Download presentation

Presentation is loading. Please wait.

Published byKendall Scholes Modified over 3 years ago

1
**Introduction to Variational Methods and Applications**

Chunming Li Institute of Imaging Science Vanderbilt University URL:

2
**Outline Brief introduction to calculus of variations Applications:**

4/13/2017 Outline Brief introduction to calculus of variations Applications: Total variation model for image denoising Region-based level set methods Multiphase level set methods

3
**A Variational Method for Image Denoising**

4/13/2017 A Variational Method for Image Denoising Original image Denoised image by TV

4
**Total Variation Model (Rudin-Osher-Fatemi)**

4/13/2017 Total Variation Model (Rudin-Osher-Fatemi) Minimize the energy functional: where I is an image. Original image I Denoised image by TV Gaussian Convolution

5
**Introduction to Calculus of Variations**

4/13/2017 Introduction to Calculus of Variations

6
**What is Functional and its Derivative?**

4/13/2017 What is Functional and its Derivative? A functional is a mapping where the domain is a space of infinite dimension Usually, the space is a set of functions with certain properties (e.g. continuity, smoothness). Can we find the minimizer of a functional F(u) by solving F’(u)=0? What is the “derivative” of a functional F(u) ?

7
4/13/2017 Hilbert Spaces A real Hilbert Space X is endowed with the following operations: Vector addition: Scalar multiplication: Inner product , with properties: Norm Basic facts of a Hilbert Space X X is complete Cauchy-Schwarz inequality where the equality holds if and only if

8
4/13/2017 Space The space is a linear space. Inner product: Norm:

9
**Linear Functional on Hilbert Space**

4/13/2017 Linear Functional on Hilbert Space A linear functional on Hilbert space X is a mapping with property: for any A functional is bounded if there is a constant c such that for all The space of all bounded linear functionals on X is called the dual space of X, denoted by X’. Linear functionals deduced from inner product: For a given vector , the functional is a bounded linear functional. Theorem: Let be a Hilbert space. Then, for any bounded linear functional , there exists a vector such that for all

10
**Directional Derivative of Functional**

4/13/2017 Directional Derivative of Functional Let be a functional on Hilbert space X, we call the directional derivative of F at x in the direction v if the limit exists. Furthermore, if is a bounded linear functional of v, we say F is Gateaux differentiable. Since is a linear functional on Hilbert space, there exists a vector such that ,then is called the Gateaux derivative of , and we write If is a minimizer of the functional , then for all , i.e (Euler-Lagrange Equation)

11
**Example Consider the functional F(u) on space defined by:**

4/13/2017 Example Consider the functional F(u) on space defined by: Rewrite F(u) with inner product For any v, compute: It can be shown that Solve Minimizer

12
**where the equality holds if and only if**

4/13/2017 A short cut Rewrite as: where the equality holds if and only if Minimizer

13
**An Important Class of Functionals**

4/13/2017 An Important Class of Functionals Consider energy functionals in the form: where is a function with variables: Gateaux derivative:

14
4/13/2017 Proof Denote by the space of functions that are infinitely continuous differentiable, with compact support. The subspace is dense in the space Compute for any Lemma: for any (integration by part)

15
4/13/2017 Let

16
4/13/2017

17
4/13/2017 Steepest Descent The directional derivative of F at in the direction of is given by What is the direction in which the functional F has steepest descent? Answer: The directional derivative is negative, and the absolute value is maximized. The direction of steepest descent

18
**Gradient Flow Gradient flow (steepest descent flow) is:**

4/13/2017 Gradient Flow Gradient flow (steepest descent flow) is: Gradient flow describes the motion of u in the space X toward a local minimum of F. For energy functional: the gradient flow is:

19
**Example: Total Variation Model**

4/13/2017 Example: Total Variation Model Consider total variation model: The procedure of finding the Gateaux derivative and gradient flow: 1. Define the Lagrangian in 2. Compute the partial derivatives of 3. Compute the Gateaux derivative

20
**Example: Total Variation Model**

4/13/2017 Example: Total Variation Model with Gateaux derivative 4. Gradient Flow

21
4/13/2017 Region Based Methods

22
**Mumford-Shah Functional**

4/13/2017 Mumford-Shah Functional Regularization term Data fidelity term Smoothing term

23
**Active Contours without Edges (Chan & Vese 2001)**

24
**Active Contours without Edges**

25
Results

26
**Multiphase Level Set Formulation (Vese & Chan, 2002)**

27
**Piece Wise Constant Model**

28
**Piece Wise Constant Model**

29
**Drawback of Piece Wise Constant Model**

Chan-Vese LBF Click to see the movie See:

30
Piece Smooth Model

31
Piece Smooth Model

32
Rerults

33
Thank you

Similar presentations

Presentation is loading. Please wait....

OK

CHAPTER 9.10~9.17 Vector Calculus.

CHAPTER 9.10~9.17 Vector Calculus.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on traction rolling stock trains Ppt on english chapters of class 10 Ppt on do's and don'ts of group discussion clip Ppt on non conventional sources of energy in india Ppt on the art of war author Ppt on phonetic transcription tool Ppt on travels and tourism Ppt on video teleconferencing equipment Ppt on kinetic molecular theory Ppt on electronic media in india