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Edge Preserving Image Restoration using L1 norm

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Title: Edge Preserving Image Restoration using L1 norm


1
Edge Preserving Image Restoration using L1 norm
  • Vivek Agarwal
  • The University of Tennessee, Knoxville

2
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

3
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
4
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.

5
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.

6
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
7
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?
8
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.

9
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
10
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.

11
Why Regularization?
  • Most of the restoration is based on Least
    Squares. But if the problem is ill-posed then
    least squares method fails.

12
Regularization
  • The general formulation for regularization
    techniques is
  • Where
  • is the Error term
  • is the regularization parameter
  • is the penalty term

13
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.

14
Tikhonov Regularization
Original Image
Blurred Image
15
Tikhonov Regularization - Restoration
16
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

17
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.
18
Computation Challenges
Total Variation
Gradient
Non-Linear PDE
19
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.

20
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
21
Computation of Regularization Operator
  • Discretization of Total variation function
  • Gradient of Total Variation is given by

22
Regularization Operator
  • The regularization operator is computer using the
    expression
  • Where

23
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.

24
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.

25
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
26
L1 norm regularization - Restoration
Synthetic Images
Input Image
Blurred and Noisy Image
27
L1 norm regularization - Restoration
Total Variation Restoration
LASSO Restoration
28
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

29
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
30
Cross Section of Restoration
Different degrees Of Blurring
Total Variation Regularization
LASSO Regularization
31
Cross Section of Restoration
Different levels of Noise
Total Variation Regularization
LASSO Regularization
32
Comparison of Algorithms
Original Image
LASSO Restoration
Tikhonov Restoration
Total Variation Restoration
33
Effect of Different Levels of Noise and Blurring
LASSO Restoration
Blurred and Noisy Image
Total Variation Restoration
Tikhonov Restoration
34
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
35
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
36
Graphical Representation 5 Real Images
Different degrees of Blur
Restoration Time
Residual Error
37
Graphical Representation - 5 Real Images
Different levels of Noise
Restoration Time
Residual Error
38
Effect of Blurring and Noise
39
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

40
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
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