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# This file is part of the Astrometry.net suite.
# Licensed under a 3-clause BSD style license - see LICENSE
from __future__ import print_function
import sys
#from numpy import array, matrix, linalg
from numpy import *
from numpy.random import *
from numpy.linalg import *
from matplotlib.pylab import figure, plot, xlabel, ylabel, loglog, clf
from matplotlib.pylab import semilogy
#from pylab import *
class Transform(object):
scale = None
rotation = None
incenter = None
outcenter = None
def apply(self, X):
#print X
dx = X - self.incenter
#print dx
dx = dx * self.scale
#print dx
dx = self.rotation * dx
#print dx
dx = dx + self.outcenter
#print dx
return dx
def __str__(self):
s = ('<Transform: tin (%f,%f) scale (%f) rot (%f, %f; %f, %f) tout (%f, %f)>' %
(self.incenter[0], self.incenter[1], self.scale,
self.rotation[0,0], self.rotation[0,1], self.rotation[1,0], self.rotation[1,1],
self.outcenter[0], self.outcenter[1]))
return s
def procrustes(X, Y):
T = Transform()
sx = X.shape
if sx[0] != 2:
print('X must be 2xN')
sy = Y.shape
if sy[0] != 2:
print('Y must be 2xN')
N = sx[1]
mx = X.mean(axis=1).reshape(2,1)
my = Y.mean(axis=1).reshape(2,1)
#print 'mean(X) is\n', mx
#print 'mean(Y) is\n', my
T.incenter = mx
T.outcenter = my
#print 'X-mx is\n', X-mx
#print '(X-mx)^2 is\n', (X-mx)*(X-mx)
varx = sum(sum((X - mx)*(X - mx)), axis=1)
vary = sum(sum((Y - my)*(Y - my)), axis=1)
#print 'var(X) is', varx
#print 'var(Y) is', vary
T.scale = sqrt(vary / varx)
#print 'scale is', T.scale
C = zeros((2,2))
for i in [0,1]:
for j in [0,1]:
C[i,j] = sum((X[i,:] - mx[i]) * (Y[j,:] - my[j]))
#print 'cov is\n', C
U,S,V = svd(C)
U = matrix(U)
V = matrix(V)
#print 'U is\n', U
#print 'U\' is\n', U.transpose()
#print 'V is\n', V
R = V * U.transpose()
#print 'R is\n', R
T.rotation = R
return T
def test_procrustes_1():
# Create a Transform, apply it to some points, then run procrustes to see if we
# recover the Transform exactly.
t1 = Transform()
t1.scale = 3.0
A = 48.0 * pi/180.0
t1.rotation = matrix([[sin(A), cos(A)], [-cos(A), sin(A)]])
t1.incenter = array([42, 500]).reshape(2,1)
t1.outcenter = array([600, -12]).reshape(2,1)
N = 4
pts = zeros((2,N))
tpts = zeros((2,N))
for i in range(N):
pts[0,i] = t1.incenter[0] + ((i % 2) - 0.5) * 200
pts[1,i] = t1.incenter[1] + (((i/2) % 2) - 0.5) * 200
for i in range(N):
pt = pts[:,i].reshape(2,1)
tpts[:,i] = t1.apply(pt).reshape(1,2)
t2 = procrustes(pts, tpts)
print('pts:', pts)
print('tpts:', tpts)
print('t1 is', t1)
print('t2 is', t2)
def draw_sample(inoise=1, fnoise=0, iqnoise=-1,
dimquads=4, quadscale=100, imgsize=1000,
Rsteps=10, Asteps=36):
# Stars that compose the field quad.
fquad = zeros((2,dimquads))
fquad[0,0] = imgsize/2 - quadscale/2
fquad[1,0] = imgsize/2
fquad[0,1] = imgsize/2 + quadscale/2
fquad[1,1] = imgsize/2
for i in range(2, dimquads):
fquad[0,i] = imgsize/2 + randn(1) * quadscale
fquad[1,i] = imgsize/2 + randn(1) * quadscale
# Index quad is field quad plus jitter.
iquad = fquad + randn(*fquad.shape)
# Solve for transformation
T = procrustes(iquad, fquad)
# Put the index quad stars through the transformation
itrans = zeros(fquad.shape)
for i in range(dimquads):
fq = fquad[:,i].reshape(2,1)
itrans[:,i] = T.apply(fq).transpose()
# Field quad center...
qc = mean(fquad, axis=1)
