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import numpy as np 

from scipy.optimize import least_squares 

from scipy.spatial import cKDTree 

 

import lsst.pipe.base as pipeBase 

 

 

def _rotation_matrix_chi_sq(flattened_rot_matrix, 

pattern_a, 

pattern_b, 

max_dist_rad): 

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

 

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

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

points in the two patterns after the rotation. 

 

Parameters 

---------- 

flattened_rot_matrix : float array 

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

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

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

pattern_a : float array of 3 vectors 

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

to transform into the frame of pattern_b. 

pattern_b : float array of 3 vectors 

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

would like to transform pattern_a into. 

max_dist_rad : float 

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

used as the standard error for the resultant chi values. 

 

Returns 

------- 

float array 

Array of differences between the vectors representing of the source 

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

used to minimize in a least squares fitter. 

""" 

# Unflatten the rotation matrix 

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

# Compare the rotated source pattern to the references. 

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

diff_pattern_a_to_b = rot_pattern_a - pattern_b 

# Compute the matrix inverse and compare the opposite transform to better 

# constrain the matrix. 

rot_pattern_b = np.dot(np.linalg.inv(rot_matrix), 

pattern_b.transpose()).transpose() 

diff_pattern_b_to_a = rot_pattern_b - pattern_a 

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

# squares fitter. 

return np.concatenate( 

(diff_pattern_a_to_b.flatten() / max_dist_rad, 

diff_pattern_b_to_a.flatten() / max_dist_rad)) 

 

 

class PessimisticPatternMatcherB: 

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

B (OPMb) from Tabur 2007. The class loads and stores the reference object 

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

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

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

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

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

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

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

-------------------------------------------------------------------------- 

Attributes: 

reference_array : float array 

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

pattern matching. 

log : an lsst.log instance 

pair_id_array : int array 

Internal lookup table. Given an id in the reference array, return 

an array of the id pair that contains this object's id sorted on 

the distance to the pairs. 

pair_delta_array : float array 

Internal lookup table. Given an id in the reference array, return 

an array of the 3 vector deltas of all other pairs sorted on their 

distance. 

pair_dist_array : float array 

Internal lookup table. Given an id in the reference return an array 

of pair distances of all other pairs sorted on distance. 

dist_array : float array 

Array of all pairs of objects in the reference array sorted on 

distance. 

id_array : int array 

Array of id pairs that lookup into the reference array sorted 

on pair distance. 

delta_array : float array 

Array of 3 vector deltas for each pair in the reference array 

sorted on pair distance. 

""" 

 

def __init__(self, reference_array, log): 

""" 

Arguments 

--------- 

reference_array : float array 

Array of spherical points x, y, z to use as reference objects. 

log : lsst.log 

logger object for reporting warnings and failures. 

""" 

self._reference_array = reference_array 

self._n_reference = len(self._reference_array) 

self.log = log 

 

self._build_distances_and_angles() 

 

def _build_distances_and_angles(self): 

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

match in. 

 

Throughout this function and the rest of the 

 

class we use id to reference the position in the input reference 

catalog and index to 'index' into the arrays sorted on distance. 

 

Parameters 

---------- 

self 

 

Returns 

------- 

None 

""" 

 

# Initialize the arrays we will need for quick look up of pairs once 

# have a candidate spoke center. 

self._pair_id_array = np.empty( 

(self._n_reference, self._n_reference - 1), 

dtype=np.uint32) 

self._pair_delta_array = np.empty( 

(self._n_reference, self._n_reference - 1, 3), 

dtype=np.float64) 

self._pair_dist_array = np.empty( 

(self._n_reference, self._n_reference - 1), 

dtype=np.float64) 

 

# Create empty lists to temporarily store our pair information per 

# reference object. These will be concatenated into our final arrays. 

sub_id_array_list = [] 

sub_delta_array_list = [] 

sub_dist_array_list = [] 

 

# Loop over reference objects and store pair distances, ids, and 

# 3 vector deltas. 

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

 

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

sub_id_array = np.zeros((self._n_reference - 1 - ref_id, 2), 

dtype=np.uint32) 

sub_id_array[:, 0] = ref_id 

sub_id_array[:, 1] = np.arange(ref_id + 1, self._n_reference, 

dtype=np.uint32) 

 

# Compute the vector deltas for each pair of reference objects 

# and compute and store the distances. 

sub_delta_array = (self._reference_array[ref_id + 1:, :] - 

ref_obj) 

sub_dist_array = np.sqrt(sub_delta_array[:, 0] ** 2 + 

sub_delta_array[:, 1] ** 2 + 

sub_delta_array[:, 2] ** 2) 

 

