International Journal of Computational Mathematics

Volume 2015, Article ID 860263, 17 pages

http://dx.doi.org/10.1155/2015/860263

## A New Study of Blind Deconvolution with Implicit Incorporation of Nonnegativity Constraints

^{1}Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, University of Liverpool, Liverpool L69 L7L, UK^{2}Department of Eye and Vision Science, University of Liverpool, Liverpool L69 3GA, UK

Received 1 July 2014; Revised 29 December 2014; Accepted 12 January 2015

Academic Editor: David Defour

Copyright © 2015 Ke Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The inverse problem of image restoration to remove noise and blur in an observed image was extensively studied in the last two decades. For the case of a known blurring kernel (or a known blurring type such as out of focus or Gaussian blur), many effective models and efficient solvers exist. However when the underlying blur is unknown, there have been fewer developments for modelling the so-called blind deblurring since the early works of You and Kaveh (1996) and Chan and Wong (1998). A major challenge is how to impose the extra constraints to ensure quality of restoration. This paper proposes a new transform based method to impose the positivity constraints automatically and then two numerical solution algorithms. Test results demonstrate the effectiveness and robustness of the proposed method in restoring blurred images.

#### 1. Introduction

Among image preprocessing problems is the reconstruction of an image from a given degraded image, such as images corrupted by noise [1, 2] or blur [3, 4] or images with missing or damaged portions [5]. Such tasks have been widely studied in the last few decades; see [6] for decoupling noise and blur modeling, [7] for imposing box constraints, [8] for a fast iterative solver for noise and blur modeling, and [9, 10] for general surveys. However there are still many outstanding issues to be addressed, especially when both the noise type and the blur type are unknown. Image restoration is closely related to higher level tasks such as segmentation [11] and registration [12, 13]. It should be remarked that models for the latter tasks often become ineffective if the underlying image is blurred.

Adopting the usual notation from the literature, assume an observed image function in domain has been contaminated with additive noise and convolution (blur) operator : where is an unknown Gaussian white noise with zero mean and is the image to be restored. When is known, there exist many effective studies to reconstruct the restoration; see, for example, [14, 15] for fixed point methods, [16] for a Krylov conjugate gradient method, and [17] for a multilevel method. However when is unknown, simultaneous restoration of is of great interest.

A model for blind deblurring by recovering both the kernel and the image* simultaneously* with no a priori information was given by You and Kaveh [18] and later improved by Chan and Wong [3]. In the latter paper, they proposed an energy minimising model, derived partial differential equations by minimising with respect to the image and the kernel , and presented an alternate minimisation scheme for solving the model. The model has extra constraints, including nonnegativity constraints, which were crucial but not exactly implemented. It is fair to say that the model by [3] is not yet reliable for general use (as remarked by [19, 20]) and there are no substantial improvements of it since 1998. There are several works trying to adapt for and extend to specific applications; see, for example, [20, 21] for using multichannel images to restore a single image, [22] for using two blurred images to restore an image, [23] for a nonvariational method, and [24] for implementing [3] by a splitting method. The particular ideas of leaving out the constraints altogether and trusting that the model will give a good result are unfortunately unreliable since they do not lead to good results.

However it is known [19, 20] that the general model [3] can only deal with very special images where the kernel can be accurately estimated. Recognizing the importance of nonnegativity constraints, Miura [25] generalized a one dimensional idea of using square functions from [26] to image deblurring and attempted to solve blind deconvolution in Fourier domain (but without any regularization). Šroubek and Milanfar [20] generalized the model [3] to the case of having multichannel blurred images of the same true image and incorporated nonnegativity into an minimization energy.

In this paper, we focus on image deblurring in the blind case where the blur operator is unknown. In this case, we are aiming to reconstruct the true image and the cause of the degradation with no prior information. Of course, if extra information of the blur operator is available, it should be used to derive the so-called semiblind models [27, 28]. Although there exist other approaches [23, 29–31] for deblurring, we shall focus on the variational framework to model the single channel image deconvolution through satisfying the nonnegativity constraints exactly and implicitly. The end product is a robust image deblurring model; we also present two methods of solving it.

The rest of this paper is organised as follows. Section 2 reviews four related variational models. Two test examples are shown to illustrate and highlight the problems and challenges faced by the first and earlier model by [3]. Section 3 first introduces our transformation approach where both the image and the blurring function are reconstructed with nonnegativity imposed implicitly and then describes the numerical solution of the model. Section 4 presents some experimental results. Section 5 concludes the paper.

#### 2. The Inverse Problem of Deblurring and Some Current Models

Here we review some blind deconvolution models before we introduce our method in the next section. As seen shortly, imposing the constraint of nonnegativity is crucial for such models.

Before proceeding, we remark that for the traditional image restoration applying a projection is the simplest idea of imposing nonnegativity . The same projection idea can be applied to impose a box constraint to ensure . See [7, 32].

There have been several other related ideas for enforcing nonnegativity in image processing [32–35]. One such example was given by [32] where a model was proposed for image reconstruction using nonnegative constraints for astronomical imaging by minimising a regularised Poisson likelihood functional while the idea of backprojection is similarly used in [33, 35]. The case of a Tikhonov regularisation (a much simpler regulariser than what we use here) was considered in [36]. The method of [34] ensured a positive kernel by considering a parametric model and optimizing a scalar which is the standard deviation of the point spread function.

A more sophisticated idea by Biraud [26] is to use the transform , with , in restoring one dimensional signal from the model , where is the* known* blur function and the noise, or after Fourier transform. The central idea here is that any or its Fourier transform can lead to nonnegative restoration . For with some cut-off frequency , so . Noting that leads to , the method of Biraud [26] is
To solve (2), a parametric iterative approach is proposed by for . See [26]. Once a good approximation is obtained, an inverse transform would yield and then a nonnegative restoration .

##### 2.1. The Earlier Blind Deconvolution Work

You and Kaveh [18] proposed a model for simultaneous recovery of both the degradation function and the image in a variational framework, by solving the problem where the fitting term is a common choice for (1) and there is freedom to choose the regularisation terms and ; You and Kaveh used the seminorm for these two terms.

Chan and Wong [3] proposed an improvement to this using the total variation (TV) seminorm for regularisation given by and , hence solving where . Minimising (4) with respect to the image and the kernel , we obtain the coupled partial differential equations given by where (similarly for also) with a small positive parameter introduced to avoid division by zero, which is set to for experimental results for both the image and blur funcion in order to avoid overly smooth reconstructions resulting from too high and the staircasing effect arising from setting too low. It is worth noting that as alternatives to the total variation seminorm we may also consider other regularisation terms such as regularisation, given by , the nonlocal TV [37, 38], the total generalised variation (TGV) [39–41], or the mean curvature [42, 43], as well as others [44, 45].

In order to solve the system, an alternate minimisation scheme was proposed to recover the kernel and the image , including the following constraints which aim to deal with the lack of a unique solution since the system is not jointly convex. This leads to imposing the constraints that the image and kernel should both be positive, and , the kernel should be symmetric (), and the kernel should have a unit integral . These constraints are imposed exactly but only after each alternate minimisation step. The complete algorithm is given in Algorithm 1.