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Introduction to Variational Methods
and Applications
Chunming Li
Institute of Imaging Science
Vanderbilt University
URL: www.vuiis.vanderbilt.edu/~licm
E-mail: [email protected]
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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
Original image
Denoised image by TV
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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 Variations
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What is Functional and its Derivative?
• A functional is a mapping
space of infinite dimension
where the domain
is a
• 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:
1.
Vector addition:
2. Scalar multiplication:
3. Inner product
, with properties:
4. Norm
Basic facts of a Hilbert Space X
1.
X is complete
2. Cauchy-Schwarz inequality
holds if and only if
where the equality
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Space
The space
•
Inner product:
•
Norm:
is a linear 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
that
is bounded if there is a constant c such
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
linear functional
that
be a Hilbert space. Then, for any bounded
, there exists a vector
such
for all
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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
all
is a minimizer of the functional
, then
, i.e.
. (Euler-Lagrange Equation)
for
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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
• 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
given by
• What is the direction
steepest descent?
in the direction of
is
in which the functional F has
• Answer:
The directional derivative
and the absolute value
is negative,
is maximized.
The direction of 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
• 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
with
Gateaux derivative
4. Gradient Flow
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Region 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
c2
c3
c4
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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: http://vuiis.vanderbilt.edu/~licm/research/LBF.html
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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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