# Append to our arrays to the output lists for later 

# concatenation. 

sub_id_array_list.append(sub_id_array) 

sub_delta_array_list.append(sub_delta_array) 

sub_dist_array_list.append(sub_dist_array) 

 

# Fill the pair look up arrays row wise and then column wise. 

self._pair_id_array[ref_id, ref_id:] = sub_id_array[:, 1] 

self._pair_delta_array[ref_id, ref_id:, :] = sub_delta_array 

self._pair_dist_array[ref_id, ref_id:] = sub_dist_array 

 

# Don't fill the array column wise if we are on the last as 

# these pairs have already been filled by previous 

# iterations. 

if ref_id < self._n_reference - 1: 

self._pair_id_array[ref_id + 1:, ref_id] = ref_id 

self._pair_delta_array[ref_id + 1:, ref_id, :] = \ 

sub_delta_array 

self._pair_dist_array[ref_id + 1:, ref_id] = sub_dist_array 

 

# Sort each row on distance for fast look up of pairs given 

# the id of one of the objects in the pair. 

sorted_pair_dist_args = self._pair_dist_array[ref_id, :].argsort() 

self._pair_dist_array[ref_id, :] = self._pair_dist_array[ 

ref_id, sorted_pair_dist_args] 

self._pair_id_array[ref_id, :] = self._pair_id_array[ 

ref_id, sorted_pair_dist_args] 

self._pair_delta_array[ref_id, :, :] = self._pair_delta_array[ 

ref_id, sorted_pair_dist_args, :] 

 

# Concatenate our arrays together. 

unsorted_id_array = np.concatenate(sub_id_array_list) 

unsorted_delta_array = np.concatenate(sub_delta_array_list) 

unsorted_dist_array = np.concatenate(sub_dist_array_list) 

 

# Sort each array on the pair distances for the initial 

# optimistic pattern matcher lookup. 

sorted_dist_args = unsorted_dist_array.argsort() 

self._dist_array = unsorted_dist_array[sorted_dist_args] 

self._id_array = unsorted_id_array[sorted_dist_args] 

self._delta_array = unsorted_delta_array[sorted_dist_args] 

 

return None 

 

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

max_n_patterns, max_shift, max_rotation, max_dist, 

min_matches, pattern_skip_array=None): 

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

 

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

attempt to match a pinwheel like pattern between these input sources 

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

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

objects into the frame of the references. 

 

Parameters 

---------- 

source_array: float array 

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

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

of shape (N, 4). 

n_check : int value 

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

checked if n_match criteria is before looping through all n_check 

objects. 

n_match : int value 

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

n_agree: int value 

Number of found patterns that must agree on their shift and 

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

expected behavior of Optimistic Pattern Matcher B. 

max_n_patters : int value 

Number of patterns to create from the input source objects to 

attempt to match into the reference objects. 

max_shift: float value 

Maximum allowed shift to match patterns in arcseconds. 

max_rotation: float value 

Maximum allowed rotation between patterns in degrees. 

max_dist: float value 

Maximum distance in arcseconds allowed between candidate spokes in 

the source and reference objects. Also sets that maximum distance 

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

final verify steps. 

pattern_skip_array: int array 

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

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

This assumes the ordering of the source objects is the same 

between different runs of the matcher which, assuming no object 

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

 

Returns 

------- 

output_struct : pipe.base.struct 

A lsst.pipe.base struct containing the following outputs. 

 

matches : int array 

(N, 2) array of matched ids for pairs. Empty list if no 

match found. 

distances_rad : float array 

Radian distances between the matched objects. Empty list 

if no match found. 

pattern_idx : int 

Index of matched pattern. None if no match found. 

shift : float 

Magnitude for the shift between the source and 

reference objects in arcseconds. None if no match found. 

""" 

 

# Given our input source_array we sort on magnitude. 

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

n_source = len(sorted_source_array) 

 

# Initialize output struct. 

output_match_struct = pipeBase.Struct( 

match_ids=[], 

distances_rad=[], 

pattern_idx=None, 

shift=None,) 

 

if n_source <= 0: 

self.log.warn("Source object array is empty. Unable to match. " 

"Exiting matcher.") 

return None 

 

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

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

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

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

 

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

# compare the different rotations we find. 

rot_vect_list = [] 

 

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

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

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

max_dist_rad = np.radians(max_dist / 3600.) 

