Heat Equation and its applications in imaging processing and mathematical biology Yongzhi Xu
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Heat Equation and its applications in
imaging processing and mathematical biology
Yongzhi XuDepartment of Mathematics
University of LouisvilleLouisville, KY 40292
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1. Introduction
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System of ‘heat equations’
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2.1. Random walks
2. Derivation of heat equation
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2.2. Fick’s law
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Then the balance law implies
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3. Applications of heat equation
1. Heat equation in image processing2. Heat equation in cancer model and spatial
ecological model
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Sampling an image: f(xi)
3.1. Heat equation in image processing
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Three components of image processing:
1. Image Compression
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2. Image Denoising
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3. Image Analysis
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One common need:
Smoothing
•Smoothing is a necessary part of image formation.
•An image can be correctly represented by a discrete set of values, the samples, only if it has been previously smoothed.
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How to smooth a image? ------ Convolution with a ‘bell-shaped’ function
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We blur the image!
Questions:1. What does it have to do with heat equation? 2. How will it be helpful for image processing?
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•
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Answer to question 1:
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Modified heat equations and applications:
• Deblur an image by reversing time in the heat equation:
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• Smoothing to detect edges
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Level curves
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• Let U(x,t) and V(x,t) be the density functions of two chemicals or species which interact or react
3.2: Reaction-diffusion Models and Pattern Formations
Acknowledgement: Pictures of animal patterns are from Junping Shi’s website.
Alan Turing, “The Chemical Basis of Morphogenesis,” Phil. Trans. Roy. Soc. (1952).
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Mathematical Biology by James Murray
Emeritus Professor University of Washington, SeattleOxford University, Oxfordhttp://www.amath.washington.edu/people/faculty/murray/
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Why do animals’ coats have different patterns?
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Murray’s theory
Murray suggests that a single mechanism could be responsible for generating all of the common patterns observed. This mechanism is based on a reaction-diffusion system of the morphogen prepatterns, and the subsequent differentiation of the cells to produce melanin simply reflects the spatial patterns of morphogen concentration.
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Reaction-diffusion systems
•Domain: rectangle (0, a) X (0,b)
•Boundary conditions: head and tail (no flux), body side (periodic)
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“Theorem 1”: Snakes always have striped (ring) patterns, but not spotted patterns.
Turing-Murray Theory: snake is the example of b/a is large.
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Snake pictures (stripe patterns)
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“Theorem 2”: There is no animal with striped body and spotted tail, but there is animal with spotted body andstriped tail.
Turing-Murray theory: The same reaction-diffusion mechanism should be responsible for the patterns on both body and tail. The body is always wider than the tail. If the body is striped, then the tail must also be striped.
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More examples: Spotted body and striped tail
Genet (left), Giraffe (right)
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Free boundary problem model of DCIS
Let the tumor to be within a rigid cylinder occupying a region
Cancer model:
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--- the growing boundary of the tumor
--- the dimensionless nutrient concentration in the surrounding
--- the rate of surrounding transfer per unit length
--- the transfer of nutrient from surrounding
--- the nutrient consumption rate
--- the nutrient concentration
--- the ratio of the nutrient diffusion time scaleTo the tumor growth time scale. C<<1.
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Micropapillary DCIS
Cribiform DCIS
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Solid DCIS (1-D model)
Cribiform DCIS(Spread evenly)
Papillary & MicroPapillary
DCIS (Baby tree)
Moving, but not growing
Stable
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(I) mixed models
More sophisticate models may be considered.
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(II) segregated models
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Inverse Problem 1 ---- Use one incisional biopsy
Inverse Problems in cancer modeling
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Inverse Problem 2--- use a sequence of needle biopsy
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Inverse Problem 3 --- use a sequence of tomography
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Acknowledgement:
Some of the graphs and pictures are copied from the manuscript of Frederic Guichard and Jean-Michel Morel, and from the website of Junping Shi.
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Thank You!