An Energy Regularization Method for the Backward Diffusion Problem and its Applications to Image Deblurring

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1 COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol. 4, No. 1, pp Commun. Comput. Phys. July 8 An Energy Regularization Method for the Backward Diffusion Problem and its Applications to Image Deblurring Houde Han, Ming Yan and Chunlin Wu Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 36, China. Received 5 June 7; Accepted in revised version 5 November 7 Available online 7 February 8 Abstract. For the backward diffusion equation, a stable discrete energy regularization algorithm is proposed. Existence and uniqueness of the numerical solution are given. Moreover, the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be estimated. Some numerical experiments illustrate the efficiency of the method, and its application in image deblurring. AMS subject classifications: 65M1, 65M3, 86A, 94A8 Key words: Energy regularization method, inverse problem, heat equation, backward diffusion equation, image deblurring, error estimate, ill-posed, well-posed. 1 Introduction Let Ω be a bounded domain in R n and let Ω be its boundary. Then Σ Ω,T is a bounded domain in R n+1. We are interested in finding the numerical solution of the following backward diffusion problem: n u t x k u x u x l or u ν ux,t gx, x Ω, cxu, in Σ,, on Ω [,T, 1.1 Corresponding author. addresses: hhan@math.tsinghua.edu.cn H. D. Han, yanmingy@mail.ustc. edu.cn M. Yan,wucl@ustc.edu.cn C. L. Wu c 8 Global-Science Press

2 178 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp where cx is a given non-negative smooth function on Ω, gx defines homogeneous boundary conditions on Ω, i.e., gx or g on Ω. 1. ν Moreover, n u ν x u n x k, 1.3 l where {n k } are the components of the unit normal vector on the boundary Ω and { x} is smooth on Ω satisfying, for all x Ω, xa lk x, 1 k, l n, n α k1ζ n k n xζ k ζ l α 1 ζ k, ζζ 1,,ζ n R n, k1 1.4 where < α < α 1 are two constants. The problem 1.1 is reduced to the isotropic heat diffusion problem if we let c δ kl, where c is a positive constant and δ kl is the Kronecker delta defined by { 1, when kl, δ kl, when k l. 1.5 The backward diffusion problem 1.1 is a typical ill-posed problem in the sense of Hadamard [9, 16]. The uniqueness of the given problem 1.1 can be found in [16], but the solution of problem 1.1 does not depend continuously on the given final data gx, and in general for any given function gx with the vanishing boundary condition 1., there is no solution satisfying 1.1. In 1935, Tikhonov [1] obtained the backward diffusion problem by a geophysical interpretation, namely recovering the geothermal prehistory from contemporary data. The problem 1.1 has been considered by many authors since the last century. After adding a priori information about the solution of the problem, such as smoothness or bounds on the solution in a given norm, we can restore stability and construct efficient numerical algorithms. Regularization methods are used by most authors to construct a solution of the ill-posed Cauchy problem for the backward diffusion equation. The main idea of most algorithms is solving a well-posed problem which is perturbed from the ill-posed one, and approximating the solution of the original problem with the solution of the well-posed one. A number of perturbations have been proposed, including the method of quasi-reversibility [3], pseudo-parabolic regularization [4], hyperbolic regularization [15]. Only the differential equation is perturbed in these methods. In [8], Showalter perturbed the initial condition rather than the differential equation, which has a better stability estimate than the previous ones.

