-
-
Notifications
You must be signed in to change notification settings - Fork 2.2k
/
fit.py
876 lines (689 loc) · 29.1 KB
/
fit.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
import math
import numpy as np
from numpy.linalg import inv, pinv
from scipy import optimize
from .._shared.utils import check_random_state
def _check_data_dim(data, dim):
if data.ndim != 2 or data.shape[1] != dim:
raise ValueError('Input data must have shape (N, %d).' % dim)
def _check_data_atleast_2D(data):
if data.ndim < 2 or data.shape[1] < 2:
raise ValueError('Input data must be at least 2D.')
def _norm_along_axis(x, axis):
"""NumPy < 1.8 does not support the `axis` argument for `np.linalg.norm`."""
return np.sqrt(np.einsum('ij,ij->i', x, x))
class BaseModel(object):
def __init__(self):
self.params = None
class LineModelND(BaseModel):
"""Total least squares estimator for N-dimensional lines.
In contrast to ordinary least squares line estimation, this estimator
minimizes the orthogonal distances of points to the estimated line.
Lines are defined by a point (origin) and a unit vector (direction)
according to the following vector equation::
X = origin + lambda * direction
Attributes
----------
params : tuple
Line model parameters in the following order `origin`, `direction`.
Examples
--------
>>> x = np.linspace(1, 2, 25)
>>> y = 1.5 * x + 3
>>> lm = LineModelND()
>>> lm.estimate(np.array([x, y]).T)
True
>>> tuple(np.round(lm.params, 5))
(array([1.5 , 5.25]), array([0.5547 , 0.83205]))
>>> res = lm.residuals(np.array([x, y]).T)
>>> np.abs(np.round(res, 9))
array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0.])
>>> np.round(lm.predict_y(x[:5]), 3)
array([4.5 , 4.562, 4.625, 4.688, 4.75 ])
>>> np.round(lm.predict_x(y[:5]), 3)
array([1. , 1.042, 1.083, 1.125, 1.167])
"""
def estimate(self, data):
"""Estimate line model from data.
This minimizes the sum of shortest (orthogonal) distances
from the given data points to the estimated line.
Parameters
----------
data : (N, dim) array
N points in a space of dimensionality dim >= 2.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
_check_data_atleast_2D(data)
origin = data.mean(axis=0)
data = data - origin
if data.shape[0] == 2: # well determined
direction = data[1] - data[0]
norm = np.linalg.norm(direction)
if norm != 0: # this should not happen to be norm 0
direction /= norm
elif data.shape[0] > 2: # over-determined
# Note: with full_matrices=1 Python dies with joblib parallel_for.
_, _, v = np.linalg.svd(data, full_matrices=False)
direction = v[0]
else: # under-determined
raise ValueError('At least 2 input points needed.')
self.params = (origin, direction)
return True
def residuals(self, data, params=None):
"""Determine residuals of data to model.
For each point, the shortest (orthogonal) distance to the line is
returned. It is obtained by projecting the data onto the line.
Parameters
----------
data : (N, dim) array
N points in a space of dimension dim.
params : (2, ) array, optional
Optional custom parameter set in the form (`origin`, `direction`).
Returns
-------
residuals : (N, ) array
Residual for each data point.
"""
_check_data_atleast_2D(data)
if params is None:
if self.params is None:
raise ValueError('Parameters cannot be None')
params = self.params
if len(params) != 2:
raise ValueError('Parameters are defined by 2 sets.')
origin, direction = params
res = (data - origin) - \
((data - origin) @ direction)[..., np.newaxis] * direction
return _norm_along_axis(res, axis=1)
def predict(self, x, axis=0, params=None):
"""Predict intersection of the estimated line model with a hyperplane
orthogonal to a given axis.
Parameters
----------
x : (n, 1) array
Coordinates along an axis.
axis : int
Axis orthogonal to the hyperplane intersecting the line.
params : (2, ) array, optional
Optional custom parameter set in the form (`origin`, `direction`).
Returns
-------
data : (n, m) array
Predicted coordinates.
Raises
------
ValueError
If the line is parallel to the given axis.
