1. Computer Vision - A Modern Approach
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...)
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

2. Fitting and the Hough Transform
Purports to answer all three questions
in practice, answer isn’t usually all that much help
We do for lines only
A line is the set of points (x, y) such that
Different choices of q, d>0 give different lines
For any (x, y) there is a one parameter family of lines through this point, given by
Each point gets to vote for each line in the family; if there is a line that has lots of votes, that should be the line passing through the points
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

3. Computer Vision - A Modern Approach
Figure 15.1, top half. Note that most points in the vote array are very dark, because they
get only one vote.
tokens
votes
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

4. 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
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

5. Computer Vision - A Modern Approach
This is 15.1 lower half
tokens
votes
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

6. Computer Vision - A Modern Approach
15.2; main point is that lots of noise can lead to large peaks in the array
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

7. Computer Vision - A Modern Approach
This is the number of votes that the real line of 20 points gets with increasing noise (figure15.3)
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

8. Computer Vision - A Modern Approach
Figure 15.4; as the noise increases in a picture without a line, the number of points in the max cell goes
up, too
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

9. Computer Vision - A Modern Approach
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.
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

10. 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!)
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

11. Computer Vision - A Modern Approach
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.
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

12. Computer Vision - A Modern Approach
You should point out that a little additional cleverness will be required to avoid having lines
that pass through one point, etc.
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

13. Computer Vision - A Modern Approach
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
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

14. Computer Vision - A Modern Approach
Least squares fit to the red points
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

15. Computer Vision - A Modern Approach
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).
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

16. Computer Vision - A Modern Approach
Same problem as above
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

17. Computer Vision - A Modern Approach
Detail of the previous slide - the line is actually quite far from the points. (All this is fig 15.7)
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

18. Computer Vision - A Modern Approach
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
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
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.
Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth

19. Computer Vision - A Modern Approach
Set: Fitting
Slides by D.A. Forsyth