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1 Introduction to Variational Methods and Applications Chunming Li Institute of Imaging Science Vanderbilt University URL:

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2 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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3 A Variational Method for Image Denoising Denoised image by TV Original image

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4 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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5 Introduction to Calculus of Variations

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6 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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7 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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8 Space Inner product: Norm: The space is a linear space.

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9 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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10 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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11 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 SolveMinimizer

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

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13 An Important Class of Functionals Consider energy functionals in the form: where is a function with variables: Gateaux derivative:

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14 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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15 Let

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17 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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18 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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19 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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20 Example: Total Variation Model with Gateaux derivative 4. Gradient Flow

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21 Region Based Methods

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22 Mumford-Shah Functional Regularization termData fidelity termSmoothing term

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23 Active Contours without Edges (Chan & Vese 2001)

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24 Active Contours without Edges

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25 Results

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26 Multiphase Level Set Formulation (Vese & Chan, 2002) c1 c2 c4 c3

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27 Piece Wise Constant Model

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28 Piece Wise Constant Model

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29 Drawback of Piece Wise Constant Model Chan-VeseLBF See: Click to see the movie

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30 Piece Smooth Model

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31 Piece Smooth Model

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32 Rerults

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33 Thank you

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