"""
if params is None:
if self.params is None:
raise ValueError('Parameters cannot be None')
params = self.params
if len(params) != 2:
raise ValueError('Parameters are defined by 2 sets.')
origin, direction = params
if direction[axis] == 0:
# line parallel to axis
raise ValueError('Line parallel to axis %s' % axis)
l = (x - origin[axis]) / direction[axis]
data = origin + l[..., np.newaxis] * direction
return data
def predict_x(self, y, params=None):
"""Predict x-coordinates for 2D lines using the estimated model.
Alias for::
predict(y, axis=1)[:, 0]
Parameters
----------
y : array
y-coordinates.
params : (2, ) array, optional
Optional custom parameter set in the form (`origin`, `direction`).
Returns
-------
x : array
Predicted x-coordinates.
"""
x = self.predict(y, axis=1, params=params)[:, 0]
return x
def predict_y(self, x, params=None):
"""Predict y-coordinates for 2D lines using the estimated model.
Alias for::
predict(x, axis=0)[:, 1]
Parameters
----------
x : array
x-coordinates.
params : (2, ) array, optional
Optional custom parameter set in the form (`origin`, `direction`).
Returns
-------
y : array
Predicted y-coordinates.
"""
y = self.predict(x, axis=0, params=params)[:, 1]
return y
class CircleModel(BaseModel):
"""Total least squares estimator for 2D circles.
The functional model of the circle is::
r**2 = (x - xc)**2 + (y - yc)**2
This estimator minimizes the squared distances from all points to the
circle::
min{ sum((r - sqrt((x_i - xc)**2 + (y_i - yc)**2))**2) }
A minimum number of 3 points is required to solve for the parameters.
Attributes
----------
params : tuple
Circle model parameters in the following order `xc`, `yc`, `r`.
Examples
--------
>>> t = np.linspace(0, 2 * np.pi, 25)
>>> xy = CircleModel().predict_xy(t, params=(2, 3, 4))
>>> model = CircleModel()
>>> model.estimate(xy)
True
>>> tuple(np.round(model.params, 5))
(2.0, 3.0, 4.0)
>>> res = model.residuals(xy)
>>> np.abs(np.round(res, 9))
array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0.])
"""
def estimate(self, data):
"""Estimate circle model from data using total least squares.
Parameters
----------
data : (N, 2) array
N points with ``(x, y)`` coordinates, respectively.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
_check_data_dim(data, dim=2)
x = data[:, 0]
y = data[:, 1]
# http://www.had2know.com/academics/best-fit-circle-least-squares.html
x2y2 = (x ** 2 + y ** 2)
sum_x = np.sum(x)
sum_y = np.sum(y)
sum_xy = np.sum(x * y)
m1 = np.array([[np.sum(x ** 2), sum_xy, sum_x],
[sum_xy, np.sum(y ** 2), sum_y],
[sum_x, sum_y, float(len(x))]])
m2 = np.array([[np.sum(x * x2y2),
np.sum(y * x2y2),
np.sum(x2y2)]]).T
a, b, c = pinv(m1) @ m2
a, b, c = a[0], b[0], c[0]
xc = a / 2
yc = b / 2
r = np.sqrt(4 * c + a ** 2 + b ** 2) / 2
self.params = (xc, yc, r)
return True
def residuals(self, data):
"""Determine residuals of data to model.
For each point the shortest distance to the circle is returned.
Parameters
----------
data : (N, 2) array
N points with ``(x, y)`` coordinates, respectively.
Returns
-------
residuals : (N, ) array
Residual for each data point.
"""
_check_data_dim(data, dim=2)
xc, yc, r = self.params
x = data[:, 0]
y = data[:, 1]
return r - np.sqrt((x - xc)**2 + (y - yc)**2)
def predict_xy(self, t, params=None):
"""Predict x- and y-coordinates using the estimated model.
Parameters
----------
t : array
Angles in circle in radians. Angles start to count from positive
x-axis to positive y-axis in a right-handed system.
params : (3, ) array, optional
Optional custom parameter set.
Returns
-------
xy : (..., 2) array
Predicted x- and y-coordinates.
"""
if params is None:
params = self.params
xc, yc, r = params
x = xc + r * np.cos(t)
y = yc + r * np.sin(t)
return np.concatenate((x[..., None], y[..., None]), axis=t.ndim)
class EllipseModel(BaseModel):
"""Total least squares estimator for 2D ellipses.
