Applications of the Moore‐Penrose Inverse in Digital Image Restoration

Chountasis, Spiros; Katsikis, Vasilios N.; Pappas, Dimitrios

doi:10.1155/2009/170724

Cited by 40 publications

(36 citation statements)

References 9 publications

(2 reference statements)

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“…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]. …”

mentioning

confidence: 86%

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

Chountasis¹,

Katsikis²,

Pappas³

2012

MATLAB - A Fundamental Tool for Scientific Computing and Engineering Applications - Volume 1

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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.

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mentioning

confidence: 86%

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

Chountasis¹,

Katsikis²,

Pappas³

2012

MATLAB - A Fundamental Tool for Scientific Computing and Engineering Applications - Volume 1

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“…the Moore-Penrose solution) of the matrix equation (2.2), and it is investigated in [4,5]. Furthermore, we have observed in [19,20] that the operator E(Y) frequently gives a better improvement of blurred image with respect to restoration contained in Y.…”

Section: Mathematical Modelsmentioning

confidence: 99%

“…This approach was firstly used in [4,5]. A simplified implementation of the partitioning method on this class of Toeplitz matrices is described in [21].…”

Section: Introductionmentioning

confidence: 99%

On removing blur in images using least squares solutions

Stanimirović

Stojanović

Pappas

et al. 2016

Filomat

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Abstract.The further investigation of the image restoration method introduced in [19,20] is presented in this paper. Continuing investigations in that area, two additional applications of the method are investigated. More precisely, we consider the possibility to replace the available matrix in the method by the restoration obtained applying the Tikhonov regularization method or the Truncated Singular Value decomposition method. Additionally, statistical analysis of numerical results generated by applying the proposed improvement of image restoration methods is presented. Previously performed numerical experiments as well as new numerical results and the statistical analysis confirm that the least squares approach can be used as a useful tool for improving restored images obtained by other image restoration methods.

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“…The methods of image restoration, based on the usage of the Moore-Penrose inverse, have been exploited in many recent papers [6][7][8]. Several methods for computing the Moore-Penrose inverse have been introduced in [2].…”

Section: Introductionmentioning

confidence: 99%

“…The unique vectorf satisfying (3) represents a row of the restored image [5][6][7][8], and it is defined bỹ…”

Section: Introductionmentioning

confidence: 99%

Application of the pseudoinverse computation in reconstruction of blurred images

Miljković¹,

Miladinović²,

Stanimirović³

et al. 2012

Filomat

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Abstract. We present a direct method for removing uniform linear motion blur from images. The method is based on a straightforward construction of the Moore-Penrose inverse of the blurring matrix for a given mathematical model. The computational load of the method is decreased significantly with respect to other competitive methods, while the resolution of the restored images remains at a very high level. The method is implemented in the programming package MATLAB and respective numerical examples are presented.

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Applications of the Moore‐Penrose Inverse in Digital Image Restoration

Cited by 40 publications

References 9 publications

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

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

On removing blur in images using least squares solutions

Application of the pseudoinverse computation in reconstruction of blurred images

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