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Introduction to Variational Methods and Applications

Chunming Li Institute of Imaging Science Vanderbilt University URL:

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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

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A Variational Method for Image Denoising

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

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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

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Introduction to Calculus of Variations

4/13/2017 Introduction to Calculus of Variations

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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) ?

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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

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

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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

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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)

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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

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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

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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:

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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)

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4/13/2017 Let

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4/13/2017

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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

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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:

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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

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Example: Total Variation Model

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

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4/13/2017 Region Based Methods

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Mumford-Shah Functional

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

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Active Contours without Edges (Chan & Vese 2001)

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Active Contours without Edges

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Results

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Multiphase Level Set Formulation (Vese & Chan, 2002)

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Piece Wise Constant Model

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Piece Wise Constant Model

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Drawback of Piece Wise Constant Model

Chan-Vese LBF Click to see the movie See:

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Piece Smooth Model

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Piece Smooth Model

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Rerults

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Thank you

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