Edge Preserving Image Restoration using L1 norm

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Edge Preserving Image Restoration using L1 norm

• Vivek Agarwal
• The University of Tennessee, Knoxville

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Outline

• Introduction
• Regularization based image restoration
• L2 norm regularization
• L1 norm regularization
• Tikhonov regularization
• Total Variation regularization
• Least Absolute Shrinkage and Selection Operator
  (LASSO)
• Results
• Conclusion and future work

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Introduction -Physics of Image formation
Imaging system
g(x,y)
K(x,y,x,y)
f(x,y)
Registration system
g(x,y)noise
noise
Reverse Process
Forward Process
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Image Restoration

• Image restoration is a subset of image
  processing.
• It is a highly ill-posed problem.
• Most of the image restoration algorithms uses
  least squares.
• L2 norm based algorithms produces smooth
  restoration which is inaccurate if the image
  consists of edges.
• L1 norm algorithms preserves the edge information
  in the restored images. But the algorithms are
  slow.

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Well-Posed Problem

• In 1923, the French mathematician Hadamard
  introduced the
• notion of well-posed problems.
• According to Hadamard a problem is called
  well-posed if
• A solution for the problem exists (existence).
• This solution is unique (uniqueness).
• This unique solution is stable under small
  perturbations in the data, in other words small
  perturbations in the data should cause small
  perturbations in the solution (stability).
• If at least one of these conditions fails the
  problem is called ill or
• incorrectly posed and demands a special
  consideration.

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Existence

• To deal with non-existence we have to enlarge
  the domain where
• the solution is sought.
• Example A quadratic equation ax2 bx c 0 in
  general form has
• two solutions

There is a solution
Real Domain No SolSution
Complex domain
Non-existence is Harmfull
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Uniqueness

• Non-uniqueness is usually caused by the lack or
  absence of
• information about underlying model.
• Example Neural networks. Error surface has
  multiple local minima
• and many of these minima fit training data very
  well, however
• Generalization capabilities of these different
  solution (predictive
• models) can be very different, ranging from poor
  to excellent. How
• to pick up a model which is going to generalize
  well?

Solution 3 Bad or good?
Solution 1 Bad or good?
Solution 2 Bad or good?
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Uniqueness

• Non-uniqueness is not always harmful. It depends
  on what we are looking for. If we are looking for
  a desired effect, that is we know how the good
  solution looks like then we can be happy with
  multiple solutions just picking up a good one
  from a variety of solution.
• The non-uniqueness is harmful if we are looking
  for an observed effect, that is we do not know
  how good solution looks like.
• The best way to combat non-uniqueness is just
  specify a model
• using prior knowledge of the domain or at
  least restrict the space
• where the desired model is searched.

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Instability

• Instability is caused by an attempt to reverse
  cause-effect
• relationships.
• Nature always solves just for forward problem,
  because of the
• arrow of time. Cause always goes before effect.
• In practice very often we have to reverse the
  relationships, that is
• to go from effect to cause.
• Example Convolution-deconvolution, Fredhold
  integral equations
• of the first kind.

Forward Operation
Effect
Cause
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L1 and L2 Norms

• The general expression for norm is given as
• L2 norm is the
  Euclidean distance or vector
• distance.
• L1 norm is also known as
  Manhattan norm because
• it corresponds to the sum of the distances along
  the coordinate
• axes.

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Why Regularization?

• Most of the restoration is based on Least
  Squares. But if the problem is ill-posed then
  least squares method fails.

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Regularization

• The general formulation for regularization
  techniques is
• Where
• is the Error term
• is the regularization parameter
• is the penalty term

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Tikhonov Regularization

• Tikhonov is a L2 norm or classical regularization
  technique.
• Tikhonov regularization technique produces
  smoothing effect on the restored image.
• In zero order Tikhonov regularization, the
  regularization operator (L) is identity matrix.
• The expression that can be used to compute,
  Tikhonov regularization is
• In Higher order Tikhonov, L is either first order
  or second order differentiation matrix.

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Tikhonov Regularization
Original Image
Blurred Image
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Tikhonov Regularization - Restoration
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Total Variation

• Total Variation is a deterministic approach.
• This regularization method preserve the edge
  information in the restored images.
• TV regularization penalty function obeys the L1
  norm.
• The mathematical expression for TV regularization
  is given as

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Difference between Tikhonov regularization and
Total Variation
S.No Tikhonov Regularization Total Variation regularization
1.
2. Assumes smooth and continuous information Smoothness is not assumed.
3. Computationally less complex Computationally more complex
4. Restored image is smooth Restored image is blocky and preserves the edges.
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Computation Challenges
Total Variation
Gradient
Non-Linear PDE
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Computation Challenges (Contd..)

