1. Generalized Hough Transform

2. The Generalized Hough Transform

3. From Standard to Generalized HT
Standard Hough Transform requires parametric representation for desired curve
This idea is generalized in the Generalized Hough Transform

4. Example: Human Face recognition
Is there some attribute of the structure of the head that we can exploit to help estimate pose estimation?
Is this attribute invariant under change in pose? Or
“Can we model how this attribute varies with pose?”

5. Hough Transform in General
Technique to isolate curves of a given shape in an image
Standard Hough Transform (HT) uses parametric formulation of curves
Generalized Hough Transform (GHT) extends for arbitrary curves

6. Key Idea to improve correlation by voting
4/23/2017
When we compute the correlation by voting, we spend most of the time casting bad votes.
Idea is to use extra shape information (e.g. gradients) to cast fewer votes:
O(n) complexity: For each of O(n) points on the boundary, cast O(1) votes.

7. General Hough Algorithm Idea
1. explicitly list points on shape
2. make table for all edge pixels for target
3. for each pixel store its position relative to some reference point on the shape
‘if I’m pixel i on the boundary, the reference point is at ref[i]’

8. The Generalized Hough Transform
Technique to find arbitrary curves in a given image
Parametric equation no longer required
Look-up table used as transform mechanism
Two phases:
R-Table Generation phase
Object Detection phase

9. The Generalized Hough Transform
Standard Techniques allow for invariance to scale and rotation in the plane
In general, objects in the real world are 3-dimensional
Hence a single silhouette provides no invariance to pose (i.e. rotation out of the plane).
No pose estimation.
This is generalized to Surface Normal Hough Transform

10. Building the R-Table in GHT

11. GHT: Building the R-Table
1. We are given the shape we want to localize
2. We build a lookup table for this shape, called R-Table
It will replace the need for a parametric equation in the transform stage

12. GHT: Building the R-Table

13. GHT: Building the R-Table

14. GHT: Building the R-Table GHT: Building the R-Table

15. Object Localization in the R-Table in GHT

16. GHT: Object Localization

17. GHT: Object Localization

18. GHT: Object Localization

19. Conclusions on GHT Conclusions on GHT
Standard Techniques allow for invariance to scale and rotation in the plane
In general, objects in the real world are 3-dimensional
Hence a single silhuette provides no invariance to pose (i.e. rotation out of the plane).
No pose estimation.
Now show more details

20. Generalized Hough Transform Algorithm

21. Algorithm of the General Hough Transform

22. Hough Transform for Curves
The H.T. can be generalized to detect any curve that can be expressed in parametric form:
Y = f(x, a1,a2,…ap)
a1, a2, … ap are the parameters
The parameter space is p-dimensional
The accumulating array is LARGE!

23. Generalized Hough Transform
algorithm
Find all desired points in image
For each feature point
for each pixel i on target boundary
get relative position of reference point from i
add this offset to position of i
increment that position in accumulator
Find local maxima in accumulator
Map maxima back to image to view

24. Generalizing the H.T.
The H.T. can be used even if the curve has not a simple analytic form!
(xc,yc) fi ri Pi ai
Pick a reference point (xc,yc)
For i = 1,…,n :
Draw segment to Pi on the boundary.
Measure its length ri, and its orientation ai.
Write the coordinates of (xc,yc) as a function of ri and ai
Record the gradient orientation fi at Pi.
Build a table with the data, indexed by fi .
xc = xi + ricos(ai)
yc = yi + risin(ai)

25. Suppose, there were m different gradient orientations:
Generalizing the H.T.
Suppose, there were m different gradient orientations:
(m <= n) fi ri ai fj rj aj f1 f2 . fm
(r11,a11),(r12,a12),…,(r1n1,a1n1)
(r21,a21),(r22,a12),…,(r2n2,a1n2)
(rm1,am1),(rm2,am2),…,(rmnm,amnm)
(xc,yc) Pi
xc = xi + ricos(ai)
yc = yi + risin(ai)
H.T. table