3 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Regularization techniques have been well developed for numerically solving the backward diffusion problem [, 5, 6, 13, 14, 17]. The difference of all these approaches lies in the functional selected to be minimized or the perturbation. In [17], we used an energy bounded solution as a regularization, and presented a possible formulation for the backward diffusion equation. The effectiveness is shown by examples in [17], while no order of convergence is proved in [17], and the present work may be considered as a discrete version of it. Similarly to [17], a given energy functional is minimized in order to obtain the regularizing approximation to solution of the original problem. The numerical examples in [17] demonstrate that the approach is very well suited to numerically solve the ill-posed problem. Furthermore, the error between the solution of the original initial boundary problem and the discrete regularizing solution can be estimated. The image deblurring is a important subject in image reconstruction [19 ], and the backward diffusion equation can be applied to image deblurring. As is well known, image blurring is regarded as an image degrading procedure which can be described by convolutions. Therein Gaussian convolution, also known as Gaussian blur, is the most frequent. The Gaussian blur of an image u can be viewed as the solution of the linear heat equation with u as the initial value [18]. In image deblurring manner, one is to find the true image before degrading from the blur one. This is equivalent to solving the backward diffusion equation, particularly the backward heat equation for Gaussian blur. In fact, the backward heat equation has been widely investigated in [7, 1, 1, 19, 1, ] for image deblurring and enhancement etc.. In this paper we consider the application of a general backward diffusion equation based on its energy regularization method. The outline of the paper is as follows. In the next section, we add a priori information on the solution of the original problem, and obtain some stability properties in discrete form on the solution. In Section 3, we propose a stable discrete energy regularization method for the backward diffusion equation, existence and uniqueness of the method are given. The error estimates between the numerical solution and the solution of the original problem are given in this section also. In Section 4, we provide some numerical experiments, which show the efficiency of the given method and its use in image deblurring. Finally, we end this paper with a short concluding section. A finite difference scheme and its stability analysis For the sake of simplicity, in the problem 1.1, we take n, T 1, Ω,1,1, Σ Ω,1. Then problem 1.1 is reduced to the following -D backward diffusion problem: u t x k x u cxu, in Σ, x l u or u ν, on Ω [,1, ux,1 gx, x Ω..1

4 18 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp We will discuss only the vanishing Dirichlet boundary condition in this paper; for the other boundary condition, we can get similar results without any difficulty. Suppose that problem.1 has an unique solution ux,t. The main concern in this paper is to find the numerical approximation of ux,t, the solution of problem.1. We now construct a finite difference scheme for problem.1. Let I, J, and N be three positive integers and let h 1 1/I,h 1/J,1/N be the three mesh sizes. We introduce the following notations: Ω h {x1 i,xj x i 1 ih 1, x j jh, i I, j J}, Σ h, {x1 i,xj,tn x1 i,xj Ω h, t n n, n N}. For any given mesh function w{w n, i I, j J, n N} on Σ h,, we define: D 1 wn wn wn i 1,j 1 h 1, D + 1 wn wn i+1,j wn 1 h 1, D wn wn wn 1 1 h, D + wn wn +1 wn 1 h. Furthermore, we have the following definitions of corresponding discrete functions of gx, x and cx: g gih 1,jh, c cih 1,jh, akl ih 1,jh. Having the above definitions, we can obtain the following finite difference scheme for backward diffusion problem.1: u n+1 u n D u n k D+ +un+1 u l c n +un+1, 1 i I 1, 1 j J 1, n N 1,. u n,j un I,j un i, un i,j, u N g, i I, j J. i I, j J, n N 1, It is easy to see that the finite difference scheme. is unstable, and the solution {u } is not continuously dependent on the final data {g }. Let us define V n un+1 u n :, W n Wn+1 : +W n We can easily get the following finite difference scheme for {V n }: V n+1 V n D+ l V n c V n,..3 1 i I 1, 1 j J 1, n N,.4 V n,j Vn I,j Vn i, Vn i,j, i I, j J, n N 1.

