Neurocomputing 74 (2011) 2464–2474

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/neucom

Image deblurring with filters learned by extreme learning machine

Liang Wang a,�, Yaping Huang a, Xiaoyue Luo b, Zhe Wang a, Siwei Luo a

a School of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, Chinab Department of Mathematics, Linfield College, OR 97128, USA

a r t i c l e i n f o

Available online 12 May 2011

Keywords:

Image processing

Inverse problem

Calculus of variations

Partial differential equation (PDE)

Machine learning

Natural image priors

12/$ - see front matter & 2011 Elsevier B.V. A

016/j.neucom.2010.12.035

esponding author.

ail address: wangliang.bjtu@gmail.com (L. W

a b s t r a c t

Image deblurring is a basic and important task of image processing. Traditional filtering based image

deblurring methods, e.g. enhancement filters, partial differential equation (PDE) and etc., are limited by

the hypothesis that natural images and noise are with low and high frequency terms, respectively.

Noise removal and edge protection are always the dilemma for traditional models.

In this paper, we study image deblurring problem from a brand new perspective—classification.

And we also generalize the traditional PDE model to a more general case, using the theories of calculus

of variations. Furthermore, inspired by the theories of approximation of functions, we transform the

operator-learning problem into a coefficient-learning problem by means of selecting a group of basis,

and build a filter-learning model. Based on extreme learning machine (ELM) [1–4], an algorithm is

designed and a group of filters are learned effectively. Then a generalized image deblurring model,

learned filtering PDE (LF-PDE), is built.

The experiments verify the effectiveness of our models and the corresponding learned filters. It is

shown that our model can overcome many drawbacks of the traditional models and achieve much

better results.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Image deblurring, or restoration, is a classical and importanttask of image processing. Nowadays there are large numbers oftheories and algorithms for image deblurring. As a type of famousand effective image deblurring methods, partial differentialequation (PDE) models, e.g. total variation (TV) model [7],Mumford–Shah model [8], Perona–Malik (P–M) PDE [9] and etc.,improve the traditional linear filtering methods, e.g. constrainedleast square model [5], Weiner filter [6], and play a veryimportant role in image processing. See [6,10] for a comprehen-sive and detailed introduction.

However, the limitations of traditional PDE based methods arealso obvious: frequency characteristic is not a good way todistinguish the features in the natural images from noise. Sonoise removal and edge protection are always the dilemma fortraditional filtering based methods. Essentially, this is caused bythe lack of statistic information of image category, i.e. the imagepriors. Statistic learning methods have proposed an effective wayto achieve the image priors from the samples of image category.Many existing models and algorithms, e.g. [13–17], can be used tolearn good image priors.

Some studies have also been made on using a neural classifierto learn an image deblurring model. Basu and Su proposed the

ll rights reserved.

ang).

projection pursuit learning network (PPLN) based method [11],and a 3-stage hybrid learning system [12] for blind deconvolu-tion. Different from the existing methods, our model, which isderived form PDE models, focuses on the continuous case. It is anextension of traditional learning models and closely connectedwith the theories of PDE and inverse problems.

In this paper, we revisit traditional PDE models from aclassification point of view, and build a theoretical frameworkunifying both PDE models and learning methods. Based on thetheories of approximation of functions [18,19], we skillfullytransform the operator (or functions) learning problem into acoefficient-learning problem with a group of basis.

Then, we design a learning algorithm based on extremelearning machine (ELM) [1–4]. And a group of filters are achievedfor the effective classification of noise and natural images.Furthermore, we propose a new and effective image deblurringmodel: learned filtering PDE (LF-PDE) model.

Experimental results show that our model can overcome manydrawbacks of the traditional PDE models, e.g. spots caused byisolated noise points, ‘‘piecewise-constant’’ characters and etc.,and achieve much better results.

2. Image restoration

Image deblurring, or restoration, is a typical inverse problemin image processing area. Due to the ill-posed character, regular-ization is necessary for a stable inverse process. We first propose a

L. Wang et al. / Neurocomputing 74 (2011) 2464–2474 2465

mathematic description of image restoration, then build anoptimization model – energy functional model, and then derivethe algorithm model – PDE model. Finally, we point out thesubstantial limitations of traditional models.

2.1. Image blurring and restoration

For linear and shift invariant (LSI) system, the blurred imagevðxÞ is modeled as the original image u0ðxÞ convoluted by a pointspread function (PSF) GðxÞ, and added by the additional noise e.Mathematically, it is written as

vðxÞ ¼ ðAu0ÞðxÞþe¼ GðxÞ,u0ðxÞþe, ð1Þ

where x¼ðx1,x2ÞAO�R2 denotes a two-dimensional (2D) vari-able, A denotes a convolution operator, , denotes the 2Dconvolution, and GðxÞ is also called the kernel of A. e is supposedto be the independent and identically distributed (i.i.d.) Gaussiannoise in this paper.

Image deblurring, or deconvolution, is the problem of restoringthe original sharp image u0ðxÞ from the blurred image vðxÞ. It is aninverse problem of image blurring, and can be modeled as anoptimization problem:

infuA I

JðAuÞðxÞ�vðxÞJ2, ð2Þ

where I denotes the functional space, e.g. C½O�, L1½O�, L2½O� and

etc., JAu�vJ :¼ ðROðAu�vÞ2 dxÞ1=2. Due to the ill-posed character of

the inverse problem, the solution of Eq. (2) may not be unique;even it is unique, it may not depend on the blurred image in acontinuous way. Therefore, some regularization methods shouldbe used to stabilize the inverse process, i.e. restricting I byadditional constrains or priors of u1:

JRðuÞJC :¼

ZOCðRðuÞÞ dxrC0, ð3Þ

where R : I-IS denotes an operator, IS denotes the feature space,i.e. the range of R, and J�JC denotes a special energy form defined

by C. For instance, let CðsðxÞÞ :¼ s2ðxÞ, then J�JC ¼ J�J22 denotes

the traditional energy form defined by the square of l2-norm;let CðsðxÞÞ :¼ jsðxÞj, then J�JC ¼ J�J1 denotes the energy defined byl1-norm.

