Coverage for python/lsst/meas/astrom/pessimistic_pattern_matcher_b_3D.py: 6%

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1 

2import numpy as np 

3from scipy.optimize import least_squares 

4from scipy.spatial import cKDTree 

5from scipy.stats import sigmaclip 

6 

7from .pessimisticPatternMatcherUtils import check_spoke 

8import lsst.pipe.base as pipeBase 

9 

10 

11def _rotation_matrix_chi_sq(flattened_rot_matrix, 

12 pattern_a, 

13 pattern_b, 

14 max_dist_rad): 

15 """Compute the squared differences for least squares fitting. 

16 

17 Given a flattened rotation matrix, one N point pattern and another N point 

18 pattern to transform into to, compute the squared differences between the 

19 points in the two patterns after the rotation. 

20 

21 Parameters 

22 ---------- 

23 flattened_rot_matrix : `numpy.ndarray`, (9, ) 

24 A flattened array representing a 3x3 rotation matrix. The array is 

25 flattened to comply with the API of scipy.optimize.least_squares. 

26 Flattened elements are [[0, 0], [0, 1], [0, 2], [1, 0]...] 

27 pattern_a : `numpy.ndarray`, (N, 3) 

28 A array containing N, 3 vectors representing the objects we would like 

29 to transform into the frame of pattern_b. 

30 pattern_b : `numpy.ndarray`, (N, 3) 

31 A array containing N, 3 vectors representing the reference frame we 

32 would like to transform pattern_a into. 

33 max_dist_rad : `float` 

34 The maximum distance allowed from the pattern matching. This value is 

35 used as the standard error for the resultant chi values. 

36 

37 Returns 

38 ------- 

39 noralized_diff : `numpy.ndarray`, (9,) 

40 Array of differences between the vectors representing of the source 

41 pattern rotated into the reference frame and the converse. This is 

42 used to minimize in a least squares fitter. 

43 """ 

44 # Unflatten the rotation matrix 

45 rot_matrix = flattened_rot_matrix.reshape((3, 3)) 

46 # Compare the rotated source pattern to the references. 

47 rot_pattern_a = np.dot(rot_matrix, pattern_a.transpose()).transpose() 

48 diff_pattern_a_to_b = rot_pattern_a - pattern_b 

49 # Return the flattened differences and length tolerances for use in a least 

50 # squares fitter. 

51 return diff_pattern_a_to_b.flatten() / max_dist_rad 

52 

53 

54class PessimisticPatternMatcherB: 

55 """Class implementing a pessimistic version of Optimistic Pattern Matcher 

56 B (OPMb) from Tabur 2007. See `DMTN-031 <http://ls.st/DMTN-031`_ 

57 

58 Parameters 

59 ---------- 

60 reference_array : `numpy.ndarray`, (N, 3) 

61 spherical points x, y, z of to use as reference objects for 

62 pattern matching. 

63 log : `lsst.log.Log` or `logging.Logger` 

64 Logger for outputting debug info. 

65 

66 Notes 

67 ----- 

68 The class loads and stores the reference object 

69 in a convenient data structure for matching any set of source objects that 

70 are assumed to contain each other. The pessimistic nature of the algorithm 

71 comes from requiring that it discovers at least two patterns that agree on 

72 the correct shift and rotation for matching before exiting. The original 

73 behavior of OPMb can be recovered simply. Patterns matched between the 

74 input datasets are n-spoked pinwheels created from n+1 points. Refer to 

75 DMTN #031 for more details. http://github.com/lsst-dm/dmtn-031 

76 """ 

77 

78 def __init__(self, reference_array, log): 

79 self._reference_array = reference_array 

80 self._n_reference = len(self._reference_array) 

81 self.log = log 

82 

83 if self._n_reference > 0: 

84 self._build_distances_and_angles() 

85 else: 

86 raise ValueError("No reference objects supplied") 

87 

88 def _build_distances_and_angles(self): 

89 """Create the data structures we will use to search for our pattern 

90 match in. 

91 

92 Throughout this function and the rest of the class we use id to 

93 reference the position in the input reference catalog and index to 

94 'index' into the arrays sorted on distance. 

95 """ 

96 # Create empty arrays to store our pair information per 

97 # reference object. 

98 self._dist_array = np.empty( 

99 int(self._n_reference * (self._n_reference - 1) / 2), 

100 dtype="float32") 

101 self._id_array = np.empty( 

102 (int(self._n_reference * (self._n_reference - 1) / 2), 2), 

103 dtype="uint16") 

104 

105 startIdx = 0 

106 # Loop over reference objects storing pair distances and ids. 

107 for ref_id, ref_obj in enumerate(self._reference_array): 

108 # Set the ending slicing index to the correct length for the 

109 # pairs we are creating. 

110 endIdx = startIdx + self._n_reference - 1 - ref_id 

111 

112 # Reserve and fill the ids of each reference object pair. 

113 # 16 bit is safe for the id array as the catalog input from 

114 # MatchPessimisticB is limited to a max length of 2 ** 16. 

115 self._id_array[startIdx:endIdx, 0] = ref_id 

116 self._id_array[startIdx:endIdx, 1] = np.arange(ref_id + 1, 

117 self._n_reference, 

118 dtype="uint16") 

119 

120 # Compute the vector deltas for each pair of reference objects. 

121 # Compute and store the distances. 

122 self._dist_array[startIdx:endIdx] = np.sqrt( 

123 ((self._reference_array[ref_id + 1:, :] 

124 - ref_obj) ** 2).sum(axis=1)) 

125 # Set startIdx of the slice to the end of the previous slice. 

126 startIdx = endIdx 

127 

128 # Sort each array on the pair distances for the initial 

129 # optimistic pattern matcher lookup. 

130 sorted_dist_args = self._dist_array.argsort() 

131 self._dist_array = self._dist_array[sorted_dist_args] 

132 self._id_array = self._id_array[sorted_dist_args] 

133 

134 def match(self, source_array, n_check, n_match, n_agree, 

135 max_n_patterns, max_shift, max_rotation, max_dist, 

136 min_matches, pattern_skip_array=None): 

137 """Match a given source catalog into the loaded reference catalog. 

138 

139 Given array of points on the unit sphere and tolerances, we 

140 attempt to match a pinwheel like pattern between these input sources 

141 and the reference objects this class was created with. This pattern 

142 informs of the shift and rotation needed to align the input source 

143 objects into the frame of the references. 

144 

145 Parameters 

146 ---------- 

147 source_array : `numpy.ndarray`, (N, 3) 

148 An array of spherical x,y,z coordinates and a magnitude in units 

149 of objects having a lower value for sorting. The array should be 

150 of shape (N, 4). 

151 n_check : `int` 

152 Number of sources to create a pattern from. Not all objects may be 

153 checked if n_match criteria is before looping through all n_check 

154 objects. 

155 n_match : `int` 

156 Number of objects to use in constructing a pattern to match. 

157 n_agree : `int` 

158 Number of found patterns that must agree on their shift and 

159 rotation before exiting. Set this value to 1 to recover the 

160 expected behavior of Optimistic Pattern Matcher B. 

161 max_n_patters : `int` 

162 Number of patterns to create from the input source objects to 

163 attempt to match into the reference objects. 

164 max_shift : `float` 

165 Maximum allowed shift to match patterns in arcseconds. 

166 max_rotation : `float` 

167 Maximum allowed rotation between patterns in degrees. 

168 max_dist : `float` 

169 Maximum distance in arcseconds allowed between candidate spokes in 

170 the source and reference objects. Also sets that maximum distance 

171 in the intermediate verify, pattern shift/rotation agreement, and 

172 final verify steps. 

173 pattern_skip_array : `int` 

174 Patterns we would like to skip. This could be due to the pattern 

175 being matched on a previous iteration that we now consider invalid. 

