图像修复技术

Inpainting algorithm for Jacquared Image Based on Phase-Field Model

Zhilin Feng1, Jianwei Yin2, Jianan Zhou3

1. College of Zhijiang, Zhejiang University of Technology, Hangzhou, 310024, China

2. State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou, 310027, China

3. Department of Information and Technology, Zhejiang Vocational College of Commerce,

Hangzhou, 310053, China

zjhzjacky@https://www.360docs.net/doc/453851772.html,, zjuyjw@https://www.360docs.net/doc/453851772.html,, pearl@https://www.360docs.net/doc/453851772.html,

Abstract

Jacquard image inpainting is an interesting new research topic in pattern preprocessing for jacquard CAD. Phase field model has been well acknowledged as an important method for image inpainting. This paper discussed the problem of jacquard image inpainting by approaching the phase field paradigm from a numerical approximation perspective. The basic idea is to represent the damaged pattern of interest in implicit form, and fill-in the damaged parts with a system of geometric partial differential equations derived from phase-field model. The novelty of our approach lies primarily in exploiting explicitly the constraint enforced by the numerical solving of the sequential evolving of phase-field model. Extensive experiments are carried out in order to validate and compare the algorithm both quantitatively and qualitatively. They show the advantages of our algorithm and its readily application to jacquard texture.

1. Introduction

CAD technique has been broadly used in jacquard texture industry. One of the most important aspects of the jacquard CAD system is to simulate the appearance of jacquard texture during output[1]. Automatic inpainting and restoration are closely related to jacquard CAD system[2]. Jacquard image inpainting is to restore a damaged image with missing information, so it is needed to determine which parts of the image the computer needs to retouch and in many cases the missing delineation of objects yields valuable information[3]. Jacquard image inpainting has become an indispensable process to quantitative analysis of images for jacquard CAD system. The process of inpainting is challenging due to poor image contrast and artifacts that result in missing or diffuse pattern boundaries. Thus, this task involves incorporating as much prior information as possible into a single framework. Traditionally, jacquard image inpainting techniques require some form of expert human supervision to provide accurate and consistent identification of pattern structures of interest[4].

A key difficulty associated with digital inpainting is to set up a measure of visual sensitivity towards defects which can be used in computer code. Most inpainting mechanisms use a singular resolution approach on the extrapolation or interpolation of pixels. Oliveira et al. introduced a simple and faster mechanism to filling the damaged area[4]. This algorithm can inpainting an image in just a few seconds, it can be used for interactive construction of tight masks. Bertalmio et.al decomposes the original image into two components, one of which is processed by inpainting and the other by texture synthesis[5]. The output image is the sum of the two processed components. This approach still remains limited to the removal of small image gaps, however, as the diffusion process continues to blur the filled region. Chan and Shen develop inpainting schems from the viewpoint of variational principles and image prior mode [6]. The method explains successfully some aspects of the human disocclusion process in vision psychology. Esedoglu et al. [7] have presented a technique for filling image regions based on a texture-segmentation step and a tensor-voting algorithm for the smooth linking of structures across holes.

In the last decades, many algorithms that deal with image processing using phase-field models have been presented in the literatures [8-11]. The range of applications of phase field models in image processing includes noise removal, image segmentation and shape optimization problems. What is common to all these models is that they are all solved by minimization of an

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energy functional which is designed to appropriately describe the behavior of the associated model. Moreover, phase-field approach can be seen as the deterministic reflection of the Bayesian framework whereas the energy corresponds to the negative log-likelihood of the probability of a certain feature configuration.

Feng et al. [8] introduced the Allen-Cahn equation in phase transition theory to remove noise from jacquard images. For nonlocal Allen-Cahn equation can generate an area-preserving motion by mean curvature flow, it can perfectly preserve shapes of the fabric texture while in the process of denoising. Benes et al. [9] presented an algorithm of image segmentation based on the level set solution of phase field equation. The approach can be understood as a regularization of the level set motion by means of curvature, where a special forcing term is imposed to enforce the initial level set closely surrounding the curves of patterns with different shapes. The phase field method is also a robust and rigorous framework for topology optimization problems.

However, from a numerical point of view, it is not easy to compute a minimizer for the non-convex functional of the phase field. Barrett et al. [10] presented a fully practical finite element approximation of a phase field system with a degenerate mobility and a logarithmic free energy. Provatas et al. [11] developed a large-scale parallel adaptive mesh code for solving phase-field type equations in two and three dimensions. For the minimization of phase field model can be defined as an optimization process, we consider the model as an optimization problem, and apply the iteration theory to solve it.