5 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Furthermore, for a given integer m 1 m N 1, take integer m < m min{m, N m}, and we define ϕ n um 1 n, m m n m. Then we get the following finite difference scheme for {ϕ n }: ϕn ϕn 1 D+ l ϕ n 1 c ϕ n 1, 1 i I 1, 1 j J 1, m m+1 n m,.5 ϕ,j n ϕn I,j ϕn i, ϕn i,j, i I, j J, m m n m. Theorem.1. If the energy norm is defined by u m I 1,J 1 i,j then we have the following estimate: Proof. By the definition of ϕ n D+ l u m D+ k um +c u m um h 1 h,.6 u m u m 1 u m+1, 1 m N 1..7 m 1 nm m m 1 nm m m 1 nm m m 1 nm m and Vn, one has ϕ n ϕ n V n+1 V n h 1 h D+ l V n c V n h 1 h V n V n ϕ n+1 D+ l ϕ n c ϕ n h 1 h ϕ n h 1 h,.8 where the summation for is for 1 i I 1, 1 j J 1. Moving the right-hand side terms to the left-hand side, we get m 1 nm m m 1 nm m V n+1 ϕ n V n h 1 h + ϕ n+1 V n+1 ϕ n Vn m 1 nm m V n ϕ n+1 ϕ n h 1 h h 1 h..9

6 18 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp If we let m1, we can get m 1 nm m ϕ m Vm u m 1 ϕ n+1 V n+1 ϕm 1 V m 1 u m Rearranging the right-hand side gives m 1 ϕ n+1 V n+1 nm m u m u m 1 u m+1 ϕ n Vn h 1 h h 1 h D+ l u m c u m u m 1 V m um Vm 1 h 1 h h 1 h D+ l u m 1 c u m 1 h 1 h ϕ n Vn h 1 h D+ l u m 1 c u m 1 u m I 1,J 1 i,j + 1 I 1,J 1 i,j h 1 h D+ l u m 1 c u m 1 h 1 h D+ l u m 1 c u m 1 h 1 h D+ l u m u m c h 1 h D+ l u m 1 D + k um+1 +c u m 1 D+ l u m D+ k um +c u m um u m+1 h 1 h h 1 h..1 Using the definition of the energy norm.6 and the Cauchy-Schwarz inequality gives the desired estimate.7. Following some analysis we can get the following Hölder-type estimate: Theorem.. With the solution u of. and the functional defined in.6, the following estimates holds: u n u N n/n u 1 n/n, n,,n..11

7 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Proof. We prove it in three steps. Step 1. In the case u, from.7 we have u n u n 1 u n+1, 1 n N 1. So we obtain u n for all n, which implies that u n. Consequently, the estimate.11 holds. Step. In the case u, we define Obviously G and we shall prove that G n :ln{ u n / u }, n,,n..1 G n t s+t G n+s+ s s+t G n t.13 for integers s, t and < s N n, < t n. If the estimate.13 is proved, we choose s N n and tn in the inequality.13, to get G n n N G N+ N n N G n N G N..14 The above equality in.14 is from the fact that G. Substituting the definition of G n in.14, we obtain ln un u n [ ] N ln un u ln un u n/n un u u n/n u u n u N n/n u 1 n/n..15 Step 3. The proof of.13. In order to prove.13, we use the method of mathematical induction. We first observe that it is satisfied when s t 1: This follows directly from.7, and therefore, G n G n G n u n u n 1 u n+1,.17 G n ln un u 1 ln un 1 u n+1 u 1 ln un+1 u + 1 ln un 1 u 1 G n+1+ 1 G n 1..18

8 184 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Then let us suppose that.13 holds for all s,t such that s+t m, s>, t>, where m is a positive integer. We have to prove that it holds for all s+1,t and s,t+1. From.13, Therefore, 1 t s+t which gives that G n t s+t G n+s+ s s+t G n t t s+t s s+1 G n+s+1+ 1 s+1 G n 1 G n s+1 G n s+ts+1 s +st+s t s+t t s+t + s s+t G n t..19 s s+1 G n+s+1+ s s+t G n t, s s+1 G n+s+1+ s s+t G n t t s+t+1 G n+s+1+ s+1 s+t+1 G n t.. Similarly we can get the following inequality for the case s,t+1: Based on what is proved above, we finally obtain G n This completes the proof of.11. G n t+1 s+t+1 G n+s+ s s+t+1 G n t 1..1 t s+t G n+s+ s s+t G n t, s, t, < t n, < s N n.. Based on the definition of the energy norm, we have the following lemma. Lemma.1. The solution of. satisfies: Proof. Using the definition of the functional, we get that u N u N 1 n N 1 n u n+1 u n { I 1,J 1 I 1,J 1 i,j i,j u N u..3 D+ l u n+1 D + k un+1 +c u n+1 D+ l u n D+ k un +c u n un u n+1 h 1 h }. h 1 h

