yongzhi xu department of mathematics university of...
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Heat Equation and its applicationsin
imaging processing and mathematical biology
Yongzhi XuDepartment of Mathematics
University of LouisvilleLouisville, KY 40292
3. Applications of heat equation
1. Heat equation in image processing2. Heat equation in cancer model and spatial
ecological model
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.
We blur the image!
Questions:1. What does it have to do with heat equation? 2. How will it be helpful for image processing?
3.2: Reaction-diffusion Models and Pattern Formations
• Let U(x,t) and V(x,t) be the density functions of two chemicals or species which interact or react
Alan Turing, “The Chemical Basis of Morphogenesis,” Phil. Trans. Roy. Soc. (1952).
Acknowledgement: Pictures of animal patterns are from Junping Shi’s website.
Mathematical Biology by James Murray
Emeritus ProfessorUniversity of Washington, SeattleOxford University, Oxfordhttp://www.amath.washington.edu/people/faculty/murray/
Murray’s theory
Murray suggests that a singlemechanism could be responsible for generating all of the common patterns observed. This mechanism is based on a reaction-diffusion system of the morphogenprepatterns, and the subsequent differentiation of the cells to produce melanin simply reflects the spatial patterns of morphogenconcentration.
Reaction-diffusion systems
•Domain: rectangle (0, a) X (0,b)
•Boundary conditions: head and tail (no flux), body side (periodic)
“Theorem 1”: Snakes always have striped (ring) patterns, but not spotted patterns.
Turing-Murray Theory: snake is the example of b/a is large.
“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.
Cancer model:
Free boundary problem model of DCISLet the tumor to be within a rigid cylinder occupying a region
--- 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 ratio of the nutrient diffusion time scaleTo the tumor growth time scale. C<<1.
--- the nutrient concentration
Solid DCIS (1-D model)
Cribiform DCIS(Spread evenly)
Papillary & MicroPapillary
DCIS (Baby tree)
Moving, but not growing
Stable
Acknowledgement:
Some of the graphs and pictures are copied from the manuscript of FredericGuichard and Jean-Michel Morel, and from the website of Junping Shi.