def test_zoom_samesize_recentered(imsize,
upsample_factor,
offset,
doplot=False,
ndim=2):
"""
Test that zooming in by some factor with the same input & output sizes
allow for non-centered input images, AND zoom back in on the center
"""
inds = np.indices([imsize] * ndim)
rr = ((inds[0] - (imsize - 1) / 2. - offset[0])**2 +
(inds[1] - (imsize - 1) / 2. - offset[1])**2)**0.5
gg = gaussian(rr)
xz, zz = zoom.zoomnd(gg,
usfac=upsample_factor,
return_xouts=True,
offsets=offset)
xr = ((xz[0] - (imsize - 1) / 2. - offset[0])**2 +
(xz[1] - (imsize - 1) / 2. - offset[1])**2)**0.5
expected_accuracy = (
(upsample_factor**1.1) * 6.2e-4 * (imsize % 2 == 1) + # odd case
(upsample_factor**2) * 1.5e-4 * (imsize % 2 == 0) + # even case
0.002) # constant offset because the above is a fit
assert ((gaussian(xr) - zz)**2).sum() < expected_accuracy
def test_inds(imsize, ndim):
"""
Make sure output indices are correct (don't care about zoomed values for this test)
"""
upsample_factor = 1
inds = np.indices([imsize]*ndim)
rr = np.sum([(ind - (imsize-1)/2.)**2 for ind in inds],axis=0)**0.5
gg = gaussian(rr)
xz,zz = zoom.zoomnd(gg,usfac=upsample_factor,return_xouts=True)
assert np.all(inds==xz)
def test_inds(imsize, ndim):
"""
Make sure output indices are correct (don't care about zoomed values for this test)
"""
upsample_factor = 1
inds = np.indices([imsize] * ndim)
rr = np.sum([(ind - (imsize - 1) / 2.)**2 for ind in inds], axis=0)**0.5
gg = gaussian(rr)
xz, zz = zoom.zoomnd(gg, usfac=upsample_factor, return_xouts=True)
assert np.all(inds == xz)
def measure_zoom_fullsize(imsize, upsample_factor,doplot=False,ndim=2):
"""
"""
inds = np.indices([imsize]*ndim)
rr = ((inds-(imsize-1)/2.)**2).sum(axis=0)**0.5
gg = gaussian(rr)
outshape = [s*upsample_factor for s in gg.shape]
xz,zz = zoom.zoomnd(gg,upsample_factor,outshape=outshape,return_xouts=True)
xr = ((xz - (imsize-1.)/2.)**2).sum(axis=0)**0.5
return ((gaussian(xr)-zz)**2).sum()
def measure_difference_zoom_samesize(imsize, upsample_factor,doplot=False,ndim=2):
"""
Test that zooming in by some factor with the same input & output sizes
works
"""
inds = np.indices([imsize]*ndim)
rr = ((inds-(imsize-1)/2.)**2).sum(axis=0)**0.5
gg = gaussian(rr)
xz,zz = zoom.zoomnd(gg,upsample_factor,return_xouts=True)
xr = ((xz - (imsize-1.)/2.)**2).sum(axis=0)**0.5
return ((gaussian(xr)-zz)**2).sum()
def test_inds(imsize, upsample_factor, ndim):
offset = [0]*ndim
inds = np.indices([imsize]*ndim)
rr = np.sum([(ind - (imsize-1)/2.)**2 for ind in inds],axis=0)**0.5
gg = gaussian(rr)
outsize = imsize*upsample_factor
xz,zz = zoom.zoomnd(gg,usfac=upsample_factor,outshape=[outsize]*ndim, return_xouts=True,offsets=offset)
# wrong newinds = np.indices([imsize]*ndim)/float(imsize*upsample_factor-1)*(imsize-1)
newinds = np.linspace(-0.5+1./upsample_factor/2.,(imsize-1)+0.5-1./upsample_factor/2.,outsize)
for dnum in range(ndim):
slices = [np.newaxis]*(dnum) + [slice(None)] + [np.newaxis]*(ndim-dnum-1)
np.testing.assert_array_almost_equal(newinds[slices]*np.ones(xz[dnum].shape),xz[dnum])
def test_zoom_fullsize(imsize, upsample_factor,doplot=False,ndim=2):
"""
Test that zooming in by some factor with output size = input size * upsample factor
"""
inds = np.indices([imsize]*ndim)
rr = ((inds-(imsize-1)/2.)**2).sum(axis=0)**0.5
gg = gaussian(rr)
outshape = [s*upsample_factor for s in gg.shape]
xz,zz = zoom.zoomnd(gg,usfac=upsample_factor,outshape=outshape,return_xouts=True)
xr = ((xz - (imsize-1.)/2.)**2).sum(axis=0)**0.5
expected_accuracy = ( (upsample_factor**2) * 1.2e-4 + # odd case
(upsample_factor**2) * 4.9e-4 * (imsize%2==0) + # even case
0.002 ) # constant offset because the above is a fit
assert ((gaussian(xr)-zz)**2).sum() < expected_accuracy
def test_zoom_samesize(imsize, upsample_factor,doplot=False,ndim=2):
"""
Test that zooming in by some factor with the same input & output sizes
works
"""
inds = np.indices([imsize]*ndim)
rr = ((inds-(imsize-1)/2.)**2).sum(axis=0)**0.5
