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

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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

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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 N=4, r=1/3, 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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D=1.26186, n=3

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D=1.26816, n=5

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Random curve, D=1.26816, n=3

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

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

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D=1.17327, n=3

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D=1.59803, n=3

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Quadratic Koch Island, D=1.5

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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 fernSierpinskis triangleKoch 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

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