The functional model of the ellipse is::
xt = xc + a*cos(theta)*cos(t) - b*sin(theta)*sin(t)
yt = yc + a*sin(theta)*cos(t) + b*cos(theta)*sin(t)
d = sqrt((x - xt)**2 + (y - yt)**2)
where ``(xt, yt)`` is the closest point on the ellipse to ``(x, y)``. Thus
d is the shortest distance from the point to the ellipse.
The estimator is based on a least squares minimization. The optimal
solution is computed directly, no iterations are required. This leads
to a simple, stable and robust fitting method.
The ``params`` attribute contains the parameters in the following order::
xc, yc, a, b, theta
Attributes
----------
params : tuple
Ellipse model parameters in the following order `xc`, `yc`, `a`, `b`,
`theta`.
Examples
--------
>>> xy = EllipseModel().predict_xy(np.linspace(0, 2 * np.pi, 25),
... params=(10, 15, 4, 8, np.deg2rad(30)))
>>> ellipse = EllipseModel()
>>> ellipse.estimate(xy)
True
>>> np.round(ellipse.params, 2)
array([10. , 15. , 4. , 8. , 0.52])
>>> np.round(abs(ellipse.residuals(xy)), 5)
array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0.])
"""
def estimate(self, data):
"""Estimate circle model from data using total least squares.
Parameters
----------
data : (N, 2) array
N points with ``(x, y)`` coordinates, respectively.
Returns
-------
success : bool
True, if model estimation succeeds.
References
----------
.. [1] Halir, R.; Flusser, J. "Numerically stable direct least squares
fitting of ellipses". In Proc. 6th International Conference in
Central Europe on Computer Graphics and Visualization.
WSCG (Vol. 98, pp. 125-132).
"""
# Original Implementation: Ben Hammel, Nick Sullivan-Molina
# another REFERENCE: [2] http://mathworld.wolfram.com/Ellipse.html
_check_data_dim(data, dim=2)
x = data[:, 0]
y = data[:, 1]
# Quadratic part of design matrix [eqn. 15] from [1]
D1 = np.vstack([x ** 2, x * y, y ** 2]).T
# Linear part of design matrix [eqn. 16] from [1]
D2 = np.vstack([x, y, np.ones(len(x))]).T
# forming scatter matrix [eqn. 17] from [1]
S1 = D1.T @ D1
S2 = D1.T @ D2
S3 = D2.T @ D2
# Constraint matrix [eqn. 18]
C1 = np.array([[0., 0., 2.], [0., -1., 0.], [2., 0., 0.]])
try:
# Reduced scatter matrix [eqn. 29]
M = inv(C1) @ (S1 - S2 @ inv(S3) @ S2.T)
except np.linalg.LinAlgError: # LinAlgError: Singular matrix
return False
# M*|a b c >=l|a b c >. Find eigenvalues and eigenvectors
# from this equation [eqn. 28]
eig_vals, eig_vecs = np.linalg.eig(M)
# eigenvector must meet constraint 4ac - b^2 to be valid.
cond = 4 * np.multiply(eig_vecs[0, :], eig_vecs[2, :]) \
- np.power(eig_vecs[1, :], 2)
a1 = eig_vecs[:, (cond > 0)]
# seeks for empty matrix
if 0 in a1.shape or len(a1.ravel()) != 3:
return False
a, b, c = a1.ravel()
# |d f g> = -S3^(-1)*S2^(T)*|a b c> [eqn. 24]
a2 = -inv(S3) @ S2.T @ a1
d, f, g = a2.ravel()
# eigenvectors are the coefficients of an ellipse in general form
# a*x^2 + 2*b*x*y + c*y^2 + 2*d*x + 2*f*y + g = 0 (eqn. 15) from [2]
b /= 2.
d /= 2.
f /= 2.