 

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

# chunk of n_check each time. 

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

n_source - n_match))): 

 

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

# now want to skip, we do so here. 

if pattern_skip_array is not None and \ 

np.any(pattern_skip_array == pattern_idx): 

self.log.debug( 

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

pattern_idx) 

continue 

# Grab the sources to attempt to create this pattern. 

pattern = sorted_source_array[ 

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

 

# Construct a pattern given the number of points defining the 

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

# match our source pattern into the reference objects. 

construct_return_struct = \ 

self._construct_pattern_and_shift_rot_matrix( 

pattern, n_match, max_cos_shift, max_cos_rot_sq, 

max_dist_rad) 

 

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

if construct_return_struct.ref_candidates is None or \ 

construct_return_struct.shift_rot_matrix is None or \ 

construct_return_struct.cos_shift is None or \ 

construct_return_struct.sin_rot is None: 

continue 

 

# Grab the output data from the Struct object. 

ref_candidates = construct_return_struct.ref_candidates 

shift_rot_matrix = construct_return_struct.shift_rot_matrix 

cos_shift = construct_return_struct.cos_shift 

sin_rot = construct_return_struct.sin_rot 

 

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

# pattern. 

if len(ref_candidates) < n_match: 

continue 

 

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

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

# use. 

tmp_rot_vect_list = [] 

for test_vect in test_vectors: 

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

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

# their lengths are mostly preserved under the transform. 

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

max_dist_rad): 

continue 

 

tmp_rot_vect_list.append(pattern_idx) 

rot_vect_list.append(tmp_rot_vect_list) 

 

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

# are in optimistic mode. 

if self._test_rotation_agreement(rot_vect_list, max_dist_rad) < \ 

n_agree - 1: 

continue 

 

# Perform final verify. 

match_sources_struct = self._match_sources(source_array[:, :3], 

shift_rot_matrix) 

 

n_matched = len(match_sources_struct.match_ids[ 

match_sources_struct.distances_rad < max_dist_rad]) 

 

# Check that we have enough matches. 

if n_matched >= min_matches: 

# Convert the observed shift to arcseconds 

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

# Print information to the logger. 

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

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

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

np.degrees(np.arcsin(sin_rot))) 

 

# Fill the struct and return. 

output_match_struct.match_ids = \ 

match_sources_struct.match_ids 

output_match_struct.distances_rad = \ 

match_sources_struct.distances_rad 

output_match_struct.pattern_idx = pattern_idx 

output_match_struct.shift = shift 

return output_match_struct 

 

self.log.warn("Failed after %i patterns." % (pattern_idx + 1)) 

return output_match_struct 

 

def _compute_test_vectors(self, source_array): 

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

of the input source catalog. 

 

Parameters 

---------- 

source_array : float array (N, 3) 

array of 3 vectors representing positions on the unit 

sphere. 

 

Returns 

------- 

float array of 3 vectors 

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

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

the code finds to test for agreement from different patterns 

when the code is running in pessimistic mode. 

""" 

 

# Get the center of source_array. 

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

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

"This could end badly.") 

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

 

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

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

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

center_vect[2]], dtype=np.float64) 

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

center_vect[2]], dtype=np.float64) 

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

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

 

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

center_vect[2]], dtype=np.float64) 

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

center_vect[2]], dtype=np.float64) 

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

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

 

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

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

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

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

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

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

 

# Return our list of vectors for later rotation testing. 

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

zbtm_vect, ztop_vect]) 

 

def _construct_pattern_and_shift_rot_matrix(self, src_pattern_array, 

n_match, max_cos_theta_shift, 

max_cos_rot_sq, max_dist_rad): 

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

 

Returns the candidate matched patterns and their 

implied rotation matrices or None. 

 

Parameters 

---------- 

src_pattern_array : float array 

Sub selection of source 3 vectors to create a pattern from 

n_match : int 

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

>= len(src_pattern_array) 

max_cos_theta_shift : float 

Maximum shift allowed between two patterns' centers. 

max_cos_rot_sq : float 

Maximum rotation between two patterns that have been shifted 

to have their centers on top of each other. 

max_dist_rad : float 

Maximum delta distance allowed between the source and reference 

pair distances to consider the reference pair a candidate for 

the source pair. Also sets the tolerance between the opening 

angles of the spokes when compared to the reference. 

 

Return 

------- 

lsst.pipe.base.Struct 

Return a Struct containing the following data: 

 

ref_candidates : list of ints 

ids of the matched pattern in the internal reference_array 

object. 

src_candidates : list of ints 

Pattern ids of the sources matched. 

shift_rot_matrix_src_to_ref : float array 

3x3 matrix specifying the full shift and rotation between the 

reference and source objects. Rotates 

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

shift_rot_matrix_ref_to_src : float array 

3x3 matrix specifying the full shift 

and rotation of the reference and source objects. Rotates 

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

cos_shift : float 

Magnitude of the shift found between the two patten 

centers. None if match is not found. 

sin_rot : float value of the rotation to align the already shifted 

source pattern to the reference pattern. None if no match 

found. 