9 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Rearranging the right-hand side with some standard tricks gives u N u N 1 n u n { N 1 n u n+1 u n N 1 n { u n+1 D+ l D+ l u n u n+1 V n un Vn This completes the proof of this lemma. u n c u n D+ l u n+1 } h 1 h D+ l u n c u n } c u n h 1 h N 1 h 1h n h 1 h u c n+1 h 1 h V n Vn h 1h..4 Theorem.3. If the solution u of. also satisfies u M, where M is a constant greater than, and we define two functionals u N 1, n1 u n, un u n h 1 h,.5 then we have the following stability estimates for the solution in these two given functionals: u 1, M ε 1, lnm lnε 1.6 u n M ε 1 ε +, lnm lnε 1 n1,,n 1.7 u M ε 1 ε + +M, lnm lnε 1.8 where ε : u N, ε 1 : u N. Proof. Summing up.11, we obtain N n1 u n N n1 u u N n/n u 1 n/n N n1 u N n/n u u N n1 e nlnã/n,

10 186 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp where ã : u N / u <1, lnã<. Using the geometrical meaning of the definite integrals gives N n1 u n u 1 u ã 1 1 u N t e tlnã dt lnã un u ln u N ln u u t dt 1 u N t M1 t dt un M ln u N lnm M ε lnm lnε..9 So the stability estimate.6 is proved. Next, we have to prove the second stability estimate. For k,,n 1, we have u N u k N 1 N 1 nk nk u n+1 u n u n+1 u n N 1 h 1 h nk u n+1 u n Using the definition of the difference scheme in., we get u N u k N 1 nk 1 N 1 nk Furthermore, we obtain u n+1 +u n D+ l u n c u n u n+1 +u n h 1 h u n +u n+1 N 1 nk u k u N N 1 + So that, for k1,,n 1, we can get nk u n + u n+1. u k u N N 1 + u n+1 ε 1 + u 1, n h 1 h. u n + u n+1..3 M ε ε lnm lnε

11 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Moreover, for k, we have u u N N 1 + u n+1 + u ε 1 + u 1, + u n M ε ε 1 + +M..3 lnm lnε This completes the proof of this theorem. If there is a bound on u for the solution, then we can see that the solution {u n } is continuously dependent on the given data {g } for n1,,n. 3 An energy regularization method and the error estimates Based on the stability estimates.6-.8 proved in the last section, we will propose an energy regularization method for the numerical solution of the problem.1 which is an ill-posed problem and no classical numerical method in partial differential equations can be used to get a numerical approximation of it. Instead of considering the backward diffusion problem, let us focus on the following finite difference scheme for the forward diffusion problem: v n+1 v n D+ l v n c v n, 1 i I 1, 1 j J 1, n N 1, 3.1 v,j n vn I,j vn i, vn i,j, i I, j J, 1 n N, v For any given grid function given, i I, j J. v h {v : v,j v I,j v i, v i,j, i I, j J}, the problem 3.1 has an unique solution {v n, i I, j J, n N}. Let vn h denote the grid function {v N, i I, j J}. Then we obtain a mapping B, B : Bv h vn h. 3. It is easy to see that the operator B is bounded and linear, while the inverse problem is unstable. To overcome this difficulty, we will introduce an energy regularization method for it, finding a solution in a small set in which we have some compactness.