2.2. Energy functional model

Image restoration can be considered as a constrained optimi-zation problem, i.e. Eqs. (2) and (3). By Lagrange multipliermethod, it can be modeled as an energy functional optimizationproblem as follows:

infuA I

ZO½jv�Auj2þl �CðRðuÞÞ� dx, ð4Þ

where A is typically modeled as a low-pass filtering process, andits kernel GðxÞ varies according to the blurring process, e.g. out offocus, motion blur, Gaussian blur and etc. l is called theregularization parameter. The restored image uoptðxÞ is achievedby solving Eq. (4).

Eq. (4) is the generalized form of the energy functional modelfor image deblurring. Traditional linear or nonlinear models, e.g.Tikhonov method (R¼r, Cð�Þ ¼ J�J2

2) [6], TV model (R¼r,Cð�Þ ¼ J�J1) [7] and etc., are the specialized cases of Eq. (4).

R and C decide the existence and uniqueness of uopt (i.e. thesolution of Eq. (4)), and play a very important role in imagerestoration.

1 In the view of statistic, JRðuÞJC is assumed to obey the exponential family of

distributions, and the constrain Eq. (3) is equivalent to a prior term of u.

2.3. PDE model

Energy functional model Eq. (4) is the mathematical descrip-tion of the task of image restoration, yet, for the implementationof the task, the corresponding PDE model should be derived. Thenumerical solution method of the PDE model is the imagerestoration algorithm.

By calculus of variations theory, if ( uoptðxÞAI such that uopt isthe solution of Eq. (4), then uopt must satisfy the equation:

ddu

ZOjv�Auj2þl �CðRðuÞÞ dx

� �¼ 0: ð5Þ

Let sðxÞ ¼ ðRðuÞÞðxÞ. From Eq. (5), one can derive the general Eulerequation:

A�Au�A�vþl � ðRuÞ� @

@sCðsÞ

����s ¼ RðuÞ

!¼ 0, ð6Þ

where Ru denotes the Frechet derivative. An and ðRuÞ� denote the

adjoint operator of A and Ru, respectively. In general, Eq. (6) ishard to solve. Solving the variational gradient flow (VGF) PDE:

@u

@t¼� A�Au�A�vþl � ðRuÞ

� @

@sCðsÞ

����s ¼ RðuÞ

!" #ð7Þ

is a way to approach uopt. VGF PDE is appropriate for derivingiterative algorithms. There are many numerical methods forsolving VGF PDE, e.g. finite difference, finite element and etc.

Let R¼r, then the VGF PDE Eq. (7) can be specified as thetraditional linear (with Cð�Þ ¼ J�J2

2) heat diffusion PDE:

@u

@t¼ lnuþ f ðuÞ ð8Þ

and the nonlinear (with Cð�Þ ¼ J�J1) heat diffusion PDE:

@u

@t¼ l � div

1

jrujru

� �þ f ðuÞ, ð9Þ

where f ðuÞ ¼ A�v�A�Au, and n :¼ divðrÞ ¼ @2=@x21þ@

2=@x22.

2.4. Limitations of traditional models

Now let us reconsider the effect of R and C more deeply.Suppose that u0A I, such that any function uAI can be decom-posed as u¼ u0þud, where ud denotes the disparity between u0

and u. If uda0 but udAnullðAÞ, then Au¼ Au0, thus u and u0 areinseparable by Eq. (2). In order to separate u and u0, additionalcriterion, i.e. Eq. (3), is necessary. The validity of the regulariza-tion method relies on the choice of R and C such that for anyu0,uAI, if udAnullðAÞ but uda0, then JRðuÞJC4JRðu0ÞJC.

Let I0 � I denotes the subspace consisting of all feasiblesolutions u0. Let ðI0Þ

c denotes the complement of I0, consistingof all infeasible images. Then the regularization is essentially theseparation of I0 and ðI0Þ

c , and the ideal result of R is to map I0 andðI0Þ

c into different parts (denoted by I0S and ðI0

S Þc , respectively) in

the feature space IS. I0S and ðI0

S Þc are separated by the range of

J�JC. If uAI0, JRðuÞJC will be small, otherwise JRðuÞJC willbe large.

Then let us review the existing PDE models. The basichypothesis of Tikhonov method is that I0 � CO such that I0

S � L2O

for all feasible uAI0. But as is known to all, many natural imagescontain jumps and edges, i.e. I0JCO, thus, many uAI0 areexcluded by Tikhonov method.

As an improvement, TV model breaks the hypothesis ofTikhonov methods by releasing the constrains from I0

S � L2O to

I0S � L1

O. I0 is extended to contain some uðxÞ with finite jumps andedges. For example, if uðxÞ contains jumps and edges, then jruj

will contain the Dirac function dðxÞ. JdðxÞJ22 ¼1, but JdðxÞJ1 ¼ 1.

