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1. 1
CONTOURING
By
Ashish V. Deshpande
Electrical Engineering Department
The University of Texas at Arlington
June 21st
2004

2. 2
OVERVIEW
• Geometry of Curves
• Contour Properties
• Curve Fitting

3. 3
CONTOUR
• Edges are linked into a representation for a
region boundary called contour
• Closed Contour
• correspond to region boundaries
• Open Contour
• part of a region boundary
• gap in boundary : Edge detection threshold too
high , weak contrast

4. 4
Representing a Contour
• Criteria for good representation
• Efficiency: Simple, compact representation.
• Accuracy: Fit image features accurately.
• Effectiveness: Suitable for the operations performed in the
later stages of the application
• Two ways
• Ordered list of edges
• Curve
(mathematical model of contour eg.Line segments , Cubic splines etc.)

5. 5
Representing a Contour
• Ordered List
• Efficiency : least compact & simplest
• Accuracy : as accurate as estimate of edge locations
• Effectiveness : not good
• Curves
• Efficiency : increases
• Accuracy : as errors in edge detection are averaged
• Effectiveness : increases

6. 6
Some Definitions
• A curve interpolates a set of points if it passes
through the points
• Approximation is fitting a curve passing close to the
points but not necessarily through the points
• An edge list is an ordered set of edge points.
• A contour is an edge list or the curve that has been
used to represent the edge list.
• A boundary is the closed contour that surrounds a
region.

7. 7
Geometry of Curves
• Representing a point
• Explicit form y = f(x)
• Implicit form f(x,y)=0
• Parametric form (x(u), y(u))
u is the parameter
• Length of an arc
∫ 





+





2
1
22u
u
du
du
dy
du
dx

8. 8
Contour Properties
• Let pi = (xi,yi) be the coordinates of edge i in the edge list
• Estimate the slope using edge points that are not adjacent :
• Left k-slope: direction from pi-k to pi
• Right k-slope: direction from pito pi+k
• K-curvature: diff. between the left and right k-slopes.
• Length of digital curve
• Distance between end points
• S/D ratio (covered in curve fitting)
∑=
−− −+−=
n
i
iiii yyxxS
2
2
1
2
1 )()(
2
1
2
1 )()( xxyyD nn −+−=

9. 9
Representation using lists
• Chain Codes : specifies direction of contour at
each edge along the edge list
• Directions are quantized
• Starting at first edge and going clockwise around the
contour the direction is specified using one of the
directions

10. 10
Chain Codes
• Example
• Limitations
• Limited set of directions
• Sensitive to noise

11. 11
Slope Representation
• Continuous version of Chain code , also called y-s
plot
• Algorithm:
1.Start at the first edge and go clockwise
2.Estimate slope (y) and arc length (s)
3.Plot the slope Versus the arc length
• Horizontal line segments in y-s space correspond to
segments in the contour
• Line segments at other orientations correspond to
circular arcs
• Portions of the plot that are not straight line segments
correspond to other curve primitives

12. 12
An Example

13. 13
Centroidal Profile Representation
• Plot the distance from the centroid to the
boundary as a function of angle

14. 14
Curve Fitting
• Four Models
• Line Segments
• Circular arcs
• Conic sections
• Cubic splines
• Note:
We proceed with the basic assumption that the edge
locations are sufficiently accurate that selected edge points
can be used to determine the fit. We will later consider
methods based on approximation that can handle such
errors.

15. 15
Goodness of a fit
• Let di be the distance of edge point i from line.
• Maximum absolute error : measures how much the
points deviate from the curve in the worst case.
• Mean squared error : gives an overall measure of
the deviation of the curve from the edge points.

16. 16
Goodness of a fit contd…
• Normalized maximum absolute error: the ratio of the
maximum absolute error to the length of the curve (error
becomes independent of the length of the curve ,i.e unitless ).
• Ratio of curve length to the end point distance: a good
measure of the complexity of the curve.
• Number of Sign Changes: in the error is a good indicator of
the appropriateness of the curve as a model for the edges in
the contour.

17. 17
Curve fitting methods
• The use of straight line segments (polylines) is
appropriate if the image consists of straight lines. It
is the starting point for fitting other models.
• Circular arcs are a useful representation for
estimating curvature
• Conic sections provide a convenient way to
represent sequences of line segments and circular as
well as elliptic and hyperbolic arcs .
• Cubic splines are good to model smooth curves

18. 18
Polyline Representation
• A polyline is a sequence of line segments joined end to end.
The ends of each line are edge points in the original edge list.
The points where line segments are joined are called vertices.
• A quick revision
• Equation of line passing from two points (x1, y1), (x2, y2):
• Distance d of edge point (u, v) from the line:
• D is the distance between (x1, y1) and (x2, y2).

