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

2. 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

3. 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.

5. 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

6. 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.

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 =

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 standard images
Snowflake/Random curves
Single shape/Group
quadratic Koch island

21. D= , n=3

22. D= , n=5

23. Random curve, D= , n=3

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

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

26. D= , n=3

27. D= , 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. Thank you!
Questions?

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