This document discusses edge linking and boundary detection techniques. It begins by describing how ideal edge detection would yield only boundary pixels, but in reality noise and other factors mean not all edges are detected. Edge detection is followed by linking procedures to assemble edge pixels into meaningful boundaries. Local processing analyzes pixel neighborhoods to link similar pixels based on gradient strength and direction. The Hough transform is introduced as a global processing technique that links points by determining if they lie on a specified shape curve through a parameter space representation.

1. 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.

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

3. 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.

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

5. Local Processing
• In other words:
(x’,y’) and (x,y) are similar if:
T
y
x
f
y
x
f 




 )
,
(
)
,
(
where T is a nonnegative threshold.
  2
/
1
2
2
)
( y
x G
G
F
mag
f 



 or |
|
|
| y
x G
G
f 


1.

6. Local Processing
• In other words (cont.):








 
x
y
G
G
y
x 1
tan
)
,
(

2.
A
y
x
y
x 


 )
,
(
)
,
( 

where A is an angle threshold.
(x’,y’) and (x,y) are similar if:

7. Image Segmentation

8. 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.

9. 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.

10. 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

11. Global Processing via
the Hough Transform

12. 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.

13. 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).

14. 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:


 
 sin
cos y
x

15. 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:
2
3
2
2
2
1 )
(
)
( c
c
y
c
x 


