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. Vision Problems (intro). Such as stereo matching, optical flow estimation, de-noising, segmentation, are typically ill-posed problems .

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### Various Regularization Methods in Computer Vision

Min-Gyu Park

Computer Vision Lab.

School of Information and Communications

GIST

Vision Problems (intro)
• 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 (intro)
• 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.
Vision Problems (intro)
• This seminar is about imposing our prior knowledge to the solution or to the scene.
• There are various kinds of approaches,
• Total variation,
• Piecewise smooth models,
• Stochastic approaches,
• With either L1 or L2 data fidelity terms.
• We will study about the properties of different priors.
Bayesian Inference & Probabilistic Modeling
• 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,
Bayesian Inference & Probabilistic Modeling
• 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)

De-noising Problem
• 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

De-noising Problem
• Formulation.

Quadratic smoothness of a first order derivatives.

First order: flat surface

Second order: quadratic surface

De-noising Problem
• 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!!!

How to minimize the energy function?
• Directly solve the Euler-Lagrange equations.
• Because the solution space is convex!(having a globally unique solution)
The Result of a Quadratic Regularizer

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

Input (noisy) image

The result is not satisfactory….

Why?
• 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.

Pros & Cons
• 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.
Introduction to Total Variation
• If we use L1-norm for the smoothness prior,
• Furthermore, if we assume the variance is 1 then,
Introduction to Total Variation
• 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

Characteristics of Total Variation
• No bias against discontinuities.
• Contrast invariant without explicitly modeling the light condition.
• Robust under impulse noise.
• Objective functions are non-convex.
• Lie between convex and non-convex problems.
How to solve it?
• 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
• To solve original non-convex problems.
Variational Methods
• 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).

Variational Methods
• Unconstrained (Lagrangian) model
• Can be solved by explicit time matching scheme as,
Variational Methods
• 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.

Variational Methods
• Comparison among variational methods in terms of explicit time marching scheme.

L2-L2

TV-L2

TV-L1

Where the degeneracy comes from.

Variational Methods
• In L2-L2 case,

where

Duality-based Approach
• 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,
Duality-based Approach
• Deeper understandings of duality in variational methods will be given in the next seminar.
Applying to Other Problems
• Optical flow (Horn and Schunck – L2-L2)
• Stereo matching (TV-L1)
• Segmentation (TV-L2)