Abstract
In this paper, a new model for image restoration under Poisson noise based on a high-order total bounded variation is proposed. Existence and uniqueness of its solution are proved. To find the global optimal solution of our strongly convex model, a split Bregman algorithm is introduced. Furthermore, a rigorous convergence theory of the proposed algorithm is established. Experimental results are provided to demonstrate the effectiveness and efficiency of the proposed method over the classic total bounded variation-based model.
1. Introduction
Being signal-dependent, Poisson noise is inevitable in various applications such as positron emission tomography [1], electronic microscopy [2], and astronomical imaging [3]. In particular, the degree of Poisson noise depends on the peak value of pixel intensity values, in the sense that the smaller peak value yields the higher intensity of Poisson noise. Moreover, the Poisson noise magnitude in an image increases with the pixel intensity of the region of this image. Since Poisson noise is different from the Gaussian noise, the restoration models and related algorithms proposed for Gaussian white noise are not effective in removing Poisson noise.
A traditional method for Poisson denoising was proposed by Richardson and Lucy (RL) [4, 5]. However, the RL algorithm converges slowly with noise remained in restored images. A standard approach to remove the noise involves incorporating a regularization term, such as Tikhonov regularization [6], total variation (TV) regularization [7], and high-order regularization [8]. In particular, the TV-based Poisson denoising model [9] can be described aswhere is a bounded open domain in and is a regularization parameter. For this model (1), a number of fast and efficient numerical methods have been proposed, for instance, the dual algorithm [10], the augmented Lagrangian method [11, 12], the split Bregman algorithm [13], the multilevel algorithm [14], and a semi-implicit gradient descent method [15]. These algorithms have also been successfully applied to remove noisy images corrupted by Gaussian, impulse, and multiplicative noises [16–21]. In [11, 13], the following TV-based Poisson restoration model has also been studied:For solving this model, in [11], the authors proposed an augmented Lagrangian method called PIDAL. However, it needs to solve a TV denoising subproblem and thus needs an inner iteration loop. Furthermore, the nonnegative property of the sequence cannot be guaranteed. To deal with these problems, Setzer et al. [13] proposed a split Bregman algorithm called PIDSplit+. Subsequently, Wen et al. [22] proposed primal-dual algorithms for model (2). Their algorithms require less memory and no need to solve any linear systems, and thus are fast. However, they did not state related theories for this Poisson restoration model in their papers.
Replacing the TV regularization term in (2) by [23], Liu and Huang [24] proposed the total bounded variation-based Poisson restoration model as follows:where are parameters, and is a bounded linear operator representing the convolution kernel. Owing to the strongly convex property of , uniqueness of the solution of this model can be guaranteed. Further, the convergence of the proposed algorithm was proved in [24].
The TV-based restoration model performs very well for preserving edges while removing noise. However, it often causes staircase effects in flat regions. To overcome the staircase effects, some high-order models [25–28] have been introduced in image restoration. For restoring blurred images corrupted by Poisson noise, the hybrid regularizations combining TV with high-order variation were also considered [29, 30]. In this paper, we propose the following model by replacing the TV term in (3) with second order term :where is a Banach space. Its definition and related properties were introduced in [17]. For this model, we prove existence and uniqueness of its solution. We use a split Bregman algorithm to compute the resulting minimization problem by introducing new auxiliary variables. Besides, we prove the convergence of our proposed algorithm. The experimental results show that our model (4) avoids undesirable staircase effects. Compared with the split Bregman algorithm for solving the total bounded variation-based Poisson restoration model (3), our method is faster.
The organization of this paper is as follows. In Section 2, we show existence and uniqueness of the solution of our proposed model (4). Section 3 presents a split Bregman algorithm for solving this model. We also study the convergence of the proposed algorithm in this section. In Section 4, experiments are carried out to show the efficiency of our proposed algorithm. We end the paper with some conclusions in Section 5.
2. Existence and Uniqueness
In this section, we prove the existence and uniqueness of the solution for proposed model (4).
Theorem 1. Model (4) has a minimizer in . Furthermore, if is injective, the minimizer is unique.
