1. Model fitting ECE 847: Digital Image Processing Stan Birchfield Clemson University

2. Acknowledgment Many slides are courtesy of Frank Dellaert, Georgia Tech Jay Summet et al. from http://www-static.cc.gatech.edu/classes/AY2007/cs4495_fall/html/materials.html

3. 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...)

4. Line fitting Least squares Minimize vertical distance Minimize perpendicular distance

5. Fitting using SVD Many 2D curves that can be represented using linear equations (in the coeff of the curve) ax+by+c=0 Conics: x’Ax=0 –includes parabola, hyperbola, ellipses

6. Fitting and the Hough Transform Purports to answer all three questions We explain for lines One representation: a line is the set of points (x, y) such that: (cos  X + (sin  Y = r Different choices of , r>0 give different lines For any token (x, y) there is a one parameter family of lines through this point, given by (cos  X + (sin  Y = r 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

7. Tokens Votes Theta = 45º = 0.785 rad r = (1√2) / 2 = 0.707 Theta: 0 to 3.14 (rad) r: 0 to 1.55 Brightest point = 20 votes

8. Mechanics of the Hough transform Construct an array representing , r For each point, render the curve ( , r) 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

9. Hough transform image spaceline space Notice that ( ,  ) and (  + ,-  ) are the same line That’s why we get two peaks Solution: Let 0 <=  <  Note symmetry: -flip vertical - then slide by 

10. Tokens Votes Brightest point = 6 votes  r

12. Noise Lowers the Peaks

13. Noise Increases the Votes in Spurious Accumulator Elements

14. Optimizations to the Hough Transform Noise: If the orientation of tokens (pixels) is known, only accumulator elements for lines with that general orientation are voted on. (Most edge detectors give orientation information.) Speed: The accumulator array can be coarse, then repeated in areas of interest at a finer scale.

15. Real World Example Original Edge Detection Found Lines Parameter Space

16. Fitting other objects The Hough transform can be used to fit points to any object that can be paramatized. (e.g. Circle, elipse) Objects of arbitrary shape can be parameterized by building an R-Table. (Assumes orientation information for each token is available.) R and beta value(s) are obtained from the R-Table, based upon omega (orientation)

17. Circle Example With no orientation, each token (point) votes for all possible circles. With orientation, each token can vote for a smaller number of circles.

18. Real World Circle Examples Crosshair indicates results of Hough transform, bounding box found via motion differencing.

19. Finding Coins Original Edges (note noise)

20. Finding Coins (Continued) PennyQuarters

21. Finding Coins (Continued) Coin finding sample images from: Vivik Kwatra Note that because the quarters and penny are different sizes, a different Hough transform (with separate accumulators) was used for each circle size.

22. Generalized Hough transform

23. Conclusion Finding lines and other parameterized objects is an important task for computer vision. The (generalized) Hough transform can detect arbitrary shapes from (edge detected) tokens. Success rate depends directly upon the noise in the edge image. Downsides: Can be slow, especially for objects in arbitrary scales and orientations (extra parameters increase accumulator space exponentially).

24. Tensor voting Another voting technique

25. Line fitting can be max. likelihood - but choice of model is important Line fitting

26. 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!)

29. 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

36. Too small

37. Too large

38. 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

40. 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

41. K-means Recall line fitting Now suppose that you have more than one line But you do not know which points belong to which line

42. K-Means Choose a fixed number of clusters (K) Choose cluster centers and point-cluster allocations to minimize error can’t do this by search, because there are too many possible allocations. Algorithm –fix cluster centers; allocate points to closest cluster –fix allocation; compute best cluster centers x could be any set of features for which we can compute a distance (careful about scaling) * From Marc Pollefeys COMP 256 2003

43. K-Means * From Marc Pollefeys COMP 256 2003

44. K-Means compute mean of data points assign data points to clusters

45. Image Segmentation by K- Means Select a value of K Select a feature vector for every pixel (color, texture, position, or combination of these etc.) Define a similarity measure between feature vectors (Usually Euclidean Distance). Apply K-Means Algorithm. Apply Connected Components Algorithm (to enforce spatial continuity). Merge any components of size less than some threshold to an adjacent component that is most similar to it. * From Marc Pollefeys COMP 256 2003

46. Example Image Clusters on intensity Clusters on color

47. Idea Data generated from mixture of Gaussians Latent (hidden) variables: Correspondence between Data Items and Gaussians

