This paper introduces an efficient deblurring image method based on a convolution-based and an iterative concept. Our method does not require specific conditions on images, so it can be widely applied for unspecific generic images. The kernel estimation is firstly performed and then will be used to estimate a latent image in each iteration. The final deblurred image is obtained from the convolution of the blurred image with the final estimated kernel. However, image deblurring is an ill-posed problem due to the nonuniqueness of solutions. Therefore, we propose a smoothing function, unlike previous approaches that applied piecewise functions on estimating a latent image. In our approach, we employ _{2}

An image deblurring is a recovering process that recovers a sharp latent image from a blurred image, which is caused by camera shake or object motion. It has widely attracted attention in image processing and computer vision fields. A number of algorithms have been proposed to address the image deblurring problem. The most common approaches to blur removal are to treat blur as a noisy convolution operated with a blur kernel. The mathematical form of the blurred image is then usually modelled as

A number of approaches have been proposed to address deblurring image problem, and one of popular approaches was based on the deconvolution-based concept. There are two types of approaches: nonblind and blind deconvolutions. The main difference between these two types is that the blur kernel must be known as prior in the case of nonblind deconvolution. The blur kernel estimation is an essentially important step in obtaining a high-quality sharp image. Most approaches use statistical priors on natural images and selection of salient edges for the blur kernel estimation [

In the blind deconvolution, the goal is to estimate both the corresponding latent image and the blur kernel. The blind deconvolution is an ill-posed problem since it has many pairs of solutions. To deal with the blind deblurring problem well-posed, most blind image deblurring methods tend to formulate the problem as a minimization problem. The blur kernel and the latent image are usually solved in an alternating fashion. The algorithm will converge quickly if an appropriately useful initialization is well chosen. However, various assumptions or regularizations are demanded. Pan et al.[_{0}_{0}

Many researchers proposed numerous image-priors underlining optimization techniques to resolve the ill-posed problem in generic images. Pan et al. [

Yan et al. [

All proposed algorithms [

In this paper, we present an efficient optimization algorithm based on the quadratic splitting method. The splitting method guarantees that each subproblem has a closed-form solution and ensures fast convergence. Additionally, we introduce a novel smoothing function for updating pixel values and a sigmoid function for scaling parameters. Our smoothing function based on _{2}

This paper is organized as follows. Our L_{2}-regularization method is described in Section

We first formulate the blind image deblurring to a minimization problem and introduce a new efficient regularization term for convergence to a solution of deblurring. As we mentioned, equation (

Let

Now, the matrix

Therefore,

Note that

Hence,

Our image deblurring method is an iterative approach based on a convolution-based concept. In each iteration, the kernel estimation is performed, and then it is used to estimate a latent image. The estimated latent image is then to be as input in the next kernel estimation. The final deblurred image is obtained from a convolution of the blurred image with the final estimated kernel. A system overview of our proposed method is shown in Figure

System overview of our proposed method.

To estimate the kernel and the latent image, equation (

A solution directly from intensity values is often inaccurate [

We use an efficient alternating minimization method [

The details of the two subproblems are described in the following sections.

We introduce an efficient alternating minimization method to solve equation (

To solve equation (

Based on a least square minimization problem, we perform

To update the variable

Note that equations (

It is worthwhile to note that, in each iteration, the weighted values of

To estimate

The closed form of

The proposed method, similar to the state-of-the-art methods, is used as a coarse-to-fine processing strategy for handling the blur kernel estimation. The algorithm is typically efficiently implemented using an image pyramid [

initialize

r

solve for

r

solve for

solve for

solve for

In the final step, we restore the final deblurred image using nonblind deconvolution, given the final kernel from equation (

We then compute a difference map between these two estimated images with different sigmoid parameters. Finally, we subtract the restored image with a difference map to suppress a ringing artifact.

To evaluate the performance of our proposed method, we tested our approach on the real-image datasets [

First, we tested our approach on the image deblurring dataset [

Quantitative evaluations on Kohler et al.’s dataset [

Samples of restored image by the best three algorithms on [

We then evaluated our approach on the synthesized images with blur kernel. The comparison of results on the first synthesized blurred image is shown in Figure

Comparison of results using a synthesized blurred image. (a) Blurred image. (b) Xu and Jia [

Moreover, we demonstrated the performance of the proposed method on synthesized blurred images corrupted by different realizations of the white Gaussian noises, 5–30

Comparison of results using a synthesized blurred image corrupted by noise 30

The PSNR values for the synthesized images corrupted by various white Gaussian noise levels are shown in Figure

The PSNR values of restore images on the synthesized blurred image, corrupted by white Gaussian noises.

The next step, we evaluated the proposed method on the specific domain images such as face and text images. We compared with the advanced deblurring methods in each domain. We then evaluated our approach on face images against the method of Pan et al. [

Comparison of results using a face image. (a) Blurred image. (b) Xu and Jia [

Comparison of results using a face image. (a) Blurred image. (b) Xu and Jia [

Next, we evaluated the proposed method on selected real and challenging text image for qualitative evaluation as shown in Figures

Comparison of results using a text image [

Comparison of results using a text image [

We evaluated the robustness of our approach to the blurred image with Gaussian noise. The comparison of results on a blurred image with Gaussian noise is shown in Figure

Comparison of results on a blurred image with Gaussian noises [

We propose a novel method for the deblurring image, especially for blind deconvolution. We present _{2} regularization in the deblurring image formed as an optimization problem. This method does not require specific conditions on images, so it can be widely applied for unspecific generic images. It applies smooth functions on estimating a latent image in each iteration. It keeps more details in the restored image than that of existing approaches. To evaluate the performance of our approach, we performed experiments on both synthesized and real-world images. We discovered that our approach outperforms the previous approaches, especially in the image, which contains Gaussian noise. Moreover, our restored images give more details and are more natural than the restored images produced by other methods.

Data are available at

The authors declare that they have no conflicts of interest.

This work was supported by the Pattani Campus Research Fund (SAT63030065) from Prince of Songkla University in Thailand.