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