# Sample stars on a R^2, theta grid.
#rads = sqrt((array(range(Rsteps))+1) / float(Rsteps)) * imgsize/2
N = Rsteps * Asteps
rads = sqrt((array(range(Rsteps))+0.5) / float(Rsteps)) * imgsize/2
thetas = array(range(Asteps)) / float(Asteps) * 2.0 * pi
fstars = zeros((2,N))
for r in range(Rsteps):
for a in range(Asteps):
fstars[0, r*Asteps + a] = sin(thetas[a]) * rads[r] + qc[0]
fstars[1, r*Asteps + a] = cos(thetas[a]) * rads[r] + qc[1]
# Put them through the transformation...
istars = zeros((2,N))
for i in range(N):
fs = fstars[:,i].reshape(2,1)
istars[:,i] = T.apply(fs).transpose()
R = sqrt((fstars[0,:] - qc[0])**2 + (fstars[1,:] - qc[1])**2)
E = sqrt(sum((fstars - istars)**2, axis=0))
# Fit to a linear model...
xfit = R**2
yfit = E**2
A = zeros((2,N))
A[0,:] = 1
A[1,:] = xfit.transpose()
(C,resids,rank,s) = lstsq(A.transpose(), yfit)
return (fquad, iquad, T, itrans, qc, fstars, istars,
R, E, C)
if __name__ == '__main__':
test_procrustes_1()
sys.exit(0)
#N = 1000
N = 100
C = zeros((2,N))
QD = zeros((N))
for i in range(N):
(fquad, iquad, T, itrans, qc, fstars, istars, R, E, c) = draw_sample()
C[:,i] = c
QD[i] = sqrt(sum((iquad - fquad)**2) / 4.0)
C0 = C[0,:]
C1 = C[1,:]
figure(1)
clf()
loglog(C0, C1, 'b.')
xlabel('E^2 vs R^2 - Fit coefficient 0')
ylabel('E^2 vs R^2 - Fit coefficient 1')
figure(2)
clf()
semilogy(QD, C1, 'b.')
xlabel('Field-to-Index Quad Mean Distance')
ylabel('E^2-vs-R^2 fit linear coefficient')
#semilogy(QD, C1, 'bo')
#xlabel('Quad Distance')
#ylabel('C1')
#figure(1)
#I=[0,2,1,3,0];
#plot(fquad[0,I], fquad[1,I], 'bo-', itrans[0,I], itrans[1,I], 'ro-')
#figure(2)
#plot(fstars[0,:], fstars[1,:], 'b.', istars[0,:], istars[1,:], 'r.')
#figure(1)
#I=[0,2,1,3,0];
#plot(fquad[0,I], fquad[1,I], 'bo-',
# itrans[0,I], itrans[1,I], 'ro-',
# fstars[0,:], fstars[1,:], 'b.',
# istars[0,:], istars[1,:], 'r.')
#figure(2)
#plot(R, E, 'r.')
#xlabel('R')
#ylabel('E')
#figure(3)
#plot(R**2, E**2, 'r.')
#xlabel('R^2')
#ylabel('E^2')
#print 'Fit coefficients are', C
#figure(2)
#xplot = array(range(101)) / 100.0 * max(xfit)
#plot(R**2, E**2, 'r.',
# xplot, C[0] + C[1]*xplot, 'b-')
#xlabel('R^2')
#ylabel('E^2')
#show()
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