• Iterative method is necessary to solve.
• TV function is non-differential at zero.
• The is non-linear operator.
• The ill conditioning of the operator
  causes numerical difficulties.
• Good Preconditioning is required.

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Computation of Regularization Operator

• Total Variation is computed using the
  formulation.
• The total variation is obtained after
  minimization of the

Total Variation Penalty function (L)
Least Square Solution
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Computation of Regularization Operator

• Discretization of Total variation function
• Gradient of Total Variation is given by

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Regularization Operator

• The regularization operator is computer using the
  expression
• Where

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Lasso Regression

• Lasso for Least Absolute Shrinkage and Selection
  Operator is a shrinkage and selection method for
  linear regression introduced by Tibshirani 1995.
• It minimizes the usual sum of squared errors,
  with a bound on the sum of the absolute values of
  the coefficients.
• The computation of solution for Lasso is a
  quadratic programming problem that can be best
  solved by least angle regression algorithm.
• Lasso also uses L1 penalty norm.

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Ridge Regression and Lasso Equivalence

• The cost function of ridge regression is given as
• Ridge regression is identical to Zero Order
  Tikhonov regularization
• Analytical Solution of Ridge and Tikhonov are
  similar
• The bias introduced favors solution with small
  weights and the effect is to smooth the output
  function.

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Ridge Regression and Lasso Equivalence

• Instead of single value of ?, different values
  of ? can be used for different pixels.
• It should provide same solution as lasso
  regression (regularization).
• Thus we establish relation between lasso and Zero
  Order Tikhonov, there is a relation between Total
  Variation and Lasso

Tikhonov
Our Aim To Prove
Proved
Lasso
Total Variation
Both are L1 Norm penalties
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L1 norm regularization - Restoration
Synthetic Images
Input Image
Blurred and Noisy Image
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L1 norm regularization - Restoration
Total Variation Restoration
LASSO Restoration
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L1 norm regularization - Restoration
I Deg of Blur
III Deg of Blur
II Deg of Blur
Blurred and Noisy Images


Total Variation Regularization
LASSO Regularization

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L1 norm regularization - Restoration
I level of Noise
III level of Noise
II level of Noise
Blurred and Noisy Images


Total Variation Regularization

LASSO Regularization
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Cross Section of Restoration
Different degrees Of Blurring
Total Variation Regularization
LASSO Regularization
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Cross Section of Restoration
Different levels of Noise
Total Variation Regularization
LASSO Regularization
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Comparison of Algorithms
Original Image
LASSO Restoration
Tikhonov Restoration
Total Variation Restoration
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Effect of Different Levels of Noise and Blurring
LASSO Restoration
Blurred and Noisy Image
Total Variation Restoration
Tikhonov Restoration
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Numerical Analysis of Results - Airplane
First Level of Noise
Plane PD Iteration CG Iteration Lambda Blurring Error () Residual Error () Restoration Time (min)
Total Variation 2 10 2.05e-02 81.4 1.74 2.50
LASSO Regression 1 6 1.00e-04 81.4 1.81 0.80
Tikhonov Regularization -- -- 1.288e-10 81.4 9.85 0.20
Second Level of Noise
Plane PD Iteration CG Iteration Lambda Blurring Error () Residual Error () Restoration Time (min)
Total Variation 1 15 1e-03 83.5 3.54 1.4
LASSO Regression 1 2 1e-03 83.5 4.228 0.8
Tikhonov Regularization -- -- 1.12e-10 83.5 11.2 0.30
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Numerical Analysis of Results - Airplane
Shelves PD Iteration CG Iteration Lambda Blurring Error () Residual Error () Restoration Time (min)
Total Variation 2 11 1.00e-04 84.1 2.01 2.00
LASSO Regression 1 8 1.00e-06 84.1 1.23 0.90
Plane PD Iteration CG Iteration Lambda Blurring Error () Residual Error () Restoration Time (min)
Total Variation 2 10 1.00e-03 81.2 3.61 2.10
LASSO Regression 1 14 1.00e-03 81.2 3.59 1.00
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Graphical Representation 5 Real Images
Different degrees of Blur
Restoration Time
Residual Error
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Graphical Representation - 5 Real Images
Different levels of Noise
Restoration Time
Residual Error
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Effect of Blurring and Noise
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Conclusion

• Total variation method preserves the edge
  information in the restored image.
• Restoration time in Total Variation
  regularization is high
• LASSO provides an impressive alternative to TV
  regularization
• Restoration time of LASSO regularization is two
  times less than restoration time of RV
  regularization
• Restoration quality of LASSO is better or equal
  to the restoration quality of TV regularization

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Conclusion

• Both LASSO and TV regularization fails to
  suppress the noise in the restored images.
• Analysis shows increase in degree of blur
  increases the restoration error
• Increase in the noise level does not have a
  significant influence on the restoration time but
  effects the residual error