176 This assumes the ordering of the source objects is the same 

177 between different runs of the matcher which, assuming no object 

178 has been inserted or the magnitudes have changed, it should be. 

179 

180 Returns 

181 ------- 

182 output_struct : `lsst.pipe.base.Struct` 

183 Result struct with components 

184 

185 - ``matches`` : (N, 2) array of matched ids for pairs. Empty list if no 

186 match found (`numpy.ndarray`, (N, 2) or `list`) 

187 - ``distances_rad`` : Radian distances between the matched objects. 

188 Empty list if no match found (`numpy.ndarray`, (N,)) 

189 - ``pattern_idx``: Index of matched pattern. None if no match found 

190 (`int`). 

191 - ``shift`` : Magnitude for the shift between the source and reference 

192 objects in arcseconds. None if no match found (`float`). 

193 """ 

194 

195 # Given our input source_array we sort on magnitude. 

196 sorted_source_array = source_array[source_array[:, -1].argsort(), :3] 

197 n_source = len(sorted_source_array) 

198 

199 # Initialize output struct. 

200 output_match_struct = pipeBase.Struct( 

201 match_ids=[], 

202 distances_rad=[], 

203 pattern_idx=None, 

204 shift=None, 

205 max_dist_rad=None,) 

206 

207 if n_source <= 0: 

208 self.log.warning("Source object array is empty. Unable to match. Exiting matcher.") 

209 return None 

210 

211 # To test if the shifts and rotations we find agree with each other, 

212 # we first create two test points situated at the top and bottom of 

213 # where the z axis on the sphere bisects the source catalog. 

214 test_vectors = self._compute_test_vectors(source_array[:, :3]) 

215 

216 # We now create an empty list of our resultant rotated vectors to 

217 # compare the different rotations we find. 

218 rot_vect_list = [] 

219 

220 # Convert the tolerances to values we will use in the code. 

221 max_cos_shift = np.cos(np.radians(max_shift / 3600.)) 

222 max_cos_rot_sq = np.cos(np.radians(max_rotation)) ** 2 

223 max_dist_rad = np.radians(max_dist / 3600.) 

224 

225 # Loop through the sources from brightest to faintest, grabbing a 

226 # chunk of n_check each time. 

227 for pattern_idx in range(np.min((max_n_patterns, 

228 n_source - n_match))): 

229 

230 # If this pattern is one that we matched on the past but we 

231 # now want to skip, we do so here. 

232 if pattern_skip_array is not None and \ 

233 np.any(pattern_skip_array == pattern_idx): 

234 self.log.debug( 

235 "Skipping previously matched bad pattern %i...", 

236 pattern_idx) 

237 continue 

238 # Grab the sources to attempt to create this pattern. 

239 pattern = sorted_source_array[ 

240 pattern_idx: np.min((pattern_idx + n_check, n_source)), :3] 

241 

242 # Construct a pattern given the number of points defining the 

243 # pattern complexity. This is the start of the primary tests to 

244 # match our source pattern into the reference objects. 

245 construct_return_struct = \ 

246 self._construct_pattern_and_shift_rot_matrix( 

247 pattern, n_match, max_cos_shift, max_cos_rot_sq, 

248 max_dist_rad) 

249 

250 # Our struct is None if we could not match the pattern. 

251 if construct_return_struct.ref_candidates is None or \ 

252 construct_return_struct.shift_rot_matrix is None or \ 

253 construct_return_struct.cos_shift is None or \ 

254 construct_return_struct.sin_rot is None: 

255 continue 

256 

257 # Grab the output data from the Struct object. 

258 ref_candidates = construct_return_struct.ref_candidates 

259 shift_rot_matrix = construct_return_struct.shift_rot_matrix 

260 cos_shift = construct_return_struct.cos_shift 

261 sin_rot = construct_return_struct.sin_rot 

262 

263 # If we didn't match enough candidates we continue to the next 

264 # pattern. 

265 if len(ref_candidates) < n_match: 

266 continue 

267 

268 # Now that we know our pattern and shift/rotation are valid we 

269 # store the the rotated versions of our test points for later 

270 # use. 

271 tmp_rot_vect_list = [] 

272 for test_vect in test_vectors: 

273 tmp_rot_vect_list.append(np.dot(shift_rot_matrix, test_vect)) 

274 # Since our test point are in the source frame, we can test if 

275 # their lengths are mostly preserved under the transform. 

276 if not self._test_pattern_lengths(np.array(tmp_rot_vect_list), 

277 max_dist_rad): 

278 continue 

279 

280 tmp_rot_vect_list.append(pattern_idx) 

281 rot_vect_list.append(tmp_rot_vect_list) 

282 

283 # Test if we have enough rotations, which agree, or if we 

284 # are in optimistic mode. 

285 if self._test_rotation_agreement(rot_vect_list, max_dist_rad) < \ 

286 n_agree - 1: 

287 continue 

288 

289 # Run the final verify step. 

290 match_struct = self._final_verify(source_array[:, :3], 

291 shift_rot_matrix, 

292 max_dist_rad, 

293 min_matches) 

294 if match_struct.match_ids is None or \ 

295 match_struct.distances_rad is None or \ 

296 match_struct.max_dist_rad is None: 

297 continue 

298 

299 # Convert the observed shift to arcseconds 

300 shift = np.degrees(np.arccos(cos_shift)) * 3600. 

301 # Print information to the logger. 

302 self.log.debug("Succeeded after %i patterns.", pattern_idx) 

303 self.log.debug("\tShift %.4f arcsec", shift) 

304 self.log.debug("\tRotation: %.4f deg", 

305 np.degrees(np.arcsin(sin_rot))) 

306 

307 # Fill the struct and return. 

308 output_match_struct.match_ids = \ 

309 match_struct.match_ids 

310 output_match_struct.distances_rad = \ 

311 match_struct.distances_rad 

312 output_match_struct.pattern_idx = pattern_idx 

313 output_match_struct.shift = shift 

314 output_match_struct.max_dist_rad = match_struct.max_dist_rad 

315 return output_match_struct 

316 

317 self.log.debug("Failed after %i patterns.", pattern_idx + 1) 

318 return output_match_struct 

319 

320 def _compute_test_vectors(self, source_array): 

321 """Compute spherical 3 vectors at the edges of the x, y, z extent 

322 of the input source catalog. 

323 

324 Parameters 

325 ---------- 

326 source_array : `numpy.ndarray`, (N, 3) 

327 array of 3 vectors representing positions on the unit 

328 sphere. 

329 

330 Returns 

331 ------- 

332 test_vectors : `numpy.ndarray`, (N, 3) 

333 Array of vectors representing the maximum extents in x, y, z 

334 of the input source array. These are used with the rotations 

335 the code finds to test for agreement from different patterns 

336 when the code is running in pessimistic mode. 

337 """ 

338 

339 # Get the center of source_array. 

340 if np.any(np.logical_not(np.isfinite(source_array))): 

341 self.log.warning("Input source objects contain non-finite values. " 

342 "This could end badly.") 

343 center_vect = np.nanmean(source_array, axis=0) 

344 

345 # So that our rotation test works over the full sky we compute 

346 # the max extent in each Cartesian direction x,y,z. 

