Hough Tranform

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Hough Tranform
Lecture 12
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Hough Transform

• Elegant method for direct object recognition
• Edges need not be connected
• Complete object need not be visible
• Key Idea Edges VOTE for the possible model

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Line detection

• Mathematical model of a line

Y mx n
Y1m x1n
Y2m x2n
P(x1,y1)
P(x2,y2)
YNm xNn
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Image and Parameter Spaces
Y mx n
Image Space
Line in Img. Space Point in Param. Space
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Image space
Fix (m,n), Vary (x,y) - Line
Y mx n
Fix (x1,y1), Vary (m,n) Lines thru a Point
Y1m x1n
P(x1,y1)
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Parameter space
Y1m x1n
Can be re-written as
n -x1 m Y1
Fix (-x1,y1), Vary (m,n) - Line
n -x1 m Y1
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Img-Param Spaces

• Image Space
• Lines
• Points
• Collinear points

• Parameter Space
• Points
• Lines
• Intersecting lines

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Hough Transform Technique

• Given an edge point, there is an infinite number
  of lines passing through it (Vary m and
  n).
• These lines can be represented as a line in
  parameter space.

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Hough Transform Technique

• Given a set of collinear edge points, each of
  them have associated a line in parameter space.
• These lines intersect at the point (m,n)
  corresponding to the parameters of the line in
  the image space.

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n (-x1) m y1
y
n (-x2) m y2
p
q
P(x1,y1)
Q(x2,y2)
x
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Hough Transform Technique

• At each point of the (discrete) parameter space,
  count how many lines pass through it.
• Use an array of counters
• Can be thought as a parameter image
• The higher the count, the more edges are
  collinear in the image space.
• Find a peak in the counter array
• This is a bright point in the parameter image
• It can be found by thresholding

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Practical Issues

• The slope of the line is -?ltmlt?
• The parameter space is INFINITE
• The representation y mx n does not express
  lines of the form x k

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Solution
Y mx n
? x cos?y sin?
? - Is the line orientation
? - Is the distance between the origin and the
line
?
?
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New Parameter Space

• Use the parameter space (?, ?)
• The new space is FINITE
• 0 lt ? lt D , where D is the image diagonal.
• 0 lt ? lt ?
• The new space can represent all lines
• Y k is represented with ? k, ?90
• X k is represented with ? k, ?0

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Consequence

• A Point in Image Space is now represented as a
  SINUSOID
• (x,y) ? ? x cos?y sin?

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Image space
Votes Horizontal axis is ?, vertical is ?.
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Image space
votes
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• Basic Hough transform algorithm
• Initialize Hd, ?0
• For each edge point Ix,y in the image
• for ? 0 to 180
• Hd, ? 1
• Find the value(s) of (d, ?) where Hd, ? is
  maximum
• The detected line in the image is given by

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Hough Transform Algorithm

• Input edge image (E(i,j)1 for edgels)
• Discretize ? and ? in increments of d? and d?.
  Let A(R,T) be an array of integer accumulators,
  initialized to 0.
• For each pixel E(i,j)1 and h1,2,T do
• ? i cos(h d? ) j sin(h d? )
• Find closest integer k corresponding to r
• Increment counter A(h,k) by one
• Find local maxima in A(R,T)

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Hough Transform Speed Up

• If we know the orientation of the edge usually
  available from the edge detection step
• We fix ? in the parameter space and increment
  only one counter!
• For each pixel E(i,j)1 compute
• ? i cos(? ) j sin(? )
• Find closest integer k corresponding
  to r
• Increment counter A(h,k) by one

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• Extension
• give more votes for stronger edges

Difficulties how big should the cells be? (too
big, and we merge quite different lines too
small, and noise causes lines to be missed)
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Image Transforms - Hough Transform
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Real World Example
Original
Found Lines
Edge Detection
Parameter Space
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????? ?????
????? ????? ???????
???? ?????? ?????? ?? ??????
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Finding Circles by Hough Transform
Equation of Circle
If radius is known
(2D Hough Space)
Accumulator Array
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Finding Circles by Hough Transform
Equation of Circle
If radius is not known 3D Hough Space! Use
Accumulator array
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Using Gradient Information

• Gradient information can save lot of
  computation

Edge Location Edge Direction
Assume radius is known
Need to increment only one point in Accumulator!!
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Finding Coins
Original
Edges (note noise)
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Finding Coins (Continued)
Penny
Quarters
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Finding Coins (Continued)
Note that because the quarters and penny are
different sizes, a different Hough transform
(with separate accumulators) was used for each
circle size.
Coin finding sample images from Vivek Kwatra
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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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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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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.