Generalized Hough Transform

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Generalized Hough Transform
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The Generalized Hough Transform
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From Standard to Generalized HT

1. Standard Hough Transform requires parametric
   representation for desired curve
2. This idea is generalized in the Generalized Hough
   Transform

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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?

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Hough Transform in General

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

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Key Idea to improve correlation by voting

1. When we compute the correlation by voting, we
   spend most of the time casting bad votes.
2. Idea is to use extra shape information (e.g.
   gradients) to cast fewer votes
3. O(n) complexity For each of O(n) points on the
   boundary, cast O(1) votes.

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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 Im pixel i on the boundary, the reference
  point is at refi

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The Generalized Hough Transform

1. Technique to find arbitrary curves in a given
   image
2. Parametric equation no longer required
3. Look-up table used as transform mechanism
4. Two phases
5. R-Table Generation phase
6. Object Detection phase

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The Generalized Hough Transform

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

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Building the R-Table in GHT
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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
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GHT Building the R-Table
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GHT Building the R-Table
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GHT Building the R-Table
GHT Building the R-Table
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Object Localization in the R-Table in GHT
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GHT Object Localization
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GHT Object Localization
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GHT Object Localization
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Conclusions on GHT
Conclusions on GHT

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

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Generalized Hough Transform Algorithm
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Algorithm of the General Hough Transform
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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!

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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

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Generalizing the H.T.
The H.T. can be used even if the curve has not a
simple analytic form!

1. Pick a reference point (xc,yc)
2. For i 1,,n
3. Draw segment to Pi on the boundary.
4. Measure its length ri, and its orientation ai.
5. Write the coordinates of (xc,yc) as a function of
   ri and ai
6. Record the gradient orientation fi at Pi.
7. Build a table with the data, indexed by fi .

xc xi ricos(ai)
yc yi risin(ai)
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Generalizing the H.T.
Suppose, there were m different gradient
orientations (m lt n)
(xc,yc)
Pi
xc xi ricos(ai)
yc yi risin(ai)
H.T. table
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Generalized H.T. Algorithm
Finds a rotated, scaled, and translated version
of the curve

1. Form an A accumulator array of possible reference
   points (xc,yc), scaling factor S and Rotation
   angle q.
2. For each edge (x,y) in the image
3. Compute f(x,y)
4. For each (r,a) corresponding to f(x,y) do
5. For each S and q
6. xc xi r(f) S cosa(f) q
7. yc yi r(f) S sina(f) q
8. A(xc,yc,S,q) A(xc,yc,S,q) 1
9. Find maxima of A.

fj
aj
q
Srj
Pj
(xc,yc)
xc xi ricos(ai)
yc yi risin(ai)
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Another variant of the Generalized Hough Transform
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
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Generalize HT applied for circuits
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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.

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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.
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By sweeping the tangent angle (ø) over the
range (0,2p) 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.
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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....

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Generalized HT in biologically motivated robotics
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Bimodal Active Stereo
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Many simultaneous problems in robotics
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Research Philosophy
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The main concept of Radon Transform
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The main concept of Radon Transform
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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

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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.

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