347 xbtm_vect = np.array([np.min(source_array[:, 0]), center_vect[1], 

348 center_vect[2]], dtype=np.float64) 

349 xtop_vect = np.array([np.max(source_array[:, 0]), center_vect[1], 

350 center_vect[2]], dtype=np.float64) 

351 xbtm_vect /= np.sqrt(np.dot(xbtm_vect, xbtm_vect)) 

352 xtop_vect /= np.sqrt(np.dot(xtop_vect, xtop_vect)) 

353 

354 ybtm_vect = np.array([center_vect[0], np.min(source_array[:, 1]), 

355 center_vect[2]], dtype=np.float64) 

356 ytop_vect = np.array([center_vect[0], np.max(source_array[:, 1]), 

357 center_vect[2]], dtype=np.float64) 

358 ybtm_vect /= np.sqrt(np.dot(ybtm_vect, ybtm_vect)) 

359 ytop_vect /= np.sqrt(np.dot(ytop_vect, ytop_vect)) 

360 

361 zbtm_vect = np.array([center_vect[0], center_vect[1], 

362 np.min(source_array[:, 2])], dtype=np.float64) 

363 ztop_vect = np.array([center_vect[0], center_vect[1], 

364 np.max(source_array[:, 2])], dtype=np.float64) 

365 zbtm_vect /= np.sqrt(np.dot(zbtm_vect, zbtm_vect)) 

366 ztop_vect /= np.sqrt(np.dot(ztop_vect, ztop_vect)) 

367 

368 # Return our list of vectors for later rotation testing. 

369 return np.array([xbtm_vect, xtop_vect, ybtm_vect, ytop_vect, 

370 zbtm_vect, ztop_vect]) 

371 

372 def _construct_pattern_and_shift_rot_matrix(self, src_pattern_array, 

373 n_match, max_cos_theta_shift, 

374 max_cos_rot_sq, max_dist_rad): 

375 """Test an input source pattern against the reference catalog. 

376 

377 Returns the candidate matched patterns and their 

378 implied rotation matrices or None. 

379 

380 Parameters 

381 ---------- 

382 src_pattern_array : `numpy.ndarray`, (N, 3) 

383 Sub selection of source 3 vectors to create a pattern from 

384 n_match : `int` 

385 Number of points to attempt to create a pattern from. Must be 

386 >= len(src_pattern_array) 

387 max_cos_theta_shift : `float` 

388 Maximum shift allowed between two patterns' centers. 

389 max_cos_rot_sq : `float` 

390 Maximum rotation between two patterns that have been shifted 

391 to have their centers on top of each other. 

392 max_dist_rad : `float` 

393 Maximum delta distance allowed between the source and reference 

394 pair distances to consider the reference pair a candidate for 

395 the source pair. Also sets the tolerance between the opening 

396 angles of the spokes when compared to the reference. 

397 

398 Return 

399 ------- 

400 output_matched_pattern : `lsst.pipe.base.Struct` 

401 Result struct with components: 

402 

403 - ``ref_candidates`` : ids of the matched pattern in the internal 

404 reference_array object (`list` of `int`). 

405 - ``src_candidates`` : Pattern ids of the sources matched 

406 (`list` of `int`). 

407 - ``shift_rot_matrix_src_to_ref`` : 3x3 matrix specifying the full 

408 shift and rotation between the reference and source objects. 

409 Rotates source into reference frame. `None` if match is not 

410 found. (`numpy.ndarray`, (3, 3)) 

411 - ``shift_rot_matrix_ref_to_src`` : 3x3 matrix specifying the full 

412 shift and rotation of the reference and source objects. Rotates 

413 reference into source frame. None if match is not found 

414 (`numpy.ndarray`, (3, 3)). 

415 - ``cos_shift`` : Magnitude of the shift found between the two 

416 patten centers. `None` if match is not found (`float`). 

417 - ``sin_rot`` : float value of the rotation to align the already 

418 shifted source pattern to the reference pattern. `None` if no match 

419 found (`float`). 

420 """ 

421 

422 # Create our place holder variables for the matched sources and 

423 # references. The source list starts with the 0th and first indexed 

424 # objects as we are guaranteed to use those and these define both 

425 # the shift and rotation of the final pattern. 

426 output_matched_pattern = pipeBase.Struct( 

427 ref_candidates=[], 

428 src_candidates=[], 

429 shift_rot_matrix=None, 

430 cos_shift=None, 

431 sin_rot=None) 

432 

433 # Create the delta vectors and distances we will need to assemble the 

434 # spokes of the pattern. 

435 src_delta_array = np.empty((len(src_pattern_array) - 1, 3)) 

436 src_delta_array[:, 0] = (src_pattern_array[1:, 0] 

437 - src_pattern_array[0, 0]) 

438 src_delta_array[:, 1] = (src_pattern_array[1:, 1] 

439 - src_pattern_array[0, 1]) 

440 src_delta_array[:, 2] = (src_pattern_array[1:, 2] 

441 - src_pattern_array[0, 2]) 

442 src_dist_array = np.sqrt(src_delta_array[:, 0]**2 

443 + src_delta_array[:, 1]**2 

444 + src_delta_array[:, 2]**2) 

445 

446 # Our first test. We search the reference dataset for pairs 

447 # that have the same length as our first source pairs to with 

448 # plus/minus the max_dist tolerance. 

449 ref_dist_index_array = self._find_candidate_reference_pairs( 

450 src_dist_array[0], self._dist_array, max_dist_rad) 

451 

452 # Start our loop over the candidate reference objects. 

453 for ref_dist_idx in ref_dist_index_array: 

454 # We have two candidates for which reference object corresponds 

455 # with the source at the center of our pattern. As such we loop 

456 # over and test both possibilities. 

457 tmp_ref_pair_list = self._id_array[ref_dist_idx] 

458 for pair_idx, ref_id in enumerate(tmp_ref_pair_list): 

459 src_candidates = [0, 1] 

460 ref_candidates = [] 

461 shift_rot_matrix = None 

462 cos_shift = None 

463 sin_rot = None 

464 # Test the angle between our candidate ref center and the 

465 # source center of our pattern. This angular distance also 

466 # defines the shift we will later use. 

467 ref_center = self._reference_array[ref_id] 

468 cos_shift = np.dot(src_pattern_array[0], ref_center) 

469 if cos_shift < max_cos_theta_shift: 

470 continue 

471 

472 # We can now append this one as a candidate. 

473 ref_candidates.append(ref_id) 

474 # Test to see which reference object to use in the pair. 

475 if pair_idx == 0: 

476 ref_candidates.append( 

477 tmp_ref_pair_list[1]) 

478 ref_delta = (self._reference_array[tmp_ref_pair_list[1]] 

479 - ref_center) 

480 else: 

481 ref_candidates.append( 

482 tmp_ref_pair_list[0]) 

483 ref_delta = (self._reference_array[tmp_ref_pair_list[0]] 

484 - ref_center) 

485 

486 # For dense fields it will be faster to compute the absolute 

487 # rotation this pair suggests first rather than saving it 

488 # after all the spokes are found. We then compute the cos^2 

489 # of the rotation and first part of the rotation matrix from 

490 # source to reference frame. 

491 test_rot_struct = self._test_rotation( 

492 src_pattern_array[0], ref_center, src_delta_array[0], 

493 ref_delta, cos_shift, max_cos_rot_sq) 

494 if test_rot_struct.cos_rot_sq is None or \ 

495 test_rot_struct.proj_ref_ctr_delta is None or \ 

496 test_rot_struct.shift_matrix is None: 

497 continue 

498 

499 # Get the data from the return struct. 

500 cos_rot_sq = test_rot_struct.cos_rot_sq 

501 proj_ref_ctr_delta = test_rot_struct.proj_ref_ctr_delta 

502 shift_matrix = test_rot_struct.shift_matrix 

503 

504 # Now that we have a candidate first spoke and reference 

505 # pattern center, we mask our future search to only those 

506 # pairs that contain our candidate reference center. 

507 tmp_ref_id_array = np.arange(len(self._reference_array), 

508 dtype="uint16") 

509 tmp_ref_dist_array = np.sqrt( 

510 ((self._reference_array 

511 - self._reference_array[ref_id]) 

512 ** 2).sum(axis=1)).astype("float32") 

513 tmp_sorted_args = np.argsort(tmp_ref_dist_array) 

514 tmp_ref_id_array = tmp_ref_id_array[tmp_sorted_args] 

515 tmp_ref_dist_array = tmp_ref_dist_array[tmp_sorted_args] 

516 

517 # Now we feed this sub data to match the spokes of 

518 # our pattern. 