Proof. Let with and . For the given , . Thus is strictly convex. Then it holds . So we can choose a minimizing sequence for . By using Hölder’s inequality and Jensen’s inequality, we haveand thus is bounded. Meanwhile, considering that and the boundedness property of , we know that is a bounded sequence in . By the compactness theory, there is such that a subsequence converges to a.e. By the lower semicontinuity of the norm, and . Further, by using Fatou’s lemma, we getThus is the minimizer of model (4).
Obviously, if is injective, the high-order total variation term and the fidelity term in (4) are convex. Since is a strongly convex function for a given constant , is strongly convex. Therefore, its minimizer is unique.
3. The Split Bregman Algorithm
We describe the split Bregman algorithm (SBA) for solving the proposed model (4), together with convergence analysis. The SBA is an efficient method, which was initially introduced by Goldstein and Osher to solve the general -regularized problems [31]. It has been successfully applied to minimization problems that involve nondifferentiable and high-order functionals [19, 24, 32].
3.1. Algorithm Description
It is straightforward that the proposed model (4) is equivalent to the following constrained optimization problem:
To solve this problem, we convert it into an unconstrained problem by applying the penalty method here:where are two penalty parameters. Consequently, the SBA for solving (4) can be described as follows:
For the -subproblem, the corresponding Euler-Lagrange equation isWe assume periodic boundary condition, and hence (10) can be solved efficiently by the FFT, i.e.,where and represent the adjoint operators of and , respectively, and denotes the Fourier transform of .
There are closed-form solutions for both and subproblems. In particular, the solution of -subproblem is given by the shrinkage operator, and the solution of -subproblem is obtained by using the quadratic formula, i.e.,
We summarize the overall algorithm in Algorithm 2.
Algorithm 2 (SBA for solving (4)).
Step 1. Input . Initialization: , and . Let .
Step 2. Compute by (11).
Step 3. Compute by (12).
Step 4. Compute by (13).
Step 5. Update by the last two equations of (9).
Step 6. Stop if the stopping criterion is satisfied. Otherwise, let and go to Step 2.
3.2. Convergence Analysis
Since our model (4) is strictly convex, there exists a unique solution as guaranteed by Theorem 1. Below we give a rigorous proof that the proposed algorithm (Algorithm 2) converges to the unique solution. Note that the following analysis is stemmed from [24, 33], where the analysis of the unconstrained split Bregman iteration was presented in detail.
Theorem 3. Assume that K is injective and . Then the sequence generated by Algorithm 2 converges to the unique solution of model (4).
Proof. Let be the sequence generated by Algorithm 2. It follows from the first-order optimality conditions at thatwhere . On the other hand, the Karush-Kuhn-Tucker (KKT) conditions for (4) read aswhere with and with . We further denote so as to get the following set of conditions:By comparing (14) and (16), we can conclude that is a fixed point of (9). We then introduce new variables to represent the errors, i.e., and By subtracting each equation in (16) from the corresponding one in (14), we getThen for (17), taking the inner product of the first three equations with respect to , and , respectively, and squaring both sides of the last two equations, we obtain Obviously, the last two equations of (19) can be rewritten as By combining (20) with the first three equations in (19), we haveSumming (21) bilaterally from to yieldsTherefore, it is straightforward to haveSince is convex, , and , we haveAnd note that is a bounded sequence in . Therefore, all terms in (23) are nonnegative. Letting , inequality (23) suggests the following conclusions:which implies thatFor any convex function , the Bregman distance satisfiesThis together with the second equality of (26) and the nonnegativity of the Bregman distance implies thatRecall that . And by the fourth equality of (26), we obtain In the same way, we also have Combining this with (29) yields Finally, by this and equality (15), we get
4. Numerical Experiments
We present experimental results to show the performance of our proposed method (4), referred to as BLLTSB, in comparison with the total bounded variation-based Poisson restoration model (3), referred to as BTVSB. All the experiments were conducted in MATLAB environment on a PC with a 3.4GHz CPU processor and 16G RAM. To assess the restoration performance qualitatively, we use the peak signal to noise ratio (PSNR) defined bywhere and are the pixel values of the original and restored images, respectively. We also use the structural similarity (SSIM) [34] for quantitative comparison. The stopping criterion in all the experiments isor the iteration reaches 500.
