Deblurring Images via Partial Differential Equations

Transcription

1 Deblurring Images via Parial Dierenial Equaions Sirisha L. Kala Mississippi Sae Universiy Advisor: Seh F. Oppenheimer Absrac: Image deblurring is one o he undamenal problems in he ield o image processing and compuer vision. Deblurring an image and sharpening he eaures wihin he image improves he deail and consequenly he uiliy o an image. In his paper we presen a PDE based model o solve he problem using numerical echniques. We resric ourselves o one dimensional case. The blurred image is obained by convolving he original image wih he Gaussian-like poin spread uncion. Deblurring can be reaed as running he diusion equaion backwards in ime, wih he blurred image as he inpu. We ormulae he problem as a linear PDE and look or soluion operaors o ge back he desired original image and analyze he graphical resuls o evaluae he perormance o he echnique. The backward diusion problem is a very ill-posed problem and direcly solving he inverse equaion can lead o very bad resuls. We will use he mehod o quasireversibiliy o sabilize our numerical approach. Using he soware Malab, we presen animaions o he deblurring process wih varying parameers.

2 1. Inroducion Image deblurring is one o he undamenal problems in he ield o image processing and compuer vision. Deblurring an image and sharpening he eaures wihin he image improves he deail and consequenly he uiliy o an image. In he ield o image processing one oen needs o enhance an image ha was blurred or smoohed by some known operaor. I is possible, in some cases, o inver he blurring operaor, hereby deblurring or reconsrucing he original image. The principal diiculy in carrying ou his approach is he nooriously ill-condiioned marices obained while rying o solve he unsable backward diusion problem. This requires some echnique o smoohing o obain a reasonable soluion. I is such an approach which we presen in his paper. We resric our sudy o 3 dieren discree images on he inie domain [,1].. Problem Formulaion Given an image or rue scene inensiy s, he sysem inroduces some amoun o degradaion. This is oen modeled as a Gaussian blurring operaion. The blurred image or image inensiy is obained by convolving he rue scene inensiy s wih a poin spread uncion h. P h u s u du 1 The poin spread uncion in he orm o a Gaussian is given by h u 1 u ep 4σπ 4σ See reerence [1]. The values o P are usually known a only a inie number o poins. These poins are called piels and he value o P he piel inensiy. Thereore, we have only parial inormaion abou he uncion P. No only ha, his palry amoun o daa is oen pollued wih noise. We can see a connecion beween he Gaussian blurring and he hea equaion here. This is he soluion o he hea equaion which can be ormulaed as ollows, s, 1 4π u ep s u du 4 Now i we se σ, we have, σ P

3 So recovering s is equivalen o solving or, in he problem given below:, P σ However, we will ind ha i we work on a inie domain and se zero Neumann condiions, we will sill ge good resuls., 1,, P σ. 3. Two Approaches 3.1 Numerical Approach Alhough he original problem is posed on he enire real line, as noed above, we will inver he problem on an inerval conaining he image o ineres by imposing zero Neumann boundary condiions. The orward diusion problem on [ ],1 wih zero Neumann boundary condiion is represened by he ollowing se o equaions., 1,, s. 3 We will call he soluion operaor or his se o be T. Tha is, he soluion is, s T. In principle we can recover he original image s by solving he problem 1 P T s σ. An approimaion or σ T is as ollows: K A A s T cos ep cos ep π π π π where he A are he Fourier cosine coeiciens o s, he original.

4 Since we know he values o P a inie number o poins, we will have K P i ep π σ A cos π i Plo 1 is he graph o he oupus rom Malab program run on Image1. As we can see rom he plo, we did ge a beer reconsrucion o our original image. 3. Regularizaion using Quasireversibiliy Using he mehod o quasireversibiliy, we perurb he operaor o obain a well-posed problem which approimaes he original image. We remodeled he problem by changing he inal condiion which, now, depends on a small parameerα., 1, α, +, σ P and he discree problem becomes α I + T σ, P K i α + ep π σ cos π i. P A Plos, 3 and 4 are he oupus rom Malab programs run on images 1, and 3. We see ha using quasireversibiliy we ge a beer approimaion o our images around paricular α andσ values. 3.3 Adding Random Noise Random noise wih zero mean and sandard deviaion 5% is added o he deblurred image and he perormance o boh he echniques discussed above is esed. The resuls, Plos 5 and 6, show ha even in he presence o noise, he quasireversibiliy mehod gave beer resuls. 3.4 Opimal alpha selecion As a inal es, we have wrien an algorihm o ge he opimal α value where we can ge bes reconsrucion o he blurred image, assuming we have parial inormaion abou he original image. The in-buil Malab uncion minsearch is used o ind he opimal α value such he mean-squared error beween hose piels o original and deblurred image is low.

5 4. Plos Plo 1 showing original, blurred, and deblurred oupus or Image1 Plo showing oupus or Image1 wih Quasi reversibiliy

6 Plo 3 showing original, blurred, and deblurred oupus or Image Plo 4 showing original, blurred, and deblurred oupus or Image3

7 Plo 5 showing oupus wih noise added o Blurred Image1 Plo 6 showing oupus wih noise added o Blurred Image1 using Quasireversibiliy

8 Reerences [1] Rober A. Hummel, B. Kimia and Seven W. Zucker, Deblurring Gaussian Blur, Compuer Vision, Graphics, and Image Processing 38, [] Gordon W. Clark and Seh F. Oppenheimer, Quasireversibiliy Mehods or Non- Well-Posed Problems, Elecronic Journal o Dierenial Equaions. [3] Sacey Levine, Yunmei Chen and Jon Sanich, Funcionals wih Nonsandard Growh in Image Processing, [4] Dennis G. Zill and Michael R. Cullen, Dierenial Equaions wih Boundary-Value Problems, Fourh Ediion.