1. Fitting

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Fitting
Choose a parametric object/some objects to represent a set of tokens
Most interesting case is when criterion is not local
can’t tell whether a set of points lies on a line by looking only at each point and the next.
Three main questions:
what object represents this set of tokens best?
which of several objects gets which token?
how many objects are there?
(you could read line for object here, or circle, or ellipse or...)
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3. Hough transform : straight lines
implementation :
1. the parameter space is discretised
2. a counter is incremented at each cell
where the lines pass
3. peaks are detected 
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4. Hough transform : straight lines
problem : unbounded parameter domain
vertical lines require infinite a
alternative representation:
Each point will add a cosine function in the
(,) parameter space 
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tokens
Figure 15.1, top half. Note that most points in the vote array are very dark, because they
get only one vote.
votes
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6. Hough transform : straight lines
Square :
Circle : 
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7. Hough transform : straight lines
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8. Mechanics of the Hough transform
Construct an array representing q, d
For each point, render the curve (q, d) into this array, adding one at each cell
Difficulties
how big should the cells be? (too big, and we cannot distinguish between quite different lines; too small, and noise causes lines to be missed)
How many lines?
count the peaks in the Hough array
Who belongs to which line?
tag the votes
Hardly ever satisfactory in practice, because problems with noise and cell size defeat it
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This is 15.1 lower half
tokens
votes
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15.2; main point is that lots of noise can lead to large peaks in the array
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This is the number of votes that the real line of 20 points gets with increasing noise (figure15.3)
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Figure 15.4; as the noise increases in a picture without a line, the number of points in the max cell goes
up, too
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Hough Transform (Summary)
HT is a voting algorithm: each point votes for all combinations of parameters which may have produced it if it were part of the target curve.
The array of counters in parameter space can be regarded as a histogram.
The final total of votes c(m) in a counter of coordinate m indicates the relative likelihood of the hypothesis “a curve with parameter set m exists in the image”.
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Hough Transform (Summary)
HT can also be regarded as pattern matching: the class of curves identified by the parameter space is the class of patterns.
HT is more efficient than direct template matching which compares all possible appearances of the pattern with the image.
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Hough Transform Advantages
First all points are processed independently, it copes well with occlusion (if the noise does not result in peaks as high as those created by the shortest true lines).
Second it is relatively robust to noise, as spurious points are unlikely to contribute consistently to any single bin, just generate background noise.
Third, it detects multiple instances of a model in a single pass.
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16. likelihood - but choice of model is important
standard least-squares
Line fitting can be max.
likelihood - but choice of
model is important
This is figure 15.5; I didn’t have the energy to write the algebra around that figure out on a slide,
but the point is well worth making.
total least-squares
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17. Who came from which line?
Assume we know how many lines there are - but which lines are they?
easy, if we know who came from which line
Three strategies
Incremental line fitting
K-means
Probabilistic (later!)
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Incremental line fitting is a really useful idea; it’s the easiest way to fit a set of lines to a curve, and is
often quite useful.
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You should point out that a little additional cleverness will be required to avoid having lines
that pass through one point, etc.
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Robustness
As we have seen, squared error can be a source of bias in the presence of noise points
One fix is EM - we’ll do this shortly
Another is an M-estimator
Square nearby, threshold far away
A third is RANSAC
Search for good points
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Least squares fit to the red points
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Also a least squares fit. The problem is the single point on the right; the error for that point is so
large that it drags the line away from the other points (the total error is lower if these points are some way from the line, because then the point on the line is not so dramatically far away).
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Same problem as above
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Detail of the previous slide - the line is actually quite far from the points. (All this is fig 15.7)
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M-estimators
Generally, minimize
where is the residual
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Figure The issue here is that one wants to replace (distance)^2 with something that looks like
distance^2 for small distances, and is about constant for large distances. We use d^2/(d^2+s^2)
for some parameter s, for different values of which the curve is plotted.
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Fit to earlier data with an appropriate choice of s
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Too small
Here the parameter is too small, and the error for every point looks like distance^2
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Too large
Here the parameter is too large, meaning that the error value is about constant for every point, and there
is very little relationship between the line and the points.
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RANSAC
Issues
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
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
This algorithm contains concepts that are hugely useful; in my experience, everyone with some
experience of applied problems who doesn’t yet know RANSAC is almost immediately able to use it to simplify some problem they’ve seen.
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46. Distance threshold Choose t so probability for inlier is α (e.g. 0.95)
Often empirically
Zero-mean Gaussian noise σ then follows
distribution with m=codimension of model
(dimension+codimension=dimension space)
Codimension
Model
t 2 1
line,F
3.84σ2 2
H,P
5.99σ2 3 T
7.81σ2
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47. How many samples?
Choose N so that, with probability p, at least one random sample is free from outliers. e.g. p=0.99
proportion of outliers e s 5%
10%
20%
25%
30%
40%
50% 2 3 5 6 7 11 17 4 9 19 35 13 34 72 12 26 57
146 16 24 37 97
293 8 20 33 54
163
588 44 78
272
1177
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48. Acceptable consensus set?
Typically, terminate when inlier ratio reaches expected ratio of inliers
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49. Fitting curves other than lines
In principle, an easy generalisation
The probability of obtaining a point, given a curve, is given by a negative exponential of distance squared
In practice, rather hard
It is generally difficult to compute the distance between a point and a curve
More detail in the book. I typically draw on the whiteboard and wave my hands for this particular topic,
rather than work from slides.
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