L. Wang et al. / Neurocomputing 74 (2011) 2464–24742466

Meanwhile, some infeasible functions uAðI0Þc , e.g. the isolated

noise points, will be mapped into I0S .

Now let us reconsider the regularization method from a newpoint of view: classification. The objective is to find an effectiveway, by choosing R and C, to classify all uAI into two classes I0

and ðI0Þc . Traditional PDE methods implement the classification in

frequency domain with a high-pass filter r. But limited by thefrequency hypothesis, r is a good classifier for smooth image andhigh oscillation, but not a good classifier for edges and noise.Therefore, the dilemma between noise removal and edge protec-tion always exists, and cannot be solved essentially just bychanging energy definition form C.

3. Filter learning method

Compared to C, a suitable mapping R is also very, or evenmore, important for a successful classification of natural imageand noise. Traditional PDE based methods (with R chosen as r)lack of suitable priors of the natural image category I0, thus,learning an effective R, i.e. good natural image priors, from naturalimage samples is key to an effective regularization method.

3.1. Learning model establishment

There are many learning methods nowadays, e.g. backwardpropagation (BP) neural network [22], support vector machine(SVM) [26], Bayes inference [27] and etc. However, they are usedonly in discrete cases, i.e. for parameters learning problems. Butour problem is to learn a continuous operator R.

Based on the theories of approximation of functions [18], wetransform the operator-learning problem into a coefficient-learn-ing problem with a group of basis. Thus, the learning problem canbe solved by the existing methods.

3.1.1. Classification model

The key idea of the learning methods is to improve thetraditional PDE models by replacing r with a group of learnedfilters. Thus, in this paper, the operator R is defined as

RðuÞ ¼jðw1,w2, . . . ,wNÞ ¼jðJ1,u,J2,u, . . . ,JN,uÞ: ð10Þ

The efficiency of R means that if uA I0, JRðuÞJC will be small,otherwise, JRðuÞJC will be large. We choose M1 natural images (orimage patches) fuþk1

g as positive samples, and M2 noise images (orimage patches) fu�k2

g as negative samples, then the learning orclassification model can be established as

minJ1 ,...,JN

G1

XM1

k1 ¼ 1

R uþk1

� ���� ���C

0@

1A�m �G2

XM2

k2 ¼ 1

R u�k2

� ���� ���C

0@

1A

8<:

9=;, ð11Þ

where G1 : Rþ-Rþ and G2 : R

þ-Rþ are both nonnegativefunctions. With Eq. (10), the concrete expression of Eq. (11) isgiven by

minJ1 ,...,JN

G1

XM1

k1 ¼ 1

JjðJ1,uþk1,J2,uþk1

, . . . ,JN,uþk1ÞJC

0@

1A

8<:�m � G2

XM2

k2 ¼ 1

JjðJ1,u�k2,J2,u�k2

, . . . ,JN,u�k2ÞJC

0@

1A9=;: ð12Þ

Then learning an effective classifier R coincides with learning agroup filters fJiðxÞg. It is a functional minimization problem.

3.1.2. Parameter learning model

Let JiðxÞAIJ for i¼1,y,N. IJ denote a separable Banach space.

Then JiðxÞ can be linearly represented by a group of denumerable

basis fTjðxÞg (i.e. JiðxÞ ¼P1

j ¼ 0 aðiÞj TjðxÞ), and can be approached by

the linear combination of the first n principle bases, i.e.

JiðxÞ �Xn�1

j ¼ 0

aðiÞj � TjðxÞ: ð13Þ

Due to the linearity of convolution operator, from Eq. (13), we canderive:

JiðxÞ,uðxÞ �Xn�1

j ¼ 0

aðiÞj ðTjðxÞ,uðxÞÞ: ð14Þ

Ref. [19] has discussed some details of the approaching method,

e.g. error bound, convergence rate and etc. Let dðk1Þ

j ¼ Tj,uþk1and

eðk2Þ

j ¼ Tj,u�k2, then, with Eq. (14), the learning model Eq. (12) can

be approached by:

minfaðiÞ

jg

G1

XM1

k1 ¼ 1

jXn�1

j ¼ 0

að1Þj dðk1Þ

j , . . . ,Xn�1

j ¼ 0

aðNÞj dðk1Þ

j

0@

1A

������������C

0@

1A

8<:�m � G2

XM2

k2 ¼ 1

jXn�1

j ¼ 0

að1Þj eðk2Þ

j , . . . ,Xn�1

j ¼ 0

aðNÞj eðk2Þ

j

0@

1A

������������C

0@

1A9=;: ð15Þ

Then learning fJiðxÞg coincides with learning the parameter set

faðiÞj g in Eq. (15). Eq. (15) is the parameter learning model.

3.2. Learning strategy

Let w¼ ðw1,w2, . . . ,wNÞT denote an N dimensional variable,

then s¼jðwÞ. Let A denote a Nn matrix with aðiÞj as its element

in the i-th row and j-th column. Let dðk1Þ ¼ ðdðk1Þ

0 ,dðk1Þ

1 , . . . ,dðk1Þ

n�1ÞT

and eðk2Þ ¼ ðeðk2Þ

0 ,eðk2Þ

1 , . . . ,eðk2Þ

n�1ÞT denote the n dimensional variable

with dðk1Þ

j ¼ Tj,uþk1and eðk2Þ

j ¼ Tj,u�k2, respectively. Let D and E

denote the set with dðk1Þ and eðk2Þ as their element, respectively,

i.e. D¼ fdðk1ÞgM1

k1 ¼ 1 and E¼ feðk2ÞgM2

k2 ¼ 1. Then Eq. (15) can be simply

written as a vector form given by

minA

G1

XM1

k1 ¼ 1

JjðAdðk1ÞÞJC

0@

1A�m � G2

XM2

k2 ¼ 1

JjðAeðk2ÞÞJC

0@

1A

8<:

9=;: ð16Þ

Eq. (16) is a typical classification problem with two classes: small,for the norm or energy of the filtering results of the image sample

set fuþk1g, and large, for the norm or energy of the filtering results

of the noise sample set fu�k2g. The interface is the norm or energy

ball J�JC ¼ C. Learning the parameter set A is equivalent tooptimizing the elements, or geometrically viewed as moving thepoints, in the data set D and E for a good classification result. Thefollowing task is to find an effective way to solve Eq. (16).

