at y and y + z are averaged with the429

weighting function ω(x − z, y)g(z). The pixels located at y could be far away from430

x, these pixels can be regarded as nonlocal ones. Since z only can be taken in a431

local neighborhood centered at 0, y + z are local positions of y. Furthermore, please432

note that the patch centered at y with large weight would be very similar to the one433

centered at x, then u(y+z) can be regarded to have the similar local pixels of u(x). In434

this sense, the block nonlocal diffusion are both local and nonlocal. Please see figure435

2 for the ideas.436

The equation (14) also can be reformulated as437

un+1(x) =

∫Ω

∫Brun(y)ω(x− z, y − z)g(z)dzdy∫

Ω

∫Brg(z)ω(x− z, y − z)dzdy

438

with some appropriate boundary condition. From this viewpoint, the above formula-439

tion means that one pixel u(x) may be contained in many overlapping patches in the440

block based mode, thus u(x) would be estimated for many times. Our formulation441

give a weighting function to average all the repeatedly estimated u(x). Thus, the442

block based operators are more robust for noise. It can be found in the following443

experiments.444

In the first experiment, we show the superiority of block diffusion operators.445

The clean original “barbara“ image in the figure 3(a) was destroyed by the additive446

Gaussian white noise with standard deviation σ = 80, and the noisy image was447

shown in the figure 3(b). We recover it with different denoising methods. For each448

algorithm, we try different parameters and select the best reconstructed results (with449



y

y + z

x − z

x

ω(x, y)

ω(x − z, y)

Fig. 2: The schematic plot of the Gauss-Seidel iteration for −4gωu

the highest PSNR) for comparison. Let us first show the differences of local, nonlocal450

and block based nonlocal operators. To see this clearly, we fix the weighting functions451

and do not update them in the related nonlocal methods. The result produced by452

TV [23] was displayed in the figure 3(c), one can find this restored image is almost453

piecewise constant and loses a lot of texture structures. The PSNR for TV is 22.08454

dB. Compared with TV, the NLTV [11] can keep more texture details since the455

repeated structure information was used in this model. However, some recovered456

textures demonstrated in the figure 3(d) are still not very clear since the noise is very457

heavy. Please find the texture details comparison in the figure 4 which contains the458

related enlargements of white rectangle areas in figure 3. The PSNR of NLTV is459

22.45 dB, which is higher than TV’s. In the figure 3(e), we show the result provided460

by the proposed block based nonlocal method BNLTV. One can find it can produce461

better texture restorations with PSNR 22.90 dB(please see figure 4 for texture details)462

under heavy noise because both of the local and nonlocal information are used in this463

algorithm. In this experiment, the fidelity parameters for TV, NLTV, and BNLTV464

are 3.6, 1.0 and 2.0, respectively.465

In the next, the advantages of the weighting function updating will be illustrated.466

In figure 6, we restore the same noisy image in figure 4 by using the NLTV and BNLTV467

(algorithm 4.2) with 10 times iteration. The related results are shown as in the figure468

5(a) and 5(b), respectively. Obviously, the reconstructed images have a much better469

quality than the previous ones, please also see the details comparison in figure 4. It is470

reasonable because the weighting function could be improved by some new estimated471

images. We display the PSNR values of the BNLN method during the iteration in472

the figure 6. The final PSNR of NLTV and BNLTV are 23.22 dB and 23.62 dB,473

respectively. Compared to the previous results without weighting function updating,474

the PSNRs are improved more than 0.7 dB. Based on some experimental experience,475

we find that it would be better to choose a big fidelity parameter to prevent the476

recovered image over smooth, thus we set the fidelity parameter λ to be 4.0 for NLTV477



σImages Methods 10 30 50 80 100 120Barbara NLTV [11] 33.08 27.41 24.70 22.45 21.52 20.95

(512× 512) BNLTV 33.53 28.36 25.76 23.62 22.92 22.15Lena NLTV [11] 33.42 27.50 25.16 23.00 21.99 21.21

(256× 256) BNLTV 33.69 28.39 25.77 24.31 23.52 22.79House NLTV [11] 35.20 30.20 27.30 24.63 23.46 22.65

(256× 256) BNLTV 35.30 30.79 28.42 26.21 25.29 24.35

Table 1: PSNRs (dB) for NLTV and the proposed BNLTV under Gaussian noise withdifferent levels.