12 188 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Suppose that the smooth function u x,t C 4, Ω [,1] is the unique solution of the continuous parabolic equation.1. Then u x,t satisfies: u t x k x u cxu, x,t Σ, x l u, on Ω [,1] 3.3 u t u x,. If u x, is given, the problem 3.3 is a well-posed problem. Let {u h n, i I, j J, n N} be the solution of the finite difference problem 3.1 with the initial condition v u ih 1,jh,. Furthermore, we know that {u h n, i I, j J, n N} is a finite difference approximation of u x,t. Using the standard methods, we can get the following error estimates between the grid function u {u n, i I, j J, n N} here u n u ih 1,jh,n of u x,t and u h : u u h 1, Oh, 3.4 u N u h N g h u h N Oh, 3.5 u n u h n Oh 4, n,,n 1, 3.6 where hmaxh 1,h, and g h is the discrete grid function of g in., g h {g, i I, j J}. From 3.5, we can assume that there exists a constant C 1 > such that g h u h N C 1 h, h>. Then we consider the finite difference scheme of the backward diffusion problem. Since the inverse problem is unstable and we can not find the exact grid function v h from vn h g h, we have to find an approximation of v h in a small set. For any given ε C 1 h < ε 1, we define the set K ε,h : { v h Bv h g h ε}, 3.7 which is a non-empty closed convex subset. The set is non-empty because g h u h N ε, and u h belongs to K ε,h. Now we consider the following energy regularization problem: Find u h K ε,h, such that u h min v h Kε,h v h. 3.8 For fixed ε>c 1 h, problem 3.8 is a well-posed problem, there exists a unique solution u h K ε,h, and the associated grid functions at all time steps {u n h, n N} are obtained. We now claim that u n h is an approximation of the solution u x,t of the continuous problem.1.

13 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp Theorem 3.1. Let C 1 h <ε 1 be any given constant. Suppose that u x,t is the solution of the continuous problem.1, u h is the solution of the finite difference problem 3.1 with the initial condition v u ih 1,jh,, and u h is the solution of the minimization problem 3.8 with {un h } the grid functions at all time steps. Then the solution u h {u n h } is an approximation of u h and satisfies the following error estimates: u u h 1, C h M ε +8 lnm lnε, 3.9 u n u n h C 5 h 4 M ε +16 ln M lnε +C 4ε, n1,,n u u h C 5 h 4 M ε +16 ln M lnε +C 4ε+8M, 3.11 where u h M and C, C 4, C 5 are constants independent of h and ε. Proof. For u h and the associated grid functions at all time steps un h, one has Set Then u h u h M. 3.1 M F : u h u h, ε F : u h N u N h M F 4M, ε F u h N g h u N h g h 4ε. Using the inequality.6 proved in Section, we obtain Using the representation u h u h 1, 4M ε F ln4m lnε F we finally obtain M ε 1 F M s ε 1 s F ds, 3.15 lnm lnε F u h u h 1, 4M ε 1 F 4M s ε 1 s F ds ln4m lnε F 1 4M s 4ε 1 s ds 4M 4ε ln4m ln4ε 4 M ε lnm lnε Combining the estimate 3.16 with the error estimate 3.4 together with the triangle inequality gives u u h 1, u u h 1, + u h u h 1,. 3.17

