Curve Fitting

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Curve Fitting
Lecture 11
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• Three main questions
• what object (line, circle, ellipse,) represents
  this set of points best?
• how many objects are there?
• which of several objects gets which point?

What criteria of fitting?
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Fitting lines
Number of lines 1
Error ? min
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d
d
d f(x) - y
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Least Square Error
Maximum Absolute Error
Mean Square Error
Normalized Maximum Error
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?r²i,j ?min
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Assume we do not know how many lines there are.
Polyline fitting

1. Splitting
2. Merging

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Polyline splitting
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Polyline splitting
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Polyline splitting
dmax
dmax lt Thresh
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• Polyline Splitting
• Input L- list of pixels (edges), Thresh
• L(x1,y1),(x2,y2),,(xn,yn)
• Output Polyline
• Fit a line between the first and last edges in
  the list
• If the ? gt Thresh, split the list by the point
  of maximum error (xi,yi) L1(x1,y1),(x2,y2),,(
  xi,yi) and L2(xi,yi),(xi1,yi1),,(xn,yn)
• 2.1 Polyline Splitting(L1,Thresh)
• 2.2 Polyline Splitting(L2,Thresh)

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Incremental line fitting (Merging)
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Incremental line fitting (Merging)
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Incremental line fitting (Merging)
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Hop-Along Polyline fitting (SplitMerge)
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Hop-Along Polyline fitting (SplitMerge)

1. Start with the first k edges from the list
2. Fit a line between the first and the last edges
   in the sublist
3. If NME gt Thresh , shorten the sublist 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 the previous
   line segment. If the lines have similar
   orientation, replace the two line segments with a
   single line.
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 the
   step 2.

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If the number of lines is known
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Robustness
A computer vision algorithm is called robust if
it can tolerate outliers (data which does not
obey the assumed model).
Robustness is the ability to extract the visual
information of relevance for a specific task,
even when this information is carried only by a
small subset of the data.
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LSE criteria
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LSE criteria
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Least-median-squares
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RANSAC

• Choose a small subset uniformly at random
• Fit to that
• Anything that is close to result is signal all
  others are noise
• Refit
• Do this many times and choose the best

• How many times?
• Often enough that we are likely to have a good
  line
• How big a subset?
• Smallest possible
• What does close mean?
• Depends on the problem
• What is a good line?
• One where the number of nearby points is so big
  it is unlikely to be all outliers

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Fitting curves other than lines
Circular arcs
P1(x1,y1),P2(x2,y2),P3(x3,y3)
x x - x1, y y - y1
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q
d q - r