1 / 26

# Various Regularization Methods in Computer Vision - PowerPoint PPT Presentation

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 .

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Various Regularization Methods in Computer Vision' - aliya

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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 (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.

• 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)

• 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

• 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)

The Result of a Quadratic Regularizer

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.

• 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

• 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.

• 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

• 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)