Fitting Curve Models to Edges
Most contours can be well described by combining
several “round” elements (e.g., circular arcs) and
“straight” elements (straight lines).
• First we will study a method for fitting connected
straight lines to edge data.
• Then we will discuss how to fit a circular arc to a
given set of points.
• Later we will apply a very general and sophisticated
method (the Hough transform) for fitting straight
lines to edge points.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
1
Fitting Curve Models to Edges
Assume that you found edge data that seem to
belong to one contour.
The shape of these data is somewhat non-linear, but
you would like to use multiple connected straight lines
(polylines) to describe the contour.
Then you can use the technique of polyline splitting.
If you have an open contour, start with a straight line
connecting the two (estimated) end points.
If you have a closed contour, start with an
appropriate polygon (a rectangle usually works well).
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
2
Fitting Polylines to Edges
Let us look at an example for an open contour:
First, use a single straight line to connect the end
points.
Then find the edge point with the greatest distance
from this straight line.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
3
Fitting Polylines to Edges
Let us look at an example for an open contour:
Then split the straight line in two straight lines that
meet at this point.
Now repeat this process with each of the two new
lines.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
4
Fitting Polylines to Edges
Let us look at an example for an open contour:
Recursively repeat this process until the maximum
distance of any point to the polyline falls below a
certain threshold.
The result is the fitting polyline.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
5
Fitting Circular Arcs to Edges
Now we will discuss how to fit a circular arc to a set
of edge points.
Consider three points p1 = (x1, y1), p2 = (x2, y2), and
p3 = (x3, y3), which are pairwise different and do not
lie on the same straight line.
Then we already know that there is exactly one
circle that goes through all of these points.
We will now describe how to find this circle, i.e.,
determine its center and radius.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
6
Fitting Circular Arcs to Edges
The points (x, y) of a circle centered at (x0, y0) and
with radius r are given by the following equation:
(x – x0)2 + (y – y0)2 = r2.
To simplify the computation, we will use a coordinate
system whose origin is in point p1 = (x1, y1).
In the new coordinate system we have:
x’ = x – x1
y’ = y – y1
(x’ – x’0)2 + (y’ – y’0)2 = r2.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
7
Fitting Circular Arcs to Edges
Since the equation (x’ – x’0)2 + (y’ – y’0)2 = r2 is true
for all points (x’, y’) on the circle, it must also apply to
p1, p2, and p3.
Substituting (x’, y’) with p1, p2, and p3 gives us:
x’02 + y’02 - r2 = 0
x’22 - 2x’2x’0 + x’02 + y’22 – 2y’2y’0 + y’02 - r2 = 0
x’32 - 2x’3x’0 + x’02 + y’32 – 2y’3y’0 + y’02 - r2 = 0
Subtract the first equation from the other two:
2x’2x’0 + 2y’2y’0 = x’22 + y’22
2x’3x’0 + 2y’3y’0 = x’32 + y’32
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
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Fitting Circular Arcs to Edges
This gives us two linear equations with two unknown
variables x’0 and y’0.
From these we can easily determine the center
(x’0, y’0) of the circle.
The radius of the circle can then be determined by:
r2 = x’02 + y’02.
We simply add (x1, y1) to (x’0, y’0) to get the center
(x0, y0) of the circle in the original coordinate system.
Then we have completely determined the circle going
through points p1, p2, and p3.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
9
Fitting Circular Arcs to Edges
In practice we usually have more than three edge
points through which we want to fit a circular arc.
One possible way to do the fitting in such a case is to
take the two end points of those edges as points p1
and p2.
Then we successively take every other edge point as
point p2 and fit a circular arc to each generated set of
points p1, p2, and p3.
In each case, we measure the deviation of the arc
from all edge points.
The arc with minimum deviation is chosen as the
curve model for these edges.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
10
Hough Transform
The Hough transform is a very general technique for
feature detection.
In the present context, we will use it for the detection
of straight lines as contour descriptors in edge point
arrays.
We could use other variants of the Hough transform
to detect circular and other shapes.