3.2.1. Extreme learning machine

Eq. (16) is a single-hidden layer feedforward neural networks(SLFN) model and can be solved by traditional gradient-basediterative learning algorithms, e.g. backward propagation neuralnetwork (BPNN) [22]. However, due to the slow learning speed,gradient-based learning algorithms are impractical and are oftheoretical value only when the data sets D and E are with largescale. Therefore, finding an effective and fast learning algorithm iskey to the learning model. Extreme learning machine (ELM)provides a very fast learning method for classification andregression [1–4]. Furthermore, ELM has the universal approxima-tion capability [23–25]. In this paper, we will use ELM to learn Aeffectively.

Fig. 1. The 88 DCT basis. Each patch denotes a discrete cosine function T j ,

and j¼ ðj1 ,j2Þ denotes the frequency. The horizontal frequency j1 increases along

the right direction, the vertical frequency j2 increases along the downward

direction.

L. Wang et al. / Neurocomputing 74 (2011) 2464–2474 2467

Let jðwÞ ¼ JwJ2 :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2

1þw22þ � � �w

2N

q, Cð�Þ ¼ j�j, and G1ð�Þ ¼

G2ð�Þ ¼ �, then Eq. (16) will be specified as

minfaðiÞ

jg

XM1

k1 ¼ 1

ZO

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn�1

j ¼ 0

að1Þj dðk1Þ

j

0@

1A2

þ � � � þXn�1

j ¼ 0

aðNÞj dðk1Þ

j

0@

1A2

vuuut dx

8><>:

�XM2

k2 ¼ 1

ZO

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn�1

j ¼ 0

að1Þj eðk2Þ

j

0@

1A

2

þ � � � þXn�1

j ¼ 0

aðNÞj eðk2Þ

j

0@

1A

2vuuut dx

9>=>;: ð17Þ

Let B¼ ðb0,b1, . . . ,bn�1ÞT , T1 ¼ n1 � 1, T2 ¼ n2 � 1, and

H1 ¼

gðwð1Þ0 Jdð1Þ0 J, . . . ,wðNÞ0 Jdð1Þ0 JÞ � � � gðwð1Þn�1Jdð1Þn�1J, . . . ,wðNÞn�1Jdð1Þn�1JÞ

gðwð1Þ0 Jdð2Þ0 J, . . . ,wðNÞ0 Jdð2Þ0 JÞ � � � gðwð1Þn�1Jdð2Þn�1J, . . . ,wðNÞn�1Jdð2Þn�1JÞ

^ � � � ^

gðwð1Þ0 JdðM1Þ

0 J, . . . ,wðNÞ0 JdðM1Þ

0 JÞ � � � gðwð1Þn�1JdðM1Þ

n�1 J, . . . ,wðNÞn�1JdðM1Þ

n�1 JÞ

0BBBBB@

1CCCCCA,

H2 ¼

gðwð1Þ0 Jeð1Þ0 J, . . . ,wðNÞ0 Jeð1Þ0 JÞ � � � gðwð1Þn�1Jeð1Þn�1J, . . . ,wðNÞn�1Jeð1Þn�1JÞ

gðwð1Þ0 Jeð2Þ0 J, . . . ,wðNÞ0 Jeð2Þ0 JÞ � � � gðwð1Þn�1Jeð2Þn�1J, . . . ,wðNÞn�1Jeð2Þn�1JÞ

^ � � � ^

gðwð1Þ0 JeðM2Þ

0 J, . . . ,wðNÞ0 JeðM2Þ

0 JÞ � � � gðwð1Þn�1JeðM2Þ

n�1 J, . . . ,wðNÞn�1JeðM2Þ

n�1 JÞ

0BBBBB@

1CCCCCA,

where M1þM2bn, n1 is a small number, n2 is a large number, and1 denotes an n-by-1 vector with all elements as 1, gðxÞ : RN-R

denotes the active function, and fwðiÞj g are with random values.ELM is to solve the equation:

H1B¼ T1 and H2B¼ T2: ð18Þ

Let H ¼ ðHT1 ,HT

2ÞT and T ¼ ðTT

1,TT2Þ

T , then Eq. (18) can be simplywritten as HB¼ T and its solution can be achieved by

~B ¼HyT ¼ ðHT HÞ�1HT T , ð19Þ

where Hy ¼ ðHT HÞ�1HT is known as the Moor–Penrose general-ized inverse of H. And the parameter set A is approximated by

aðiÞj ¼wðiÞj � bj.

