1. Hough transform and geometric transform
Jen-Chang Liu, Spring 20056

2. Motivation for edge linking
Edge detection: find pixels lying on edges
=> How to link edge points ?
Sobel: horizontal
Sobel: vertical
original
Goal:找出車牌方塊

3. Global processing Given n edge points in an image
=> Find subsets of the points that lie on straight lines
Step 1: for a pair of points, determine
their line function
Step 2: determine whether there are
points that are close to the line
Step 3: repeat step 1 until all pair of
points are tested

4. Global processing: complexity
n edge points n
=> C2
~ n2 lines n
=> (n-2)C2
~ n3 tests to determine
whether other points are near
this line
Any faster method?

5. Hough transform Hough [1962] How to determine a line ? yi = axi + b
Tow point (xi, yi) and (xj, yj)
determine a line
(ak, bk)
(a, b)
yi = axi + b
yj = axj + b
Line with parameters (a, b)
* Many lines pass a point
* (xi, yi) and (xj, yj) both have a line
with parameters (a, b) passing through

6. Idea: Parameter space of line
A line is determined by (a, b) B
(ak, bk)
(a, b)
B = -Axi + yi
(ak, bk)
(a, b)
yi = Axi + B
B = -Axj + yj A

7. Example: find all lines [i.e. parameter (a,b)] pass a pointy
a + b = 1
(a,b)=(, ?)
(1,1) b
(0,1)
(a,b)=(1,0)
(1,0)
(a,b)=(0,1) x a
Replace in y = ax+b
1 = a + b
Problem: How to define the
horizontal lines?

8. Problem with y=ax+b Solution for line function with slope a = 
Normal representation of a line
垂直向量 Inner product = 0
(x, y)

9. Implementing Hough transform
1. decide on a discrete set of values of θ and  to use
2. For each edge point, calculate the values of  = xcosθ+ysinθfor all discrete θ
3. Create an accumulator array whose size are the number of angles θ and values 
4. Updating the accumulator array as we go

10. Limit θ and 
1. Restrict   0 y -3 3 x

11. Limit θ and  (cont.) 2. -90θ180 Line in the 2nd quadrant
will not appear in the image y x

12. Example
128
128
128
-90
180

13. Example:
Edge image: Gradient => threshold
original

14. Project 2 Topic: Generalize Hough transform to detect circles
1-3 persons a team
Prepare PowerPoint slides to explain your methods
Write the MATLAB function to perform it
Give me team members on 4/29
Present and demo on 5/6

15. Image rotation
Rotation matrix
(x’, y’)
(x, y) θ

16. Real situation when rotating
Boundary of image changes
Rotated pixels are not on the
integer image grid

17. Real situation when rotating (cont.)
Take nearest integer grid
What are the gray values?
Rotated back and
interpolation !!!

18. Gray-level interpolation
Nearest neighbor
Bilinear interpolation
Use 4 nearest neighbors
Cubic convolution interpolation
Fit a surface over a large number of neighbors

19. Interpolation 內插法 – 1D
Nearest neighbor
Linear interpolation

20. Interpolation 內插法 – 2D Bilinear interpolation Interpolate for (x,y’)
and (x+1, y’) first
2. Interpolate for (x’,y’)

21. High order interpolation
Cubic interpolation
weighting
f(x’)=R3(-1-)f(x1)+ R3(-)f(x2)+R3(1-)f(x3)+ R3(2-)f(x4)

22. Example: Anamorphosis

23. Spatial transformations
r(x,y)
s(x,y)
tiepoints
f(x,y)
g(x’,y’)
distorted
original

24. Spatial transformations (cont.)
Within the quadrilateral region, geometric transformation is modeled by a pair of bilinear equation
4 pairs of tiepoints to solve the equations
Restored image:

25. Gray-level interpolation
Non-integer coordinate x’ or y’

26. Tiepoints after original distortion restored distorted
Nearest neighbor
restored
distorted
Bilinear interp.
restored
distorted

27. original
distorted
(Using
Previous
Slide)
restored
Diff.