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Topology Based Method of Segmentation of Gray Scale Images Peter Saveliev Marshall University , USA. Outline. 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.
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Topology Based Method of Segmentation of Gray Scale Images Peter SavelievMarshall University,USA
Outline • 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.
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.
Inclusion tree 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 trees 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.
Combined inclusion tree 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 + = • 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
Topology graph • 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.
The topology of a binary image • 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.
Cell decomposition 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.
Cycles Cycles are used as a tool of image segmentation. 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.
Topology graph 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.
Outline of the algorithm • 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.
Performance 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).
Processing time Intel Core 2 Dual CPU T7500 2.2GHz
Filtering cycles If a 0-cycle is an ancestor of another, only one of them is taken into account.
Filtering cycles If a 0-cycle is an ancestor of another, only one of them is taken into account.
Summary • 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.
