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

Various Regularization Methods in Computer Vision

Min-Gyu Park

Computer Vision Lab.

School of Information and Communications

GIST


Vision problems intro

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 intro1

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 intro2

Vision Problems (intro)

  • 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.


Bayesian inference probabilistic modeling

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 modeling1

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

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 problem1

De-noising Problem

  • Formulation.

Quadratic smoothness of a first order derivatives.

First order: flat surface

Second order: quadratic surface


De noising problem2

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

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

The Result of a Quadratic Regularizer

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

Input (noisy) image

The result is not satisfactory….


Various regularization methods in computer vision

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

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

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 variation1

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

Characteristics of Total Variation

  • 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.


How to solve it

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

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 methods1

Variational Methods

  • Unconstrained (Lagrangian) model

  • Can be solved by explicit time matching scheme as,


Variational methods2

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 methods3

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 methods4

Variational Methods

  • In L2-L2 case,

where


Duality based approach

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 approach1

Duality-based Approach

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


Applying to other problems

Applying to Other Problems

  • Optical flow (Horn and Schunck – L2-L2)

  • Stereo matching (TV-L1)

  • Segmentation (TV-L2)


Q a discussion

Q&A / Discussion


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