48. Expectation-Maximization (Generalized K-Means) Notice: –Given the mixture model, it’s easy to calculate the correspondence –Given the correspondence it’s easy to estimate the mixture models K-Means involves –Model (hypothesis space): Mixture of N Gaussians –Latent variables: Correspondence of data and Gaussians Replace hard assignments with soft assignments  EM EM is guaranteed to converge (EM steps do not decrease likelihood)

49. Learning a Gaussian Mixture (with known covariance) M-Step E-Step

50. Generalized K-Means (EM) compute weighted mean of data points compute weights of assigning data points to clusters

51. EM Clustering: Results 

52. Application: Clustering Flow Clustered Flow ML correspondence

53. Missing variable problems In many vision problems, if some variables were known the maximum likelihood inference problem would be easy –fitting; if we knew which line each token came from, it would be easy to determine line parameters –segmentation; if we knew the segment each pixel came from, it would be easy to determine the segment parameters –fundamental matrix estimation; if we knew which feature corresponded to which, it would be easy to determine the fundamental matrix –etc. This sort of thing happens in statistics, too

54. Missing variable problems Strategy –estimate appropriate values for the missing variables –plug these in, now estimate parameters –re-estimate appropriate values for missing variables, continue eg –guess which line gets which point –now fit the lines –now reallocate points to lines, using our knowledge of the lines –now refit, etc. We’ve seen this line of thought before (k means)

55. Missing variables - strategy We have a problem with parameters, missing variables This suggests: Iterate until convergence –replace missing variable with expected values, given fixed values of parameters –fix missing variables, choose parameters to maximise likelihood given fixed values of missing variables e.g., iterate till convergence –allocate each point to a line with a weight, which is the probability of the point given the line –refit lines to the weighted set of points Converges to local extremum Somewhat more general form is available

56. EM E-step: –calculate errors –calculate probabilities M-step: –re-calculate RGB clusters Demo with fixed sigma=20 segment01.m

57. MATLAB code essentials % init models sigma=20; m(:,g)=128+50*randn(3,1); pi(g)=1/nrClusters; % E-step % calculate errors E{g}=zeros(h,w); e=(I(:,:,c)-m(c,g))/sigma; E{g}=E{g}+e.*e; % unnormalized probabilities q{g}=pi(g)*exp(-0.5*E{g}); % normalize probabilities p{g}=q{g}./sumq; % M-step P=p{g}; R=P.*I(:,:,1); G=P.*I(:,:,2); B=P.*I(:,:,3); pi(g)=sum(P(:)); m(:,g) = [sum(R(:)); sum(G(:)); sum(B(:))]/pi(g); pi=pi/sum(pi)

58. Expectation-Maximization See my TR online We have parameters  and data U Also: hidden “nuisance” variables J We want to find optimal 

59. Example: Mixture 2 Components Gaussian

60. Posterior in Parameter Space 2D !

61. EM Successive lower bounds

62. Line Fitting Parameters  =( , c) c = distance to origin c

63. Lines and robustness We have one line, and n points Some come from the line, some from “noise” This is a mixture model: We wish to determine –line parameters –p(comes from line)

64. Estimating the mixture model Introduce a set of hidden variables, , one for each point. They are one when the point is on the line, and zero when off. If these are known, the negative log-likelihood becomes (the line’s parameters are  c): Here K is a normalising constant, kn is the noise intensity (we’ll choose this later).

65. Substituting for delta We shall substitute the expected value of , for a given  recall  =( , c, ) E(  _i)=1. P(  _i=1|  )+0.... Notice that if kn is small and positive, then if distance is small, this value is close to 1 and if it is large, close to zero

66. Algorithm for line fitting Obtain some start point Now compute  ’s using formula above Now compute maximum likelihood estimate of – , c come from fitting to weighted points – comes by counting Iterate to convergence

68. The expected values of the deltas at the maximum (notice the one value close to zero).

69. Closeup of the fit

70. Other examples Segmentation –a segment is a gaussian that emits feature vectors (which could contain colour; or colour and position; or colour, texture and position). –segment parameters are mean and (perhaps) covariance –if we knew which segment each point belonged to, estimating these parameters would be easy –rest is on same lines as fitting line Fitting multiple lines –rather like fitting one line, except there are more hidden variables –easiest is to encode as an array of hidden variables, which represent a table with a one where the i’th point comes from the j’th line, zeros otherwise –rest is on same lines as above