3.2.2. Learning algorithm

In this paper, fTjðxÞg is chosen as cosine function (CT) basisgiven by

TjðxÞ ¼ c1c2cosðj1px1Þcosðj2px2Þ, ð20Þ

where xA ½0,1�2, and the coefficients c1 and c2 are given by

c1 ¼1, j1 ¼ 0,ffiffiffi

2p

, j1a0,

(c2 ¼

1, j2 ¼ 0,ffiffiffi2p

, j2a0:

(

For discrete image u, i.e. the matrix form of uðxÞ, the basis fTjðxÞgshould be sampled as the corresponding matrix form given by

ðT jÞi1 ,i2 ¼c1c2ffiffiffiffiffiffiffiffiffi‘1‘2p cos

j1ð2i1þ1Þp2‘1

� �cos

j2ð2i2þ1Þp2‘2

� �, ð21Þ

where ðT jÞi1 ,i2 denotes the i1-th row and i2-th column element ofmatrix T j, and i1 ¼ 0,1, . . . ,‘1�1, i2 ¼ 0,1, . . . ,‘2�1. fT jg is alsoknown as the famous discrete cosine transform (DCT) basis whichis widely used in image processing. Fig. 1 shows the 88, i.e.with ‘1 ¼ ‘2 ¼ 8, DCT basis.

In this paper, we use 13 nature images which are also used asthe training set of Independent Component Analysis (ICA) to

derive 200 image patches as the positive samples fuþk1g. Then we

sample from both Gaussian and Random noise images to form

200 image patches as negative samples fu�k2g. uþk1

and u�k2are with

2020 pixels and gray values normalized in [0,1]. Fig. 2(a) showssome random selected images and sampled image patches.

Basis fT jg are chosen as 88 DCT basis. Data set D and E, i.e.the filtering results of fuþk1

g and fu�k2g, are both consisting of

12 800 image patches. Fig. 2(b) and (c) show some randomlyselected samples of D and E, respectively.

Comparing Fig. 2(b) to Fig. 2(c), one can find that some

filtering results of uþk1and u�k2

are very similar while others are

not. Some filters T j are very sensitive to uþk1and u�k2

while others

are not. The learning algorithms is to select the good bases thatcan distinguish the image patches from noises effectively.

Then based on the ELM method, the filter group fJ ig, i.e. thediscrete form of fJiðxÞgwith 88 pixels, can be learned effectively.And the learning algorithm is summarized in Table 1.

Some skills can also be used to improve the learning result. Forexample, one can use the result of ELM (i.e. the result of step 3 inTable 1) as the initial value of the gradient descent, iterate S times(S is a small number, e.g. S¼5), and use the iterative result togenerate filter group fJig. The result of ELM approximates thetheoretical solution of Eq. (17). Without ELM (i.e. using randommatrix A as the initial value), the gradient descent method dosenot reach convergence after 400 iterations.

3.3. Learning results

Let N¼25 and n¼64. The learning results are shown in Fig. 3.Seen from Fig. 3, each J i has several principle components, and

just several T j are the principle components on the whole viewof all fJig. The principle components are most, but not all,high-frequency terms. The learned filters fJig have noise-likecharacteristic.

There are many existing high-pass filters, e.g. the 33 Prewittfilter:

Px1¼

�1 0 1

�1 0 1

�1 0 1

0B@

1CA, Px2

¼

1 1 1

0 0 0

�1 �1 �1

0B@

1CA,

Fig. 2. Training set and the corresponding filtering results: (a) shows some selected natural images and the sampled 2020 image patches; (b) shows some randomly

selected samples in D; and (c) shows some randomly selected samples in E.

Table 1The learning algorithm.

1. Initialization: Choose a group of orthonormal basis fT jg, and then obtain the

training sets fuþk1g and fu�k2

g sampled from the natural images and noise

images, respectively.

2. Preprocessing: Generate data sets D and E, from training sets fuþk1g and fu�k2

g,

with element dðk1Þ

j ¼ T j,uþk1

and eðk2 Þ

j ¼ T j,u�k2, respectively.

3. ELM algorithm: Generate matrix H and T , then solve B by Eq. (19); and then

generate the parameter set A by aðiÞj ¼wðiÞj � bj.

4. Generate the learned filters fJ ig by Eq. (13).

L. Wang et al. / Neurocomputing 74 (2011) 2464–24742468

which is the discrete form of the gradient operator r, or the4-point neighborhood Laplace filter:

L¼

0 �1 0

�1 4 �1

0 �1 0

0B@

1CA,

which is the minus discrete form of the Laplace operator n.Compared to the above, the learned filter group fJ ig can capturecomplex image features including scale and orientation informa-tion. Therefore, fJ ig can match nature images much better thantraditional filters.

4. Learned filtering PDE model

By the ELM based learning algorithm, a group of filters fJ ig

have been learned from the natural image samples. Then we willutilize fJ ig to build the corresponding image deblurring model.

The image deblurring model established in this paper is anonlinear filtering process with filters learned from the imagesamples. Therefore, we call the model learned filtering PDE(LF-PDE) model.

4.1. Integral equation and PDE model

According to Eq. (10), one can derive the Frechet derivativeRu, i.e.

Ruð�Þ ¼XN

i ¼ 1

@j@wi� ðJi,�Þ

wi ¼ Ji,u

: ð22Þ

Then, from Eq. (22), one can derive the adjoint operator of Ru, i.e.

