Image Denoising Algorithms

Transcription

1 Image Denoising Algorithms Xiang Hao School of Compting, University of Utah, USA, Abstract. This is a report of an assignment of the class Mathematics of Imaging. In this assignment, we first implement different image denoising algorithms. Namely, H1 reglarization, Total Variation (TV) Primal form, TV Dal form, and TV Primal-Dal form, and then we test and compare these algorithms on different images. Keywords: Image Denoising, H1 Reglarization, Total Variation 1 Introdction Image restoration is an image processing step, in which both the inpt and otpt are images. In image restoration, one tries to improve the qality of an image. For example, remove the noise from an image, make a blrred image sharper, or fill some missing portion of an image. It s a important process since it sally improves the performance of other image processing step, sch as image segmentation and image registration. Image denoising is a research field belonging to image restoration, which tries to remove the noise from the image. There are different approaches to accomplish this job, one of them is the variational approach, which tries to minimize a fnctional. In this report, we will discss different algorithms of the variational approach. The first one is called H1 reglarization algorithm. This algorithm want to minimize the L 2 norm of the gradient of the image. It has some good properties, for example, it has niqe soltion, and it is easy to implement, bt it does not accept contor discontinities, casing the obtained soltion to be smooth. Rdin, Osher, and Fatemi (ROF) introdced total variation based methods in order to preserve sharp edges. While the ROF model can preserve edges, it also has some nmerical drawbacks. A lot of researchers proposed many different soltions, and we will discss some of them, sch as, total variation dal form, and total variation primal-dal form.

2 2 Xiang Hao 2 Methods The image denoising problem can be formlated as the following. Given an observed image f, we know f is the addition of the ideal image and some noise with mean 0 and variance σ 2. Or goal is to compte the ideal image, which shold satisfy the constraint, f 2 2 Ω σ 2, The problem is nderdetermined and obviosly an ill posted problem, since given f and σ, there exists infinite satisfying the above constraint. The sal method to solve a nderdetermined system is to add more constraints to the system. Usally, the constraint can be represented as minimizing a fnction g(). For example, if we prefer the with minimm l p norm among all possible. It can be represented as min g() = min Ω p. There might be several with the minimm norm, depending what the constraint is, and which norm is sed. In addition, different constraints mean different preference, and therefore adding different constraints will probably obtain different. Now, the problem becomes min g() s.t. f 2 2 Ω σ 2, (1) This is an constrained problem. Using Lagrange mltiplier, it s eqivalent to the following nconstrained minimization problem with appropriate λ, which is a variable. min g() + λ( f 2 2 Ω σ 2 ) (2) λ is determined by σ. In other words, there is a one to one corresponding between λ and σ. If we know the standard deviation σ, we will know λ. We will be able to solve both the constraint problem (1) and the nconstrained problem (2). Bt the problem is that σ is sally nknown, so we can not solve neither the constraint problem (1) nor the nconstraint problem (2). However, observing that the above minimization problem is eqivalent to the following problem, if λ is known. min g() + λ f 2 2 (3) We can solve this problem by the following two steps. First, we pick a λ, and then we will solve the minimization problem (3). The vale we pick for λ is important, so sally we need to test different λ and find the best one. Now, we will consider the constraint fnction g(). As mentioned before, different constraints have different meaning. One type of the sitable choice of the constraint is abot the gradient of the image. For example,in H1 reglarization, g() = Ω x 2 dx; in Total Variation Primal form, g() = Ω x dx. We will list the eqations we want to solve for each of different constraints and discss the performance of each algorithm in the next section.