19. 19
Polyline Representation types
• Splitting
• Merging
• Split and Merge Algorithm
• Hop-Along Algorithm

20. 20
Splitting (Top Down)
• Algorithm :
(1) Take the line segment connecting the end points of the
contour (if the contour is closed, consider the line
segment connecting the two farthest points)
(2) Find the farthest edge point from the line segment
(3) If the normalized maximum error of that point from the
line segment is above a threshold, split the segment into
two segments at that point (i.e., new vertex)
(4)Repeat the same procedure for each of the two
subsegments until the error is very small.

21. 21
Example (Open Contour)

22. 22
Example (Closed Contour)

23. 23
Merging (Bottom Up)
• Algorithm :
(1) Use the first two edge points to define a line segment.
(2) Add a new edge point if it does not deviate too far from
the current line segment.
(3) Update the parameters of the line segment using least-
squares.
(4) Start a new line segment when edge points deviate too
far from the line segment.

24. 24
Tolerance Band
• Algorithm:
(1) Two line segments that are parallel to the line segment
approximating the edge at a distance ε from the center
are computed.
(2) Edges are added to the current line segment as long as
the new edges are inside the tolerance band.
(3) The parameters of the line segment are recomputed as
the edges are added to the segment.
(4) The vertex at the end of the segment is the starting point
for the next segment

25. 25
Example

26. 26
Split and Merge
• The accuracy of line segment approximations can be
improved by interleaving merge and split
operations.
• Algorithm:
(1) After recursive subdivision (split), allow
adjacent segments to be replaced by a single
segment (merge).
(2) Alternate applications of split and merge until
no segments are merged or split.

27. 27
Example

28. 28
Hop Along Algorithm
1. Start with the first k edges from the list.
2. Fit a line segment between the first and the last edges in the
sub list.
3. If the normalized maximum error is too large, shorten the
sub list to the point of maximum error. Return to step 2.
4. If the line fit succeeds, compare the orientation of the current
line segment with that of the previous line segment. If the
lines have similar orientations, replace the two line segments
with a single line segment.
5. Make the current line segment the previous line segment and
advance the window of edges so that there are k edges in the
sublist. Return to step 2.

29. 29
Circular arcs
• Subsequences of the line segments can be
replaced by circular arcs if desired. Replacing
by arcs involves fitting circular arcs through
the end points of two or more line segments,
i.e. fitting is done on vertices in the polyline
• The circular arc is a fit between the first and
last end points and one other point. There
can be several methods depending on how
the middle point is chosen .

30. 30
Replacing line segments with arcs
• Algorithm:
1.Initialize the window of vertices to the 3 end points of the
first 2 line segments in the polyline.
2.Compute the ratio of the length of the part of the contour
corresponding to the 2 line segments to the distance
between the end points. If the ratio is large then leave the
first line segment unchanged, advance the window of
vertices by one vertex, and repeat this step.
3.Fit a circle through the 3 vertices.
4.Calculate the normalized error and number of sign
changes.

31. 31
Contd…
5. If the normalized maximum error is too large or the
number of sign changes is too small then leave the first
line segment unchanged, advance the window of vertices
and return to step 2.
6.If the circle fit succeeds, then try to include the next line
segment in the circular arc. Repeat this step until no more
line segments can be sub-summed by this circular arc.
7.Advance the window to the next 3 polyline vertices after
the end of the circular arc and return to step 2.

32. 32
Error Calculation
• To calculate error in fitting a circular arc , define
distance of point Q (x1, y1) from the circle along a
line passing through the center (x0, y0)
• q =
The distance from Q to the circular arc is d=q - r
2
01
2
01 )()( yyxx −+−

33. 33
Conic Sections & Cubic Splines
• Conic sections correspond to hyperbolas, parabolas,
and ellipses (2nd degree polynomials).
• Cubic splines correspond to 3rd degree polynomials.
• Conic sections and cubic splines allow more
complex contours to be represented using fewer
segments.
• Conic sections can be fit between three vertices in
the polyline approximation.