519 pattern_spoke_struct = self._create_pattern_spokes( 

520 src_pattern_array[0], src_delta_array, src_dist_array, 

521 self._reference_array[ref_id], ref_id, proj_ref_ctr_delta, 

522 tmp_ref_dist_array, tmp_ref_id_array, max_dist_rad, 

523 n_match) 

524 

525 # If we don't find enough candidates we can continue to the 

526 # next reference center pair. 

527 if len(pattern_spoke_struct.ref_spoke_list) < n_match - 2 or \ 

528 len(pattern_spoke_struct.src_spoke_list) < n_match - 2: 

529 continue 

530 

531 # If we have the right number of matched ids we store these. 

532 ref_candidates.extend(pattern_spoke_struct.ref_spoke_list) 

533 src_candidates.extend(pattern_spoke_struct.src_spoke_list) 

534 

535 # We can now create our full rotation matrix for both the 

536 # shift and rotation. Reminder shift, aligns the pattern 

537 # centers, rotation rotates the spokes on top of each other. 

538 shift_rot_struct = self._create_shift_rot_matrix( 

539 cos_rot_sq, shift_matrix, src_delta_array[0], 

540 self._reference_array[ref_id], ref_delta) 

541 # If we fail to create the rotation matrix, continue to the 

542 # next objects. 

543 if shift_rot_struct.sin_rot is None or \ 

544 shift_rot_struct.shift_rot_matrix is None: 

545 continue 

546 

547 # Get the data from the return struct. 

548 sin_rot = shift_rot_struct.sin_rot 

549 shift_rot_matrix = shift_rot_struct.shift_rot_matrix 

550 

551 # Now that we have enough candidates we test to see if it 

552 # passes intermediate verify. This shifts and rotates the 

553 # source pattern into the reference frame and then tests that 

554 # each source/reference object pair is within max_dist. It also 

555 # tests the opposite rotation that is reference to source 

556 # frame. 

557 fit_shift_rot_matrix = self._intermediate_verify( 

558 src_pattern_array[src_candidates], 

559 self._reference_array[ref_candidates], 

560 shift_rot_matrix, max_dist_rad) 

561 

562 if fit_shift_rot_matrix is not None: 

563 # Fill the struct and return. 

564 output_matched_pattern.ref_candidates = ref_candidates 

565 output_matched_pattern.src_candidates = src_candidates 

566 output_matched_pattern.shift_rot_matrix = \ 

567 fit_shift_rot_matrix 

568 output_matched_pattern.cos_shift = cos_shift 

569 output_matched_pattern.sin_rot = sin_rot 

570 return output_matched_pattern 

571 

572 return output_matched_pattern 

573 

574 def _find_candidate_reference_pairs(self, src_dist, ref_dist_array, 

575 max_dist_rad): 

576 """Wrap numpy.searchsorted to find the range of reference spokes 

577 within a spoke distance tolerance of our source spoke. 

578 

579 Returns an array sorted from the smallest absolute delta distance 

580 between source and reference spoke length. This sorting increases the 

581 speed for the pattern search greatly. 

582 

583 Parameters 

584 ---------- 

585 src_dist : `float` 

586 float value of the distance we would like to search for in 

587 the reference array in radians. 

588 ref_dist_array : `numpy.ndarray`, (N,) 

589 sorted array of distances in radians. 

590 max_dist_rad : `float` 

591 maximum plus/minus search to find in the reference array in 

592 radians. 

593 

594 Return 

595 ------ 

596 tmp_diff_array : `numpy.ndarray`, (N,) 

597 indices lookup into the input ref_dist_array sorted by the 

598 difference in value to the src_dist from absolute value 

599 smallest to largest. 

600 """ 

601 # Find the index of the minimum and maximum values that satisfy 

602 # the tolerance. 

603 start_idx = np.searchsorted(ref_dist_array, src_dist - max_dist_rad) 

604 end_idx = np.searchsorted(ref_dist_array, src_dist + max_dist_rad, 

605 side='right') 

606 

607 # If these are equal there are no candidates and we exit. 

608 if start_idx == end_idx: 

609 return np.array([], dtype="int") 

610 

611 # Make sure the endpoints of the input array are respected. 

612 if start_idx < 0: 

613 start_idx = 0 

614 if end_idx > ref_dist_array.shape[0]: 

615 end_idx = ref_dist_array.shape[0] 

616 

617 # Now we sort the indices from smallest absolute delta dist difference 

618 # to the largest and return the vector. This step greatly increases 

619 # the speed of the algorithm. 

620 tmp_diff_array = np.fabs(ref_dist_array[start_idx:end_idx] - src_dist) 

621 return tmp_diff_array.argsort() + start_idx 

622 

623 def _test_rotation(self, src_center, ref_center, src_delta, ref_delta, 

624 cos_shift, max_cos_rot_sq): 

625 """ Test if the rotation implied between the source 

626 pattern and reference pattern is within tolerance. To test this 

627 we need to create the first part of our spherical rotation matrix 

628 which we also return for use later. 

629 

630 Parameters 

631 ---------- 

632 src_center : `numpy.ndarray`, (N, 3) 

633 pattern. 

634 ref_center : `numpy.ndarray`, (N, 3) 

635 3 vector defining the center of the candidate reference pinwheel 

636 pattern. 

637 src_delta : `numpy.ndarray`, (N, 3) 

638 3 vector delta between the source pattern center and the end of 

639 the pinwheel spoke. 

640 ref_delta : `numpy.ndarray`, (N, 3) 

641 3 vector delta of the candidate matched reference pair 

642 cos_shift : `float` 

643 Cosine of the angle between the source and reference candidate 

644 centers. 

645 max_cos_rot_sq : `float` 

646 candidate reference pair after shifting the centers on top of each 

647 other. The function will return None if the rotation implied is 

648 greater than max_cos_rot_sq. 

649 

650 Returns 

651 ------- 

652 result : `lsst.pipe.base.Struct` 

653 Result struct with components: 

654 

655 - ``cos_rot_sq`` : magnitude of the rotation needed to align the 

656 two patterns after their centers are shifted on top of each 

657 other. `None` if rotation test fails (`float`). 

658 - ``shift_matrix`` : 3x3 rotation matrix describing the shift needed to 

659 align the source and candidate reference center. `None` if rotation 

660 test fails (`numpy.ndarray`, (N, 3)). 

661 """ 

662 

663 # Make sure the sine is a real number. 

664 if cos_shift > 1.0: 

665 cos_shift = 1. 

666 elif cos_shift < -1.0: 

667 cos_shift = -1. 

668 sin_shift = np.sqrt(1 - cos_shift ** 2) 

669 

670 # If the sine of our shift is zero we only need to use the identity 

671 # matrix for the shift. Else we construct the rotation matrix for 

672 # shift. 

673 if sin_shift > 0: 

674 rot_axis = np.cross(src_center, ref_center) 

675 rot_axis /= sin_shift 

676 shift_matrix = self._create_spherical_rotation_matrix( 

677 rot_axis, cos_shift, sin_shift) 

678 else: 

679 shift_matrix = np.identity(3) 

680 

681 # Now that we have our shift we apply it to the src delta vector 

682 # and check the rotation. 

683 rot_src_delta = np.dot(shift_matrix, src_delta) 

684 proj_src_delta = (rot_src_delta 

685 - np.dot(rot_src_delta, ref_center) * ref_center) 

686 proj_ref_delta = (ref_delta 

687 - np.dot(ref_delta, ref_center) * ref_center) 

688 cos_rot_sq = (np.dot(proj_src_delta, proj_ref_delta) ** 2 

689 / (np.dot(proj_src_delta, proj_src_delta) 

690 * np.dot(proj_ref_delta, proj_ref_delta))) 

691 # If the rotation isn't in tolerance return None. 