""" 

 

# Create our place holder variables for the matched sources and 

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

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

# the shift and rotation of the final pattern. 

output_matched_pattern = pipeBase.Struct( 

ref_candidates=[], 

src_candidates=[], 

shift_rot_matrix=None, 

cos_shift=None, 

sin_rot=None) 

 

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

# spokes of the pattern. 

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

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

src_pattern_array[0, 0]) 

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

src_pattern_array[0, 1]) 

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

src_pattern_array[0, 2]) 

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

src_delta_array[:, 1]**2 + 

src_delta_array[:, 2]**2) 

 

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

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

# plus/minus the max_dist tolerance. 

ref_dist_index_array = self._find_candidate_reference_pairs( 

src_dist_array[0], self._dist_array, max_dist_rad) 

 

# Start our loop over the candidate reference objects. 

for ref_dist_idx in ref_dist_index_array: 

# We have two candidates for which reference object corresponds 

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

# over and test both possibilities. 

tmp_ref_pair_list = self._id_array[ref_dist_idx] 

for pair_idx, ref_id in enumerate(tmp_ref_pair_list): 

src_candidates = [0, 1] 

ref_candidates = [] 

shift_rot_matrix = None 

cos_shift = None 

sin_rot = None 

# Test the angle between our candidate ref center and the 

# source center of our pattern. This angular distance also 

# defines the shift we will later use. 

ref_center = self._reference_array[ref_id] 

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

if cos_shift < max_cos_theta_shift: 

continue 

 

# We can now append this one as a candidate. 

ref_candidates.append(ref_id) 

ref_delta = self._delta_array[ref_dist_idx] 

# If the candidate reference center we found is second in 

# this pair we need to reverse the direction of the 

# corresponding pair's delta vector. 

if pair_idx == 0: 

ref_candidates.append( 

tmp_ref_pair_list[1]) 

else: 

ref_candidates.append( 

tmp_ref_pair_list[0]) 

ref_delta *= -1 

 

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

# rotation this pair suggests first rather than saving it 

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

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

# source to reference frame. 

test_rot_struct = self._test_rotation( 

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

ref_delta, cos_shift, max_cos_rot_sq) 

if test_rot_struct.cos_rot_sq is None or \ 

test_rot_struct.proj_ref_ctr_delta is None or \ 

test_rot_struct.shift_matrix is None: 

continue 

 

# Get the data from the return struct. 

cos_rot_sq = test_rot_struct.cos_rot_sq 

proj_ref_ctr_delta = test_rot_struct.proj_ref_ctr_delta 

shift_matrix = test_rot_struct.shift_matrix 

 

# Now that we have a candidate first spoke and reference 

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

# pairs that contain our candidate reference center. 

tmp_ref_delta_array = self._pair_delta_array[ref_id] 

tmp_ref_dist_array = self._pair_dist_array[ref_id] 

tmp_ref_id_array = self._pair_id_array[ref_id] 

 

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

# our pattern. 

pattern_spoke_struct = self._create_pattern_spokes( 

src_pattern_array[0], src_delta_array, src_dist_array, 

self._reference_array[ref_id], ref_id, proj_ref_ctr_delta, 

tmp_ref_delta_array, tmp_ref_dist_array, 

tmp_ref_id_array, max_dist_rad, n_match) 

 

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

# next reference center pair. 

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

len(pattern_spoke_struct.src_spoke_list) < n_match - 2: 

continue 

 

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

ref_candidates.extend(pattern_spoke_struct.ref_spoke_list) 

src_candidates.extend(pattern_spoke_struct.src_spoke_list) 

 

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

# shift and rotation. Reminder shift, aligns the pattern 

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

shift_rot_struct = self._create_shift_rot_matrix( 

cos_rot_sq, shift_matrix, src_delta_array[0], 

self._reference_array[ref_id], ref_delta) 

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

# next objects. 

if shift_rot_struct.sin_rot is None or \ 

shift_rot_struct.shift_rot_matrix is None: 

continue 

 

# Get the data from the return struct. 

sin_rot = shift_rot_struct.sin_rot 

shift_rot_matrix = shift_rot_struct.shift_rot_matrix 

 

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

# passes intermediate verify. This shifts and rotates the 

# source pattern into the reference frame and then tests that 

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

# tests the opposite rotation that is reference to source 

# frame. 

fit_shift_rot_matrix = self._intermediate_verify( 

src_pattern_array[src_candidates], 

self._reference_array[ref_candidates], 

shift_rot_matrix, max_dist_rad) 

 

if fit_shift_rot_matrix is not None: 

# Fill the struct and return. 

output_matched_pattern.ref_candidates = ref_candidates 

output_matched_pattern.src_candidates = src_candidates 

output_matched_pattern.shift_rot_matrix = \ 

fit_shift_rot_matrix 

output_matched_pattern.cos_shift = cos_shift 

output_matched_pattern.sin_rot = sin_rot 

return output_matched_pattern 

 

return output_matched_pattern 

 

def _find_candidate_reference_pairs(self, src_dist, ref_dist_array, 

max_dist_rad): 

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

within a spoke distance tolerance of our source spoke. 

