Colorado School of Mines
Computer Vision
Professor William Hoff
Dept of Electrical Engineering &Computer Science
Colorado School of Mines
Computer Vision
http://inside.mines.edu/~whoff/
1
RANSAC
Colorado School of Mines
Computer Vision
2
Identifying Correct Matches
•
•
•
•
Let’s say that we want to match points from one image to another
We detect feature points in each image (e.g., SIFT points) and then match
their descriptors
Now we have a set of matching pairs, but some are incorrect
How do we identify the correct matches?
Graffiti image dataset from http://kahlan.eps.surrey.ac.uk/featurespace/web/data.htm
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Computer Vision
3
SIFT Points from each image
Number of features detected: 1741
Colorado School of Mines
Number of features detected: 2052
Computer Vision
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14
143
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34255
162 30
103
440 252
164
385
33
91
23
237
240
263
325
Tentative matches (based on feature descriptors): 442
80
2279
68 23075
20
32
79
48 395
275
86
258
51
129
133
290
Tentative matches
278
333120426
41
38
338
128
208
204
202
125
124
201 427
47
216
215
130
132
376
49
53
340
270
269
102
330
396
429
161 401 98
101
182 329
341
257
256
99
415
16
441 85 386
65
289
274
221 131
399
398
418
354
293137 10
247
139
405 365
428
434
358165
223
226 378
225
93
420
419
346
314
260 267
220
362
288
304
262
196
59 138
58
244166 322 179109 261
203
197
430 309 27
432
183
423 259
332
371
54
152
438
437
175
174
277
276
191
339
181
281
41422
413
178
306
173
271
92
214
74
242
241
189114
439347
194 199273
123
105
285
384
424
94
76
228
344
400
146 21
31
108
200 394
145
144
233 151
308
307
282
388 361442
207
335
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302 153
397
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392
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45 213
291
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360 172
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193 425
352 412
379
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431
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368
326
206372
421
287
78
406
217
297
296
407
42
389
60
19
353 17
380
337
110
87 163
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243
186
369
81
375
410
245167
170323
169
112
370
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50
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62
35
37119
248
61
115
2
70140
134
390 364
212
435
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18
324
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192 433
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298
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402
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417
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135350
345
436
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411 150 24
373342374283
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158357
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319 177
318
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377
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239 356
238
295
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89312359
83
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393 336 408
96
95
84
210 343
106
198
272
43
8890
147 383
116
11
280
279
235
156 25
205
211
66
231381
40
160 316 317
184
382
127
294
310
209
26
97
299
118122
351
321
64 229
246
254
1 67
29
224
315
44
149
292
73 77
52
219
387
168
253
320
28
232 303
250
249
222
416
15
57
301
328
154
3 300141 148
121
55
251
171
409
113
159
8
7
155
404
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634 71 12
234
195
72
266
331
56
Colorado School of Mines
164
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Computer Vision
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59 138
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240385
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395
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102 258
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270
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182329
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134
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165
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105
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125
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93
314
377
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405365
74 413
22 430309
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228
414152
274 427 340
166362
27 244
223
306
333
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322179 260
76 21 384
145
438
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109261
429 396
146
434
262267
92
144
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242
241
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175432
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423183 196
173
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181259
428
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308 400 94
307
197
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106
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191
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189
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194199
352
379
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271 276
442
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388361
277 371
412
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297
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108
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123
339
78
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353 19
380
236
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172 392
61
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439
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2 70
285
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132
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421
87
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326 268
207 282
163 389
243
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36 435
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176 110 368
167
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135
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248 323
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350 142
407 206
372213
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112 369337
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311 390
370
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411 436
364
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115 37
417
324
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150 355
402
287
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180 265
192433
111
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126 212
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363 107185
391
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403
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294
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382 235
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5
% Demonstrate the matching of a plane, from one image to another.
clear variables
close all
%rng(0);
% Reset random number generator
Find SIFT points in
images (1 of 3)
% Read in test images.