gg = gaussian(rr)
xz,zz = zoom.zoomnd(gg,usfac=upsample_factor,return_xouts=True)
xr = ((xz - (imsize-1.)/2.)**2).sum(axis=0)**0.5
expected_accuracy = ( (upsample_factor**1.1) * 6.2e-4 * (imsize%2==1) + # odd case
(upsample_factor**2) * 1.5e-4 * (imsize%2==0) + # even case
0.002 ) # constant offset because the above is a fit
assert ((gaussian(xr)-zz)**2).sum() < expected_accuracy
def test_zoom_samesize(imsize, upsample_factor, doplot=False, ndim=2):
"""
Test that zooming in by some factor with the same input & output sizes
works
"""
inds = np.indices([imsize] * ndim)
rr = ((inds - (imsize - 1) / 2.)**2).sum(axis=0)**0.5
gg = gaussian(rr)
xz, zz = zoom.zoomnd(gg, usfac=upsample_factor, return_xouts=True)
xr = ((xz - (imsize - 1.) / 2.)**2).sum(axis=0)**0.5
expected_accuracy = (
(upsample_factor**1.1) * 6.2e-4 * (imsize % 2 == 1) + # odd case
(upsample_factor**2) * 1.5e-4 * (imsize % 2 == 0) + # even case
0.002) # constant offset because the above is a fit
assert ((gaussian(xr) - zz)**2).sum() < expected_accuracy
def test_zoom_samesize_uncentered(imsize, upsample_factor, offset, doplot=False,ndim=2):
"""
Test that zooming in by some factor with the same input & output sizes
allow for non-centered input images
"""
inds = np.indices([imsize]*ndim)
rr = ((inds[0] - (imsize-1)/2. - offset[0])**2 + (inds[1] - (imsize-1)/2. - offset[1])**2)**0.5
gg = gaussian(rr)
xz,zz = zoom.zoomnd(gg,usfac=upsample_factor,return_xouts=True)
xr = ((xz[0] - (imsize-1)/2. - offset[0])**2 + (xz[1] - (imsize-1)/2. - offset[1])**2)**0.5
expected_accuracy = ( (upsample_factor**1.1) * 6.2e-4 * (imsize%2==1) + # odd case
(upsample_factor**2) * 1.5e-4 * (imsize%2==0) + # even case
0.002 ) # constant offset because the above is a fit
assert ((gaussian(xr)-zz)**2).sum() < expected_accuracy
def test_zoom_fullsize(imsize, upsample_factor, doplot=False, ndim=2):
"""
Test that zooming in by some factor with output size = input size * upsample factor
"""
inds = np.indices([imsize] * ndim)
rr = ((inds - (imsize - 1) / 2.)**2).sum(axis=0)**0.5
gg = gaussian(rr)
outshape = [s * upsample_factor for s in gg.shape]
xz, zz = zoom.zoomnd(gg,
usfac=upsample_factor,
outshape=outshape,
return_xouts=True)
xr = ((xz - (imsize - 1.) / 2.)**2).sum(axis=0)**0.5
expected_accuracy = (
(upsample_factor**2) * 1.2e-4 + # odd case
(upsample_factor**2) * 4.9e-4 * (imsize % 2 == 0) + # even case
0.002) # constant offset because the above is a fit
assert ((gaussian(xr) - zz)**2).sum() < expected_accuracy
def test_inds(imsize, upsample_factor, ndim):
offset = [0] * ndim
inds = np.indices([imsize] * ndim)
rr = np.sum([(ind - (imsize - 1) / 2.)**2 for ind in inds], axis=0)**0.5
gg = gaussian(rr)
outsize = imsize * upsample_factor
xz, zz = zoom.zoomnd(gg,
usfac=upsample_factor,
outshape=[outsize] * ndim,
return_xouts=True,
offsets=offset)
# wrong newinds = np.indices([imsize]*ndim)/float(imsize*upsample_factor-1)*(imsize-1)
newinds = np.linspace(-0.5 + 1. / upsample_factor / 2.,
(imsize - 1) + 0.5 - 1. / upsample_factor / 2.,
outsize)
for dnum in range(ndim):
slices = [np.newaxis] * (dnum) + [slice(None)
] + [np.newaxis] * (ndim - dnum - 1)
np.testing.assert_array_almost_equal(
newinds[slices] * np.ones(xz[dnum].shape), xz[dnum])
def chi2_shift(im1,
im2,
err=None,
upsample_factor='auto',
boundary='wrap',
nthreads=1,
use_numpy_fft=False,
zeromean=False,
nfitted=2,
verbose=False,
return_error=True,
return_chi2array=False,
max_auto_size=512,
max_nsig=1.1):
"""
Find the offsets between image 1 and image 2 using the DFT upsampling method
(http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation/content/html/efficient_subpixel_registration.html)
combined with :math:`\chi^2` to measure the errors on the fit
Equation 1 gives the :math:`\chi^2` value as a function of shift, where Y
is the model as a function of shift:
.. math::
\chi^2(dx,dy) & = & \Sigma_{ij} \\frac{(X_{ij}-Y_{ij}(dx,dy))^2}{\sigma_{ij}^2} \\\\
..