 

Returns an array sorted from the smallest absolute delta distance 

between source and reference spoke length. This sorting increases the 

speed for the pattern search greatly. 

 

Parameters 

---------- 

src_dist : float radians 

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

the reference array in radians. 

ref_dist_array : float array 

sorted array of distances in radians. 

max_dist_rad : float 

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

radians. 

 

Return 

------ 

int array 

indices lookup into the input ref_dist_array sorted by the 

difference in value to the src_dist from absolute value 

smallest to largest. 

""" 

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

# the tolerance. 

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

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

side='right') 

 

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

if start_idx == end_idx: 

return [] 

 

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

if start_idx < 0: 

start_idx = 0 

if end_idx > ref_dist_array.shape[0]: 

end_idx = ref_dist_array.shape[0] 

 

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

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

# the speed of the algorithm. 

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

return tmp_diff_array.argsort() + start_idx 

 

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

cos_shift, max_cos_rot_sq): 

""" Test if the rotation implied between the source 

pattern and reference pattern is within tolerance. To test this 

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

which we also return for use later. 

 

Parameters 

---------- 

src_center : float array3 

pattern. 

ref_center : float array 

3 vector defining the center of the candidate reference pinwheel 

pattern. 

src_delta : float array 

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

the pinwheel spoke. 

ref_delta : float array 

3 vector delta of the candidate matched reference pair 

cos_shift : float 

Cosine of the angle between the source and reference candidate 

centers. 

max_cos_rot_sq : float 

candidate reference pair after shifting the centers on top of each 

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

greater than max_cos_rot_sq. 

 

Returns 

------- 

lsst.pipe.base.Struct 

Return a pipe.base.Struct containing the following data. 

 

cos_rot_sq : float 

magnitude of the rotation needed to align the two patterns 

after their centers are shifted on top of each other. 

None if rotation test fails. 

shift_matrix : float array 

3x3 rotation matrix describing the shift needed to align 

the source and candidate reference center. 

None if rotation test fails. 

""" 

 

# Make sure the sine is a real number. 

if cos_shift > 1.0: 

cos_shift = 1. 

elif cos_shift < -1.0: 

cos_shift = -1. 

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

 

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

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

# shift. 

if sin_shift > 0: 

rot_axis = np.cross(src_center, ref_center) 

rot_axis /= sin_shift 

shift_matrix = self._create_spherical_rotation_matrix( 

rot_axis, cos_shift, sin_shift) 

else: 

shift_matrix = np.identity(3) 

 

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

# and check the rotation. 

rot_src_delta = np.dot(shift_matrix, src_delta) 

proj_src_delta = (rot_src_delta - 

np.dot(rot_src_delta, ref_center) * ref_center) 

proj_ref_delta = (ref_delta - 

np.dot(ref_delta, ref_center) * ref_center) 

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

(np.dot(proj_src_delta, proj_src_delta) * 

np.dot(proj_ref_delta, proj_ref_delta))) 

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

if cos_rot_sq < max_cos_rot_sq: 

return pipeBase.Struct( 

cos_rot_sq=None, 

proj_ref_ctr_delta=None, 

shift_matrix=None,) 

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

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

return pipeBase.Struct( 

cos_rot_sq=cos_rot_sq, 

proj_ref_ctr_delta=proj_ref_delta, 

shift_matrix=shift_matrix,) 

 

def _create_spherical_rotation_matrix(self, rot_axis, cos_rotation, 

sin_rotion): 

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

axis. 

 

Parameters 

---------- 

rot_axis : float array 

3 vector defining the axis to rotate about. 

cos_rotation : float 

cosine of the rotation angle. 

sin_rotation : float 

sine of the rotation angle. 

 

Return 

------ 

float array 

3x3 spherical, rotation matrix. 

""" 

 

rot_cross_matrix = np.array( 

[[0., -rot_axis[2], rot_axis[1]], 

[rot_axis[2], 0., -rot_axis[0]], 

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

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

sin_rotion*rot_cross_matrix + 

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

 

return shift_matrix 

 

def _create_pattern_spokes(self, src_ctr, src_delta_array, src_dist_array, 

ref_ctr, ref_ctr_id, proj_ref_ctr_delta, 

ref_delta_array, ref_dist_array, 

ref_id_array, max_dist_rad, n_match): 

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

shift and rotation are within tolerance. 

