Viktoria TaroudakiAdvisor: Prof. Dianne P. O’Leary

IntroductionAndromeda nebulae, picture from NASA

Hurricane Alex, picture from NOAA

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Blurring

Machine errors in transforming the image into data Background (Light , contrast) Motion of the object, motion of the camera

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Clear Image Blurred Image

Example 1

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Clear Image Blurred Image

Example 2

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Pixel Values [0,255]

0: black255: white

Pixels

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Symbol Size Explanation

mxn Matrix defined through the Point Spread function (PSF) in thecase of a linear problem

X Original Clear Image

nx1 Vector containing the values corresponding to the pixels of theimage X

B The blurred image we measure

mx1 Vector which contains the values of the pixels of the blurredimage B

mx1 Vector of noise

Notation

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K

x

b

e

Model

In the case of linear Point Spread Function

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eKxb +=

Problem

Given , matrix and a distribution for the noise with mean value 0 and variance a matrix ,

we need to compute confidence intervals for the quantities , for where

are given vectors.

2S

kw

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b Ke

xw ⋅= Tk*kφ pk ,...,2,1=

Confidence intervals Intervals into which the values of a parameter fall, with

a certain probability.

Example: confidence interval, One at a time confidence intervals

Simultaneous confidence intervalspk=α uφl k

*kk ,...,2,1,}Pr{ =≤≤

α ,…,p, , k=uφl k*kk ≥≤≤ }21Pr{

1≤α

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α%100

Suppose that the noise is normally distributed, and that is an unbiased estimate of the true solution . Then, given in (0,1), there is a probability that the true value of is contained in the interval where and ,

Where and is such that for the probability , i.e., for confidence

where is the normal

distribution of with mean value 0 and variance 1.

x̂

xwT [ ]l,u{ }22)ˆ(:min κl T =−=

SxxxKxw

{ }22)ˆ(:max κ=−=Sx

xxKxwTu

[ ] [ ])ˆ()ˆ()ˆ( 2T2 xxKSxxKxxKS

−−=− −

∫−

=κ

κ

α xx dn )1,0;( )1,0;(xn

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α α%100

κα α%100

x

With the same assumptions , and nonsingular and symmetric, the probability that is contained in the interval is greater than or equal to where

and

, , , and is the probability density function for the chi-squared distribution with degrees of freedom.

0≥x

{ }0,)ˆ(:min ≥≤−= xxxKxwSx

µTl

{ }0,)ˆ(:max ≥≤−= xxxKxwSx

µTu

∫=2

0

2 )(γ

ρρχα dq2

0 )ˆ(min Sx xxK −=r2

02 γµ += r

2qχ

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S*φ

[ ]l,u α

q=)(rank K

q

The matrix K The matrix K is defined through the Point Spread

function (PSF) and can be determined by using a single point source in a pixel of an image and then moving it to other pixels of the image to obtain more blurred images. Knowing the clear image and the blurred image for all the pixels we can compute the matrix K.

In the following, the matrix K will be a given nxnsquare matrix.

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Sub-matrices We want to reduce the size of the matrix (and so of the

image) to facilitate the computations. Suppose that . For a sub-image rxc we proceed as follows: We have two matrices and where has rc columns of unit vectors and has n-rc columns of unit vectors. Then,

where is the vector corresponding to the sub-image of the original image. Ignoring the second term of the right hand side, we obtain

EE

( ) ( ) ttsst

stsT

TTT xKxK

xx

KKxEE

EEKEKxE +=

=

= ][

sx

sss xKb =

E E

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Kxb =

Implementation Matlab (IPT- Image Processing toolbox)

Case of spatially invariant blurSmaller Problems- same matrixParallel Computing (Open MP or MPI)

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Databases http://sipi.usc.edu/database/USC-SIPI Image DatabaseSignal & Image Processing Institute of the University of Southern California http://www.imageprocessingplace.com/root_files_V3/

image_databases.htmImage Processing Place

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Validation Known data Expected results

Take images Blur images Add noise Use code to compute the confidence intervals for these images Deblur images Count samples in the computed intervals

If close to 95% fall in the 95% confidence intervals, then the code is validated.

If not… check code, correspondence with theory and special circumstances

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TestingGood code: Right results in relatively short time

Depends on Size of image Format of image Various Point Spread Functions

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Schedule-Milestones September: literature, image processing toolbox of Matlab, Project Proposal

October: literature, problem, code

November: code, validation

Early December: midyear report

Late January: testing

February: parallel computing

March: testing, validation

April: validation, final report

May: final presentation

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REFERENCES[1] Tony F. Chan and Jianhong (Jackie) Shen, “Image Processing and Analysis”, SIAM, Philadelphia, 2005[2] Per Christian Hansen, James G. Nagy and Dianne P. O’Leary, “Deblurring Images Matrices, Spectra, and Filtering”, SIAM, Philadelphia, 2006[3] James G. Nagy and Dianne P. O’Leary, “Image Restoration through Subimages and Confidence Images”, Electronic Transactions on Numerical Analysis, 13, 2002, p 22-37 [4] Dianne P. O’Leary and Bert W. Rust, “Confidence Intervals for inequality constrained least squares problems, with applications to ill-posed problems”, SIAM Journal on Scientific and Statistical Computing, 7, 1986, p 473-489[5] Bert W. Rust and Dianne P. O’Leary, “Confidence intervals for discrete approximations to ill-posed problems”, The Journal of Computational and Graphical Statistics, 3, 1994, p 67-96

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Thank you

IMAGE DEBLURRING - COMPUTATION OF CONFIDENCE INTERVALS�Introduction�BlurringExample 1Example 2PixelsNotationModelProblemConfidence intervalsSlide Number 11Slide Number 12The matrix KSub-matricesImplementationDatabasesValidationTestingSchedule-MilestonesREFERENCES�[1] Tony F. Chan and Jianhong (Jackie) Shen, “Image Processing and Analysis”, SIAM, Philadelphia, 2005�[2] Per Christian Hansen, James G. Nagy and Dianne P. O’Leary, “Deblurring Images Matrices, Spectra, and Filtering”, SIAM, Philadelphia, 2006�[3] James G. Nagy and Dianne P. O’Leary, “Image Restoration through Subimages and Confidence Images”, Electronic Transactions on Numerical Analysis, 13, 2002, p 22-37 �[4] Dianne P. O’Leary and Bert W. Rust, “Confidence Intervals for inequality constrained least squares problems, with applications to ill-posed problems”, SIAM Journal on Scientific and Statistical Computing, 7, 1986, p 473-489�[5] Bert W. Rust and Dianne P. O’Leary, “Confidence intervals for discrete approximations to ill-posed problems”, The Journal of Computational and Graphical Statistics, 3, 1994, p 67-96�Slide Number 21