& = & \Sigma_{ij} \left[ X_{ij}^2/\sigma_{ij}^2 - 2X_{ij}Y_{ij}(dx,dy)/\sigma_{ij}^2 + Y_{ij}(dx,dy)^2/\sigma_{ij}^2 \\right] \\\\
Equation 2-4:
.. math::
Term~1: f(dx,dy) & = & \Sigma_{ij} \\frac{X_{ij}^2}{\sigma_{ij}^2} \\\\
f(dx,dy) & = & f(0,0) , \\forall dx,dy \\\\
Term~2: g(dx,dy) & = & -2 \Sigma_{ij} \\frac{X_{ij}Y_{ij}(dx,dy)}{\sigma_{ij}^2} = -2 \Sigma_{ij} \left(\\frac{X_{ij}}{\sigma_{ij}^2}\\right) Y_{ij}(dx,dy) \\\\
Term~3: h(dx,dy) & = & \Sigma_{ij} \\frac{Y_{ij}(dx,dy)^2}{\sigma_{ij}^2} = \Sigma_{ij} \left(\\frac{1}{\sigma_{ij}^2}\\right) Y^2_{ij}(dx,dy)
The cross-correlation can be computed with fourier transforms, and is defined
.. math::
CC_{m,n}(x,y) = \Sigma_{ij} x^*_{ij} y_{(n+i)(m+j)}
which can then be applied to our problem, noting that the cross-correlation
has the same form as term 2 and 3 in :math:`\chi^2` (term 1 is a constant,
with no dependence on the shift)
.. math::
Term~2: & CC(X/\sigma^2,Y)[dx,dy] & = & \Sigma_{ij} \left(\\frac{X_{ij}}{\sigma_{ij}^2}\\right)^* Y_{ij}(dx,dy) \\\\
Term~3: & CC(\sigma^{-2},Y^2)[dx,dy] & = & \Sigma_{ij} \left(\\frac{1}{\sigma_{ij}^2}\\right)^* Y^2_{ij}(dx,dy) \\\\
Technically, only terms 2 and 3 has any effect on the resulting image,
since term 1 is the same for all shifts, and the quantity of interest is
:math:`\Delta \chi^2` when determining the best-fit shift and error.
Parameters
----------
im1 : np.ndarray
im2 : np.ndarray
The images to register.
err : np.ndarray
Per-pixel error in image 2
boundary : 'wrap','constant','reflect','nearest'
Option to pass to map_coordinates for determining what to do with
shifts outside of the boundaries.
upsample_factor : int or 'auto'
upsampling factor; governs accuracy of fit (1/usfac is best accuracy)
(can be "automatically" determined based on chi^2 error)
return_error : bool
Returns the "fit error" (1-sigma in x and y) based on the delta-chi2
values
return_chi2_array : bool
Returns the x and y shifts and the chi2 as a function of those shifts
in addition to other returned parameters. i.e., the last return from
this function will be a tuple (x, y, chi2)
zeromean : bool
Subtract the mean from the images before cross-correlating? If no, you
may get a 0,0 offset because the DC levels are strongly correlated.
verbose : bool
Print error message if upsampling factor is inadequate to measure errors
use_numpy_fft : bool
Force use numpy's fft over fftw? (only matters if you have fftw
installed)
nthreads : bool
Number of threads to use for fft (only matters if you have fftw
installed)
nfitted : int
number of degrees of freedom in the fit (used for chi^2 computations).
