Image Deblurring using Radial Basis functions for Interpolating Weights of Spatially Variant Blur Público

Abstract

Even though technology in applications from microscopy to astronomy has been improving significantly and producing much better pictures, the image deblurring problem is still relevant as a post-processing technique to improve image quality. A substantial amount of work has been done for spatially invariant blurs, but relatively little for the spatially variant case. One approach is to assume the blur is locally approximately spatially invariant, and use interpolation of the local blurring operators to obtain a global approximation of the spatially variant blur. By using different types of functions, piecewise constant, piecewise linear, and radial basis functions (RBF) for the interpolation, we found that piecewise linear functions and RBF outperform piecewise constant functions.  We then conduct further experiments concerning the number of regions the image should be partitioned into and the type of interpolated functions used to represent the variation of the spatially variant blur in the x- and y-directions.

Table of Contents

1 Introduction..........................................................................................1

1.1 Numerating images.............................................................................1

1.2 Ax=b problem ....................................................................................3

2 The Construction of the Blurring Function..............................................4

2.1 The blurring matrix A.........................................................................5

2.2 Boundary conditions..........................................................................7

2.3 Solving the Ax=b problem...................................................................9

2.4 Sparse matrix-vector multiplication...................................................15

2.5 Spatially invariant and variant Gaussian blur......................................17

2.6 The interpolation matrix Di...............................................................18

2.6.1 Piecewise constant interpolation weight..........................................18

2.6.2 Piecewise linear interpolation weight..............................................19

3 Radial Basis Function (RBF)..................................................................22

3.1 A brief summary of RBF.....................................................................22

3.2 Choosing the parameter γ.................................................................23

3.3 Choosing the weight w......................................................................24

3.4 Number of interpolated neighboring regions......................................25

4 Numerical Experiments......................................................................30

4.1 Spatially variant separable Gaussian blur...........................................31

4.1.1 Construction of variant functions...................................................32

4.1.2 Experimenting with images...........................................................34

4.2 Number of partitioned image regions and its neighbor regions............41

4.2.1 Number of partitioned image regions with ∆r = 1.............................41

4.2.2 Number of interpolated neighbor regions........................................46

5 Concluding Remarks...........................................................................49

Bibliography.........................................................................................51

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics Computer Science Mathematics
Palabra Clave	image restoration spatially variant point spread function radial basis functions convolution numerical linear algebra discrete ill-posed problems
Committee Chair / Thesis Advisor	Nagy, James G., Emory University
Committee Members	Jacobson, Jeremy, Emory University Ruthotto, Lars, Emory University

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