Outline

A graph, non-tree representation of the topology of a gray scale image Peter SavelievMarshall University, USA

A graph, non-tree representation of the topology of a gray scale image Peter Saveliev

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OutlineA new (directed) graph representation of the

topology of a gray scale image is presented. This graph combines the inclusion trees of

the lower and upper level sets of the gray level function.

These sets are captures by cycles: simple closed curves.

Algorithm is based on cell decomposition: the image is represented as a combination of pixels as well as edges and vertices;

A graph representation of the topology of a color image is presented.


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Topology of binary images via cycles

Cycles capture components and holes.


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Topology of gray scale images

Cycles capture components and holes of the lower level sets


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Topology of gray scale images


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Gray scale function

The connected components of upper and lower level sets are building blocks of segmentation.

Peter

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Inclusion treeThe connected components

of the lower level sets form a tree structure based on inclusion.


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Lower level set inclusion tree


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Lower and upper level inclusion trees

.


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Inclusion treesTo represent the topology of the image

we need both inclusion trees, combined in some way.


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Combined inclusion treeBased on inclusion of the contours.

+ = + =

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

The gray levels are also mixed.


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The topology graph of the image

+ = + =


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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.


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Topology graph

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

object B has hole A, provided A and B correspond to consecutive gray levels.

object B has hole A, provided A and B correspond to the same gray level.

And vice versa.


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Cell decomposition A binary image is a rectangle

covered by black and white pixels arranged in a grid.

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


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Cell decomposition

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


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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.


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Cycles in 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 an object;

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


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Image segmentation via cycles


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Outline of the algorithmAll pixels in the image are ordered according to the

gray level. Following this order, each pixel is processed:

A. add its vertices, unless those are already present as parts of other pixels;

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

C. add the face of the pixel.At every step, the graph is given a new node and

arrows to represent the merging and the splitting of the cycles:D. adding a new vertex creates a new component;E. adding a new edge may connect two components, or

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


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Adding an edge


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Stages of analysis


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PerformanceN is the number of pixels in the

image. The memory is O(N). The complexity is O(N2).


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Pixcavator

See demo session


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Analysis of color images

If only one of the three primary color is changing, the topology is the same as of a gray scale image

Thresholding - RGB

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A graph, non-tree representation of the topology of a gray scale

image Peter Saveliev

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A graph, non-tree representation of the topology of a gray scale

image Peter Saveliev

Color imagesThe RGB color space: S={ (r, g, b) : 0 ≤ r, g, b ≤ 255}Partial order on S: (r, g, b) ≤ (r’,

g’, b’) ifr ≤ r’, g ≤ g’, b ≤ b’.

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Outline of the algorithmAll pixels in the image are ordered according to

partial order of the RGB space. 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.

A graph, non-tree representation of the topology of a gray scale image

Peter Saveliev

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Topology graph of a color image

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

object B has hole A, provided A and B correspond to consecutive colors.

object B has hole A, provided A and B correspond to the same color.

And vice versa. A graph, non-tree representation of the topology of a gray scale image

Peter Saveliev

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Thank you

A graph, non-tree representation of the topology of a gray scale image

Peter Saveliev

Outline

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