ðRuÞ�ð�Þ ¼

XN

i ¼ 1

J�i ,@j@wi�

� � wi ¼ Ji,u

, ð23Þ

where J�i ðx1,x2Þ ¼ J�i ð�x1,�x2Þ is called the adjoint filter of Jiðx1,x2Þ,and J�i ðx1,x2Þ denotes the conjugate of Jiðx1,x2Þ. For discrete form,J�i is achieved by 1801 rotation of Ji and replacing each value of Ji

1 3 4

5 8

2

14

6

9

16

12

15

7

10 11

13

1516 14 13

3 24 1

5678

9101112

Fig. 4. Illustration of the relationship between the filter J i and its adjoint filter J�i : (a) shows the process of generating J�i . For convenience, J i is chosen as a 44 matrix

with number 1–16. ‘‘2’’ denotes exchanging for each other and (b) shows the exchanging result J�i , which is exactly the 1801 rotation of J i.

−0.5

0

0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Fig. 3. The learning results: (a) shows the learned DCT coefficients. Each patch, a 88 matrix, denotes the 2D DCT coefficients of the filter J i and (b) shows the

corresponding filter group fJ ig.

L. Wang et al. / Neurocomputing 74 (2011) 2464–2474 2469

by its conjugate. If J i consists of real numbers only, the adjointfilter J�i can be derived by just 1801 rotation of J i. Fig. 4 illustratesthis process.

By Eq. (23) the general VGF PDE (i.e. Eq. (7)) can be written as

@u

@t¼�l

XN

i ¼ 1

J�i ,@C@s

@j@wi

� �þ f ðuÞ, ð24Þ

where f ðuÞ ¼ A�v�A�Au. Eq. (24) is the LF-PDE model. Let

CðsÞ ¼ jsj, and jðwÞ ¼ JwJ2 :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i ¼ 1 w2i

q, then the LF-PDE model

(24) will be specified as

@u

@t¼�l

XN

i ¼ 1

J�i ,Ji,uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i ¼ 1ðJi,uÞ2q

0B@

1CAþ f ðuÞ: ð25Þ

Eq. (25) will be used for the experiments in this paper. LF-PDE is aclass of models, and one can build the special LF-PDE model bychoosing other suitable forms of CðsÞ and jðwÞ.

4.2. Iterative algorithms

The image deblurring algorithm can also be viewed as aneffective numerical calculation method of the LF-PDE modelEq. (24). Approximating the derivative @u=@t by the differenceð1=DtÞðu½mþ1��u½m�Þ, then we can derive the simplest iterativealgorithm as follows:

u½mþ1� ¼ u½m�þZ � f ðu½m�Þ

�Z � lXN

i ¼ 1

J�i ,@C@s

����s ¼ Rðu½m�Þ

@j@wi

����wi ¼ Ji,u½m�

!, ð26Þ

where u½m� ¼ uðx,mDtÞ and u½mþ1� ¼ uðx,ðmþ1ÞDtÞ denote therestored image in the m-th and (mþ1)-th step of iterations,

respectively, and Z¼Dt denotes the step size of the iteration.For discrete image u, the iterative model Eq. (26) still holds, justwith the corresponding discrete form, i.e.

u½mþ1� ¼ u½m�þZ � f ðu½m�Þ

�Z � lXN

i ¼ 1

J�i ,@C@s

����s ¼ Rðu½m�Þ

@j@wi

����wi ¼ Ji,u½m�

!, ð27Þ

where s¼ RðuÞ ¼jðJ1,u,J2,u, . . . ,JN,uÞ, and wi ¼ J i,u. Theiterative algorithm corresponding to Eq. (25) is given by

u½mþ1� ¼ u½m��Z � lXN

i ¼ 1

J�i ,J i,u½m�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i ¼ 1ðJ�i ,u½m�Þ2

q0B@

1CAþZ � f ðu½m�Þ:

ð28Þ

From the view of numerical solution of PDE [28], Eq. (27) and (28)are both explicit iterative forms. Other forms, e.g. implicit form,semi-implicit, can also be used to improve the stability andprecision of the solution, but the amount of calculation increasesrapidly for the inverse operation. In this paper, we will use theexplicit iterative form Eq. (28) for the experiments.

LF-PDE model can be viewed as the optimization of traditionalPDE methods by more appropriate assumption of natural images.It replaces the low-frequency characteristic hypothesis of naturalimages by the characteristic learned from the natural imagesamples.

5. Experimental results

The test images are chosen as binary image patch and naturalimages. Traditional filtering and PDE methods, e.g. Wienerfilter, Tikhonov model, TV model [29], heat diffusion, and filters

L. Wang et al. / Neurocomputing 74 (2011) 2464–24742470

achieved by other learning methods, e.g. FOE [14] and GSM-FOE[15], are used to compare with the LF-PDE model.

Gaussian blur, which is also the basis of the Gaussian scalespace [20,21], is a very important blurring process, and is moredifficult for deblurring than motion blur and out of focus. In thispaper, we will focus on Gaussian blur. The kernel GðxÞ is chosen asthe Gaussian function given by

GðxÞ ¼1

2ps2exp �

JxJ2

2s2

!: ð29Þ

G is chosen as the discrete form of Eq. (29) (with s¼ 4 pixels). Theblock size of G is 2525 pixels, and the sum of all elements in Gis restricted to be 1. e is chosen as the i.i.d. Gaussian noise withvarious standard deviation se.