I1 = imread('img1.png');
I2 = imread('img3.png');
% If color, convert them to grayscale
if size(I1,3)>1
I1 = rgb2gray(I1);
if size(I2,3)>1
I2 = rgb2gray(I2);
end
end
% Extract SIFT features.
% First make sure the vl_sift code is in the path
if exist('vl_sift', 'file')==0
run('C:\Users\William\Documents\Research\vlfeat-0.9.20\toolbox\vl_setup');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First image
I1 = single(I1); % Convert to single precision floating point
figure(1), imshow(I1,[]);
% These parameters limit the number of features detected
peak_thresh = 0;
% increase to limit; default is 0
edge_thresh = 10;
% decrease to limit; default is 10
[f1,d1] = vl_sift(I1, ...
'PeakThresh', peak_thresh, ...
'edgethresh', edge_thresh );
fprintf('Number of frames (features) detected: %d\n', size(f1,2));
% Show all SIFT features detected
h = vl_plotframe(f1) ;
set(h,'color','y','linewidth',1.5) ;
Colorado School of Mines
Computer Vision
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Second image
I2 = single(I2);
figure(2), imshow(I2,[]);
[f2,d2] = vl_sift(I2, ...
'PeakThresh', peak_thresh, ...
'edgethresh', edge_thresh );
fprintf('Number of frames (features) detected: %d\n', size(f2,2));
Find SIFT points in
images (2 of 3)
% Show all SIFT features detected
h
= vl_plotframe(f2) ;
set(h,'color','y','linewidth',1.5) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Match features
% The index of the original match and the closest descriptor is stored in
% each column of matches and the distance between the pair is stored in
% scores.
% Define threshold for matching. Descriptor D1 is matched to a descriptor
% D2 only if the distance d(D1,D2) multiplied by THRESH is not greater than
% the distance of D1 to all other descriptors
thresh = 1.5;
% default = 1.5; increase to limit matches
[matches, scores] = vl_ubcmatch(d1, d2, thresh);
fprintf('Number of matching frames (features): %d\n', size(matches,2));
indices1 = matches(1,:);
f1match = f1(:,indices1);
d1match = d1(:,indices1);
% Get matching features
indices2 = matches(2,:);
f2match = f2(:,indices2);
d2match = d2(:,indices2);
Colorado School of Mines
Computer Vision
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Show matches
figure(3), imshow([I1,I2],[]);
o = size(I1,2) ;
line([f1match(1,:);f2match(1,:)+o], ...
[f1match(2,:);f2match(2,:)]) ;
for i=1:size(f1match,2)
x = f1match(1,i);
y = f1match(2,i);
text(x,y,sprintf('%d',i), 'Color', 'r');
end
for i=1:size(f2match,2)
x = f2match(1,i);
y = f2match(2,i);
text(x+o,y,sprintf('%d',i), 'Color', 'r');
end
Colorado School of Mines
Computer Vision
Find SIFT points in
images (3 of 3)
8
Identifying incorrect matches
• We find a transformation that explains the
movement of the greatest number of matches
– Those points that agree with that transformation are
correct (inliers)
– All other matches are outliers and are discarded
• What transformation to use?
– Depends on your application ... for a planar object, you
could use a homography (projective transform)
– To fit a homography, requires at least 4 corresponding
image points
 How to find this transformation?
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Computer Vision
9
Algorithm (RANSAC)
1) Randomly select a minimal set of points and solve
for the transformation
2) Count the number of points that agree with this
transformation
3) If this transformation has the highest number of
inliers so far, save it
4) Repeat for N trials and return the best
transformation
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Computer Vision
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Example: line fitting
We will randomly
select subsets of 2
points (since 2 is the
minimum number of
points to fit a line)
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Computer Vision
Example: line fitting
n=2
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Computer Vision
Model fitting
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Computer Vision
Measure distances
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Computer Vision
Count inliers
c=3
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Computer Vision
Another trial
c=3
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Computer Vision
The best model
c=15
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Computer Vision
RANSAC Algorithm
• “Random Sample Consensus”
– Can be used to find inliers for any type of model fitting
• Widely used in computer vision
• Requires two parameters:
– The number of trials N (how many do we need?)
