Topology project

1. Using TDA on Images
Carmen Cox

2. The Idea
Mona Lisa by Leonardo Da Vinci Two Girls Reading by Pablo Picasso

3. The Idea
Mona Lisa by Leonardo Da Vinci Two Girls Reading by Pablo Picasso

4. Big Question
• Can TDA pick up zero- and one-dimensional
persistence on images?

5. Method
• Create a simplicial complex out of a grayscale
image and feed it to rca1mfscm
• Comparison between persistence diagrams

6. The Dataset: Images
• Each image is made up of pixels with some value
associated with each pixel
▫ Thus, an image can be thought of as a giant matrix
• Color images use RGB values between 0 (white)
and 255 (black) but I used grayscale instead,
where the values are a float between 0 and 1

7. The Dataset: Images

8. The Dataset: Images

9. Creating a Simplex out of an Image
0 0.25 0.37 …
0.7
0.59
…

10. Creating a Simplex out of an Image
vertex

11. Creating a Simplex out of an Image
edge

12. Creating a Simplex out of an Image

13. Creating a Simplex out of an Image

14. Creating a Simplex out of an Image

15. Feed into rca1mfscm
• Vertex input:
▫ [index index grayscale value]
• Edge input, of edge between vertex i and vertex j:
▫ [index i index j max grayscale value of i and j]
• Create N x 3 array, S
>> [I J] = rca1mfscm(S,1.2);

16. Some Simple Examples
Zero-dimensional persistence

17. Some Simple Examples
One-dimensional persistence

18. Some Simple Examples
Zero-dimensional persistence

19. Some Simple Examples
One-dimensional persistence

20. Some Simple Examples
Zero-dimensional persistence

21. Some Simple Examples
One-dimensional persistence

22. A More Interesting Example
Zero-dimensional Persistence

23. A More Interesting Example
One-Dimensional Persistence

24. Mona Lisa
0-D
1-D

25. Two Girls Reading
0-D
1-D

26. Comparison: Zero-Dimensional
Persistence
Mona Lisa Two Girls Reading

27. Comparison: One-Dimensional
Persistence
Mona Lisa Two Girls Reading

28. Girl with a Pearl Earring
0-D
1-D

29. Comparison: Zero-Dimensional
Persistence
Mona Lisa Girl with a Pearl Earring Two Girls Reading

30. Comparison: One-Dimensional
Persistence
Mona Lisa Girl with a Pearl Earring Two Girls Reading

31. Future Directions
• Machine learning
▫ Can paintings from different art periods be
distinguished using TDA and machine learning?
Can any group of images be distinguished?
• Color images
• Addressing problem of size

32. Thank you!