26. Generalized H.T. Algorithm:
Finds a rotated, scaled, and translated version of the curve:
Form an A accumulator array of possible reference points (xc,yc), scaling factor S and Rotation angle q.
For each edge (x,y) in the image:
Compute f(x,y)
For each (r,a) corresponding to f(x,y) do:
For each S and q:
xc = xi + r(f) S cos[a(f) + q]
yc = yi + r(f) S sin[a(f) + q]
A(xc,yc,S,q) = A(xc,yc,S,q) + 1
Find maxima of A. fj aj q
Srj Pj Pi fi
Sri ai q
(xc,yc) Pk fi
Srk ak q
xc = xi + ricos(ai)
yc = yi + risin(ai)

27. Another variant of the Generalized Hough Transform
4/23/2017
Find Object Center given edges
Create Accumulator Array
Initialize:
For each edge point
For each entry in table, compute:
Increment Accumulator:
Find Local Maxima in 27

28. Generalize HT applied for circuits

29. Properties of Generalized Hough Transform
• What can we do when the curve we want to detect is not easily described parametrically?
~ By this, we mean, it cannot be captured in a relatively small number of parameters.
~ Recall, the dimensionality of the Hough space equal the number of parameters!
• The GHT constructs a parametric description of an arbitrary shape based on a learning process.
• This parametric description is not, in general, compact.
• We will begin by assuming the size, shape, and rotation (orientation) of the region is known a priori. (Or that we want only to detect instances of a given size and orientation.
~ The voting space is (equivalent to) image space, 2D, in the case of known size and rotation.
~ We will see how to deal with unknown orientation and size shortly -- with a 4D Hough space.

30. The list of ( , ) pairs, for a given and constitutes
: An arbitrary reference point inside the shape.
: The length of the j-th line from the reference point to
the shape perimeter, intersecting at a point of tangent
angle ø.
: The angle of the (current) tangent(s) to the perimeter.
: The orientation of the j-th line segment.
The list of ( , ) pairs, for a given and constitutes
a partial characterization of the shape.

31. • By sweeping the tangent angle (ø) over the range (0,2π) in
some reasonable quantization (!), we build what is called
the R-table (reference table) description of the shape.
• Each pixel x (say, a detected edge point) with local
orientation ø provides evidence (votes for) reference
points at the set of locations indicated by the list in the
R-table for that tangent direction...
• A vote is cast for each (r , ) pair in the list for that ø value.
The voting space is isomorphic to image space.
• Again, this assumes known size and orientation for all
appearances of the shape.
• After all the edge points have voted for all of their possible
reference points, we interrogate the voting space for
significant local maxima. These suggest possible
detections of the shape of interest.

32. • If we have not prenormalized for size (S) and rotation ( )
then our voting space is four dimensional and the reference location receiving the vote(s) for a given edge point and R-table entry is:
• Now, we interrogate the 4D accumulator array to recover likely locations, scale, and orientation for appearances of the shape.
• This is really a fancy form of a template match -- but one that is far more robust than a straightforward template matching algorithm.
• Selecting among multiple possible shapes requires multiple R-tables, multiple voting spaces.
But, so does looking for lines and circles in the same image....

33. Generalized HT in biologically motivated robotics

34. Bimodal Active Stereo

35. Many simultaneous problems in robotics

36. Research Philosophy

37. The main concept of Radon Transform

38. The main concept of Radon Transform

63. Hough Transform: Comments
4/23/2017
Hough Transform: Comments
Works on Disconnected Edges
Relatively insensitive to occlusion
Effective for simple shapes (lines, circles, etc)
Trade-off between work in Image Space and Parameter Space
Handling inaccurate edge locations:
Increment Patch in Accumulator rather than a single point 63

64. end H.T. Summary H.T. is a “voting” scheme
points vote for a set of parameters describing a line or curve.
The more votes for a particular set
the more evidence that the corresponding curve is present in the image.
Can detect MULTIPLE curves in one shot.
Computational cost increases with the number of parameters describing the curve.
end