NLTV [11] BNLTVImages iter. 2 iter. 4 iter. 6 iter. 8 iter. 10House Error 295.02 261.66 209.91 200.12 195.90 193.87

256× 256 CPU time (S) 1.36 24.07 47.88 70.98 93.99 117.01Lena Error 414.76 375.63 312.88 298.37 292.97 291.22

256× 256 CPU time (S) 1.45 25.95 50.93 75.67 100.29 125.23Barbara Error 1845.7 1690.1 1415.9 1367.9 1346.1 1338.6

512× 512 CPU time (S) 6.80 121.19 237.78 360.31 483.67 606.35

Table 2: The absolute error (||u − uoriginal||22) and CPU time (Seconds) for theNLTV[11] and BNLTV. The numbers of iter. in the table are the outer iterationnumbers of the BNLTV.

and 2.0 for BNLTV.478

For more comparison results of different test images under a variety of noise479

levels, please see in table 1. It can be found that the proposed BNLTV has a better480

performance than NLTV in all the cases. Generally speaking, the heavier the noise,481

the larger increasement for the PSNR.482

6. Conclusion and Discussion. In this paper, we proposed a block based non-483

local method for image restoration. In our model, the weighting function in nonlocal484

method is not fixed but adaptively determined by the cost functional itself. The485

weighting function ω(x, y) also has a statistical meaning, and it stands for a probabil-486

ity of the similarity between patches centered at x and y. By updating the weighting487

function, the quality of the restored images will be greatly improved, especially under488

the heavy noise. Besides, we also extend the existing point based nonlocal diffusion489

to a block based one. The numerical results show that the proposed block nonlocal490

operators are more robust for noise than the previous ones. It can help us to get a491

better reconstruction images.492

The proposed method can be applied in many image tasks. One of the possi-493

ble application is the inpainting and compressed sensing, in which the weights are494

more important and our method would greatly improve the results. Due to the page495

limitation, let us left them as a future work in the oncoming another paper.496

Though the proposed method can produce better result than the existing NLTV,497

the computational cost is higher than NLTV. Generally speaking, for 256×256 images,498

it would cost about 11 − 12 seconds CPU time on MacBook Pro 2011 for one outer499

iteration (including the weight calculation and restoration). In table 2, we list some500

of the reconstruction absolute error (||u− uoriginal||22) and the CPU time for BNLTV501

and NLTV [11]. In this table, all the data was obtained under the Gaussian noise502

with standard deviation 100.503



(a) Clean

(b) Noisy, σ = 80 (c) TV [23], 22.08 dB

(d) NLTV [11], 22.45 dB (e) BNLTV, 22.90 dB

Fig. 3: Comparison of point based nonlocal operator and the proposed block basednonlocal operator without weight updating.



(a) Clean (b) Noisy (c) TV (d) NLTVwithoutweightupdating

(e) BNLTVwithoutweightupdating

(f) NLTVwith weightupdating

(g) BNLTVwith weightupdating

Fig. 4: The enlargement of the white rectangle texture area in figure 3 and 5.

(a) NLTV, 23.22 dB (b) BNLTV, 23.62 dB

Fig. 5: Comparison of point based nonlocal operator and the proposed block basednonlocal operator with weight updating 10 times.

1 2 3 4 5 6 7 8 9 10

22.4

22.6

22.8

23

23.2

23.4

23.6

23.8

24

Fig. 6: The PSNR values of BNLTV during the iteration. (x-axis: iteration number,y-axis: PSNR values, dB).



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a block nonlocal tv method for image restorationmath0.bnu.edu.cn/~liujun/papers/siims_2017.pdf63...

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