# finding center of ellipse [eqn.19 and 20] from [2]
x0 = (c * d - b * f) / (b ** 2. - a * c)
y0 = (a * f - b * d) / (b ** 2. - a * c)
# Find the semi-axes lengths [eqn. 21 and 22] from [2]
numerator = a * f ** 2 + c * d ** 2 + g * b ** 2 \
- 2 * b * d * f - a * c * g
term = np.sqrt((a - c) ** 2 + 4 * b ** 2)
denominator1 = (b ** 2 - a * c) * (term - (a + c))
denominator2 = (b ** 2 - a * c) * (- term - (a + c))
width = np.sqrt(2 * numerator / denominator1)
height = np.sqrt(2 * numerator / denominator2)
# angle of counterclockwise rotation of major-axis of ellipse
# to x-axis [eqn. 23] from [2].
phi = 0.5 * np.arctan((2. * b) / (a - c))
if a > c:
phi += 0.5 * np.pi
self.params = np.nan_to_num([x0, y0, width, height, phi]).tolist()
self.params = [float(np.real(x)) for x in self.params]
return True
def residuals(self, data):
"""Determine residuals of data to model.
For each point the shortest distance to the ellipse is returned.
Parameters
----------
data : (N, 2) array
N points with ``(x, y)`` coordinates, respectively.
Returns
-------
residuals : (N, ) array
Residual for each data point.
"""
_check_data_dim(data, dim=2)
xc, yc, a, b, theta = self.params
ctheta = math.cos(theta)
stheta = math.sin(theta)
x = data[:, 0]
y = data[:, 1]
N = data.shape[0]
def fun(t, xi, yi):
ct = math.cos(t)
st = math.sin(t)
xt = xc + a * ctheta * ct - b * stheta * st
yt = yc + a * stheta * ct + b * ctheta * st
return (xi - xt) ** 2 + (yi - yt) ** 2
# def Dfun(t, xi, yi):
# ct = math.cos(t)
# st = math.sin(t)
# xt = xc + a * ctheta * ct - b * stheta * st
# yt = yc + a * stheta * ct + b * ctheta * st
# dfx_t = - 2 * (xi - xt) * (- a * ctheta * st
# - b * stheta * ct)
# dfy_t = - 2 * (yi - yt) * (- a * stheta * st
# + b * ctheta * ct)
# return [dfx_t + dfy_t]
residuals = np.empty((N, ), dtype=np.double)
# initial guess for parameter t of closest point on ellipse
t0 = np.arctan2(y - yc, x - xc) - theta
# determine shortest distance to ellipse for each point
for i in range(N):
xi = x[i]
yi = y[i]
# faster without Dfun, because of the python overhead
t, _ = optimize.leastsq(fun, t0[i], args=(xi, yi))
residuals[i] = np.sqrt(fun(t, xi, yi))
return residuals
def predict_xy(self, t, params=None):
"""Predict x- and y-coordinates using the estimated model.
Parameters
----------
t : array
Angles in circle in radians. Angles start to count from positive
x-axis to positive y-axis in a right-handed system.
params : (5, ) array, optional
Optional custom parameter set.
Returns
-------
xy : (..., 2) array
Predicted x- and y-coordinates.
"""
if params is None:
params = self.params
xc, yc, a, b, theta = params
ct = np.cos(t)
st = np.sin(t)
ctheta = math.cos(theta)
stheta = math.sin(theta)
x = xc + a * ctheta * ct - b * stheta * st
y = yc + a * stheta * ct + b * ctheta * st
return np.concatenate((x[..., None], y[..., None]), axis=t.ndim)
def _dynamic_max_trials(n_inliers, n_samples, min_samples, probability):
"""Determine number trials such that at least one outlier-free subset is
sampled for the given inlier/outlier ratio.
Parameters
----------
n_inliers : int
Number of inliers in the data.
n_samples : int
Total number of samples in the data.
min_samples : int
Minimum number of samples chosen randomly from original data.
probability : float
Probability (confidence) that one outlier-free sample is generated.
Returns
-------
trials : int
Number of trials.
"""
if n_inliers == 0:
return np.inf
nom = 1 - probability
if nom == 0:
return np.inf
inlier_ratio = n_inliers / float(n_samples)
denom = 1 - inlier_ratio ** min_samples
if denom == 0:
return 1
elif denom == 1:
return np.inf
nom = np.log(nom)
denom = np.log(denom)
if denom == 0:
return 0
return int(np.ceil(nom / denom))
def ransac(data, model_class, min_samples, residual_threshold,
is_data_valid=None, is_model_valid=None,
max_trials=100, stop_sample_num=np.inf, stop_residuals_sum=0,
stop_probability=1, random_state=None, initial_inliers=None):
"""Fit a model to data with the RANSAC (random sample consensus) algorithm.