71. Issues with EM Local maxima –can be a serious nuisance in some problems –no guarantee that we have reached the “right” maximum Starting –k means to cluster the points is often a good idea

72. Choosing parameters What about the noise parameter, and the sigma for the line? –several methods from first principles knowledge of the problem (seldom really possible) play around with a few examples and choose (usually quite effective, as precise choice doesn’t matter much) –notice that if kn is large, this says that points very seldom come from noise, however far from the line they lie usually biases the fit, by pushing outliers into the line rule of thumb; its better to fit to the better fitting points, within reason; if this is hard to do, then the model could be a problem

73. Other examples Segmentation –a segment is a gaussian that emits feature vectors (which could contain colour; or colour and position; or colour, texture and position). –segment parameters are mean and (perhaps) covariance –if we knew which segment each point belonged to, estimating these parameters would be easy –rest is on same lines as fitting line Fitting multiple lines –rather like fitting one line, except there are more hidden variables –easiest is to encode as an array of hidden variables, which represent a table with a one where the i’th point comes from the j’th line, zeros otherwise –rest is on same lines as above

74. Issues with EM Local maxima –can be a serious nuisance in some problems –no guarantee that we have reached the “right” maximum Starting –k means to cluster the points is often a good idea

75. Local maximum

76. which is an excellent fit to some points

77. and the deltas for this maximum

78. A dataset that is well fitted by four lines

79. Result of EM fitting, with one line (or at least, one available local maximum).

80. Result of EM fitting, with two lines (or at least, one available local maximum).

81. Seven lines can produce a rather logical answer

82. Figure from “Color and Texture Based Image Segmentation Using EM and Its Application to Content Based Image Retrieval”,S.J. Belongie et al., Proc. Int. Conf. Computer Vision, 1998, c1998, IEEE Segmentation with EM

83. Motion segmentation with EM Model image pair (or video sequence) as consisting of regions of parametric motion –affine motion is popular Now we need to –determine which pixels belong to which region –estimate parameters Likelihood –assume Straightforward missing variable problem, rest is calculation

84. Three frames from the MPEG “flower garden” sequence Figure from “Representing Images with layers,”, by J. Wang and E.H. Adelson, IEEE Transactions on Image Processing, 1994, c 1994, IEEE

85. Grey level shows region no. with highest probability Segments and motion fields associated with them Figure from “Representing Images with layers,”, by J. Wang and E.H. Adelson, IEEE Transactions on Image Processing, 1994, c 1994, IEEE

86. If we use multiple frames to estimate the appearance of a segment, we can fill in occlusions; so we can re-render the sequence with some segments removed. Figure from “Representing Images with layers,”, by J. Wang and E.H. Adelson, IEEE Transactions on Image Processing, 1994, c 1994, IEEE

87. Some generalities Many, but not all problems that can be attacked with EM can also be attacked with RANSAC –need to be able to get a parameter estimate with a manageably small number of random choices. –RANSAC is usually better Didn’t present in the most general form –in the general form, the likelihood may not be a linear function of the missing variables –in this case, one takes an expectation of the likelihood, rather than substituting expected values of missing variables –Issue doesn’t seem to arise in vision applications.

88. Model Selection We wish to choose a model to fit to data –e.g. is it a line or a circle? –e.g is this a perspective or orthographic camera? –e.g. is there an aeroplane there or is it noise? Issue –In general, models with more parameters will fit a dataset better, but are poorer at prediction –This means we can’t simply look at the negative log- likelihood (or fitting error)

89. Top is not necessarily a better fit than bottom (actually, almost always worse)

91. We can discount the fitting error with some term in the number of parameters in the model.

92. Discounts AIC (an information criterion) –choose model with smallest value of –p is the number of parameters BIC (Bayes information criterion) –choose model with smallest value of –N is the number of data points Minimum description length –same criterion as BIC, but derived in a completely different way

93. Cross-validation Split data set into two pieces, fit to one, and compute negative log- likelihood on the other Average over multiple different splits Choose the model with the smallest value of this average The difference in averages for two different models is an estimate of the difference in KL divergence of the models from the source of the data

94. Model averaging Very often, it is smarter to use multiple models for prediction than just one e.g. motion capture data –there are a small number of schemes that are used to put markers on the body –given we know the scheme S and the measurements D, we can estimate the configuration of the body X We want If it is obvious what the scheme is from the data, then averaging makes little difference If it isn’t, then not averaging underestimates the variance of X --- we think we have a more precise estimate than we do.