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

2. 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

3. 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.

5. 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

6. 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.

7. Julia Set

8. Richardson’s plot

9. 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

10. Generation of Koch snowflake

11. 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

12. binarize

13. Small objects deleted

14. Small holes filled

15. Boundaries found

16. Dilation method • Log-log plot: area ~ dilation count • D = 2 - slope

17. EMD(Euclidean Distance Map) method • Log-log plot: area ~ threshold level • D = 2 - slope

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

19. 3. Generating standard images • Purpose • Test and validate algorithms or their implementations like BIP • Features • Known fractal dimension • Diverse appearance • Based on Koch curve

20. 3. Generating standardimages • Snowflake/Random curves • Single shape/Group • quadratic Koch island

21. D=1.26186, n=3

22. D=1.26816, n=5

23. Random curve, D=1.26816, n=3

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

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

26. D=1.17327, n=3

27. D=1.59803, n=3

28. Quadratic Koch Island, D=1.5

29. 4. Results • Biofilm images • Standard images – discussion of algorithm

30. Result of biofilm images

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

38. Demonstration • KochGen • BIP • Biofilm3 • Julia

39. Questions? Thank you!

40. 1. Fractals and Fractal Dimension • What is in common in these three pictures? Fractal fern Sierpinski’s triangle Koch snowflake

41. Types of fractals • Iteration function system (random) • Iteration function system (deterministic) • L-system • Julia set • Mandelbrot set • Heron Map

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

45. Original image fractal compressed