Iterative Methods for Image Deblurring

Iterative Methods for Image Deblurring

Math 561

Fall, 2006

Iterative Methods for Image Deblurring

Outline

Introduction

The Computational Problem

Regularization Review

Iterative Methods

Iterative Methods for Image Deblurring

Introduction

Class of Difficult Problems

b(s) =

∫a(s, t)x(t)dt + e(s)

b = Ax + e

where

I A - known (implicitly) large, ill-conditioned matrix

I b - known, given data

I e - noise, statistical properties may be known

Goal: Compute an approximation of x.

Iterative Methods for Image Deblurring

Introduction

Applications: Image Deblurring

b = Ax + e

I Given blurred image, b,information about the blurring, A,and noise, e.

I Goal: Compute approximation oftrue image, x.

Iterative Methods for Image Deblurring

Introduction

Applications: Image Deblurring

b = Ax + e

I Given blurred image, b,information about the blurring, A,and noise, e.

I Goal: Compute approximation oftrue image, x.

Iterative Methods for Image Deblurring

Introduction

Applications: Super Resolution for Iris Recognition

Basic idea: Given set of low resolution images,

Iterative Methods for Image Deblurring

Introduction

Applications: Super Resolution for Iris Recognition

Basic idea: Given set of low resolution images,

Iterative Methods for Image Deblurring

Introduction

Applications: Super Resolution for Iris Recognition

Basic idea: Given set of low resolution images, combine to get ahigh resolution image

Iterative Methods for Image Deblurring

Introduction

Applications: Super Resolution for Iris Recognition

Basic idea: Given set of low resolution images, combine to get ahigh resolution image without distortions.

Iterative Methods for Image Deblurring

Introduction

Applications: Super Resolution for Medical Imaging

Similar to iris recognition, but

I Registration (alignment) of images more difficult.

I Solving b = Ax + e is a subproblem of a nonlinearoptimization method.

Iterative Methods for Image Deblurring

The Computational Problem

The Computational Problem

From the matrix-vector equation

b = Ax + e

I Given b and A, compute an approximation of x

I Regarding the noise, e:I It is usually not known.I However, some statistical information may be known.I It is usually small, but it cannot be ignored!

That is, solving the linear algebra problem:

Ax = b ⇒ x = A−1b

usually does not work.

Iterative Methods for Image Deblurring

The Computational Problem

The Computational Problem

I Given A and b = Ax + e

I Goal: Compute approximationof true image, x

I Naıve inverse solution

x = A−1b

= A−1 (Ax + e)

= x + A−1e

= x + error

is corrupted with noise!

Iterative Methods for Image Deblurring

The Computational Problem

The Computational Problem

I Given A and b = Ax + e

I Goal: Compute approximationof true image, x

I Naıve inverse solution

x = A−1b

= A−1 (Ax + e)

= x + A−1e

= x + error

is corrupted with noise!

Iterative Methods for Image Deblurring

The Computational Problem

The Computational Problem

I Given A and b = Ax + e

I Goal: Compute approximationof true image, x

I Naıve inverse solution

x = A−1b

= A−1 (Ax + e)

= x + A−1e

= x + error

is corrupted with noise!

Iterative Methods for Image Deblurring

Regularization Review

Regularization

Basic Idea: Modify the inversion process to compute:

xreg = A−1regb

so that

x = A−1regb

= A−1reg (Ax + e)

= A−1regAx + A−1

rege

where

A−1regAx ≈ x and A−1

rege is not too large

Iterative Methods for Image Deblurring

Regularization Review

Regularization

Examples of well known regularization methods:

I Tikhonov

I Truncated Singular Value Decomposition (TSVD)

I Total Variation

Difficulty: Computing A−1regb is often very computationally

expensive when A is large.

(In our problems, A can easily be 106 × 106)

Iterative Methods for Image Deblurring

Iterative Methods

For large matrices use iterative methods

x0 = initial estimate of xfor k = 1, 2, 3, . . .

• xk = computations involving xk−1, A, b,a preconditioner matrix, M, andother intermediate quantities

• determine if stopping criteria are satisfiedend

I xk converges to A−1bI Expensive computations at each iteration:

I Multiplying A times a vector.I Applying the preconditioner.

These computations are usually cheap for sparse andstructured matrices.

Iterative Methods for Image Deblurring

Iterative Methods

Properties of iterative methods for our problem:

I Early iterations:I xk begins to reconstruct xI error term is small

I But, eventually iteration converges to x = A−1b = x + error

I Thus, as k gets large, xk is dominated by error

I Goal is to stop iteration when:I xk is a good approximation of xI error is not too large

Iterative Methods for Image Deblurring

Iterative Methods

Properties of iterative methods for our problem:

I Early iterations:I xk begins to reconstruct xI error term is small

I But, eventually iteration converges to x = A−1b = x + error

I Thus, as k gets large, xk is dominated by error

I Goal is to stop iteration when:I xk is a good approximation of xI error is not too large

Iterative Methods for Image Deblurring

Iterative Methods

Properties of iterative methods for our problem:

I Early iterations:I xk begins to reconstruct xI error term is small

I But, eventually iteration converges to x = A−1b = x + error

I Thus, as k gets large, xk is dominated by error

I Goal is to stop iteration when:I xk is a good approximation of xI error is not too large

Iterative Methods for Image Deblurring

Iterative Methods

Properties of iterative methods for our problem:

I Early iterations:I xk begins to reconstruct xI error term is small

I But, eventually iteration converges to x = A−1b = x + error

I Thus, as k gets large, xk is dominated by error

I Goal is to stop iteration when:I xk is a good approximation of xI error is not too large

Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

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1inverse solution

Iterative Methods for Image Deblurring

Iterative Methods

Example of iterative method LSQR

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Iterative Methods for Image Deblurring

Iterative Methods

Difficulty with iterative methods

I Difficult to find an appropriate stopping iteration, kstop.

I The error plot in the previous example gives kstop = kopt.

I However, can only plot errors if true solution is known!

I Other methods can be used to estimate kstop.

I But they often choose kstop > kopt.

Thus, computed solution contains too much error!

Iterative Methods for Image Deblurring

Iterative Methods

Hybrid Method

Basic Idea:

I Use iterative method for b = Ax + e

I At each iteration:I Project very large problem to a very small problem.I Use sophisticated regularization methods to solve very small

problem.I Project solution of very small problem back to very large

problem.

I Advantage: error term does not grow!

I Thus, it is okay to have kstop > kopt.

Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

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Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

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Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

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Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

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Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

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Iterative Methods for Image Deblurring

Iterative Methods

Example of hybrid method

0 20 40 60 80 1000.3

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Iterative Methods for Image Deblurring

Iterative Methods

Hybrid Methods

In any case, the hybrid methods still have the form:

x0 = initial estimate of xfor k = 1, 2, 3, . . .

• xk = computations involving xk−1, A, b,a preconditioner matrix, M, andother intermediate quantities

• determine if stopping criteria are satisfiedend

I xk converges to A−1regb

I Expensive computations at each iteration:I Multiplying A times a vector.I Applying the preconditioner.