3 Variational Image Denoising Algorithms 3 In H1 reglarization, the fnctional we want to minimize is min x Ω 2 dx + λ f 2 2, We arrive at the Eler-Lagrange eqations + λ( f) = 0, v = 0 on Ω onω In Total Variation Primal form, the fnctional we want to minimize is min x dx + λ f Ω 2 2, We arrive at the Eler-Lagrange eqations div( ) + 2λ( f) = 0 onω, v = 0 on Ω The Dal form of the Total Variation [2]. We have the following eqivalent forms x = max x ω = max div ω, ω C0 1(Ω), ω 1 ω 1 Ω Ω Therefore, the fnctional we want to minimize is min div ω + λ f 2 2. (4) max ω C 1 0 (Ω), ω 1 Using min-max theorem, we can solve this problem by first minimizing Ω 1 min ω C0 1(Ω), ω 1 2 div ω + 2λf 2 2, and then, compte as = f + 1 2λ div ω. The Total Variation Primal-Dal [1] is jst a different nmerical algorithm to solve the dal form (4). Ω

4 4 Xiang Hao 3 Reslts and Discssion In this section, we will compare and discss the reslts of the different algorithms. 3.1 H1 reglarization Fig. 1. First row: Original and noisy image; Second row: two H1 restorations with different λ. In H1 reglarization, g() = Ω x 2 dx, and we want to minimize the L 2 norm of the gradient of the image, which is very large in an image with edges. So the H1 reglarization algorithm will blr the edges as shown in Figre 1. From Figre 1, we can also see that different λ will reslt in different restoration. This is easy to nderstand, since different λ allow different level of variance of the noise. 3.2 TV Primal form There is an nstable operation in the TV Primal from, which is div( ). When is zero, it s nstable. One soltion is to add a small constant β to the denominator. In Figre 2, we can see that withot β there are some black block in the restored image cased by the nstable operation. By introdcing β, the operation is stable, bt it also introdces some blrring to the image.

5 Variational Image Denoising Algorithms 5 Fig. 2. First row: Original and noisy image; Second row: two TV Primal restorations with (right) and withot (left) β. 3.3 TV Dal Form and TV Primal-Dal Form Since TV Dal Form and TV Primal-Dal Form are different algorithms to solve the same eqation, we will compare them together here. From Figre 3, it s hard to tell which algorithm is better. They are roghly the same. The Primal-Dal restorations may be a little bit better than the Dal restorations. Fig. 3. First colmn: Original and noisy image; Second colmn: two TV Dal restorations with different τ. Third colmn: two TV Primal-Dal restorations with different τ

6 6 Xiang Hao 3.4 Test all algorithms on Floroscopy images In this section, we test all algorithms on two Floroscopy images, one has medim noise, the other has severe noise. Fig. 4. Floroscopy image with medim noise. First row: Original and noisy image; Second row: H1 restoration (left) and TV primal restoration (right). Third row: TV Dal restoration (left) and TV Primal-Dal restoration (right) Fig. 5. Floroscopy image with severe noise. First row: Original and noisy image; Second row: H1 restoration (left) and TV primal restoration (right). Third row: TV Dal restoration (left) and TV Primal-Dal restoration (right)

7 Variational Image Denoising Algorithms SNR and Comptational complexity In Figre 6, we test the residal between the noise free image and the restored image in each iteration. For TV primal and TV primal-dal algorithms, the residal is decreasing, and TV primal is decreasing faster than TV primal-dal for this image. For H1 reglarization, however, the residal is first decreasing bt after some time, it starts to increase. I think this is cased by the blrring. We know that H1 reglarization will blr the image. At the beginning, H1 reglarization blrs the image, bt it still improve the qality of the image somehow, bt after some iteration, it blrs the image too mch and the image qality will decrease. Fig. 6. The X axis is the CPU time (seconds). The Y axis is the L2 norm of the different between the noisy image and noise free image Fig. 7. The X axis is the CPU time (seconds). The Y axis is the SNR

8 8 Xiang Hao In Figre 6, we compte the SNR of restored image in each iteration. For TV primal and TV primal-dal algorithms, the SNR is increasing, however, for H1 reglarization, the SNR is first increasing and then decreasing, which is also cased by the H1 reglarization blrring. References 1. M. Zh and T. Chan. An efficient primal-dal hybrid gradient algorithm for total variation image restoration. UCLA CAM Report, pages 08 34, Mingqiang Zh, Stephen J. Wright, and Tony F. Chan. Dality-based algorithms for total-variation-reglarized image restoration. Compt. Optim. Appl., 47: , November 2010.