2. Inpainting model for jacquard image

Let 2 R :be a bounded open set and ()g L f :represent the original image intensity. The function g has discontinuities that represent the contours of objects in the image. L et be a image field, which stands for the state of the image system at the position and the time , and (,)u u x t R x :0t t K be the set of discontinuity points of u . Here, we assume that the image field has two stable states corresponding to and . In this case, the phase-field energy of the image system is often given as follows:

1u 1u

2\1(,)(())(())2K E u K u x F u x dx H H H : 3 (1) where is a double well potential with wells at and (i.e., a non-negative function vanishing only at and 22

()(1)/4F u u 1 1 1 1 ). Here, the value can be interpreted as a phase field (or order parameter),

which is related to the structure of the pixel in such a

way that ()

u x ()1u x

corresponds to one of the two phases and ()1u x corresponds to the other. The set of discontinuity points of u parameterizes the interface between the two phases in the corresponding configuration and is denoted by a closed set K . The first term in represents the interfacial energy, which penalizes high derivatives of the function u and therefore constrains the number of interfaces separating the two phases. The second term represents the double-well potential that tries to force the values of image pixels into one phase or the other. The small

parameter (,)E u K H H

links to the width of transition layer, which controls a ratio of the contribution from the interfacial energy and the double-well potential.

The phase-field functional simultaneously measures the quality of an evloving curve K and of an approximation function u of the image: the minimum of this functional is reached if the approximating function is close to the original image, piecewise smooth and with few discontinuities. The problem is to find a partition of the image which minimizes this deviation. Obviously it is not possible to directly find the global minimum of the energy by examining the entire set of possible segmentations. The principle of our computational method is to generate local transformations of a given segmentation and keep these transformations when the resulting energy is reduced.

Heuristically, we expect solutions to Eq. (1) to be smooth and close to the image g at places x K , and K constitutes edges of the image. To show existence of solutions to Eq. (1), a weak formulation was proposed by De Giorgi et al . [12]by setting u K S (the jumps set of u ) and minimizing only over u SBV ,the space of functions of bounded variation. We recall some definitions and properties concerning functions with bounded variation.

Definition 1. Let . We say that u is a function with bounded variation in :, and we write , if the distributional derivative Du of is a vector-valued measure on : with finite total variation.

12(;)u L R :2(;)u BV R :u De inition 2. L et . We denote by the complement of the L ebesgue set of u , i.e., 1

(;)u L R :2

u S u x S if and only if for

some ()

lim |()|0n

B x u y z dy U U U

o 3

2

z R , where 2

(){:||}

B x y R y x U U

Def inition 3. L et . We say that u is a special function of bounded variation, and we write , if .

()u BV :()u SBV :0c

D u De Giorgi et al .[12] gave the weak formulation of the original problem (1) as follows:

(,)(,)u E u K E u S

(2) . They also proved that minimizers of the weak problem (2) are minimizers of the original problem (1). However, from a numerical point of view, it is not easy to compute a minimizer for Eq. (2), due to the term , and to the fact that this functional is not lower-semicontinuous with respect to . It is natural to try to approximate Eq. (2) by simpler functionals defined on SBV spaces. Ambrosio and Tortorelli [16] showed that Eq. (2) can be approximated by a sequence of elliptic functionals which are numerically more tractable. The approximation takes place in the sense of the convergence.

1()u H S u S -*Let , let be the triangulations and let (0,1)(0,1): u ()T H :H denote the greatest length of the edges in the triangulations. Moreover let be the finite element space of piecewise affine functions on the mesh and let be a sequence of

triangulations with ()V H :()T H :{}j

T H 0j H o .

Modica [13] proved that

Theorem1. L et (){():(){1,1}}BVC BV \\: :: ,and let be a continuous function such that {:, and :[0,W R o f )1} ()0}{1,z R W z 1(||1)()c z W z J d 2(||c z J d 1) for every , with z R 2J t .Then, the discrete functionals

2

\1(())(())2

(,)T T

K u x F u x dx E u T H H H :- ° ?° f

ˉ3 (3) -*converge as 0H o to the functional 0()(E u c :

)

3)u dx for every L ipschitz set :

and every function , where 12

()loc u L R

1

0c 3

, and

(4)

1()())()u H S if u BVC u otherwise - :°

) ?

f °ˉ

When the edges behind the inpainting domain are all determined, the inpainting domain is divided into smooth pieces. These smooth pieces are independent to each other, so we fill in these smooth pieces one by one.