14 19 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp We obtain the following estimate of the error between the solution and the numerical result: u u h 1, C h M ε +8 lnm lnε, 3.18 where C is a positive constant independent of h and ε. Using the equivalent norm theorem we can find a constant C 3 > such that u h N u N h u h N u N h 1 C 3 u h N u N h From the stability estimates proved in Theorem.3, we have the error estimates u h n u n M ε h C 4 ε+8, n1,,n 1 3. ln M lnε u h u M ε h C 4 ε+8 +4M, 3.1 ln M lnε where C 4 is a positive constant independent of ε. Similarly, combining the estimates with the estimates 3.6 and using the triangle inequality we have the following estimates: u n u n h u n u h n + u h n u n h, 3. u n u n h C 5 h 4 M ε +16 lnm lnε +C 4ε, n1,,n u u h C 5 h 4 M ε +16 lnm lnε +C 4ε+8M. 3.4 This completes the proof of this theorem. In fact, the upper bound of the error can be also given by u u h C h 4 +ε++ M ε, 3.5 ln M lnε from which we can easily see that the value of ǫ has a strong influence on the bound of the error. However, as demonstrated in 3.7, the value of ǫ cannot been arbitrarily small. 4 Numerical examples 4.1 Test examples The examples chosen here are such that the solutions are already known. We consider the following problem: u t α u, x,y,t,π,π,1, u x u x1 u y u y1, 4.1 u t1 sinmxsinmy, x,y,π,π,

15 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp m1, α1/4, N 16, ǫ1 4 m5, α1/1, N4, ǫ1 Figure 1: Numerical result at t. where α> is a constant. The unique solution of problem 4.1 is ux,y,te α1 tm sinmxsinmy. 4. For numerical approximations, we discretize the problem by a uniform mesh of size h π/n in the x-direction and y-direction and a time mesh of size, and obtain a numerical solution u h at t to the problem 4.1. The L norm errors u u h for m1, α1/4 are shown in Table 1. The numerical results demonstrate a convergence rate of ǫ as N is sufficiently large. Fig. 1 shows the numerical results at t m 1, α 1/4, N 16, ǫ 1 4 and m5, α1/1, N4, ǫ1. These results are found to be in good agreement with the exact solution 4.. Table 1: L norm errors for problem 4.1 m1. ǫ\ N In the diffusion equation, the higher the frequency of the signal is, the faster the attenuation. Therefore, if m is large and the diffusion time is long, the given final value will be

16 19 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp much smaller than the solution of the backward diffusion equation. This results in that the solution overflows in current computers with finite wordlength. In summery, the computation becomes more and more difficult as the frequency of the final value grows. 4. Examples in image deblurring In this subsection, we are concerned with images which are degraded by the diffusion problem 1.1. We assume that the original images are of high quality in focused images. After time T, the images are blurred by diffusion 1.1 with the second boundary condition u ν, and we obtained blurred images gx, which are what we observed at time T. The problem is how to get the original images u x from the known blurred images gx, which is equivalent to how to obtain the numerical solution of the backward diffusion problem. We use the energy regularization method derived in the last section to reconstruct the images. In Figs. and 3, we show two examples using the isotropic diffusion equation, which is also known as the heat equation. In each figure, three pictures are given. Therein a is the true image; while b is the blurred version u T x, which is from the original image a by the diffusion equation.1. The right one c is the reconstructed image obtained by solving the backward diffusion equation from the blurred image b. a b c Figure : Deblurring using the backward isotropic diffusion equation, left: original image, middle: blurred image, right: recovered image. In our last example we illustrate the effect of anisotropic blurring and solving the corresponding backward problem. In Fig. 4, a is the true image; while b is the blurred version which is obtained via anisotropic diffusion equation. Fig. 4 c is the recovered image by our method.

17 H. D. Han, M. Yan and C. L. Wu / Commun. Comput. Phys., 4 8, pp a b c Figure 3: Deblurring using the backward isotropic diffusion equation, left: original image, middle: blurred image, right: recovered image. a b c Figure 4: Deblurring using the backward anisotropic diffusion equation, left: original image, middle: blurred image, right: recovered image. 5 Conclusion We proposed a discrete energy regularization method for the backward diffusion equation by finite difference methods. We prove the existence and uniqueness of the solution to this discrete problem. In addition, the error estimates are given. Also, in the example section, we apply this algorithm in image deblurring, and the numerical examples demonstrate the efficiency of this method. Many problems, such as using this method for restoration of noisy blurred image, are still open. Acknowledgments This work is supported by National Natural Science Foundation of China No

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