692 if cos_rot_sq < max_cos_rot_sq: 

693 return pipeBase.Struct( 

694 cos_rot_sq=None, 

695 proj_ref_ctr_delta=None, 

696 shift_matrix=None,) 

697 # Return the rotation angle, the plane projected reference vector, 

698 # and the first half of the full shift and rotation matrix. 

699 return pipeBase.Struct( 

700 cos_rot_sq=cos_rot_sq, 

701 proj_ref_ctr_delta=proj_ref_delta, 

702 shift_matrix=shift_matrix,) 

703 

704 def _create_spherical_rotation_matrix(self, rot_axis, cos_rotation, 

705 sin_rotion): 

706 """Construct a generalized 3D rotation matrix about a given 

707 axis. 

708 

709 Parameters 

710 ---------- 

711 rot_axis : `numpy.ndarray`, (3,) 

712 3 vector defining the axis to rotate about. 

713 cos_rotation : `float` 

714 cosine of the rotation angle. 

715 sin_rotation : `float` 

716 sine of the rotation angle. 

717 

718 Return 

719 ------ 

720 shift_matrix : `numpy.ndarray`, (3, 3) 

721 3x3 spherical, rotation matrix. 

722 """ 

723 

724 rot_cross_matrix = np.array( 

725 [[0., -rot_axis[2], rot_axis[1]], 

726 [rot_axis[2], 0., -rot_axis[0]], 

727 [-rot_axis[1], rot_axis[0], 0.]], dtype=np.float64) 

728 shift_matrix = (cos_rotation*np.identity(3) 

729 + sin_rotion*rot_cross_matrix 

730 + (1. - cos_rotation)*np.outer(rot_axis, rot_axis)) 

731 

732 return shift_matrix 

733 

734 def _create_pattern_spokes(self, src_ctr, src_delta_array, src_dist_array, 

735 ref_ctr, ref_ctr_id, proj_ref_ctr_delta, 

736 ref_dist_array, ref_id_array, max_dist_rad, 

737 n_match): 

738 """ Create the individual spokes that make up the pattern now that the 

739 shift and rotation are within tolerance. 

740 

741 If we can't create a valid pattern we exit early. 

742 

743 Parameters 

744 ---------- 

745 src_ctr : `numpy.ndarray`, (3,) 

746 3 vector of the source pinwheel center 

747 src_delta_array : `numpy.ndarray`, (N, 3) 

748 Array of 3 vector deltas between the source center and the pairs 

749 that make up the remaining spokes of the pinwheel 

750 src_dist_array : `numpy.ndarray`, (N, 3) 

751 Array of the distances of each src_delta in the pinwheel 

752 ref_ctr : `numpy.ndarray`, (3,) 

753 3 vector of the candidate reference center 

754 ref_ctr_id : `int` 

755 id of the ref_ctr in the master reference array 

756 proj_ref_ctr_delta : `numpy.ndarray`, (3,) 

757 Plane projected 3 vector formed from the center point of the 

758 candidate pin-wheel and the second point in the pattern to create 

759 the first spoke pair. This is the candidate pair that was matched 

760 in the main _construct_pattern_and_shift_rot_matrix loop 

761 ref_dist_array : `numpy.ndarray`, (N,) 

762 Array of vector distances for each of the reference pairs 

763 ref_id_array : `numpy.ndarray`, (N,) 

764 Array of id lookups into the master reference array that our 

765 center id object is paired with. 

766 max_dist_rad : `float` 

767 Maximum search distance 

768 n_match : `int` 

769 Number of source deltas that must be matched into the reference 

770 deltas in order to consider this a successful pattern match. 

771 

772 Returns 

773 ------- 

774 output_spokes : `lsst.pipe.base.Struct` 

775 Result struct with components: 

776 

777 - ``ref_spoke_list`` : list of ints specifying ids into the master 

778 reference array (`list` of `int`). 

779 - ``src_spoke_list`` : list of ints specifying indices into the 

780 current source pattern that is being tested (`list` of `int`). 

781 """ 

782 # Struct where we will be putting our results. 

783 output_spokes = pipeBase.Struct( 

784 ref_spoke_list=[], 

785 src_spoke_list=[],) 

786 

787 # Counter for number of spokes we failed to find a reference 

788 # candidate for. We break the loop if we haven't found enough. 

789 n_fail = 0 

790 ref_spoke_list = [] 

791 src_spoke_list = [] 

792 

793 # Plane project the center/first spoke of the source pattern using 

794 # the center vector of the pattern as normal. 

795 proj_src_ctr_delta = (src_delta_array[0] 

796 - np.dot(src_delta_array[0], src_ctr) * src_ctr) 

797 proj_src_ctr_dist_sq = np.dot(proj_src_ctr_delta, proj_src_ctr_delta) 

798 

799 # Pre-compute the squared length of the projected reference vector. 

800 proj_ref_ctr_dist_sq = np.dot(proj_ref_ctr_delta, proj_ref_ctr_delta) 

801 

802 # Loop over the source pairs. 

803 for src_idx in range(1, len(src_dist_array)): 

804 if n_fail > len(src_dist_array) - (n_match - 1): 

805 break 

806 

807 # Given our length tolerance we can use it to compute a tolerance 

808 # on the angle between our spoke. 

809 src_sin_tol = (max_dist_rad 

810 / (src_dist_array[src_idx] + max_dist_rad)) 

811 

812 # Test if the small angle approximation will still hold. This is 

813 # defined as when sin(theta) ~= theta to within 0.1% of each 

814 # other. If the implied opening angle is too large we set it to 

815 # the 0.1% threshold. 

816 max_sin_tol = 0.0447 

817 if src_sin_tol > max_sin_tol: 

818 src_sin_tol = max_sin_tol 

819 

820 # Plane project the candidate source spoke and compute the cosine 

821 # and sine of the opening angle. 

822 proj_src_delta = ( 

823 src_delta_array[src_idx] 

824 - np.dot(src_delta_array[src_idx], src_ctr) * src_ctr) 

825 geom_dist_src = np.sqrt( 

826 np.dot(proj_src_delta, proj_src_delta) 

827 * proj_src_ctr_dist_sq) 

828 

829 # Compute cosine and sine of the delta vector opening angle. 

830 cos_theta_src = (np.dot(proj_src_delta, proj_src_ctr_delta) 

831 / geom_dist_src) 

832 cross_src = (np.cross(proj_src_delta, proj_src_ctr_delta) 

833 / geom_dist_src) 

834 sin_theta_src = np.dot(cross_src, src_ctr) 

835 

836 # Find the reference pairs that include our candidate pattern 

837 # center and sort them in increasing delta 

838 ref_dist_idx_array = self._find_candidate_reference_pairs( 

839 src_dist_array[src_idx], ref_dist_array, max_dist_rad) 

840 

841 # Test the spokes and return the id of the reference object. 

842 # Return None if no match is found. 

843 ref_id = check_spoke( 

844 cos_theta_src, 

845 sin_theta_src, 

846 ref_ctr, 

847 proj_ref_ctr_delta, 

848 proj_ref_ctr_dist_sq, 

849 ref_dist_idx_array, 

850 ref_id_array, 

851 self._reference_array, 

852 src_sin_tol) 

853 if ref_id < 0: 

854 n_fail += 1 

855 continue 

856 

857 # Append the successful indices to our list. The src_idx needs 

858 # an extra iteration to skip the first and second source objects. 

859 ref_spoke_list.append(ref_id) 

860 src_spoke_list.append(src_idx + 1) 

861 # If we found enough reference objects we can return early. This is 

862 # n_match - 2 as we already have 2 source objects matched into the 

863 # reference data. 

864 if len(ref_spoke_list) >= n_match - 2: 

865 # Set the struct data and return the struct. 