– The agreement threshold (how close does an inlier have to
be?)
Fischler, M. A., & Bolles, R. C. 1981. Random sampling consensus: A paradigm for model fitting
with applications to image analysis and automated cartography. Communications of the
Associations for Computing Machinery, 24(26), 381–395.
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Computer Vision
18
Number of samples (trials) required
Try example:
e = 0.5
p = 0.99
s=2
N = 10
•
Assume we know e, the probability that a point is an outlier
•
The probability that a sample of size s is all inliers is (1- e)s
•
With N samples (trials), the probability that all samples have at least one
outlier is palloutliers = (1-(1- e)s )N
•
We want at least one sample to have no outliers. The probability is
pnotalloutliers = 1 - palloutliers
•
We’ll pick pnotalloutliers to be some high value, like 0.99.
•
If we know e, we can solve for N
– So (1- e) is the probability that any point is an inlier
– So 1-(1- e)s is the probability of getting at least one outlier in that sample
log(1  pnotalloutliers )
N
log(1  (1  e ) s )
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N is the number of trials needed to
get at least one sample of all inliers,
with probability pnotalloutliers
Computer Vision
19
Number of samples required
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Computer Vision
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Better estimate for required number of trials
•
Again, assume we know e, the probability that a point is
an outlier
– So (1- e) is the probability that any point is an inlier
•
Assume we have M total points. The estimated number
of inliers is 𝑛𝑖𝑛𝑙𝑖𝑒𝑟 = 1 − 𝜀 𝑀
“n choose k” is
the number of
ways to choose k
items from a set
of n items
n
n!
  
 k  k!(n  k )!
•
The number of ways we can pick a sample of s points out
𝑀
of M points is 𝑚 =
𝑠
•
The number of ways we can choose s points that are all
𝑛𝑖𝑛𝑙𝑖𝑒𝑟
inliers is 𝑛 =
𝑠
•
The probability that any sample of s points is all inliers is
p=n/m
N is the number of trials needed to
log(1  pnotalloutliers )
N
get at least one sample of all inliers,
log(1  p )
with probability pnotalloutliers
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Computer Vision
21
Estimating outlier probability
• If you don’t know the outlier probability e in
advance, you can estimate it from the data:
– Start with a very conservative guess for e (say, 0.9)
– Compute the number of trials required ... it will be very
large
– Each time you get a fit, count the number of inliers
– If the number of inliers is the best so far, revise your
estimate of e: 𝜀 = (1 − 𝑛𝑖𝑛𝑙𝑖𝑒𝑟 )/𝑀
– Recompute the number of trials required using the new
estimate of e and continue
Colorado School of Mines
Computer Vision
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Deciding if a point is an inlier
• You can just pick a threshold for a point’s residual error, for it
to be counted as an inlier (e.g., you could measure this
empirically)
• Another strategy:
– Compute the conditional probability of the residual assuming that it is
an inlier, and compare it to the conditional probability of the residual
assuming that it is an outlier
– Namely, test if 𝑃 𝑟 𝑖𝑛𝑙𝑖𝑒𝑟 > 𝑃(𝑟|𝑜𝑢𝑡𝑙𝑖𝑒𝑟)
– Typically, we assume inliers have Gaussian distributed noise (with
standard deviation s), and outliers have uniform distributed noise
– This approach is called MSAC (“m-estimator sample consensus”)
P. H. S. Torr , A. Zisserman, "MLESAC: A new robust estimator with application to estimating image
geometry" , Computer Vision and Image Understanding, vol 78 Issue 1, April 2000, pp 138 - 156.