We could even use it outside of computer vision, for
example in data mining applications.
So understanding the Hough transform may benefit
you in many situations.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
11
Hough Transform
The Hough transform is a voting mechanism.
In general, each point in the input space votes for
several combinations of parameters in the output
space.
Those combinations of parameters that receive the
most votes are declared the winners.
We will use the Hough transform to fit a straight line
to edge position data.
To keep the description simple and consistent, let us
assume that the input image is continuous and
described by an x-y coordinate system.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
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Hough Transform
A straight line can be described by the equation:
y = mx + c
The variables x and y are the parameters of our input
space, and m and c are the parameters of the output
space.
For a given value (x, y) indicating the position of an
edge in the input, we can determine the possible
values of m and c by rewriting the above equation:
c = -xm + y
You see that this represents a straight line in m-c
space, which is our output space.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
13
Hough Transform
Example: Each of the three points A, B, and C on a straight line
in input space are transformed into straight lines in output space.
y
c
C
C
winner
parameters
B
B
A
0
A
x
input space
0
m
output space
The parameters of their crossing point (which would be the
winners) are the parameters of the straight line in input space.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
14
Hough Transform
Hough Transform Algorithm:
1. Quantize input and output spaces appropriately.
2. Assume that each cell in the parameter (output)
space is an accumulator (counter). Initialize all
cells to zero.
3. For each point (x, y) in the image (input) space,
increment by one each of the accumulators that
satisfy the equation.
4. Maxima in the accumulator array correspond to
the parameters of model instances.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
15
Hough Transform
The Hough transform does not require
preprocessing of edge information such as ordering,
noise removal, or filling of gaps.
It simply provides an estimate of how to best fit a
straight line (or other curve model) to the available
edge data.
If there are multiple straight lines in the image, the
Hough transform will result in multiple peaks. You can
search for these peaks to find the parameters for all
the corresponding straight lines.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
16
Improved Hough Transform
Here is some practical advice for doing the Hough
transform (e.g., for some future assignment).
The m-c space described on the previous slides is
simple but not very practical. It cannot represent
vertical lines, and the closer the orientation of a line
gets to being vertical, the greater is the change in m
required to turn the line significantly.
We are going to discuss an alternative output space
that requires a bit more computation but avoids the
problems of the m-c space.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
17
Improved Hough Transform
As we said before, it is problematic to use m (slope)
and c (intercept) as an output space.
Instead, it is a good idea to use the orientation  and
length d of the normal of a straight line to describe it.
The normal n of a straight line l is perpendicular to l and
connects l with the origin of the coordinate system.
The range of  is from 0 to 360, and the range of d is
from 0 to the length of the image diagonal.
Note that we can skip the  interval from 180 to 270,
because it would require a negative d.
Let us assume that the image is 450×450 units large.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
18
Improved Hough Transform
line to be
described
450
636
y
representation
of same line in
output space
d
d

0
0
x
input space
0
450
0

output space
360
The parameters  and d form the output space for our Hough
transform.
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
19
Improved Hough Transform
For any edge point (x0, y0) indicated by our Sobel
edge detector, we have to find all parameters  and
d for those straight lines that pass through (x0, y0).
We will then increase the counters in our output
space located at every (, d) by the edge strength,
i.e., the magnitude provided by the Sobel detector.
This way we will find out which parameters (, d) are
most likely to indicate the clearest lines in the image.
But first of all, we have to discuss how to find all the
parameters (, d) for a given point (x0, y0).
October 8, 2013
Computer Vision
Lecture 11: The Hough Transform
20
Improved Hough Transform
By varying  from 0 to 360 we can find all lines crossing (x0, y0):
450
But how can we
compute parameter d
for each value of ?
(x0, y0)
y
Idea: Rotate (x0, y0)
around origin by - so
that it lands on x-axis.
Then the x-coordinate
of the rotated point is
the value of d.
d2
d1
2
1
0
0
October 8, 2013
d3
3
x
450
Computer Vision
Lecture 11: The Hough Transform
21