✅ Học Price Action | Pin Bar chuyên sâu – Bài 14
✅ Học Price Action | Pin Bar chuyên sâu – Bài 14

Download presentation

Presentation is loading. Please wait.

Published byKendall Scholes Modified over 8 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

© 2023 SlidePlayer.com Inc.
All rights reserved.

You are watching: Introduction to Variational Methods and Applications. Info created by Bút Chì Xanh selection and synthesis along with other related topics.