FRACTAL DIMENSION OF BIOFILM IMAGES Presented by Zhou Ji Major advisor: Dr. Giri Narasimhan

Outline Introduction Fractal dimension of pixel-based images Biofilm research Fractals and fractal dimension Fractal dimension of pixel-based images Generation of standard images with known fractal dimension Numerical results and conclusion

1. Introduction What is biofilm? What interests biologist? 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? What is fractal dimension? 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 Irregularity Fractal dimension 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! 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