We choose Lena (), Train (), and Barbara () as the test images, as shown in Figure 1, and two types of blurring kernels, Gaussian blur and motion blur (we use the MATLAB commands fspecial (‘gaussian’, , ) and fspecial (‘motion’, 10,45) to generate Gaussian blur and motion blur, respectively). The blurry and noisy images are simulated as follows. The original image is first convolved with the blur kernel , followed by additive Poisson noise (the Poisson noise is generated by using the MATLAB function poissrnd). Since Poisson noise is data-dependent, the noise level of the observed images depends on the pixel intensity. To test different noise levels, we use poissrnd, where is the blurred image and is set to be or 5.
The choice of parameters significantly affects the convergence rate and other characteristics of the methods. In our experiments, we study the influence of parameters on the BTVSB. Concretely speaking, after taking a reasonable guess for the parameters , we consider different values of . We observe which value of could contribute to the biggest value of PSNR, then we set it to be the value of . And in turn, we use the same strategy for the choice of other parameters. Similar strategy is adopted in our BLLTSB for the parameters. After the stopping criterion is satisfied, the algorithms are terminated for the parameter tuning.
Experiment 1. In this example, we use the Lena image as the test image. We first consider to restore the image corrupted by the Gaussian blur and Poisson noise with . For the BTVSB, we set . For our BLLTSB, we use . Then the Poisson noise with is added to the blurred image. For the BTVSB and the BLLTSB, we choose and , respectively. The numerical results are arranged in Table 1, while the responding visual results are shown in Figure 2.
Experiment 2. In our second example, the Train image is adopted as the test image. Similar to the above example, the original image is degraded by the same Gaussian blur and Poisson noise with . The parameters are set to be for the BTVSB and for the BLLTSB. Then we consider more Poisson noise with . The parameters for the BTVSB are . And for our proposed BLLTSB, are chosen. Numerical results are also listed in Table 1. The visual results are presented in Figure 3.
Since the proposed model (4) has higher order than the previous model (3), the BLLTSB spends more time per iteration than the BTVSB. But our BLLTSB needs less iterations for the same stopping criterion. Thus, it is faster as a whole, as shown in Table 1. From the plots of PSNR and relative error versus CPU time in Figures 2 and 3, we can also conclude that our BLLTSB is faster than the BTVSB. In terms of PSNR and SSIM, the restoration results by these two methods are similar. But from Figures 2 and 3, we can see that the restored images by the BTVSB suffer from staircase effects. Moreover, the higher the noise level, the more obvious the staircase effects. It is easier to observe from the face of Lena and the head of the train.
To further show the advantages of our BLLTSB, we consider to restore an image degraded by the motion blur in the following experiment.
Experiment 3. In our third example, we test the Barbara image. Different from the above two examples, the original image is corrupted by the motion blur and then Poisson noise with . Here, parameters for the BTVSB and our BLLTSB are set to be and , respectively. Results of this experiment are given in Table 1 and Figure 4.
As shown in Figures 4(a4) and 4(a5), our BLLTSB is faster than BTVSB. It can also be seen from Table 1. Besides, we can easily see from Figures 4(a2) and 4(a3) that our model successfully overcomes the staircase effects and produces relative natural result.
5. Conclusion
In this paper, we have proposed a new model based on high-order total bounded variation for restoring blurred images under Poisson noise. We proved existence and uniqueness of its solution. Furthermore, we proposed a split Bregman algorithm to solve this model and proved its convergence. Compared with the previous method for total bounded variation-based Poissonian image restoration model, experimental results show that our model can overcome the staircase effects and the proposed algorithm is fast.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We would like to thank Dr. Zhi-Feng Pang of Henan University for his useful suggestion. This work was supported by the Science and Technology Project of Jiangxi Provincial Department of Education (GJJ171015), the China Scholarship Council (201708360066), the NNSF of China Grants 61865012 and 11701260, and the Natural Science Foundation of Jiangxi Province (20151BAB207010).