Using TDA on Images

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Using TDA on Images:

Using TDA on Images Carmen Cox

The Idea:

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

The Idea:

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

Big Question:

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

Method:

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

The Dataset: Images:

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

The Dataset: Images:

The Dataset: Images

The Dataset: Images:

The Dataset: Images

Creating a Simplex out of an Image:

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

Creating a Simplex out of an Image:

Creating a Simplex out of an Image vertex

Creating a Simplex out of an Image:

Creating a Simplex out of an Image edge

Creating a Simplex out of an Image:

Creating a Simplex out of an Image

Creating a Simplex out of an Image:

Creating a Simplex out of an Image

Creating a Simplex out of an Image:

Creating a Simplex out of an Image

Feed into rca1mfscm:

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);

Some Simple Examples:

Some Simple Examples Zero-dimensional persistence

Some Simple Examples:

Some Simple Examples One-dimensional persistence

Some Simple Examples:

Some Simple Examples Zero-dimensional persistence

Some Simple Examples:

Some Simple Examples One-dimensional persistence

Some Simple Examples:

Some Simple Examples Zero-dimensional persistence

Some Simple Examples:

Some Simple Examples One-dimensional persistence

A More Interesting Example:

A More Interesting Example Zero-dimensional Persistence

A More Interesting Example:

A More Interesting Example One-Dimensional Persistence

Mona Lisa:

Mona Lisa 0-D 1-D

Two Girls Reading:

Two Girls Reading 0-D 1-D

Comparison: Zero-Dimensional Persistence:

Comparison: Zero-Dimensional Persistence Mona Lisa Two Girls Reading

Comparison: One-Dimensional Persistence:

Comparison: One-Dimensional Persistence Mona Lisa Two Girls Reading

Girl with a Pearl Earring:

Girl with a Pearl Earring 0-D 1-D

Comparison: Zero-Dimensional Persistence:

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

Comparison: One-Dimensional Persistence:

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

Future Directions:

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

Thank you!:

Thank you!

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