#### Title

#### Document Type

Project

#### Publication Date

Fall 2018

#### Keywords

Linear algebra, SVD, Spectral altering, Imaging

#### Abstract

Let the matrix *B* be a blurred version of a sharp image represented by the matrix *X*. Given *B*, we would like to recover *X*.

To accomplish this, we construct linear models of the blurring process that produced *B* from *X*. The idea is that we could then reverse the blurring to reproduce the original image.

For example, if the blurred image satisfies

* B = CXR ^{T}*

for some invertible matrices *C* and *R*, then we could recover *X* as

* X* = *C-*^{1}*B*(*R ^{T}*)

^{-1}.

However, the blurring model usually fails to account for all the blurring that actually occurred. Likely, the blurred image actually satisfies a relation like

* B* = *CXR ^{T}* +

*E*

where *E* is a matrix representing random errors and other blurring effects not accounted for by the model.

If we were to proceed as above, we would produce

* C*^{-1}*B*(*R ^{T}*)

^{-1}=

*X*+

*C*

^{-1}

*E*(

*R*

^{T}^{)-1}.

The term *C*^{-1}*E*(*R ^{T}*)

^{-1}often severely compromises the accuracy of the clear image

*X*. We will explore ways to modify the reconstruction process to produce an image close to

*X*that minimizes contamination by the error term

*E*.

This report is comprised of three parts. In the first, we examine the construction of blurring models, in the second we discuss methods of deblurring images using these models, and in the third we will work with an example photograph to illustrate the deblurring process.

The mathematical techniques we use include the singular value decomposition, matrix norms, certain matrix structures such as Kronecker products, and related theorems. The relevant details of these topics are provided in the appendix.

#### Department

Mathematics

#### Recommended Citation

McMullen, Jamie, "Deblurring Images" (2018). *WWU Honors Program Senior Projects*. 100.

https://cedar.wwu.edu/wwu_honors/100

#### Subjects - Topical (LCSH)

Algebras, Linear; Matrices; Singular value decomposition; Spectral imaging

#### Genre/Form

student projects; term papers

#### Type

Text

#### Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

#### Language

English

#### Format

application/pdf

## Comments

Advisor: Tjalling Ypma