Heat Equation and its applications in

imaging processing and mathematical biology

Yongzhi XuDepartment of Mathematics

University of LouisvilleLouisville, KY 40292

1. Introduction

System of ‘heat equations’

2.1. Random walks

2. Derivation of heat equation

2.2. Fick’s law

Then the balance law implies

3. Applications of heat equation

1. Heat equation in image processing2. Heat equation in cancer model and spatial

ecological model

Sampling an image: f(xi)

3.1. Heat equation in image processing

Three components of image processing:

1. Image Compression

2. Image Denoising

3. Image Analysis

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.

How to smooth a image? ------ Convolution with a ‘bell-shaped’ function

We blur the image!

Questions:1. What does it have to do with heat equation? 2. How will it be helpful for image processing?

•

Answer to question 1:

Modified heat equations and applications:

• Deblur an image by reversing time in the heat equation:

• Smoothing to detect edges

Level curves

• 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).

Mathematical Biology by James Murray

Emeritus Professor University of Washington, SeattleOxford University, Oxfordhttp://www.amath.washington.edu/people/faculty/murray/

Why do animals’ coats have different patterns?

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.

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.

Snake pictures (stripe patterns)

“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.

More examples: Spotted body and striped tail

Genet (left), Giraffe (right)

Free boundary problem model of DCIS

Let the tumor to be within a rigid cylinder occupying a region

Cancer model:

--- 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.

Micropapillary DCIS

Cribiform DCIS

Solid DCIS (1-D model)

Cribiform DCIS(Spread evenly)

Papillary & MicroPapillary

DCIS (Baby tree)

Moving, but not growing

Stable

(I) mixed models

More sophisticate models may be considered.

(II) segregated models

Inverse Problem 1 ---- Use one incisional biopsy

Inverse Problems in cancer modeling

Inverse Problem 2--- use a sequence of needle biopsy

Inverse Problem 3 --- use a sequence of tomography

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.

Thank You!