RANSAC is an iterative algorithm for the robust estimation of parameters
from a subset of inliers from the complete data set. Each iteration
performs the following tasks:
1. Select `min_samples` random samples from the original data and check
whether the set of data is valid (see `is_data_valid`).
2. Estimate a model to the random subset
(`model_cls.estimate(*data[random_subset]`) and check whether the
estimated model is valid (see `is_model_valid`).
3. Classify all data as inliers or outliers by calculating the residuals
to the estimated model (`model_cls.residuals(*data)`) - all data samples
with residuals smaller than the `residual_threshold` are considered as
inliers.
4. Save estimated model as best model if number of inlier samples is
maximal. In case the current estimated model has the same number of
inliers, it is only considered as the best model if it has less sum of
residuals.
These steps are performed either a maximum number of times or until one of
the special stop criteria are met. The final model is estimated using all
inlier samples of the previously determined best model.
Parameters
----------
data : [list, tuple of] (N, ...) array
Data set to which the model is fitted, where N is the number of data
points and the remaining dimension are depending on model requirements.
If the model class requires multiple input data arrays (e.g. source and
destination coordinates of ``skimage.transform.AffineTransform``),
they can be optionally passed as tuple or list. Note, that in this case
the functions ``estimate(*data)``, ``residuals(*data)``,
``is_model_valid(model, *random_data)`` and
``is_data_valid(*random_data)`` must all take each data array as
separate arguments.
model_class : object
Object with the following object methods:
* ``success = estimate(*data)``
* ``residuals(*data)``
where `success` indicates whether the model estimation succeeded
(`True` or `None` for success, `False` for failure).
min_samples : int in range (0, N)
The minimum number of data points to fit a model to.
residual_threshold : float larger than 0
Maximum distance for a data point to be classified as an inlier.
is_data_valid : function, optional
This function is called with the randomly selected data before the
model is fitted to it: `is_data_valid(*random_data)`.
is_model_valid : function, optional
This function is called with the estimated model and the randomly
selected data: `is_model_valid(model, *random_data)`, .
max_trials : int, optional
Maximum number of iterations for random sample selection.
stop_sample_num : int, optional
Stop iteration if at least this number of inliers are found.
stop_residuals_sum : float, optional
Stop iteration if sum of residuals is less than or equal to this
threshold.
stop_probability : float in range [0, 1], optional
RANSAC iteration stops if at least one outlier-free set of the
training data is sampled with ``probability >= stop_probability``,
depending on the current best model's inlier ratio and the number
of trials. This requires to generate at least N samples (trials):
N >= log(1 - probability) / log(1 - e**m)
where the probability (confidence) is typically set to a high value
such as 0.99, e is the current fraction of inliers w.r.t. the
total number of samples, and m is the min_samples value.
random_state : int, RandomState instance or None, optional
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
initial_inliers : array-like of bool, shape (N,), optional
Initial samples selection for model estimation
Returns
-------
model : object
Best model with largest consensus set.
inliers : (N, ) array
Boolean mask of inliers classified as ``True``.
References
----------
.. [1] "RANSAC", Wikipedia, https://en.wikipedia.org/wiki/RANSAC
Examples
--------
Generate ellipse data without tilt and add noise:
>>> t = np.linspace(0, 2 * np.pi, 50)
>>> xc, yc = 20, 30
>>> a, b = 5, 10
>>> x = xc + a * np.cos(t)
>>> y = yc + b * np.sin(t)
>>> data = np.column_stack([x, y])
>>> np.random.seed(seed=1234)
>>> data += np.random.normal(size=data.shape)
Add some faulty data:
>>> data[0] = (100, 100)
>>> data[1] = (110, 120)
>>> data[2] = (120, 130)
>>> data[3] = (140, 130)
Estimate ellipse model using all available data:
>>> model = EllipseModel()
>>> model.estimate(data)
True
>>> np.round(model.params) # doctest: +SKIP
array([ 72., 75., 77., 14., 1.])