Should probably always be 2.
max_auto_size : int
Maximum zoom image size to create when using auto-upsampling
Returns
-------
dx,dy : float,float
Measures the amount im2 is offset from im1 (i.e., shift im2 by -1 *
these #'s to match im1)
errx,erry : float,float
optional, error in x and y directions
xvals,yvals,chi2n_upsampled : ndarray,ndarray,ndarray,
x,y positions (in original chi^2 coordinates) of the chi^2 values and
their corresponding chi^2 value
Examples
--------
Create a 2d array,
shift it in both directions,
then use chi2_shift to determine the shift
>>> rr = ((np.indices([100,100]) - np.array([50.,50.])[:,None,None])**2).sum(axis=0)**0.5
>>> image = np.exp(-rr**2/(3.**2*2.)) * 20
>>> shifted = np.roll(np.roll(image,12,0),5,1) + np.random.randn(100,100)
>>> dx,dy,edx,edy = chi2_shift(image, shifted, upsample_factor='auto')
>>> shifted2 = image_registration.fft_tools.shift2d(image,3.665,-4.25) + np.random.randn(100,100)
>>> dx2,dy2,edx2,edy2 = chi2_shift(image, shifted2, upsample_factor='auto')
"""
chi2, term1, term2, term3 = chi2n_map(im1,
im2,
err,
boundary=boundary,
nthreads=nthreads,
zeromean=zeromean,
use_numpy_fft=use_numpy_fft,
return_all=True,
reduced=False)
ymax, xmax = np.unravel_index(chi2.argmin(), chi2.shape)
# needed for ffts
im1 = np.nan_to_num(im1)
im2 = np.nan_to_num(im2)
ylen, xlen = im1.shape
xcen = xlen / 2 - (1 - xlen % 2)
ycen = ylen / 2 - (1 - ylen % 2)
# original shift calculation
yshift = ymax - ycen # shift im2 by these numbers to get im1
xshift = xmax - xcen
if verbose:
print "Coarse xmax/ymax = %i,%i, for offset %f,%f" % (xmax, ymax,
xshift, yshift)
# below is sub-pixel zoom-in stuff
# find delta-chi^2 limiting values for varying DOFs
try:
import scipy.stats
# 1,2,3-sigma delta-chi2 levels
m1 = scipy.stats.chi2.ppf(1 - scipy.stats.norm.sf(1) * 2, nfitted)
m2 = scipy.stats.chi2.ppf(1 - scipy.stats.norm.sf(2) * 2, nfitted)
m3 = scipy.stats.chi2.ppf(1 - scipy.stats.norm.sf(3) * 2, nfitted)
m_auto = scipy.stats.chi2.ppf(1 - scipy.stats.norm.sf(max_nsig) * 2,
nfitted)
except ImportError:
