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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

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Outline

1. Brief introduction to calculus of variations

2. 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

Denoised image by TVOriginal image

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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 VariationsIntroduction to Calculus of Variations

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What is Functional and its Derivative?

• Can we find the minimizer of a functional F(u) by solving F’(u)=0?• What is the “derivative” of a functional F(u) ?

• Usually, the space is a set of functions with certain properties (e.g. continuity, smoothness).

• A functional is a mapping where the domain is a space of infinite dimension

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Hilbert Spaces

1. Vector addition:

2. Scalar multiplication:

3. Inner product , with properties:

4. Norm

A real Hilbert Space X is endowed with the following operations:

Basic facts of a Hilbert Space X

1. X is complete

2. Cauchy-Schwarz inequality where the equality holds if and only if

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Space

• Inner product:

• Norm:

The space is a linear space.

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Linear Functional on Hilbert Space

• Theorem: Let be a Hilbert space. Then, for any bounded linear functional , there exists a vector such that for all

• Linear functionals deduced from inner product: For a given vector , the functional is a bounded linear functional.

• A functional is bounded if there is a constant c such that for all

• A linear functional on Hilbert space X is a mapping with property: for any

• The space of all bounded linear functionals on X is called the dual space of X, denoted by X’.

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Directional Derivative of Functional

• 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)

• 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.

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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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A short cut

• Rewrite as:

where the equality holds if and only if

Minimizer

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An Important Class of Functionals

• Consider energy functionals in the form:

where is a function with variables:

• Gateaux derivative:

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Proof

• Lemma:

for any

• Compute for any

• Denote by the space of functions that are infinitely continuous differentiable, with compact support.

(integration by part)

• The subspace is dense in the space

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Let

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Steepest Descent

• The directional derivative of F at in the direction of is given by

• Answer:

The directional derivative is negative,

and the absolute value is maximized.

The direction of steepest descent

• What is the direction in which the functional F has steepest descent?

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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

• The procedure of finding the Gateaux derivative and gradient flow:

1. Define the Lagrangian in

• Consider total variation model:

2. Compute the partial derivatives of

3. Compute the Gateaux derivative

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Example: Total Variation Model

with

Gateaux derivative

4. Gradient Flow

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Region Based MethodsRegion Based Methods

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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)

c1

c2c4

c3

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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

See: http://vuiis.vanderbilt.edu/~licm/research/LBF.html

Click to see the movie

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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