 

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

 

Parameters 

---------- 

src_ctr : float array 

3 vector of the source pinwheel center 

src_delta_array : float array 

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

that make up the remaining spokes of the pinwheel 

src_dist_array : float array 

Array of the distances of each src_delta in the pinwheel 

ref_ctr : float array 

3 vector of the candidate reference center 

ref_ctr_id : int 

id of the ref_ctr in the master reference array 

proj_ref_ctr_delta : `float` array-like 

Plane projected 3 vector formed from the center point of the 

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

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

in the main _construct_pattern_and_shift_rot_matrix loop 

ref_delta_array : float array 

Array of 3 vector deltas that are have the current candidate 

reference center as part of the pair 

ref_dist_array : float array 

Array of vector distances for each of the reference pairs 

ref_id_array : int array 

Array of id lookups into the master reference array that our 

center id object is paired with. 

max_dist_rad : float 

Maximum search distance 

n_match : int 

Number of source deltas that must be matched into the reference 

deltas in order to consider this a successful pattern match. 

 

Returns 

------- 

lsst.pipe.base.Struct 

The Struct contains the following data: 

 

ref_spoke_list : list of ints specifying ids into the master 

reference array 

src_spoke_list : list of ints specifying indices into the current 

source pattern that is being tested. 

""" 

# Struct where we will be putting our results. 

output_spokes = pipeBase.Struct( 

ref_spoke_list=[], 

src_spoke_list=[],) 

 

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

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

n_fail = 0 

ref_spoke_list = [] 

src_spoke_list = [] 

 

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

# the center vector of the pattern as normal. 

proj_src_ctr_delta = (src_delta_array[0] - 

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

proj_src_ctr_dist_sq = np.dot(proj_src_ctr_delta, proj_src_ctr_delta) 

 

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

proj_ref_ctr_dist_sq = np.dot(proj_ref_ctr_delta, proj_ref_ctr_delta) 

 

# Loop over the source pairs. 

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

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

break 

 

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

# on the angle between our spoke. 

src_sin_tol = (max_dist_rad / 

(src_dist_array[src_idx] + max_dist_rad)) 

 

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

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

# other. This also implicitly sets a minimum spoke length that we 

# can use. 

if src_sin_tol > 0.0447: 

n_fail += 1 

continue 

 

# Plane project the candidate source spoke and compute the cosine 

# and sine of the opening angle. 

proj_src_delta = ( 

src_delta_array[src_idx] - 

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

geom_dist_src = np.sqrt( 

np.dot(proj_src_delta, proj_src_delta) * 

proj_src_ctr_dist_sq) 

 

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

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

geom_dist_src) 

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

geom_dist_src) 

sin_theta_src = np.dot(cross_src, src_ctr) 

 

# Find the reference pairs that include our candidate pattern 

# center and sort them in increasing delta 

ref_dist_idx_array = self._find_candidate_reference_pairs( 

src_dist_array[src_idx], ref_dist_array, max_dist_rad) 

 

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

# Return None if no match is found. 

ref_id = self._test_spoke( 

cos_theta_src, 

sin_theta_src, 

ref_ctr, 

ref_ctr_id, 

proj_ref_ctr_delta, 

proj_ref_ctr_dist_sq, 

ref_dist_idx_array, 

ref_delta_array, 

ref_id_array, 

src_sin_tol) 

if ref_id is None: 

n_fail += 1 

continue 

 

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

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

ref_spoke_list.append(ref_id) 

src_spoke_list.append(src_idx + 1) 

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

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

# reference data. 

if len(ref_spoke_list) >= n_match - 2: 

# Set the struct data and return the struct. 

output_spokes.ref_spoke_list = ref_spoke_list 

output_spokes.src_spoke_list = src_spoke_list 

return output_spokes 

 

return output_spokes 

 

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

proj_ref_ctr_delta, proj_ref_ctr_dist_sq, 

ref_dist_idx_array, ref_delta_array, 

ref_id_array, src_sin_tol): 

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

for the source object against the reference object. 

 

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

the comparison. 

 

Parameters 

---------- 

cos_theta_src : `float` 

Cosine of the angle between the current candidate source spoke and 

the first spoke. 

sin_theta_src : `float` 

Sine of the angle between the current candidate source spoke and 

the first spoke. 

ref_ctr : float array 

3 vector of the candidate reference center 

ref_ctr_id : int 

id lookup of the ref_ctr into the master reference array 

proj_ref_ctr_delta : `float` 

Plane projected first spoke in the reference pattern using the 

pattern center as normal. 

proj_ref_ctr_dist_sq : `float` 

Squared length of the projected vector. 

ref_dist_idx_array : int array 

Indices sorted by the delta distance between the source 

spoke we are trying to test and the candidate reference 

spokes. 

ref_delta_array : float array 

Array of 3 vector deltas that are have the current candidate 

reference center as part of the pair 

ref_id_array : int array 

Array of id lookups into the master reference array that our 

center id object is paired with. 

src_sin_tol : float 

Sine of tolerance allowed between source and reference spoke 

opening angles. 