# assume m=2 (2 degrees of freedom)
m1 = 2.2957489288986364
m2 = 6.1800743062441734
m3 = 11.829158081900793
m_auto = 2.6088233328527037 # slightly >1 sigma
# biggest scale = where chi^2/n ~ 9 or 11.8 for M=2?
if upsample_factor == 'auto':
# deltachi2 is not reduced deltachi2
deltachi2_lowres = (chi2 - chi2.min())
if verbose:
print "Minimum chi2: %g Max delta-chi2 (lowres): %g Min delta-chi2 (lowres): %g" % (
chi2.min(), deltachi2_lowres.max(),
deltachi2_lowres[deltachi2_lowres > 0].min())
sigmamax_area = deltachi2_lowres < m_auto
if sigmamax_area.sum() > 1:
yy, xx = np.indices(sigmamax_area.shape)
xvals = xx[sigmamax_area]
yvals = yy[sigmamax_area]
xvrange = xvals.max() - xvals.min()
yvrange = yvals.max() - yvals.min()
size = max(xvrange, yvrange)
else:
size = 1
upsample_factor = max_auto_size / 2. / size
if upsample_factor < 1:
upsample_factor = 1
s1 = s2 = max_auto_size
# zoom factor = s1 / upsample_factor = 2*size
zoom_factor = 2. * size
if verbose:
print "Selected upsample factor %0.1f for image size %i and zoom factor %0.1f (max-sigma range was %i for area %i)" % (
upsample_factor, s1, zoom_factor, size, sigmamax_area.sum())
else:
s1, s2 = im1.shape
zoom_factor = s1 / upsample_factor
if zoom_factor <= 1:
zoom_factor = 2
s1 = zoom_factor * upsample_factor
s2 = zoom_factor * upsample_factor
(yshifts_corrections,
xshifts_corrections), chi2_ups = zoom.zoomnd(chi2,
usfac=upsample_factor,
outshape=[s1, s2],
offsets=[yshift, xshift],
return_xouts=True)
# deltachi2 is not reduced deltachi2
deltachi2_ups = (chi2_ups - chi2_ups.min())
if verbose:
print "Minimum chi2_ups: %g Max delta-chi2 (highres): %g Min delta-chi2 (highres): %g" % (
chi2_ups.min(), deltachi2_ups.max(),
deltachi2_ups[deltachi2_ups > 0].min())
if verbose > 1:
pass
#if hasattr(term3_ups,'len'):
# print "term3_ups has shape ",term3_ups.shape," term2: ",term2_ups.shape," term1=",term1
#else:
# print "term2 shape: ",term2.shape," term1: ",term1," term3: ",term3_ups
# THE UPSAMPLED BEST-FIT HAS BEEN FOUND
# BELOW IS TO COMPUTE THE ERROR
errx_low, errx_high, erry_low, erry_high = chi2map_to_errors(
chi2_ups, upsample_factor)
yshift_corr = yshifts_corrections.flat[chi2_ups.argmin()] - ycen
xshift_corr = xshifts_corrections.flat[chi2_ups.argmin()] - xcen
shift_xvals = xshifts_corrections - xcen
shift_yvals = yshifts_corrections - ycen
returns = [-xshift_corr, -yshift_corr]
if return_error:
returns.append((errx_low + errx_high) / 2.)
returns.append((erry_low + erry_high) / 2.)
if return_chi2array:
returns.append((shift_xvals, shift_yvals, chi2_ups))
return returns
def chi2_shift_iterzoom(im1,
im2,
err=None,
upsample_factor='auto',
boundary='wrap',
nthreads=1,
use_numpy_fft=False,
zeromean=False,
verbose=False,
return_error=True,
return_chi2array=False,
zoom_shape=[10, 10],
rezoom_shape=[100, 100],
rezoom_factor=5,
mindiff=1,
**kwargs):
"""
Find the offsets between image 1 and image 2 using an iterative DFT
upsampling method combined with :math:`\chi^2` to measure the errors on the
fit
A simpler version of :func:`chi2_shift` that only computes the
:math:`\chi^2` array on the largest scales, then uses a fourier upsampling
technique to zoom in.
Parameters
----------
im1 : np.ndarray
im2 : np.ndarray
The images to register.
err : np.ndarray
Per-pixel error in image 2
boundary : 'wrap','constant','reflect','nearest'
Option to pass to map_coordinates for determining what to do with
shifts outside of the boundaries.
upsample_factor : int or 'auto'
upsampling factor; governs accuracy of fit (1/usfac is best accuracy)
(can be "automatically" determined based on chi^2 error)
zeromean : bool
Subtract the mean from the images before cross-correlating? If no, you
may get a 0,0 offset because the DC levels are strongly correlated.
verbose : bool
Print error message if upsampling factor is inadequate to measure errors
use_numpy_fft : bool
Force use numpy's fft over fftw? (only matters if you have fftw
installed)
nthreads : bool
Number of threads to use for fft (only matters if you have fftw
installed)
nfitted : int
number of degrees of freedom in the fit (used for chi^2 computations).
Should probably always be 2.
zoom_shape : [int,int]
Shape of iterative zoom image
rezoom_shape : [int,int]
Shape of the final output chi^2 map to use for determining the errors
rezoom_factor : int
Amount to zoom above the last zoom factor. Should be <=
rezoom_shape/zoom_shape
Other Parameters
----------------
return_error : bool
Returns the "fit error" (1-sigma in x and y) based on the delta-chi2
values
return_chi2_array : bool
Returns the x and y shifts and the chi2 as a function of those shifts
in addition to other returned parameters. i.e., the last return from
this function will be a tuple (x, y, chi2)
Returns
-------
dx,dy : float,float
Measures the amount im2 is offset from im1 (i.e., shift im2 by -1 *
these #'s to match im1)
errx,erry : float,float
optional, error in x and y directions
xvals,yvals,chi2n_upsampled : ndarray,ndarray,ndarray,
x,y positions (in original chi^2 coordinates) of the chi^2 values and
their corresponding chi^2 value
Examples
--------
Create a 2d array,
shift it in both directions,
then use chi2_shift_iterzoom to determine the shift
>>> np.random.seed(42) # so the doctest will pass
>>> image = np.random.randn(50,55)
>>> shifted = np.roll(np.roll(image,12,0),5,1)
>>> dx,dy,edx,edy = chi2_shift_iterzoom(image, shifted, upsample_factor='auto')
>>> shifted2 = image_registration.fft_tools.shift2d(image,3.665,-4.25)
>>> dx2,dy2,edx2,edy2 = chi2_shift_iterzoom(image, shifted2, upsample_factor='auto')
"""
chi2, term1, term2, term3 = chi2n_map(im1,
im2,
err,
boundary=boundary,
nthreads=nthreads,
zeromean=zeromean,
use_numpy_fft=use_numpy_fft,
return_all=True,
reduced=False)
