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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
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Iterative Methods for Image Deblurring
Iterative Methods
Example of iterative method LSQR
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Iterative Methods for Image Deblurring
Iterative Methods
Example of iterative method LSQR
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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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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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1
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
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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
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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.
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