FRACTAL DIMENSION OF BIOFILM IMAGES

Presented by Zhou Ji

Major advisor: Dr. Giri Narasimhan

Outline

1. Introduction– Biofilm research– Fractals and fractal dimension

2. Fractal dimension of pixel-based images

3. Generation of standard images with known fractal dimension

4. Numerical results and conclusion

1. Introduction

• What is biofilm?– A thin layer of bacteria.

• What interests biologist?– The structure and how they grow.

• What does this project want to do?– Quantify the pictures of them.

1. Introduction

• What is fractal?– Special geometrical figure that is not whole

number-dimensional, like lines, surfaces or solids

• What is fractal dimension?– Measurement of where it is in between

• How does this project use the concept?– Calculate fractal dimension from pixel image

Properties of fractal

• Self-similarity– In each tiny piece we observe the form of the

entire shape.

• Irregularity– There are no smooth boundary. Length or area

cannot be determined.

• Fractal dimension– It has not dimension of whole number.

Julia Set

Richardson’s plot

Calculating fractal dimension

• From Richardson’s plot– Log-log plot of log L vs. log

• L: length, : interval

– Formula: D = 1 - slope

• Koch curve– Generation– Formula: D = log N/log(1/r) – Koch snowflake N=4, r=1/3, D = 1.26186

Generation of Koch snowflake

2. Fractal dimension of pixel-based images

• What is special?– Detection of objects.– Lower limit of scale - pixel.– Boundary described with pixels - width.

• BIP (by Qichang Li et al)

– Preprocessing– Algorithms

• binarize

• Small objects deleted

• Small holes filled

• Boundaries found

Dilation method

• Log-log plot: area ~ dilation count• D = 2 - slope

EMD(Euclidean Distance Map) method

• Log-log plot: area ~ threshold level• D = 2 - slope

Mass radius method

• Log-log plot: average area in a circle~ radius• D = slope

3. Generating standard images

• Purpose– Test and validate algorithms or their

implementations like BIP

• Features– Known fractal dimension– Diverse appearance– Based on Koch curve

3. Generating standard images

• Snowflake/Random curves

• Single shape/Group

• quadratic Koch island

• D=1.26186, n=3

• D=1.26816, n=5

• Random curve, D=1.26816, n=3

• Group of single snowflake, D=1.26816, n=4

• Group of random curves, D=1.26816, n=4

• D=1.17327, n=3

• D=1.59803, n=3

• Quadratic Koch Island, D=1.5

4. Results

• Biofilm images

• Standard images – discussion of algorithm

• Result of biofilm images

Summary

• Powerful tools (BIP & KochGen) developed

• Comparison of Algorithms

• More correlations of fractal dimension in biofilm images are still to be found.

Demonstration

• KochGen

• BIP

• Biofilm3

• Julia

Thank you!Thank you!

Questions?

1. Fractals and Fractal Dimension

• What is in common in these three pictures?

Fractal fern Sierpinski’s triangle Koch snowflake

Types of fractals

• Iteration function system (random)

• Iteration function system (deterministic)

• L-system

• Julia set

• Mandelbrot set

• Heron Map

Application of fractals

• Simulation and model – Kidney, skeletal structure, nervous system– landscape, plant– Stock market, internet traffic– Music

• Image compression• Others

– Biofilm research

Original image fractal compressed