# at this point, the chi2 map contains ALL of the information!
# below is sub-pixel zoom-in stuff
chi2zoom, zf, offsets = iterative_zoom.iterative_zoom(chi2,
mindiff=mindiff,
zoomshape=zoom_shape,
return_zoomed=True,
verbose=verbose,
return_center=False,
**kwargs)
if np.all(chi2zoom == 0):
# if you've over-zoomed & broken things, you can zoom in by the same
# factor but with a bigger field of view
(yy, xx), chi2_rezoom = zoom.zoomnd(chi2,
usfac=zf,
offsets=offsets,
outshape=rezoom_shape,
middle_convention=np.floor,
return_xouts=True,
**kwargs)
else:
(yy, xx), chi2_rezoom = zoom.zoomnd(chi2,
usfac=zf * rezoom_factor,
offsets=offsets,
outshape=rezoom_shape,
middle_convention=np.floor,
return_xouts=True,
**kwargs)
# x and y are swapped and negative
returns = [-off for off in offsets[::-1]]
if return_error:
errx_low, errx_high, erry_low, erry_high = chi2map_to_errors(
chi2_rezoom, zf * rezoom_factor)
returns.append((errx_low + errx_high) / 2.)
returns.append((erry_low + erry_high) / 2.)
if return_chi2array:
yy = (chi2.shape[0] - 1) / 2 - yy
xx = (chi2.shape[1] - 1) / 2 - xx
returns.append((xx, yy, chi2_rezoom))
return returns
def chi2_shift(im1, im2, err=None, upsample_factor='auto', boundary='wrap',
nthreads=1, use_numpy_fft=False, zeromean=False, nfitted=2,
verbose=False, return_error=True, return_chi2array=False,
max_auto_size=512, max_nsig=1.1):
"""
Find the offsets between image 1 and image 2 using the DFT upsampling method
(http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation/content/html/efficient_subpixel_registration.html)
combined with :math:`\chi^2` to measure the errors on the fit
Equation 1 gives the :math:`\chi^2` value as a function of shift, where Y
is the model as a function of shift:
.. math::
\chi^2(dx,dy) & = & \Sigma_{ij} \\frac{(X_{ij}-Y_{ij}(dx,dy))^2}{\sigma_{ij}^2} \\\\
..
& = & \Sigma_{ij} \left[ X_{ij}^2/\sigma_{ij}^2 - 2X_{ij}Y_{ij}(dx,dy)/\sigma_{ij}^2 + Y_{ij}(dx,dy)^2/\sigma_{ij}^2 \\right] \\\\
Equation 2-4:
.. math::
Term~1: f(dx,dy) & = & \Sigma_{ij} \\frac{X_{ij}^2}{\sigma_{ij}^2} \\\\
f(dx,dy) & = & f(0,0) , \\forall dx,dy \\\\
Term~2: g(dx,dy) & = & -2 \Sigma_{ij} \\frac{X_{ij}Y_{ij}(dx,dy)}{\sigma_{ij}^2} = -2 \Sigma_{ij} \left(\\frac{X_{ij}}{\sigma_{ij}^2}\\right) Y_{ij}(dx,dy) \\\\
Term~3: h(dx,dy) & = & \Sigma_{ij} \\frac{Y_{ij}(dx,dy)^2}{\sigma_{ij}^2} = \Sigma_{ij} \left(\\frac{1}{\sigma_{ij}^2}\\right) Y^2_{ij}(dx,dy)
The cross-correlation can be computed with fourier transforms, and is defined
.. math::
CC_{m,n}(x,y) = \Sigma_{ij} x^*_{ij} y_{(n+i)(m+j)}
which can then be applied to our problem, noting that the cross-correlation
has the same form as term 2 and 3 in :math:`\chi^2` (term 1 is a constant,
with no dependence on the shift)
.. math::
Term~2: & CC(X/\sigma^2,Y)[dx,dy] & = & \Sigma_{ij} \left(\\frac{X_{ij}}{\sigma_{ij}^2}\\right)^* Y_{ij}(dx,dy) \\\\
Term~3: & CC(\sigma^{-2},Y^2)[dx,dy] & = & \Sigma_{ij} \left(\\frac{1}{\sigma_{ij}^2}\\right)^* Y^2_{ij}(dx,dy) \\\\
Technically, only terms 2 and 3 has any effect on the resulting image,
since term 1 is the same for all shifts, and the quantity of interest is
:math:`\Delta \chi^2` when determining the best-fit shift and error.