34. 34
Conic section between three points

35. 35
Curve Approximation
• Higher accuracy can be obtained by computing an approximation
that is not forced to pass through particular edge points.
• Approximation based methods use all the edge points to find a
good fit (this is contrast to the previous methods that use only the
interpolated edge points).
• Methods to approximate curves depend on the reliability with
which edge points can be grouped into contours.
(1) Least-squares regression can be used if it is certain that the
edge points grouped together belong to the same contour.
(2) Robust regression is more appropriate if there are some
grouping errors.
(3) If grouping cannot be done easily using edge linking or
following methods , then cluster analysis techniques like
Hough transform are used

36. 36
Least-squares line fit
• Linear regression fits a line to the edge points by minimizing
the sum of squares of the perpendicular distances of the edge
points from the line being fit.
∑ +−=
i
ii yxX 22
)cossin( ρθθ

37. 37
Some math…(solution to the regression problem)

38. 38
Limitations
• It works well for cases where the noise follows a
Gaussian distribution.
• It does not work well when there are outliers
(edge points incorrectly linked to the contour) in
the data (e.g., due to errors in edge linking ).

39. 39
Robust Regression
• Outliers can pull the regression line far away from
its correct location.
• Robust regression methods try various subsets of the
data points and choose the subset that provides the
best fit.

40. 40
Least - Median Suares Regression
• Algorithm:
(assume that there are n data points and p parameters in the model)
(1) Choose p points at random from the set of n data points.
(2) Compute the fit of the model to the p points.
(3) Compute the median of the square of the residuals.
(4) Repeat until the median squared error is less than some
threshold, or take the best median error result after
some number of runs.
• This algorithm can handle up to 50% outliers in the
data set.

41. 41
Hough Transform
• Voting mechanism
• Each point on the curve votes for several
combinations of parameters, the parameters that
win a majority of votes are winners
• This technique assumes that a primitive edge
detection is already performed on an image

42. 42
The technique
• Consider the equation of a straight line y= mx +c
• Rewriting the equation c= y – mx
• Let us assume that m and c are variables of interest and x
and y are constants. The above equation represents a line in
m-c space.
• A point (x,y) corresponds to line in m-c space
• If there are n points lying on the straight line (in x-y plane),
then these will correspond to a family of lines in parameter
space all passing through (m,c). This point gives parameters
of original straight line

43. 43
Hough Transform Algorithm
1. Quantize the parameter space appropriately
2. Assume that each cell in the parameter space is an
accumulator. Initialize all cells to zero.
3. For each point (x,y) in the image space , increment by 1
each of the accumulators that satisfy the equation
4. Maxima in the accumulator array correspond to the
parameters of model instances
5. If the peak in the parameter space covers more than 1
accumulator then the centroid of the region gives an
estimate of the parameters

44. 44
A Demonstration
• The data is given on the left. The result of the Hough transform is a
bright orange box on the right of the figure (the Hough space) which is
the result of a large number of counts for that accumulator.

45. 45
Limitations
• Size of discrete parameter space increases
exponentially as number of parameters
increases hence Hough transform may be
computationally very inefficient for complex
models

46. 46
Fourier Descriptors
• Fourier coefficients are used to represent a closed contour
since they are periodic.
• Boundary of an object is expressed as a sequence of
coordinates
• We can represent each co ordinate pair as a complex number
so that
• For a closed curve u(n) is periodic
• DFT
• IDFT
)](),([)( nynxnu =
1,....2,1,0),()()( −=+= Nnnjynxnu
∑
−
=
∏
−≤≤=
1
0
/2
10,)()(
N
k
Nknj
Nnekanu
∑
−
=
∏−
−≤≤=
1
0
2
10,)(
1
)(
N
n
N
nkj
Nnenu
N
ka

47. 47
Fourier Descriptors (contd..)
• The complex coefficients a(k) are called Fourier
descriptors (FD) of the boundary
• Low resolution approximation
• Only first M coefficients are used
∑
−
=
∏
−≤≤=
1
0
2
10,)()(
M
k
N
knj
Nnekanu

48. 48
Summary
Contour Representation
Using Interpolation
Using Edge list
Using Curve fitting
Using Approximation
Using Regression
Using Hough Transform
Using Fourier Descriptors

49. 49
References
• Rafael C. Gonzalez, Digital Image Processing
• Ramesh C. Jain, et al., Machine Vision
• Adrian Low, Introductory Computer Vision and
Image Processing

50. 50
Thank You !!