Image Reconstruction Methods for MATLAB Users - A Moore-Penrose Inverse Approach

Abstract

The Moore-Penrose pseudoinverse is a useful concept in dealing with optimization problems, as the determination of aŞleast squaresŤ solution of linear systems. A typical application of the Moore-Penrose inverse is its use in Image and signal Processing and Image restoration. The presented method in this article is based on the use of the Moore-Penrose generalized inverse of a matrix and provides us a fast computational algorithm for a fast and accurate digital image restoration. This article is an extension of the work presented in [7]; [8]. Theoretical background The Moore-Penrose inverseWe shall denote by R r×m the algebra of all r × m real matrices. For T ∈ R r×m , R(T) will denote the range of T and N(T) the kernel of T. The generalized inverse T † is the unique matrix that satisfies the following four Penrose equations:where T * denotes the transpose matrix of T.Let us consider the equation Tx = b, T ∈ R r×m , b ∈ R r ,whereT is singular. If T is an arbitrary matrix, then there may be none, one or an infinite number of solutions, depending on whether b ∈ R(T) or not, and on the rank of T.B u ti fb / ∈ R(T), then the equation has no solution. Therefore, another point of view of this problem is the following: instead of trying to solve the equation Tx − b = 0, we are looking for a minimal norm vector u that minimizes the norm Tu − b . Note that this vector u is unique. So, in this case we consider the equation Tx = P R(T) b,whereP R(T) is the orthogonal projection on R(T). Since we are interested in the distance between Tx and b, it is natural to make use of T 2 norm.The following two propositions can be found in [12].Proposition 0.1. Let T ∈ R r×m and b ∈ R r , b / ∈ R(T).T h e n ,f o ru∈ R m , the following are equivalent:Let B = {u ∈ R m |T * Tu = T * b}. This set of solutions is closed and convex; it therefore has a unique vector u 0 with minimal norm. In fact, B is an affine manifold; it is of the form u 0 + N (T). In the literature (c.f. [12]), B is known as the set of the least square solutions. We shall make use of this property for the construction of an alternative method in image processing inverse problems.

Application of the Least Squares Solutions in Image Deblurring

“…A short preview of main image restoration methods, used for obtaining possible reconstructions , is presented in [10]. A detailed description of these methods can be found in [13]. A recent survey book presenting all the modern image reconstruction methods is given in [21].…”

“…Removal of blur in images based on the least squares solutions is investigated in [9,10]. Particularly, an application of the least squares solution of minimal norm in image deblurring is investigated in [11][12][13]. Our main intention in this paper is further investigation and extension of the algorithms introduced in [9,10] that allows us to remove a linear or separable motion blur from images.…”

“…The least squares solution, denoted by 1 , includes the Moore-Penrose inverse of the blurring matrix as well as an arbitrary matrix . The particular least squares solution, based on the Moore-Penrose inverse, was investigated in [11][12][13].…”