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. Non-existence is Harmfull
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
Least Square Solution
Total Variation Penalty function (L)

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 λ, different values of λ 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
Total Variation
Lasso
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
II Deg of Blur
III Deg of Blur
Blurred and
Noisy Images
Total Variation
Regularization
LASSO
Regularization

29. L1 norm regularization - Restoration
I level of Noise
II level of Noise
III 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
Tikhonov Restoration
Total Variation Restoration

34. Numerical Analysis of Results - Airplane
First Level of Noise
Plane PD
Iteration CG
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
1.81
0.80
Tikhonov
Regularization --
1.288e-10
9.85
0.20
Second Level of Noise
Plane PD
Iteration CG
Lambda
Blurring Error
(%)
Residual Error
Restoration Time
(min)
Total
Variation 1 15
1e-03
83.5
3.54
1.4
LASSO
Regression 2
4.228
0.8
Tikhonov
Regularization --
1.12e-10
11.2
0.30

35. Numerical Analysis of Results - Airplane
Shelves PD
Iteration CG
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
1.23
0.90
Plane PD
Iteration CG
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
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