http://www.robots.ox.ac.uk/~vgg/publications/2000/Torr00/torr00.pdf
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Computer Vision
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fit homography to corresponding points. Use the MLESAC (m-estimator
% sample consensus) algorithm to find inliers. See the paper: "MLESAC: A
% new robust estimator with application to estimating image geometry" by
% Torr and Zisserman, at
% http://www.robots.ox.ac.uk/~vgg/publications/2000/Torr00/torr00.pdf
MLESAC code (1 of 5)
N = size(f1match,2);
% Number of corresponding point pairs
pts1 = f1match(1:2,:)'; % Get x,y coordinates from image 1, size is (N,2)
pts2 = f2match(1:2,:)'; % Get x,y coordinates from image 2, size is (N,2)
%
%
%
%
S
Estimate the uncertainty of point locations in the second image. We'll
assume Gaussian distributed residual errors, with a sigma of S pixels.
Actually, points found in higher scales will have proportionally larger
errors.
= 1.0;
% Sigma of errors of points found in lowest scale
% Create a 2xN matrix of sigmas for each point, where each column is the
% sigmax, sigmay values for that point.
sigmas = [S * f2match(3,:); S * f2match(3,:)]; % Assume same for x,y
% We will need the sigmas as a Nx2 matrix.
sigs = sigmas';
% Now, if a point is an outlier, its residual error can be anything; i.e.,
% any value between 0 and the size of the image. Let's
% assume a uniform probability distribution.
pOutlier = 1/(size(I1,1)*size(I1,2));
% Estimate the fraction of inliers. We'll actually estimate this later
% from the data, but start with a worst case scenario assumption.
inlierFraction = 0.1;
% Let's say that we want to get a good sample with probability Ps.
Ps = 0.99;
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Computer Vision
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% Determine the required number of iterations
nInlier = round(N*inlierFraction); % number of inliers among the N points
% Calculate the number of ways to pick a set of 4 points, out of N points.
% This is "N-choose-4", which is N*(N-1)*(N-2)*(N-3)/factorial(4).
m = nInlier*(nInlier-1)*(nInlier-2)*(nInlier-3)/factorial(4); % #ways to choose 4 inliers
n = N*(N-1)*(N-2)*(N-3)/factorial(4); % #ways to choose 4 points
p = m/n;
% probability that any sample of 4 points is all inliers
nIterations = log(1-Ps) / log(1 - p);
nIterations = ceil(nIterations);
fprintf('Initial estimated number of iterations needed: %d\n', nIterations);
MLESAC code (2 of 5)
sample_count = 0;
% The number of Ransac trials
pBest = -Inf;
% Best probability found so far (actually, the log)
while sample_count < nIterations
% Grab 4 matching points at random
v = randperm(N);
p1 = pts1(v(1:4),:);
p2 = pts2(v(1:4),:);
% Try fitting a homography, that transforms p1 to p2. Fitting will
% fail if points are co-linear, so do a "catch" that so the program
% doesn't quit. Also, Matlab will display a warning if the result is
% close to singular, so turn warnings temporarily off.
warning('off', 'all');
try
tform_1_2 = cp2tform(p1,p2,'projective');
catch
continue;
end
warning('on', 'all');
% Ok, we were able to fit a homography to this sample.
sample_count = sample_count + 1;
% Use that homography to transform all pts1 to pts2
pts2map = tformfwd(pts1, tform_1_2);
% Look at the residual errors of all the points.
dp = (pts2map - pts2);
% Size is Nx2
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Computer Vision
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% Compute the probability that each residual error, dpi, could be an
% inlier. This is the equation of a Gaussian; ie.
%
G(dpi) = exp(-dpi' * Ci^-1 * dpi)/(2*pi*det(Ci)^0.5)
% where Ci is the covariance matrix of residual dpi:
%
Ci = [ sxi^2 0; 0 syi^2 ]
% Note that det(Ci)^0.5 = sxi*syi
dp = dp ./ sigs;
% Scale by the sigmas
rsq = dp(:,1).^2 + dp(:,2).^2;
% residual squared distance errors
numerator = exp(-rsq/2);
% Numerator of Gaussians, size is Nx1
denominator = 2*pi*sigs(:,1).*sigs(:,2);
% Denominator, size Nx1
MLESAC code (3 of 5)
% These are the probabilities of each point, if they are inliers (ie,
% this is just the Gaussian probability of the error).