5.1. Deblurring experiments with image patch

The original binary image patch is chosen as Rect, with 5050pixels and three rectangulars. As the comparison of LF-PDE, we

Fig. 5. Deblurring experiment with binary image patch Rect: (a) is the original image p

and added by i.i.d. white Gaussian noise with standard deviation se ¼ 0, 0.01 and 0.05

filtering method Eq. (30) to (b), (c), and (d), respectively; (f), (j) and (n) are the best, by

respectively; (g), (k) and (o) are the best, by PSNR, deblurring results of the improved it

LF-PDE Eq. (28) to (b), (c), and (d), respectively.

will use the Wiener filter (W.f.) method given by [6]:

u¼ F�13ðF3GÞ� � v

JF3GJ2þl

� �: ð30Þ

Tikhonov filtering (Tik. f.) method given by [6]:

u¼ F�13

ðF3GÞ� � v

JF3GJ2þlJoJ2

� �, ð31Þ

and TV model (i.e. Eq. (9)). In Eq. (30) and (31), F3 and F�13 denote

the discrete Fourier and inverse discrete Fourier transform, l is aconstant, and o¼ ðo1,o2Þ denotes the 2D frequency.

The blurring kernel is chosen as G with s¼ 4 pixels, and e ischosen as i.i.d. Gaussian noise with se ¼ 0, 0.01 and 0.05,respectively. The experimental results are shown in Fig. 5, andthe PSNR values are given in Table 2.

From Fig. 5, one can see that TV model is not suitable for veryblurry images, even without noise. Although the PSNR of LF-PDEis smaller than TV model for Fig. 5(c) and (d), they fit thecharacteristics of natural images much better with the learnedfeatures, i.e. the texture information, which can make the restored

atch Rect; (b), (c) and (d) are the degraded image blurred by the Gaussian kernel G, respectively; (e), (i) and (m) are the best, by PSNR, deblurring results of Wiener

PSNR, deblurring results of Tikhonov filtering method Eq. (31), to (b), (c), and (d),

erative method for TV model [29]; and (h), (l) and (p) are the deblurring results of

L. Wang et al. / Neurocomputing 74 (2011) 2464–2474 2471

image look more natural. The same result can also be seen fromthe experiments with natural images (in Figs. 6 and 7).

In practice, one can cut the blurred image into image patches,restore each patch individually, and then compose the restored imagewith the restored patches. Therefore, the deblurring result of theimage patches determines the restoration quality of the whole image.

5.2. Deblurring experiments with natural images

The experiments are conducted with five natural images. Bothtraditional filtering and learning based non-blind deblurringmethods are used for the comparison. G is chosen as Gaussiankernel with s¼ 4 pixels. e is chosen as i.i.d. Gaussian noise withse ¼ 0:05. Table 3 shows the PSNR values of the results.

Seen from Table 3, learning based methods are better thantraditional filtering based methods on the whole view. TV model

Table 2PSNR value, in dB, of the deblurring results with image patch Rect.

se Wiener filter Tik. filter TV model LF-PDE

0 17.16 17.23 17.12 18.320.01 15.60 15.56 17.43 16.68

0.05 14.99 15.04 16.38 16.09

Fig. 6. Deblurring with natural image Boat: (a) is the original image Boat; (b) shows th

deblurring result of Wiener filter, i.e. Eq. (30); (e) is the result of Eq. (8), i.e. VGF PDE of

(g) is the result of FOE [14] filters; (h) is the results of GSM-FOE [15] filters; and (i) is

[29] achieves the best result for image House because House is withobviously ‘‘piecewise-constant’’ characteristic. LF-PDE achievessatisfactory results. Figs. 6 and 7 show the deblurring results ofBoat (more complex image) and Lena (simpler image), respectively.From Figs. 6 and 7, we can see that the results of LF-PDE also lookmore natural.

5.3. Blind deblurring

LF-PDE focuses on learning an effective regularization method,and restores the image with the right kernel. In practice, the PSF isusually unknown, and some PSF estimation methods, e.g. maximumlikelihood (M.L.), must be used for blind deblurring. See [30,31] fordetails. Fig. 8 shows the deblurring results of LF-PDE when theestimated PSF is chosen as Gaussian kernel with various kernel size.Fig. 9 shows the blind deblurring results of LF-PDE when the PSF isestimated by M.L. and wavelet based iterative method.

PSF plays a very important role in image deblurring. Even G ischosen as Gaussian function, the inaccurate estimation of s willaffect the deblurring result. Poor PSF estimation result can corruptthe restored image (in Fig. 9(b)) even if the learned filters J i isefficient for regularization. Therefore, the application of LF-PDE islimited. To find an effective PSF estimation method is key toextending LF-PDE, which is worth studying in the future.

e PSF (Gaussian kernel with s¼ 4 pixels); (c) is the noisy blurry image; (d) is the

Tikhonov model; (f) is the result of improved iterative method for TV model [29];

the result of the explicit iterative form of LF-PDE, i.e. Eq. (28).

Fig. 7. Deblurring with natural image Lena: (a) is the original image Lena; (b) shows the PSF (Gaussian kernel with s¼ 4 pixels); (c) is the noisy blurry image; (d) is the

deblurring result of Wiener filter, i.e. Eq. (30); (e) is the result of Eq. (8), i.e. VGF PDE of Tikhonov model; (f) is the result of improved iterative method for TV model [29];

(g) is the result of FOE [14] filters; (h) is the results of GSM-FOE [15] filters; and (i) is the result of the explicit iterative form of LF-PDE, i.e. Eq. (28).

Table 3PSNR value, in dB, of the deblurring results with five natural images. Peppers and

House are with 256256 pixels, and other 3 images are with 512512 pixels.