 

Returns 

------- 

None or int 

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

return an int id into the master reference array. 

""" 

 

# Loop over our candidate reference objects. 

for ref_dist_idx in ref_dist_idx_array: 

# Check the direction of the delta vector. 

ref_sign = 1 

if ref_id_array[ref_dist_idx] < ref_ctr_id: 

ref_sign = -1 

 

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

# current reference candidate. 

proj_ref_delta = ( 

ref_delta_array[ref_dist_idx] - 

np.dot(ref_delta_array[ref_dist_idx], ref_ctr) * ref_ctr) 

geom_dist_ref = np.sqrt(proj_ref_ctr_dist_sq * 

np.dot(proj_ref_delta, proj_ref_delta)) 

cos_theta_ref = ref_sign * ( 

np.dot(proj_ref_delta, proj_ref_ctr_delta) / 

geom_dist_ref) 

 

# Make sure we can safely make the comparison in case 

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

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

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

(1 - cos_theta_ref ** 2)) 

else: 

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

src_sin_tol ** 2) 

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

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

# small. 

if cos_sq_comparison > src_sin_tol ** 2: 

continue 

 

# The cosine tests the magnitude of the angle but not 

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

# This cross product calculation does that. 

cross_ref = ref_sign * ( 

np.cross(proj_ref_delta, proj_ref_ctr_delta) / 

geom_dist_ref) 

sin_theta_ref = np.dot(cross_ref, ref_ctr) 

 

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

# near zero. 

if abs(cos_theta_src) < src_sin_tol: 

sin_comparison = (sin_theta_src - sin_theta_ref) / src_sin_tol 

else: 

sin_comparison = \ 

(sin_theta_src - sin_theta_ref) / cos_theta_ref 

 

if abs(sin_comparison) > src_sin_tol: 

continue 

 

# Return the correct id of the candidate we found. 

return ref_id_array[ref_dist_idx] 

 

return None 

 

def _create_shift_rot_matrix(self, cos_rot_sq, shift_matrix, src_delta, 

ref_ctr, ref_delta): 

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

 

Parameters 

---------- 

cos_rot_sq : float 

cosine of the rotation needed to align our source and reference 

candidate patterns. 

shift_matrix : float array 

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

of the candidate reference pattern center. 

src_delta : float array 

3 vector delta of representing the first spoke of the source 

pattern 

ref_ctr : float array 

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

reference pattern. 

ref_delta : float array 

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

 

Returns 

------- 

lsst.pipe.base.Struct 

Struct object containing the following data: 

 

sin_rot : float sine of the amount of rotation between the 

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

signed and tells us the chirality of the rotation. 

shift_rot_matrix : float array representing the 3x3 rotation 

matrix that takes the source pattern and shifts and rotates 

it to align with the reference pattern. 

""" 

cos_rot = np.sqrt(cos_rot_sq) 

rot_src_delta = np.dot(shift_matrix, src_delta) 

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

 

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

rot_matrix = self._create_spherical_rotation_matrix( 

ref_ctr, cos_rot, sin_rot) 

 

shift_rot_matrix = np.dot(rot_matrix, shift_matrix) 

 

return pipeBase.Struct( 

sin_rot=sin_rot, 

shift_rot_matrix=shift_rot_matrix,) 

 

def _intermediate_verify(self, src_pattern, ref_pattern, shift_rot_matrix, 

max_dist_rad): 

""" Perform an intermediate verify step. 

 

Rotate the matches references into the source frame and test their 

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

tolerance. 

 

Parameters 

---------- 

src_pattern : float array 

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

pinwheel pattern. 

ref_pattern : float array 

Array of 3 vectors representing our candidate reference pinwheel 

pattern. 

shift_rot_matrix : float array 

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

onto the frame of the reference objects 

max_dist_rad : float 

Maximum distance allowed to consider two objects the same. 

 

Returns 

------- 

float array or None 

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

source pattern are within max_dist_rad of their matched reference 

objects. Returns None if this criteria is not satisfied. 

""" 

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

raise ValueError( 

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

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

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

 

if self._intermediate_verify_comparison( 

src_pattern, ref_pattern, shift_rot_matrix, max_dist_rad): 

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

# want to fit the implied transform using all points from 

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

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

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

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

# the rotated vectors to within the match tolerance. 

fit_shift_rot_matrix = least_squares( 

_rotation_matrix_chi_sq, 

x0=shift_rot_matrix.flatten(), 

args=(src_pattern, ref_pattern, max_dist_rad) 

).x.reshape((3, 3)) 

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

if self._intermediate_verify_comparison( 

src_pattern, ref_pattern, fit_shift_rot_matrix, 

max_dist_rad): 

return fit_shift_rot_matrix 

 

return None 

 

def _intermediate_verify_comparison(self, pattern_a, pattern_b, 

shift_rot_matrix, max_dist_rad): 

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

a second one. 