Parameters
----------
im1 : np.ndarray
im2 : np.ndarray
The images to register.
err : np.ndarray
Per-pixel error in image 2
boundary : 'wrap','constant','reflect','nearest'
Option to pass to map_coordinates for determining what to do with
shifts outside of the boundaries.
upsample_factor : int or 'auto'
upsampling factor; governs accuracy of fit (1/usfac is best accuracy)
(can be "automatically" determined based on chi^2 error)
return_error : bool
Returns the "fit error" (1-sigma in x and y) based on the delta-chi2
values
return_chi2_array : bool
Returns the x and y shifts and the chi2 as a function of those shifts
in addition to other returned parameters. i.e., the last return from
this function will be a tuple (x, y, chi2)
zeromean : bool
Subtract the mean from the images before cross-correlating? If no, you
may get a 0,0 offset because the DC levels are strongly correlated.
verbose : bool
Print error message if upsampling factor is inadequate to measure errors
use_numpy_fft : bool
Force use numpy's fft over fftw? (only matters if you have fftw
installed)
nthreads : bool
Number of threads to use for fft (only matters if you have fftw
installed)
nfitted : int
number of degrees of freedom in the fit (used for chi^2 computations).
Should probably always be 2.
max_auto_size : int
Maximum zoom image size to create when using auto-upsampling
Returns
-------
dx,dy : float,float
Measures the amount im2 is offset from im1 (i.e., shift im2 by -1 *
these #'s to match im1)
errx,erry : float,float
optional, error in x and y directions
xvals,yvals,chi2n_upsampled : ndarray,ndarray,ndarray,
x,y positions (in original chi^2 coordinates) of the chi^2 values and
their corresponding chi^2 value
Examples
--------
Create a 2d array,
shift it in both directions,
then use chi2_shift to determine the shift
>>> rr = ((np.indices([100,100]) - np.array([50.,50.])[:,None,None])**2).sum(axis=0)**0.5
>>> image = np.exp(-rr**2/(3.**2*2.)) * 20
>>> shifted = np.roll(np.roll(image,12,0),5,1) + np.random.randn(100,100)
>>> dx,dy,edx,edy = chi2_shift(image, shifted, upsample_factor='auto')
>>> shifted2 = image_registration.fft_tools.shift2d(image,3.665,-4.25) + np.random.randn(100,100)
>>> dx2,dy2,edx2,edy2 = chi2_shift(image, shifted2, upsample_factor='auto')
"""
chi2,term1,term2,term3 = chi2n_map(im1, im2, err, boundary=boundary,
nthreads=nthreads, zeromean=zeromean, use_numpy_fft=use_numpy_fft,
return_all=True, reduced=False)
ymax, xmax = np.unravel_index(chi2.argmin(), chi2.shape)
# needed for ffts
im1 = np.nan_to_num(im1)
im2 = np.nan_to_num(im2)
ylen,xlen = im1.shape
xcen = xlen/2-(1-xlen%2)
ycen = ylen/2-(1-ylen%2)
# original shift calculation
yshift = ymax-ycen # shift im2 by these numbers to get im1
xshift = xmax-xcen
if verbose:
print "Coarse xmax/ymax = %i,%i, for offset %f,%f" % (xmax,ymax,xshift,yshift)
# below is sub-pixel zoom-in stuff
# find delta-chi^2 limiting values for varying DOFs
try:
import scipy.stats
# 1,2,3-sigma delta-chi2 levels
m1 = scipy.stats.chi2.ppf( 1-scipy.stats.norm.sf(1)*2, nfitted )
m2 = scipy.stats.chi2.ppf( 1-scipy.stats.norm.sf(2)*2, nfitted )
m3 = scipy.stats.chi2.ppf( 1-scipy.stats.norm.sf(3)*2, nfitted )
m_auto = scipy.stats.chi2.ppf( 1-scipy.stats.norm.sf(max_nsig)*2, nfitted )
except ImportError:
