1. Edge Linking & Boundary Detection
Ms.Mona Deshmukh

3. Edge Linking & Boundary Detection
Ideal case:
Techniques detecting intensity discontinuities should yield pixels lying only on edges ( or the boundary between regions).
Real life:
The detected set of pixels very rarely describes a complete edge due to effects from: noise, breaks in the edge due to non-uniform illumination.

4. Edge Linking & Boundary Detection
Solution:
Edge-detection techniques are followed by linking and other boundary detection procedures which assemble edge pixels into meaningful boundaries.

5. Local Processing
Analyze the pixel characteristics in a small neighborhood (3x3, 5x5) about every (x,y) in an image.
Link similar points to form a edge/boundary of pixels sharing common properties.

6. Local Processing Criteria used/Properties:
The strength of the response of the gradient operator that produced the edge pixel.
The direction of the gradient vector.

7. Local Processing In other words: or 1.
(x’,y’) and (x,y) are similar if:
where T is a nonnegative threshold.

8. Local Processing In other words (cont.): 2.
(x’,y’) and (x,y) are similar if:
where A is an angle threshold.

9. Image Segmentation

10. Global Processing via the Hough Transform
Points are linked by determining whether they lie on a curve of specified shape.
Problem:
Find subsets of n points that lie on straight lines.

11. Global Processing via the Hough Transform
Solution:
Find all lines determined by every pair of points
Find all subsets of points close to particular lines
Involves: n(n-1)/2 ~ n2 lines
n(n(n-1))/2 ~ n3 computations
for comparing every point to all lines.

12. Global Processing via the Hough Transform
Better solution: Hough Transform
Equation of line passing through point (xi,yi):
yi = axi + b (a,b varies)
But: b = -xia + yi 
equation of single line on ab plane

13. Global Processing via the Hough Transform

14. Global Processing via the Hough Transform
A line in the (x,y) plane passes through several points of interest and has a set of specific (a,b) values.
A line in parameter space [(a,b) plane] denotes all lines that pass through a certain point (xi,yi) and has an infinite number of (a,b) values.

15. Global Processing via the Hough Transform
A specific line is represented by a point in the (a,b) plane.
Two lines in parameter space that meet at a certain point show points belonging to the same line (in x,y plane).

16. Global Processing via the Hough Transform
Since a,b approach infinity as a line approaches the vertical, we can use the normal representation of a line:

17. Global Processing via the Hough Transform
Hough transform is applicable to any function of the form g(v,c) = 0.
v: vector of coordinates, c: coefficients.
e.g. points lying on a circle:

18. Global Processing via Graph-Theoretic Techniques
A global approach based on representing edge segments in the form of a graph and searching the graph for low-cost paths that correspond to significant edges.

19. Global Processing via Graph-Theoretic Techniques
Advantage:
performs well in the presence of noise
Disadvantage:
complicated and requires more processing time.

20. Global Processing via Graph-Theoretic Techniques
Graph G = (N,U):
A finite, nonempty set of nodes N together with a set U of unordered pairs of distinct elements of N.
(ni,nj) of U: arc

21. Global Processing via Graph-Theoretic Techniques
Directed graph:
a graph in which arcs are directed
If ni to nj is directed, nj is a successor of its parent node ni.

22. Global Processing via Graph-Theoretic Techniques
Expansion of node:
To identify the successors of a node
Level 0 consists of a single node (start node)
Last level contains the goal nodes.

23. Global Processing via Graph-Theoretic Techniques
A sequence of nodes n1,n2,…,nk (with each ni being a successor of ni-1) is called a path from n1 to nk and its cost is:

24. Global Processing via Graph-Theoretic Techniques
Edge element:
The boundary between two pixels p & q such that p and q are 4-neighbors.
Edge:
A sequence of connected edge elements.

25. Global Processing via Graph-Theoretic Techniques
H: the highest intensity value in the image

26. Global Processing via Graph-Theoretic Techniques
(0)
(1)
(2)

27. Global Processing via Graph-Theoretic Techniques
Minimum cost path:

28. Image Segmentation

29. Image Segmentation

30. Image Segmentation