3. Discretization algorithm f or inpainting

model

Let 0(,):[0,][0,]u i j M N R u o , with [0,][0,]

M N u R R u , be a discrete 2D gray level image. From the description of manual inpainting techniques, an iterative algorithm seems a natural choice. The digital inpainting procedure will construct a family of images (,,):[0,][0,]u i j n M N N R u u o such that 0(,,0)(u i j u i

,)j and lim (,,)(,)n u i j n u i j of R , where

is the output of the algorithm (inpainted image). Any general algorithm of that form can be written as

(,)R u i j 1(,)(,)(,),(,)n n n t u i j u i j tu i j i j ' : (5) where the superindex n denotes the inpainting time,

are the pixel coordinates, is the rate of improvement and stands for the update of the

image .

(,)i j t '(,)n t u i j (,)n u i j In order to arrive at the joint minimum of Eq. (3), a scheme for the mesh adaptation is first enforced to refine and reorganize a triangular mesh to characterize the essential contour structure. Then, a conjugate gradient algorithm is applied to find the absolute minimum of the discrete version of the functional at each iteration.

(,)u T Step 1. Initialize iteration index:.

0j m Step 2. Set initial j H and j u .

Step 3. Generate an adapted triangulation j

T H by

the mesh adaptation algorithm, according to j u .Step 3.1 Compute 2L error 21(||)||j

j S u g H K H

()

||||())||

)

j j L e e S u F u H H H w :

|

21122

21/2 1/2

2()j S S T u H H K K §· ¨??1

|,.

Step 3.2 If , then goto Step 4.

()j

j S u H K K Step 3.3 Adjust the triangular mesh by error strategy, and return to Step 3.1.

Step 4. Minimize ()j

j E u H

on the triangulation j

T H by the conjugate gradient minimizing algorithm.

Step 4.1. Initialize step index:, define a initial descent direction , define a subspace 0k m k p {}k k W p where a suitable admissible minimum of ()j

j E u H .

Step 4.2. Compute the gradient and the

Hessian approximation matrix , project ()k E u H j 2()k E u H

()k E u H and j 2()k E u H on .

k W Step 4.3. If ()0k E u H , get a minimizer and goto Step 9.

k u Step 4.4. Compute the incomplete Cholesky

factorization of t HDH j 2()k E u H

, where H is lower

triangular with unit diagonal entries, and D is positive diagonal.

Step 4.5. If j 2()k E u H

is sufficiently positive definite, then compute the descent direction by

solving the linear system k d j 2()k k E u p H

()k E u H with the standard preconditioned conjugate gradient

method; If j 2()k E u H is only positive semi-definite or

almost positive semi-definite, then compute the descent direction by degenerate preconditioned conjugate gradient method.

k d Step 4.6. Compute the optimum step along by minimizing to the interval {}k t k d ()k E u k k u td for .

[0,1]t Step 4.7. Update the current index:.

1k k m Step 4.8. If 1||k k u u J , return to Setp 4.2. Otherwise, return to Step 4.1.

Step 5. Update the current index:1j j m .Step 6. Generate a new j H .

Step 7. If 1||j j H H !P , return to Step 3. Otherwise, goto Step 8. Step 8. Stop.

4. Experimental results

Now, we will present the results of experiments for the proposed model. The results have been obtained using software written in C programming language on the UNIX operating system running on a IPC SUN workstation. The tested image is represented by 256 × 256 matrices of intensity values. Fig.1 shows the results of applying our algorithm on an example of inpainting damaged jacquard images. Numerical results with our proposed algorithm are very encouraging. Fig. 1(a)-(c) are three damaged jacquard images which stained by lines and spots, and Fig. 1(d)-(f) are edges of Fig. 1(a)-(c) respectively. Fig. 2(a)-(c) illustrates the inpainting results of three jacquard images using our algorithm, and Fig. 2(d)-(f) are edges of Fig. 2(a)-(c) respectively.

The following experiments are designed for comparing efficiency of the proposed algorithm with a numerical solving method for phase field model, i.e., Oliveira’s algorithm [4]. In Fig. 3(a)-(c), although the Oliveira’s model restores most of the high frequency texture information, the result is not satisfying. As shown in Fig. 2(a)-(c), both the lines and the spots are restored in an undetectable way by our method, where the method in Fig. 3(a)-(c) fails to keep the boundary of the different structures and cause serious visible

artifacts.

