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Various Regularization Methods in Computer Vision

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Various Regularization Methods in Computer Vision

Min-Gyu Park

Computer Vision Lab.

School of Information and Communications

GIST

- Such as stereo matching, optical flow estimation, de-noising, segmentation, are typically ill-posed problems.
- Because these are inverse problems.

- Properties of well-posed problems.
- Existence: a solution exists.
- Uniqueness: the solution is unique.
- Stability: the solution continuously depends on the input data.

- Vision problems are difficult to compute the solution directly.
- Then, how to find a meaningful solution to such a hard problem?

- Impose the prior knowledge to the solution.
- Which means we constrict the space of possible solutions to physically meaningful ones.

- This seminar is about imposing our prior knowledge to the solution or to the scene.
- There are various kinds of approaches,
- Quadratic regularization,
- Total variation,
- Piecewise smooth models,
- Stochastic approaches,
- With either L1 or L2 data fidelity terms.

- We will study about the properties of different priors.

- We will see the simple de-noising problem.
- f is a noisy input image, u is the noise-free (de-noised) image, and n is Gaussian noise.

- Our objective is finding the posterior distribution,
- Where the posterior distribution can be directly estimated or can be estimated as,

- Probabilistic modeling
- Depending on how we model p(u), the solution will be significantly different.

Prior term

Likelihood term (data fidelity term)

Evidence(does not depend on u)

- Critical issue.
- How to smooth the input image while preserving some important features such as image edge.

Input (noisy) image

De-noised image via L1 regularization term

- Formulation.

Quadratic smoothness of a first order derivatives.

First order: flat surface

Second order: quadratic surface

- By combining both likelihood and prior terms,
- Thus, maximization of p(f|u)p(u) is equivalent to minimize the free energy of Gibbs distribution.

Is the exactly Gibbs function!!!

- Directly solve the Euler-Lagrange equations.
- Because the solution space is convex!(having a globally unique solution)

Noise are removed (smoothed), but edges are also blurred.

Input (noisy) image

The result is not satisfactory….

- Due to bias against discontinuities.

intensity

5

4

3

2

1

0

Discontinuity are penalized more!!!

1 2 3 4 5 6

whereas L1 norm(total variation)treats both as same.

- If there is no discontinuity in the result such as depth map, surface, and noise-free image, quadratic regularizer will be a good solution.
- L2 regulaizer is biased against discontinuities.
- Easy to solve! Descent gradient will find the solution.
- Quadratic problems has a unique global solution.
- Meaning it is a well-posed problem.
- But, we cannot guarantee the solution is truly correct.

- Quadratic problems has a unique global solution.

- If we use L1-norm for the smoothness prior,
- Furthermore, if we assume the variance is 1 then,

- Then, the free energy is defined as total variation of a function u.

Definition of total variation:

u(x)

s.t. the summation should be a finite value (TV(f) < ). Those functions have bounded variation(BV).

0

x

- Advantages:
- No bias against discontinuities.
- Contrast invariant without explicitly modeling the light condition.
- Robust under impulse noise.

- Disadvantages:
- Objective functions are non-convex.
- Lie between convex and non-convex problems.

- Objective functions are non-convex.

- With L1, L2 data terms, wecan use
- Variational methods
- Explicit Time Marching
- Linearization of Euler-Lagrangian
- Nonlinear Primal-dual method
- Nonlinear multi-grid method

- Graph cuts
- Convex optimization (first order scheme)
- Second order cone programming

- Variational methods
- To solve original non-convex problems.

- Definition.
- Informally speaking, they are based on solving Euler-Lagrange equations.

- Problem Definition (constrained problem).

The first total variation based approach in computer vision, named after Rudin, Osher and Fatemi, shortly as ROF model (1992).

- Unconstrained (Lagrangian) model
- Can be solved by explicit time matching scheme as,

- What happens if we change the data fidelity term to L1 norm as,
- More difficult to solve (non-convex), but robust against outliers such as occlusion.

This formulation is called as TV-L1 framework.

- Comparison among variational methods in terms of explicit time marching scheme.

L2-L2

TV-L2

TV-L1

Where the degeneracy comes from.

- In L2-L2 case,

where

- Why do we use duality instead of the primal problem?
- The function becomes continuously differentiable.
- Not always, but in case of total variation.

- For example, we use below property to introduce a dual variable p,

- Deeper understandings of duality in variational methods will be given in the next seminar.

- Optical flow (Horn and Schunck – L2-L2)
- Stereo matching (TV-L1)
- Segmentation (TV-L2)