Topology Based Method of Segmentation of Gray Scale Images   Peter SavelievMarshall University, USA

Goal: a graph representation of the topology of a gray scale image.

The graph represents the hierarchy of the lower and upper level sets of the gray level function.

This graph contains the inclusion trees, but it is not a tree.

The topological tools:◦ cell decomposition: the image is represented as a

combination of pixels as well as edges and vertices. ◦ cycles: both upper and lower level sets are captured

by circular sequences of edges.

Outline

Topological analysis of binary images

Gray scale function

Segmentation: capturing upper and lower level sets of the gray level function of the image.

Rationale: the connected components of these sets are building blocks of real items depicted in the image.

The connected components of the lower level sets have a clear hierarchy based on inclusion. This hierarchy provides a graph representation of the topology of the image.

Inclusion tree

The lower inclusion tree

.

The lower inclusion tree and the upper inclusion tree

The inclusion trees for upper and lower level sets, if considered separately, do not help in finding out which object has which hole.

Therefore, in order to capture the topology of the image, the two trees have to be combined in some way.

Inclusion trees

P. Monasse and F. Guichard, Fast computation of a contrast invariant image representation. IEEE Transactions on Image Processing, 9(5), pp. 860–872, 2000.

Jordan Theorem: A component of a level set encircles or is encircled by components of other level sets.

Combined inclusion tree

+ =

Combined inclusion tree

The lower level sets are mixed with the upper level sets.

The gray levels are also mixed.

An alternative way to combine the inclusion trees

+ =

The topology graph of the image

The lower and upper inclusion trees remain intact within the graph.

The graph breaks into layers that coincide with the topology graphs of the corresponding binary images.

The topology graph is not a tree in general.

Topology graph

Our goal is to capture the topological features present in the image: connected components and their holes.

We think of black objects as connected components and white objects as holes in the dark objects.

The topology of a binary image

A binary image is a rectangle covered by black and white pixels arranged in a grid.

A pixel is a square, or a tile: [n, n + 1] × [m, m + 1].

Cell decomposition

a vertex {n}×{m} is a 0-cell, an edge {n}×(m, m + 1) is a 1-cell, and a face (n, n + 1)×(m, m + 1) is a 2-cell.

Cell decomposition

Two adjacent edges are 1-cells and they share a vertex, a 0-cell;

Two adjacent faces are 2-cells and they share an edge, a 1-cell.

Cell decomposition

Cycles are used as a tool of image segmentation.

Cycles

Both connected components and holes are captured by cycles:

a 0-cycle as a sequence of vertices that follows the outer boundary of a connected component;

a 1-cycle as a sequence of edges that follows the outer boundary of a hole.

Cycles partition the image

The nodes of the topology graph are the cycles in the image and there is an arrow from node A to node B if:

0-cycle B has 0-cycle A inside, provided A and B correspond to consecutive gray levels.

0-cycle B has 1-cycle A inside, provided A and B correspond to the same gray level.

And vice versa.

Topology graph

All pixels in the image are ordered in such a way that all black pixels come before white ones.

Following this order, each pixel is processed: ◦ add its vertices, unless those are already present as parts

of other pixels; ◦ add its edges, unless those are already present as parts of

other pixels;◦ add the face of the pixel.

At every step, the graph is given a new node and arrows that connect the nodes in order to represent the merging and the splitting of the cycles:◦ adding a new vertex creates a new component;◦ adding a new edge may connect two components, or

create, or split a hole;◦ adding the face to the hole eliminates the hole.

Outline of the algorithm

Adding an edge

Suppose N is the number of pixels in the image. Then

The memory usage is O(N). The complexity of the algorithm is O(N2).

Performance

Intel Core 2 Dual CPU T7500 2.2GHz

Processing time

If a 0-cycle is an ancestor of another, only one of them is taken into account.

Filtering cycles

Filtering cycles

If a 0-cycle is an ancestor of another, only one of them is taken into account.

The approach and the method are justified by appealing to classical mathematics.

The new representation of the topology of a gray scale image is a graph that isn’t a tree in general.

This data structure allows components and holes to be treated simultaneously but kept separate.

The algorithm and its interpretation are intuitive. The algorithm is fast enough to be practical. The analysis produces meaningful results for

various gray scale images.

Summary

Thank you

Combined inclusion tree vs. topology graph