(a)(b)

(c)

(d)(e)(f)

Fig. 1(a)-(f ) Three damaged jacquard images, and

their edges respectively

(a)(b)

(c)

(d)(e)(f)

Fig. 2(a)-(f) inpainting results using our algorithm,

and their edges respectively

(a)(b)

(c)

(d)(e)(f)

Fig. 3(a)-(f

) inpainting results using Oliveira’s algorithm, and their edges respectively

Table 1 shows comparisons of the iteration times(ITs) and the average iteration time(AIT) between the two algorithms. We can see that the two algorithms spend

Table 1. Computational time comparison between the Oliveira’s algorithm and the proposed algorithm

Oliveira’s algorithm Proposed algorithm ITs AIT ITs AIT

Fig.2 (a)

150.403 Fig.3 (a) 150.398 Fig.2 (b)200.547

Fig.3 (b)

200.514

Fig.2 (c) 300.652 Fig.3 (c) 300.602

the same iteration time to accomplish the inpainting process, but the proposed algorithm consumes much less time than Oliveira’s algorithm at each iteration.

5. Conclusions

This paper has presented a novel algorithm based on phase-field model for retouching damaged patterns from jacquard texture images. The phase-field model is formulated as a variational theory of approximation in which the boundary function has a simple explicit form in terms of the approximation function. Our method improves the robustness and effectiveness by imposing some explicit smooth constrains of connection on the formation of discontinuous edges. Experiments show the proposed algorithm provides a better visual effect, especially for jacquard image inpainting.

6. Acknowledgement

The work has been supported by the National High-Tech Research and Development Program of China("863")(No. 2007AA01Z124), the Zhejiang Provincial Natural Science Foundation of China (No. Y106045,Y1080343) and the National Natural Science Foundation, China (No. 60703042).

7. References

[1] Cho C.S., Chung B.M., and Park M.J., “Development of real-time vision-based fabric inspection system”, IEEE Transactions on Industrial Electronics, Vol. 52, No. 4, pp. 1073-1079, 2005.

[2] Ngana H.Y.T., Panga G.K.H., Yungb S.P., Ngb M.K, “Wavelet based methods on patterned fabric defect detection” , Pattern Recognition, Vol. 38, No. 4, pp. 559-576, 2005.

[3] Bertalmio M., Sapiro G., Caselles V., “Image inpainting”, in: Proceeding of 2000 ACM SIGGRAPH, pp. 417-424, New Orleans, Louisianna, 2000.

[4] Oliveira M.M., Bowen B., Mckenna R., Chang Y.S., Fast digital image inpainting, in: Proceedings of the International Conference on Visualization, Imaging and Image Processing, Marbella, Spain, 2001.

[5] Bertalmio M.,Vese L.,Sapiro G., “Simultaneous structure and texture image inpainting”, IEEE Transactions on Image Processing, Vol. 12, No. 8, pp. 882-889, 2003.

[6] Chan T.F, Shen J., “Non-texture inpainting by curvature-driven diffusions”, Journal of Image Representation, Vol. 12, No. 4, pp. 436-449, 2003. [7] Esedoglu S., Shen J., “Digital inpainting based on the Mumford-Shah-Euler image model”, European Journal on Applied Mathematics, Vol. 13, No. 2, pp. 353- 370, 2002.

[8] Feng Z.L., Yin J.W., Chen G., Dong J.X., “Research on jacquard fabrics image denoising using Allen-Cahn level set model”, Journal of Zhejiang University (Engineering Science), Vol. 39, No. 2, pp. 185-189, 2005.

[9] Benes M., Chalupecky V., Mikula K., “Geometrical image segmentation by the Allen–Cahn equation”, Applied Numerical Mathematics, Vol. 51, No. 4, pp. 187-205, 2004.

[10] Barrett J.W., Nurnberg R., Styles V., “Finite Element Approximation of a Phase Field Model for Void Electromigration”, SIAM Journal on Numerical Analysis, Vol. 42 , No. 5, pp. 738-772, 2004.

[11] Provatas N., Goldenfeld N., Dantzig J., “Adaptive Mesh Refinement Computation of Solidification Microstructures using Dynamic Data Structures”, Journal of Computational Physics, Vol. 148, No. 1,pp. 265-290, 1999.

[12] De Giorgi E., Carriero M., et al., “Existence theorem for a minimum problem with free discontinuity set”, Archive for Rational Mechanics and Analysis, Vol. 11, No. 4, pp. 291-322, 1990.

[13] Modica L., “The gradient theory of phase transitions and the minimal interface criterion”, Archive for Rational Mechanics and Analysis, Vol. 98, No. 3, pp. 123-142, 1987.