Introduction to Variational Methods and Applications

1. Introduction to Variational Methodsand Applications Chunming Li Institute of Imaging Science Vanderbilt University URL: www.vuiis.vanderbilt.edu/~licm E-mail: chunming.li@vanderbilt.edu

2. 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 Denoised image by TV Original image

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

5. Introduction to Calculus of Variations

6. A functional is a mapping where the domain is a space of infinite dimension What is Functional and its Derivative? • 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. 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. Space The space is a linear space. • Inner product: • Norm:

9. 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 • 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 Linear Functional on Hilbert Space • The space of all bounded linear functionals on X is called the dual space of X, denoted by X’.

10. 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. • 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. . Directional Derivative of Functional • Furthermore, if is a bounded linear functional of v, we say F is Gateaux differentiable. (Euler-Lagrange Equation)

11. 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 Example

12. Rewrite as: where the equality holds if and only if Minimizer A short cut

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

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

15. Let

17. 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 Steepest Descent

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

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

20. with Gateaux derivative 4. Gradient Flow Example: Total Variation Model

21. Region Based Methods

22. 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. c1 c2 c3 c4 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: http://vuiis.vanderbilt.edu/~licm/research/LBF.html

30. Piece Smooth Model

31. Piece Smooth Model

32. Rerults

33. Thank you