FRACTAL DIMENSION OF BIOFILM IMAGES

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FRACTAL DIMENSION OF BIOFILM IMAGES

• Presented by Zhou Ji
• Major advisor Dr. Giri Narasimhan

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Outline

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

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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.

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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

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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.

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• Julia Set

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Richardsons plot
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Calculating fractal dimension

• From Richardsons 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 N4, r1/3, D 1.26186

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Generation of Koch snowflake
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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

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• binarize

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• Small objects deleted

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• Small holes filled

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• Boundaries found

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Dilation method

• Log-log plot area dilation count
• D 2 - slope

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EMD(Euclidean Distance Map) method

• Log-log plot area threshold level
• D 2 - slope

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Mass radius method

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

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3. Generating standard images

• Purpose
• Test and validate algorithms or their
  implementations like BIP
• Features
• Known fractal dimension
• Diverse appearance
• Based on Koch curve

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3. Generating standard images

• Snowflake/Random curves
• Single shape/Group
• quadratic Koch island

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• D1.26186, n3

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• D1.26816, n5

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• Random curve, D1.26816, n3

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• Group of single snowflake, D1.26816, n4

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• Group of random curves, D1.26816, n4

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• D1.17327, n3

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• D1.59803, n3

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• Quadratic Koch Island, D1.5

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4. Results

• Biofilm images
• Standard images discussion of algorithm

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• Result of biofilm images

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Summary

• Powerful tools (BIP KochGen) developed
• Comparison of Algorithms
• More correlations of fractal dimension in biofilm
  images are still to be found.

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Demonstration

• KochGen
• BIP
• Biofilm3
• Julia

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Thank you!

• Questions?

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1. Fractals and Fractal Dimension

• What is in common in these three pictures?
• Fractal fern Sierpinskis triangle Koch snowflake

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Types of fractals

• Iteration function system (random)
• Iteration function system (deterministic)
• L-system
• Julia set
• Mandelbrot set
• Heron Map

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Application of fractals

• Simulation and model
• Kidney, skeletal structure, nervous system
• landscape, plant
• Stock market, internet traffic
• Music
• Image compression
• Others
• Biofilm research

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Original image fractal compressed