1. Finding Lines in Images
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Finding Lines - Hough Transform

2. Finding Lines - Hough Transform
References
[1] Gonzalez and Woods – Section 10.2
11/14/2018
Finding Lines - Hough Transform

3. Finding Lines - Hough Transform
Problem Statement
Many objects in images are bounded by lines.
Lines of communication
Roads and rivers
Cables
Many man-made objects
Buildings
Vehicles
To isolate and identify the objects, it is necessary to find the lines.
Often this step is run after the segmentation step.
Can this be extended to other shapes?
11/14/2018
Finding Lines - Hough Transform

4. Finding Lines - Hough Transform
What is needed?
Typically, the images have been converted to B&W.
Any two points can be connected with a line.
Many lines are possible.
How does one select the lines of interest.
Look for some dominance characteristic.
Several points per line.
Use curve fitting?
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Finding Lines - Hough Transform

5. Finding Lines - Hough Transform
Example Image: Lines
From [1]
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Finding Lines - Hough Transform

6. Finding Lines - Hough Transform
Curve Fitting
Several (x,y) values (points).
Have a hypothesis: y = f(x) is a good model.
Test the hypothesis.
Objective function.
Sum of square of errors.
Maximum error.
Measure goodness.
Reformulate hypothesis.
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Finding Lines - Hough Transform

7. Finding Lines - Hough Transform
Parameterize the decision process.
Every line is characterized by
y = ax + b
The objective then is to find a,b such that a number of points lie on this line.
Transform this problem to the a,b domain.
b = y – ax
(x,y) are point coordinates
Often called parameter space.
Consequence:
Say 2 points lie on the same line.
In parameter space each point corresponds to a line.
The lines will intersect, at the (a,b) value!!!
11/14/2018
Finding Lines - Hough Transform

8. Finding Lines - Hough Transform
From [1]
11/14/2018
Finding Lines - Hough Transform

9. Multiple points per line
Recall we used several points per line as the means of deciding is this was a useful feature.
How do we know if there are enough points on the line?
What if the points are little off?
Motivation similar to computing Sum of the Square of the Errors.
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Finding Lines - Hough Transform

10. Parameter Space Subdivisions
Makes Hough computationally efficient.
For each point find the allowable (a,b) and insert in respective cell.
Solves both the problems on the previous slide!!
From [1]
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Finding Lines - Hough Transform

11. Finding Lines - Hough Transform
Near Vertical Lines
y = ax + b
For near vertical lines, the slope approaches infinity.
What about the parameter space?
So this representation is a problem for near vertical lines.
Lets try a normal representation for a line.
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Finding Lines - Hough Transform

12. Normal Representation of a Line
x cos  + y sin  = 
 is the length of the normal from the origin to the line
 is the angle between the normal and x-axis
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Finding Lines - Hough Transform

13. Finding Lines - Hough Transform
Normal Representation of a Line
From [1]
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Finding Lines - Hough Transform

14. Hough Transform in Normal Representation
From [1]
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Finding Lines - Hough Transform

15. Finding Lines - Hough Transform
Review of Approach
Convert from image space to parameter space.
It is easier to search in the parameter space.
Can we generalize?
If the curve has more than 2 parameters, then one has to search a higher dimensional space.
Increases computational complexity.
11/14/2018
Finding Lines - Hough Transform