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

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

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

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

Introduction to Calculus of Variations 4/13/2017 Introduction to Calculus of Variations

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) ?

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

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

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

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)

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

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

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:

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)

4/13/2017 Let

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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

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:

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

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

4/13/2017 Region Based Methods

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

Active Contours without Edges (Chan & Vese 2001)

Active Contours without Edges

Results

Multiphase Level Set Formulation (Vese & Chan, 2002)

Piece Wise Constant Model

Piece Wise Constant Model

Drawback of Piece Wise Constant Model Chan-Vese LBF Click to see the movie See: http://vuiis.vanderbilt.edu/~licm/research/LBF.html

Piece Smooth Model

Piece Smooth Model

Rerults

Thank you