866 output_spokes.ref_spoke_list = ref_spoke_list 

867 output_spokes.src_spoke_list = src_spoke_list 

868 return output_spokes 

869 

870 return output_spokes 

871 

872 def _test_spoke(self, cos_theta_src, sin_theta_src, ref_ctr, ref_ctr_id, 

873 proj_ref_ctr_delta, proj_ref_ctr_dist_sq, 

874 ref_dist_idx_array, ref_id_array, src_sin_tol): 

875 """Test the opening angle between the first spoke of our pattern 

876 for the source object against the reference object. 

877 

878 This method makes heavy use of the small angle approximation to perform 

879 the comparison. 

880 

881 Parameters 

882 ---------- 

883 cos_theta_src : `float` 

884 Cosine of the angle between the current candidate source spoke and 

885 the first spoke. 

886 sin_theta_src : `float` 

887 Sine of the angle between the current candidate source spoke and 

888 the first spoke. 

889 ref_ctr : `numpy.ndarray`, (3,) 

890 3 vector of the candidate reference center 

891 ref_ctr_id : `int` 

892 id lookup of the ref_ctr into the master reference array 

893 proj_ref_ctr_delta : `float` 

894 Plane projected first spoke in the reference pattern using the 

895 pattern center as normal. 

896 proj_ref_ctr_dist_sq : `float` 

897 Squared length of the projected vector. 

898 ref_dist_idx_array : `numpy.ndarray`, (N,) 

899 Indices sorted by the delta distance between the source 

900 spoke we are trying to test and the candidate reference 

901 spokes. 

902 ref_id_array : `numpy.ndarray`, (N,) 

903 Array of id lookups into the master reference array that our 

904 center id object is paired with. 

905 src_sin_tol : `float` 

906 Sine of tolerance allowed between source and reference spoke 

907 opening angles. 

908 

909 Returns 

910 ------- 

911 id : `int` 

912 If we can not find a candidate spoke we return `None` else we 

913 return an int id into the master reference array. 

914 """ 

915 

916 # Loop over our candidate reference objects. 

917 for ref_dist_idx in ref_dist_idx_array: 

918 # Compute the delta vector from the pattern center. 

919 ref_delta = (self._reference_array[ref_id_array[ref_dist_idx]] 

920 - ref_ctr) 

921 # Compute the cos between our "center" reference vector and the 

922 # current reference candidate. 

923 proj_ref_delta = ref_delta - np.dot(ref_delta, ref_ctr) * ref_ctr 

924 geom_dist_ref = np.sqrt(proj_ref_ctr_dist_sq 

925 * np.dot(proj_ref_delta, proj_ref_delta)) 

926 cos_theta_ref = (np.dot(proj_ref_delta, proj_ref_ctr_delta) 

927 / geom_dist_ref) 

928 

929 # Make sure we can safely make the comparison in case 

930 # our "center" and candidate vectors are mostly aligned. 

931 if cos_theta_ref ** 2 < (1 - src_sin_tol ** 2): 

932 cos_sq_comparison = ((cos_theta_src - cos_theta_ref) ** 2 

933 / (1 - cos_theta_ref ** 2)) 

934 else: 

935 cos_sq_comparison = ((cos_theta_src - cos_theta_ref) ** 2 

936 / src_sin_tol ** 2) 

937 # Test the difference of the cosine of the reference angle against 

938 # the source angle. Assumes that the delta between the two is 

939 # small. 

940 if cos_sq_comparison > src_sin_tol ** 2: 

941 continue 

942 

943 # The cosine tests the magnitude of the angle but not 

944 # its direction. To do that we need to know the sine as well. 

945 # This cross product calculation does that. 

946 cross_ref = (np.cross(proj_ref_delta, proj_ref_ctr_delta) 

947 / geom_dist_ref) 

948 sin_theta_ref = np.dot(cross_ref, ref_ctr) 

949 

950 # Check the value of the cos again to make sure that it is not 

951 # near zero. 

952 if abs(cos_theta_src) < src_sin_tol: 

953 sin_comparison = (sin_theta_src - sin_theta_ref) / src_sin_tol 

954 else: 

955 sin_comparison = \ 

956 (sin_theta_src - sin_theta_ref) / cos_theta_ref 

957 if abs(sin_comparison) > src_sin_tol: 

958 continue 

959 

960 # Return the correct id of the candidate we found. 

961 return ref_id_array[ref_dist_idx] 

962 

963 return -1 

964 

965 def _create_shift_rot_matrix(self, cos_rot_sq, shift_matrix, src_delta, 

966 ref_ctr, ref_delta): 

967 """ Create the final part of our spherical rotation matrix. 

968 

969 Parameters 

970 ---------- 

971 cos_rot_sq : `float` 

972 cosine of the rotation needed to align our source and reference 

973 candidate patterns. 

974 shift_matrix : `numpy.ndarray`, (3, 3) 

975 3x3 rotation matrix for shifting the source pattern center on top 

976 of the candidate reference pattern center. 

977 src_delta : `numpy.ndarray`, (3,) 

978 3 vector delta of representing the first spoke of the source 

979 pattern 

980 ref_ctr : `numpy.ndarray`, (3,) 

981 3 vector on the unit-sphere representing the center of our 

982 reference pattern. 

983 ref_delta : `numpy.ndarray`, (3,) 

984 3 vector delta made by the first pair of the reference pattern. 

985 

986 Returns 

987 ------- 

988 result : `lsst.pipe.base.Struct` 

989 Result struct with components: 

990 

991 - ``sin_rot`` : float sine of the amount of rotation between the 

992 source and reference pattern. We use sine here as it is 

993 signed and tells us the chirality of the rotation (`float`). 

994 - ``shift_rot_matrix`` : float array representing the 3x3 rotation 

995 matrix that takes the source pattern and shifts and rotates 

996 it to align with the reference pattern (`numpy.ndarray`, (3,3)). 

997 """ 

998 cos_rot = np.sqrt(cos_rot_sq) 

999 rot_src_delta = np.dot(shift_matrix, src_delta) 

1000 delta_dot_cross = np.dot(np.cross(rot_src_delta, ref_delta), ref_ctr) 

1001 

1002 sin_rot = np.sign(delta_dot_cross) * np.sqrt(1 - cos_rot_sq) 

1003 rot_matrix = self._create_spherical_rotation_matrix( 

1004 ref_ctr, cos_rot, sin_rot) 

1005 

1006 shift_rot_matrix = np.dot(rot_matrix, shift_matrix) 

1007 

1008 return pipeBase.Struct( 

1009 sin_rot=sin_rot, 

1010 shift_rot_matrix=shift_rot_matrix,) 

1011 

1012 def _intermediate_verify(self, src_pattern, ref_pattern, shift_rot_matrix, 

1013 max_dist_rad): 

1014 """ Perform an intermediate verify step. 

1015 

1016 Rotate the matches references into the source frame and test their 

1017 distances against tolerance. Only return true if all points are within 

1018 tolerance. 

1019 

1020 Parameters 

1021 ---------- 

1022 src_pattern : `numpy.ndarray`, (N,3) 

1023 Array of 3 vectors representing the points that make up our source 

1024 pinwheel pattern. 

1025 ref_pattern : `numpy.ndarray`, (N,3) 

1026 Array of 3 vectors representing our candidate reference pinwheel 

1027 pattern. 

1028 shift_rot_matrix : `numpy.ndarray`, (3,3) 

1029 3x3 rotation matrix that takes the source objects and rotates them 

1030 onto the frame of the reference objects 

1031 max_dist_rad : `float` 

1032 Maximum distance allowed to consider two objects the same. 

1033 

1034 Returns 

1035 ------- 

1036 fit_shift_rot_matrix : `numpy.ndarray`, (3,3) 

1037 Return the fitted shift/rotation matrix if all of the points in our 

1038 source pattern are within max_dist_rad of their matched reference 

1039 objects. Returns None if this criteria is not satisfied. 