Methods Wiener

filter

PDE

(Eq. 8)

TV model

[29]

FOE

[14]

GSM-FOE

[15]

LF-PDE

Boat 22.05 23.22 22.78 24.63 23.69 25.07Barbara 21.17 22.53 22.37 23.91 23.57 24.86House 22.12 23.67 25.81 25.13 25.24 25.77

Lena 23.76 24.02 23.28 26.29 25.83 26.91Peppers 21.97 22.74 23.87 24.38 23.49 24.36

L. Wang et al. / Neurocomputing 74 (2011) 2464–24742472

Another attempt of blind deblurring tries to learn an inverse filter,denoted by ~G , of G directly. ~G should make the filtering result~u ¼ ~G,v approximate the original sharp image u0. ~G can be learnedby neural network, e.g. BPNN [22], PPLN [11] and etc. This methodachieves good results for blind deblurring (more details in [11,12]),and provides another way to extend LF-PDE for blind deblurring.

6. Conclusions

Filter design is a basic problem for image processing. PDEbased image processing methods are extension of classical linear

and time invariant (LTI) filtering system. Designing a suitable PDEmodel with special image processing properties is of significancein study. Traditional PDE models suppose that the noise is withhigh-frequency terms and natural images are with low-frequencyterms, thus, the high-pass filter r is always used. But limited bythis inaccurate hypothesis, noise removal and edge protection arealways the dilemma for traditional PDE models.

This paper examines this classical problem from a brand newperspective—classification, and improves the traditional PDE modelby using more suitable natural image priors, i.e. better classifiers orfilters learned from natural image samples. Then we build a newand more general image deblurring model: LF-PDE, which com-bines the ideas of both learning and filtering methods. An effectivelearning algorithm based on the ELM method has been designedand a group of filters have been learned from the image samples.

The experiments verify the effectiveness of LF-PDE model andthe corresponding learned filters, and show that LF-PDE modelcan overcome several drawbacks of traditional PDE models, e.g.spots caused by isolated noise points, ‘‘piecewise-constant’’ char-acters and etc. Moreover, one can design other special forms ofLF-PDE, by specifying the forms of CðsÞ and jðwÞ in Eq. (24), withparticular image deblurring properties.

LF-PDE is a non-blind image deblurring model. In practice, PSFis not always known, thus, extending LF-PDE for blind deblurring

Fig. 8. Deblurring results of LF-PDE with various Gaussian kernel size: (a) is the original image Peppers; ((b) shows the PSF (Gaussian kernel with s¼ 4 pixels); (c) is the

noisy blurry image; (d) is the deblurring result with smaller Gaussian kernel (s¼ 2 pixels); (e) is the deblurring result with larger Gaussian kernel (s¼ 6 pixels); and (f) is

the result with the same Gaussian kernel (s¼ 4 pixels) as the PSF.

Fig. 9. Blind deblurring results of LF-PDE with various PSF estimation methods. The original and blurry image are chosen as Fig. 8(a) and (c), respectively: (a) shows the PSF

estimated by M.L. method; (b) is the deblurring result of LF-PDE using (a) as the estimated PSF. PSNR¼18.03 dB; (c) shows the PSF estimated by the wavelet based iterative

method; and (d) is the deblurring result of LF-PDE using (c) as the estimated PSF. PSNR¼22.19 dB.

L. Wang et al. / Neurocomputing 74 (2011) 2464–2474 2473

is key to the improvement of the LF-PDE model. This is worthstudying in the future.

Acknowledgment

The authors thank the support of National Nature ScienceFoundation of China (60975078, 60902058, 60805041, 60872082,60773016), Beijing Natural Science Foundation (4092033), DoctoralFoundations of Ministry of Education of China (200800041049) andDevelopment Grant for Computer Application Technology Subjectjointly Sponsored by Beijing Municipal Commission of Educationand Beijing Jiaotong University (XK100040519).

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Liang Wang was born in 1983. He received B.S. andM.S. degree in Electronic Engineering from the Schoolof Electronic and Information Engineering, BeijingJiaotong University in 2006 and 2008, respectively,majors in Analog & Digital Microelectronic Circuit andEDA software design. He is currently a Ph.D. Candidatein Computer Application Technology at School ofComputer and Information Technology, Beijing Jiao-tong University. His research interests include signaland image processing, partial differential equation,inverse problems, sparse coding, pattern recognitionand machine learning etc.

Yaping Huang was born in 1974. She received her B.S.,M.S. and Ph.D. degree from Beijing Jiaotong Universityin 1995, 1998 and 2004, respectively. Since 2005, shehas been an Associate Professor in the institute ofcomputer and information technology at Beijing Jiao-tong University. Her research interests include compu-ter vision, pattern recognition, and machine learning.

Xiaoyue Luo was born in 1977. She received B.S.degree in Beijing Jiaotong University, Beijing, China,major in Mathematics; M.S. and Ph.D. degree in Michi-gan State University, East Lansing, MI, major in Statis-tics and Applied Mathematics, respectively. Herresearch interests include inverse and ill-posed pro-blems, local regularization method, integral equation,statistical analysis, sparse coding, image processing etc.

Zhe Wang was born in 1983. He received B.S. degree inthe School of Computer Scientific and Technology, JilinUniversity in 2006, majors in Computer Science andTechnology. He is currently a Ph.D. Candidate inComputer Application Technology at Beijing JiaotongUniversity. His research interests include image pro-cessing, sparse coding, independent component analy-sis (ICA), machine learning and etc.

Siwei Luo was born on December 23, 1943. Heobtained his Ph.D. degree in Computer Science formShinshu University, Japan, in 1984. He is currently aProfessor and Doctoral Supervisor of the School ofComputer and Information Technology, Beijing Jiao-tong University. His research interests include neuro-computing, neural networks, pattern recognition, andparallel computing.