Edge linking via Hough transform

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 