1040 """ 

1041 if len(src_pattern) != len(ref_pattern): 

1042 raise ValueError( 

1043 "Source pattern length does not match ref pattern.\n" 

1044 "\t source pattern len=%i, reference pattern len=%i" % 

1045 (len(src_pattern), len(ref_pattern))) 

1046 

1047 if self._intermediate_verify_comparison( 

1048 src_pattern, ref_pattern, shift_rot_matrix, max_dist_rad): 

1049 # Now that we know our initial shift and rot matrix is valid we 

1050 # want to fit the implied transform using all points from 

1051 # our pattern. This is a more robust rotation matrix as our 

1052 # initial matrix only used the first 2 points from the source 

1053 # pattern to estimate the shift and rotation. The matrix we fit 

1054 # are allowed to be non unitary but need to preserve the length of 

1055 # the rotated vectors to within the match tolerance. 

1056 fit_shift_rot_matrix = least_squares( 

1057 _rotation_matrix_chi_sq, 

1058 x0=shift_rot_matrix.flatten(), 

1059 args=(src_pattern, ref_pattern, max_dist_rad) 

1060 ).x.reshape((3, 3)) 

1061 # Do another verify in case the fits have wondered off. 

1062 if self._intermediate_verify_comparison( 

1063 src_pattern, ref_pattern, fit_shift_rot_matrix, 

1064 max_dist_rad): 

1065 return fit_shift_rot_matrix 

1066 

1067 return None 

1068 

1069 def _intermediate_verify_comparison(self, pattern_a, pattern_b, 

1070 shift_rot_matrix, max_dist_rad): 

1071 """Test the input rotation matrix against one input pattern and 

1072 a second one. 

1073 

1074 If every point in the pattern after rotation is within a distance of 

1075 max_dist_rad to its candidate point in the other pattern, we return 

1076 True. 

1077 

1078 Parameters 

1079 ---------- 

1080 pattern_a : `numpy.ndarray`, (N,3) 

1081 Array of 3 vectors representing the points that make up our source 

1082 pinwheel pattern. 

1083 pattern_b : `numpy.ndarray`, (N,3) 

1084 Array of 3 vectors representing our candidate reference pinwheel 

1085 pattern. 

1086 shift_rot_matrix : `numpy.ndarray`, (3,3) 

1087 3x3 rotation matrix that takes the source objects and rotates them 

1088 onto the frame of the reference objects 

1089 max_dist_rad : `float` 

1090 Maximum distance allowed to consider two objects the same. 

1091 

1092 

1093 Returns 

1094 ------- 

1095 bool 

1096 True if all rotated source points are within max_dist_rad of 

1097 the candidate references matches. 

1098 """ 

1099 shifted_pattern_a = np.dot(shift_rot_matrix, 

1100 pattern_a.transpose()).transpose() 

1101 tmp_delta_array = shifted_pattern_a - pattern_b 

1102 tmp_dist_array = (tmp_delta_array[:, 0] ** 2 

1103 + tmp_delta_array[:, 1] ** 2 

1104 + tmp_delta_array[:, 2] ** 2) 

1105 return np.all(tmp_dist_array < max_dist_rad ** 2) 

1106 

1107 def _test_pattern_lengths(self, test_pattern, max_dist_rad): 

1108 """ Test that the all vectors in a pattern are unit length within 

1109 tolerance. 

1110 

1111 This is useful for assuring the non unitary transforms do not contain 

1112 too much distortion. 

1113 

1114 Parameters 

1115 ---------- 

1116 test_pattern : `numpy.ndarray`, (N, 3) 

1117 Test vectors at the maximum and minimum x, y, z extents. 

1118 max_dist_rad : `float` 

1119 maximum distance in radians to consider two points "agreeing" on 

1120 a rotation 

1121 

1122 Returns 

1123 ------- 

1124 test : `bool` 

1125 Length tests pass. 

1126 """ 

1127 dists = (test_pattern[:, 0] ** 2 

1128 + test_pattern[:, 1] ** 2 

1129 + test_pattern[:, 2] ** 2) 

1130 return np.all( 

1131 np.logical_and((1 - max_dist_rad) ** 2 < dists, 

1132 dists < (1 + max_dist_rad) ** 2)) 

1133 

1134 def _test_rotation_agreement(self, rot_vects, max_dist_rad): 

1135 """ Test this rotation against the previous N found and return 

1136 the number that a agree within tolerance to where our test 

1137 points are. 

1138 

1139 Parameters 

1140 ---------- 

1141 rot_vects : `numpy.ndarray`, (N, 3) 

1142 Arrays of rotated 3 vectors representing the maximum x, y, 

1143 z extent on the unit sphere of the input source objects rotated by 

1144 the candidate rotations into the reference frame. 

1145 max_dist_rad : `float` 

1146 maximum distance in radians to consider two points "agreeing" on 

1147 a rotation 

1148 

1149 Returns 

1150 ------- 

1151 tot_consent : `int` 

1152 Number of candidate rotations that agree for all of the rotated 

1153 test 3 vectors. 

1154 """ 

1155 

1156 self.log.debug("Comparing pattern %i to previous %i rotations...", 

1157 rot_vects[-1][-1], len(rot_vects) - 1) 

1158 

1159 tot_consent = 0 

1160 for rot_idx in range(max((len(rot_vects) - 1), 0)): 

1161 tmp_dist_list = [] 

1162 for vect_idx in range(len(rot_vects[rot_idx]) - 1): 

1163 tmp_delta_vect = (rot_vects[rot_idx][vect_idx] 

1164 - rot_vects[-1][vect_idx]) 

1165 tmp_dist_list.append( 

1166 np.dot(tmp_delta_vect, tmp_delta_vect)) 

1167 if np.all(np.array(tmp_dist_list) < max_dist_rad ** 2): 

1168 tot_consent += 1 

1169 return tot_consent 

1170 

1171 def _final_verify(self, 

1172 source_array, 

1173 shift_rot_matrix, 

1174 max_dist_rad, 

1175 min_matches): 

1176 """Match the all sources into the reference catalog using the shift/rot 

1177 matrix. 

1178 

1179 After the initial shift/rot matrix is found, we refit the shift/rot 

1180 matrix using the matches the initial matrix produces to find a more 

1181 stable solution. 

1182 

1183 Parameters 

1184 ---------- 

1185 source_array : `numpy.ndarray` (N, 3) 

1186 3-vector positions on the unit-sphere representing the sources to 

1187 match 

1188 shift_rot_matrix : `numpy.ndarray` (3, 3) 

1189 Rotation matrix representing inferred shift/rotation of the 

1190 sources onto the reference catalog. Matrix need not be unitary. 

1191 max_dist_rad : `float` 

1192 Maximum distance allowed for a match. 

1193 min_matches : `int` 

1194 Minimum number of matched objects required to consider the 

1195 match good. 

1196 

1197 Returns 

1198 ------- 

1199 output_struct : `lsst.pipe.base.Struct` 

1200 Result struct with components: 

1201 

1202 - ``match_ids`` : Pairs of indexes into the source and reference 

1203 data respectively defining a match (`numpy.ndarray`, (N, 2)). 

1204 - ``distances_rad`` : distances to between the matched objects in 

1205 the shift/rotated frame. (`numpy.ndarray`, (N,)). 

1206 - ``max_dist_rad`` : Value of the max matched distance. Either 

1207 returning the input value of the 2 sigma clipped value of the 

1208 shift/rotated distances. (`float`). 

1209 """ 

1210 output_struct = pipeBase.Struct( 

1211 match_ids=None, 

1212 distances_rad=None, 

1213 max_dist_rad=None, 

1214 ) 

1215 

1216 # Perform an iterative final verify. 