pInlier = numerator./denominator;
% Let's define inliers to be those points whose inlier probability is
% greater than the outlier probability.
indicesInliers = (pInlier > pOutlier);
nInlier = sum(indicesInliers);
% Update the number of iterations required
if nInlier/N > inlierFraction
inlierFraction = nInlier/N;
m = nInlier*(nInlier-1)*(nInlier-2)*(nInlier-3)/factorial(4);
n = N*(N-1)*(N-2)*(N-3)/factorial(4); % #ways to choose 4 points
p = m/n; % probability that any sample of 4 points is all inliers
nIterations = log(1-Ps) / log(1 - p);
nIterations = ceil(nIterations);
fprintf('(Iteration %d): New estimated number of iterations needed: %d\n', ...
sample_count, nIterations);
end
% Net probability of the data (log)
p = sum(log(pInlier(indicesInliers))) + (N-nInlier)*log(pOutlier);
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Computer Vision
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% Keep this solution if probability is better than the best so far.
if p>pBest
pBest = p;
nBest = nInlier;
indicesBest = indicesInliers;
fprintf(' (Iter %d): best so far with %d inliers, p=%f\n', ...
sample_count, nInlier, p);
%disp(find(indicesBest)');
MLESAC code (4 of 5)
% Show inliers
figure(4), imshow([I1,I2],[]);
o = size(I1,2) ;
for i=1:size(f1match,2)
x1 = f1match(1,i);
y1 = f1match(2,i);
x2 = f2match(1,i);
y2 = f2match(2,i);
if indicesBest(i)
text(x1,y1,sprintf('%d',i), 'Color', 'g');
text(x2+o,y2,sprintf('%d',i), 'Color', 'g');
else
text(x1,y1,sprintf('%d',i), 'Color', 'r');
text(x2+o,y2,sprintf('%d',i), 'Color', 'r');
end
end
pause
end
end
fprintf('Final number of iterations used: %d\n', sample_count);
fprintf('Estimated inlier fraction: %f\n', inlierFraction);
% Show the final inliers
f1Inlier = f1match(:,indicesBest);
f2Inlier = f2match(:,indicesBest);
figure, subplot(1,2,1), imshow(I1,[]);
h = vl_plotframe(f1Inlier) ;
set(h,'color','y','linewidth',1.5) ;
subplot(1,2,2), imshow(I2,[]);
h = vl_plotframe(f2Inlier) ;
set(h,'color','y','linewidth',1.5) ;
Colorado School of Mines
Computer Vision
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% Ok, refit homography using all the inliers.
p1 = pts1(indicesBest,:);
p2 = pts2(indicesBest,:);
tform_1_2 = cp2tform(p1,p2,'projective');
% Use that homography to transform all p1 to p2
p2map = tformfwd(p1, tform_1_2);
MLESAC code (5 of 5)
% Look at the residual errors of all the points.
dp = (p2map - p2);
rsq = dp(:,1).^2 + dp(:,2).^2;
% residual squared distance errors
rmsErr = sqrt( sum(rsq)/nBest );
disp('RMS error: '), disp(rmsErr);
% Get the 3x3 homography matrix, such that p2 = H*p1.
Hom_1_2 = tform_1_2.tdata.T';
I1warp = imtransform(I1, tform_1_2, ...
'XData',[1,size(I1,2)], ...
'YData',[1,size(I1,1)]);
% Overlay the images
RGB(:,:,1) = (I1warp+I2)/2;
RGB(:,:,2) = (I1warp+I2)/2;
RGB(:,:,3) = I2/2;
RGB = uint8(RGB);
figure, imshow(RGB);
% Also show a difference image
Idiff = I2 - I1warp;
figure, imshow(abs(Idiff),[]);
Colorado School of Mines
Computer Vision
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14
143
104
34255
162 30
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difference image
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fused image
Computer Vision
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