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:
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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
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A Variational Method for Image Denoising
Original image
Denoised image by TV

4. Total Variation Model (Rudin-Osher-Fatemi)
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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

5. Introduction to Calculus of Variations
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Introduction to Calculus of Variations

6. What is Functional and its Derivative?
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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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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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Space
The space is a linear space.
Inner product:
Norm:

9. Linear Functional on Hilbert Space
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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
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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:
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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

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

13. An Important Class of Functionals
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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
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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Let

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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:
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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:

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

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Region Based Methods

22. Mumford-Shah Functional
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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