Estimate ellipse model using RANSAC:
>>> ransac_model, inliers = ransac(data, EllipseModel, 20, 3, max_trials=50)
>>> abs(np.round(ransac_model.params))
array([20., 30., 5., 10., 0.])
>>> inliers # doctest: +SKIP
array([False, False, False, False, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True], dtype=bool)
>>> sum(inliers) > 40
True
RANSAC can be used to robustly estimate a geometric transformation. In this section,
we also show how to use a proportion of the total samples, rather than an absolute number.
>>> from skimage.transform import SimilarityTransform
>>> np.random.seed(0)
>>> src = 100 * np.random.rand(50, 2)
>>> model0 = SimilarityTransform(scale=0.5, rotation=1, translation=(10, 20))
>>> dst = model0(src)
>>> dst[0] = (10000, 10000)
>>> dst[1] = (-100, 100)
>>> dst[2] = (50, 50)
>>> ratio = 0.5 # use half of the samples
>>> min_samples = int(ratio * len(src))
>>> model, inliers = ransac((src, dst), SimilarityTransform, min_samples, 10,
... initial_inliers=np.ones(len(src), dtype=bool))
>>> inliers
array([False, False, False, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True, True,
True, True, True, True, True])
"""
best_model = None
best_inlier_num = 0
best_inlier_residuals_sum = np.inf
best_inliers = None
random_state = check_random_state(random_state)
# in case data is not pair of input and output, male it like it
if not isinstance(data, (tuple, list)):
data = (data, )
num_samples = len(data[0])
if not (0 < min_samples < num_samples):
raise ValueError("`min_samples` must be in range (0, <number-of-samples>)")
if residual_threshold < 0:
raise ValueError("`residual_threshold` must be greater than zero")
if max_trials < 0:
raise ValueError("`max_trials` must be greater than zero")
if not (0 <= stop_probability <= 1):
raise ValueError("`stop_probability` must be in range [0, 1]")
if initial_inliers is not None and len(initial_inliers) != num_samples:
raise ValueError("RANSAC received a vector of initial inliers (length %i)"
" that didn't match the number of samples (%i)."
" The vector of initial inliers should have the same length"
" as the number of samples and contain only True (this sample"
" is an initial inlier) and False (this one isn't) values."
% (len(initial_inliers), num_samples))
# for the first run use initial guess of inliers
spl_idxs = (initial_inliers if initial_inliers is not None
else random_state.choice(num_samples, min_samples, replace=False))
for num_trials in range(max_trials):
# do sample selection according data pairs
samples = [d[spl_idxs] for d in data]
# for next iteration choose random sample set and be sure that no samples repeat
spl_idxs = random_state.choice(num_samples, min_samples, replace=False)
# optional check if random sample set is valid
if is_data_valid is not None and not is_data_valid(*samples):
continue
# estimate model for current random sample set
sample_model = model_class()
success = sample_model.estimate(*samples)
# backwards compatibility
if success is not None and not success:
continue
# optional check if estimated model is valid
if is_model_valid is not None and not is_model_valid(sample_model, *samples):
continue
sample_model_residuals = np.abs(sample_model.residuals(*data))
# consensus set / inliers
sample_model_inliers = sample_model_residuals < residual_threshold
sample_model_residuals_sum = np.sum(sample_model_residuals ** 2)
# choose as new best model if number of inliers is maximal
sample_inlier_num = np.sum(sample_model_inliers)
if (
# more inliers
sample_inlier_num > best_inlier_num
# same number of inliers but less "error" in terms of residuals
or (sample_inlier_num == best_inlier_num
and sample_model_residuals_sum < best_inlier_residuals_sum)
):
best_model = sample_model
best_inlier_num = sample_inlier_num
best_inlier_residuals_sum = sample_model_residuals_sum
best_inliers = sample_model_inliers
dynamic_max_trials = _dynamic_max_trials(best_inlier_num,
num_samples,
min_samples,
stop_probability)
if (best_inlier_num >= stop_sample_num
or best_inlier_residuals_sum <= stop_residuals_sum
or num_trials >= dynamic_max_trials):
break
# estimate final model using all inliers
if best_inliers is not None:
# select inliers for each data array
data_inliers = [d[best_inliers] for d in data]
best_model.estimate(*data_inliers)
return best_model, best_inliers