1217 match_sources_struct = self._match_sources(source_array, 

1218 shift_rot_matrix) 

1219 cut_ids = match_sources_struct.match_ids[ 

1220 match_sources_struct.distances_rad < max_dist_rad] 

1221 

1222 n_matched = len(cut_ids) 

1223 clipped_struct = self._clip_distances( 

1224 match_sources_struct.distances_rad) 

1225 n_matched_clipped = clipped_struct.n_matched_clipped 

1226 

1227 if n_matched < min_matches or n_matched_clipped < min_matches: 

1228 return output_struct 

1229 

1230 # The shift/rotation matrix returned by 

1231 # ``_construct_pattern_and_shift_rot_matrix``, above, was 

1232 # based on only six points. Here, we refine that result by 

1233 # using all of the good matches from the “final verification” 

1234 # step, above. This will produce a more consistent result. 

1235 fit_shift_rot_matrix = least_squares( 

1236 _rotation_matrix_chi_sq, 

1237 x0=shift_rot_matrix.flatten(), 

1238 args=(source_array[cut_ids[:, 0], :3], 

1239 self._reference_array[cut_ids[:, 1], :3], 

1240 max_dist_rad) 

1241 ).x.reshape((3, 3)) 

1242 

1243 # Redo the matching using the newly fit shift/rotation matrix. 

1244 match_sources_struct = self._match_sources( 

1245 source_array, fit_shift_rot_matrix) 

1246 

1247 # Double check the match distances to make sure enough matches 

1248 # survive still. We'll just overwrite the previously used variables. 

1249 n_matched = np.sum( 

1250 match_sources_struct.distances_rad < max_dist_rad) 

1251 clipped_struct = self._clip_distances( 

1252 match_sources_struct.distances_rad) 

1253 n_matched_clipped = clipped_struct.n_matched_clipped 

1254 clipped_max_dist = clipped_struct.clipped_max_dist 

1255 

1256 if n_matched < min_matches or n_matched_clipped < min_matches: 

1257 return output_struct 

1258 

1259 # Store our matches in the output struct. Decide between the clipped 

1260 # distance and the input max dist based on which is smaller. 

1261 output_struct.match_ids = match_sources_struct.match_ids 

1262 output_struct.distances_rad = match_sources_struct.distances_rad 

1263 if clipped_max_dist < max_dist_rad: 

1264 output_struct.max_dist_rad = clipped_max_dist 

1265 else: 

1266 output_struct.max_dist_rad = max_dist_rad 

1267 

1268 return output_struct 

1269 

1270 def _clip_distances(self, distances_rad): 

1271 """Compute a clipped max distance and calculate the number of pairs 

1272 that pass the clipped dist. 

1273 

1274 Parameters 

1275 ---------- 

1276 distances_rad : `numpy.ndarray`, (N,) 

1277 Distances between pairs. 

1278 

1279 Returns 

1280 ------- 

1281 output_struct : `lsst.pipe.base.Struct` 

1282 Result struct with components: 

1283 

1284 - ``n_matched_clipped`` : Number of pairs that survive the 

1285 clipping on distance. (`float`) 

1286 - ``clipped_max_dist`` : Maximum distance after clipping. 

1287 (`float`). 

1288 """ 

1289 clipped_dists, _, clipped_max_dist = sigmaclip( 

1290 distances_rad, 

1291 low=100, 

1292 high=2) 

1293 # Check clipped distances. The minimum value here 

1294 # prevents over convergence on perfect test data. 

1295 if clipped_max_dist < 1e-16: 

1296 clipped_max_dist = 1e-16 

1297 n_matched_clipped = np.sum(distances_rad < clipped_max_dist) 

1298 else: 

1299 n_matched_clipped = len(clipped_dists) 

1300 

1301 return pipeBase.Struct(n_matched_clipped=n_matched_clipped, 

1302 clipped_max_dist=clipped_max_dist) 

1303 

1304 def _match_sources(self, 

1305 source_array, 

1306 shift_rot_matrix): 

1307 """ Shift both the reference and source catalog to the the respective 

1308 frames and find their nearest neighbor using a kdTree. 

1309 

1310 Removes all matches who do not agree when either the reference or 

1311 source catalog is rotated. Cuts on a maximum distance are left to an 

1312 external function. 

1313 

1314 Parameters 

1315 ---------- 

1316 source_array : `numpy.ndarray`, (N, 3) 

1317 array of 3 vectors representing the source objects we are trying 

1318 to match into the source catalog. 

1319 shift_rot_matrix : `numpy.ndarray`, (3, 3) 

1320 3x3 rotation matrix that performs the spherical rotation from the 

1321 source frame into the reference frame. 

1322 

1323 Returns 

1324 ------- 

1325 results : `lsst.pipe.base.Struct` 

1326 Result struct with components: 

1327 

1328 - ``matches`` : array of integer ids into the source and 

1329 reference arrays. Matches are only returned for those that 

1330 satisfy the distance and handshake criteria 

1331 (`numpy.ndarray`, (N, 2)). 

1332 - ``distances`` : Distances between each match in radians after 

1333 the shift and rotation is applied (`numpy.ndarray`, (N)). 

1334 """ 

1335 shifted_references = np.dot( 

1336 np.linalg.inv(shift_rot_matrix), 

1337 self._reference_array.transpose()).transpose() 

1338 shifted_sources = np.dot( 

1339 shift_rot_matrix, 

1340 source_array.transpose()).transpose() 

1341 

1342 ref_matches = np.empty((len(shifted_references), 2), 

1343 dtype="uint16") 

1344 src_matches = np.empty((len(shifted_sources), 2), 

1345 dtype="uint16") 

1346 

1347 ref_matches[:, 1] = np.arange(len(shifted_references), 

1348 dtype="uint16") 

1349 src_matches[:, 0] = np.arange(len(shifted_sources), 

1350 dtype="uint16") 

1351 

1352 ref_kdtree = cKDTree(self._reference_array) 

1353 src_kdtree = cKDTree(source_array) 

1354 

1355 ref_to_src_dist, tmp_ref_to_src_idx = \ 

1356 src_kdtree.query(shifted_references) 

1357 src_to_ref_dist, tmp_src_to_ref_idx = \ 

1358 ref_kdtree.query(shifted_sources) 

1359 

1360 ref_matches[:, 0] = tmp_ref_to_src_idx 

1361 src_matches[:, 1] = tmp_src_to_ref_idx 

1362 

1363 handshake_mask = self._handshake_match(src_matches, ref_matches) 

1364 return pipeBase.Struct( 

1365 match_ids=src_matches[handshake_mask], 

1366 distances_rad=src_to_ref_dist[handshake_mask],) 

1367 

1368 def _handshake_match(self, matches_src, matches_ref): 

1369 """Return only those matches where both the source 

1370 and reference objects agree they they are each others' 

1371 nearest neighbor. 

1372 

1373 Parameters 

1374 ---------- 

1375 matches_src : `numpy.ndarray`, (N, 2) 

1376 int array of nearest neighbor matches between shifted and 

1377 rotated reference objects matched into the sources. 

1378 matches_ref : `numpy.ndarray`, (N, 2) 

1379 int array of nearest neighbor matches between shifted and 

1380 rotated source objects matched into the references. 

1381 Return 

1382 ------ 

1383 handshake_mask_array : `numpy.ndarray`, (N,) 

1384 Return the array positions where the two match catalogs agree. 

1385 """ 

1386 handshake_mask_array = np.zeros(len(matches_src), dtype=bool) 

1387 

1388 for src_match_idx, match in enumerate(matches_src): 

1389 ref_match_idx = np.searchsorted(matches_ref[:, 1], match[1]) 

1390 if match[0] == matches_ref[ref_match_idx, 0]: 

1391 handshake_mask_array[src_match_idx] = True 

1392 return handshake_mask_array