 

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

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

True. 

 

Parameters 

---------- 

pattern_a : float array 

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

pinwheel pattern. 

pattern_b : float array 

Array of 3 vectors representing our candidate reference pinwheel 

pattern. 

shift_rot_matrix : float array 

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

onto the frame of the reference objects 

max_dist_rad : float 

Maximum distance allowed to consider two objects the same. 

 

 

Returns 

------- 

bool 

True if all rotated source points are within max_dist_rad of 

the candidate references matches. 

""" 

shifted_pattern_a = np.dot(shift_rot_matrix, 

pattern_a.transpose()).transpose() 

tmp_delta_array = shifted_pattern_a - pattern_b 

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

tmp_delta_array[:, 1] ** 2 + 

tmp_delta_array[:, 2] ** 2) 

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

 

def _test_pattern_lengths(self, test_pattern, max_dist_rad): 

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

tolerance. 

 

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

too much distortion. 

""" 

dists = (test_pattern[:, 0] ** 2 + 

test_pattern[:, 1] ** 2 + 

test_pattern[:, 2] ** 2) 

return np.all( 

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

dists < (1 + max_dist_rad) ** 2)) 

 

def _test_rotation_agreement(self, rot_vects, max_dist_rad): 

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

the number that a agree within tolerance to where our test 

points are. 

 

Parameters 

---------- 

rot_vects : float array 

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

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

the candidate rotations into the reference frame. 

max_dist_rad : float 

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

a rotation 

 

Returns 

------- 

int 

Number of candidate rotations that agree for all of the rotated 

test 3 vectors. 

""" 

 

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

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

 

tot_consent = 0 

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

tmp_dist_list = [] 

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

tmp_delta_vect = (rot_vects[rot_idx][vect_idx] - 

rot_vects[-1][vect_idx]) 

tmp_dist_list.append( 

np.dot(tmp_delta_vect, tmp_delta_vect)) 

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

tot_consent += 1 

return tot_consent 

 

def _match_sources(self, 

source_array, 

shift_rot_matrix): 

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

frames and find their nearest neighbor using a kdTree. 

 

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

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

external function. 

 

Parameters 

---------- 

source_array : float array 

array of 3 vectors representing the source objects we are trying 

to match into the source catalog. 

shift_rot_matrix : float array 

3x3 rotation matrix that performs the spherical rotation from the 

source frame into the reference frame. 

 

Returns 

------- 

lsst.pipe.base.Struct 

A Struct object containing the following data 

matches : a (N, 2) array of integer ids into the source and 

reference arrays. Matches are only returned for those that 

satisfy the distance and handshake criteria. 

distances : float array of the distance between each match in 

radians after the shift and rotation is applied. 

""" 

shifted_references = np.dot( 

np.linalg.inv(shift_rot_matrix), 

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

shifted_sources = np.dot( 

shift_rot_matrix, 

source_array.transpose()).transpose() 

 

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

dtype=np.uint32) 

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

dtype=np.uint32) 

 

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

dtype=np.uint32) 

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

dtype=np.uint32) 

 

ref_kdtree = cKDTree(self._reference_array) 

src_kdtree = cKDTree(source_array) 

 

ref_to_src_dist, tmp_ref_to_src_idx = \ 

src_kdtree.query(shifted_references) 

src_to_ref_dist, tmp_src_to_ref_idx = \ 

ref_kdtree.query(shifted_sources) 

 

ref_matches[:, 0] = tmp_ref_to_src_idx 

src_matches[:, 1] = tmp_src_to_ref_idx 

 

handshake_mask = self._handshake_match(src_matches, ref_matches) 

return pipeBase.Struct( 

match_ids=src_matches[handshake_mask], 

distances_rad=src_to_ref_dist[handshake_mask],) 

 

def _handshake_match(self, matches_src, matches_ref): 

"""Return only those matches where both the source 

and reference objects agree they they are each others' 

nearest neighbor. 

Parameters 

---------- 

matches_src : int array 

(N, 2) int array of nearest neighbor matches between shifted and 

rotated reference objects matched into the sources. 

matches_ref : int array 

(M, 2) int array of nearest neighbor matches between shifted and 

rotated source objects matched into the references. 

Return 

------ 

bool array 

Return the array positions where the two match catalogs agree. 

""" 

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

 

for src_match_idx, match in enumerate(matches_src): 

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

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

handshake_mask_array[src_match_idx] = True 

return handshake_mask_array