# assume m=2 (2 degrees of freedom)
m1 = 2.2957489288986364
m2 = 6.1800743062441734
m3 = 11.829158081900793
m_auto = 2.6088233328527037 # slightly >1 sigma
# biggest scale = where chi^2/n ~ 9 or 11.8 for M=2?
if upsample_factor=='auto':
# deltachi2 is not reduced deltachi2
deltachi2_lowres = (chi2 - chi2.min())
if verbose:
print "Minimum chi2: %g Max delta-chi2 (lowres): %g Min delta-chi2 (lowres): %g" % (chi2.min(),deltachi2_lowres.max(),deltachi2_lowres[deltachi2_lowres>0].min())
sigmamax_area = deltachi2_lowres<m_auto
if sigmamax_area.sum() > 1:
yy,xx = np.indices(sigmamax_area.shape)
xvals = xx[sigmamax_area]
yvals = yy[sigmamax_area]
xvrange = xvals.max()-xvals.min()
yvrange = yvals.max()-yvals.min()
size = max(xvrange,yvrange)
else:
size = 1
upsample_factor = max_auto_size/2. / size
if upsample_factor < 1:
upsample_factor = 1
s1 = s2 = max_auto_size
# zoom factor = s1 / upsample_factor = 2*size
zoom_factor = 2.*size
if verbose:
print "Selected upsample factor %0.1f for image size %i and zoom factor %0.1f (max-sigma range was %i for area %i)" % (upsample_factor, s1, zoom_factor, size, sigmamax_area.sum())
else:
s1,s2 = im1.shape
zoom_factor = s1/upsample_factor
if zoom_factor <= 1:
zoom_factor = 2
s1 = zoom_factor*upsample_factor
s2 = zoom_factor*upsample_factor
(yshifts_corrections,xshifts_corrections),chi2_ups = zoom.zoomnd(chi2,
usfac=upsample_factor, outshape=[s1,s2], offsets=[yshift,xshift],
return_xouts=True)
# deltachi2 is not reduced deltachi2
deltachi2_ups = (chi2_ups - chi2_ups.min())
if verbose:
print "Minimum chi2_ups: %g Max delta-chi2 (highres): %g Min delta-chi2 (highres): %g" % (chi2_ups.min(),deltachi2_ups.max(),deltachi2_ups[deltachi2_ups>0].min())
if verbose > 1:
pass
#if hasattr(term3_ups,'len'):
# print "term3_ups has shape ",term3_ups.shape," term2: ",term2_ups.shape," term1=",term1
#else:
# print "term2 shape: ",term2.shape," term1: ",term1," term3: ",term3_ups
# THE UPSAMPLED BEST-FIT HAS BEEN FOUND
# BELOW IS TO COMPUTE THE ERROR
errx_low,errx_high,erry_low,erry_high = chi2map_to_errors(chi2_ups, upsample_factor)
yshift_corr = yshifts_corrections.flat[chi2_ups.argmin()]-ycen
xshift_corr = xshifts_corrections.flat[chi2_ups.argmin()]-xcen
shift_xvals = xshifts_corrections-xcen
shift_yvals = yshifts_corrections-ycen
returns = [-xshift_corr,-yshift_corr]
if return_error:
returns.append( (errx_low+errx_high)/2. )
returns.append( (erry_low+erry_high)/2. )
if return_chi2array:
returns.append((shift_xvals,shift_yvals,chi2_ups))
return returns
def chi2_shift_iterzoom(im1, im2, err=None, upsample_factor='auto',
boundary='wrap', nthreads=1, use_numpy_fft=False, zeromean=False,
verbose=False, return_error=True, return_chi2array=False,
zoom_shape=[10,10], rezoom_shape=[100,100], rezoom_factor=5,
mindiff=1, **kwargs):
"""
Find the offsets between image 1 and image 2 using an iterative DFT
upsampling method combined with :math:`\chi^2` to measure the errors on the
fit
A simpler version of :func:`chi2_shift` that only computes the
:math:`\chi^2` array on the largest scales, then uses a fourier upsampling
technique to zoom in.
Parameters
----------
im1 : np.ndarray
im2 : np.ndarray
The images to register.
err : np.ndarray
Per-pixel error in image 2
boundary : 'wrap','constant','reflect','nearest'
Option to pass to map_coordinates for determining what to do with
shifts outside of the boundaries.
upsample_factor : int or 'auto'
upsampling factor; governs accuracy of fit (1/usfac is best accuracy)
(can be "automatically" determined based on chi^2 error)
zeromean : bool
Subtract the mean from the images before cross-correlating? If no, you
may get a 0,0 offset because the DC levels are strongly correlated.
verbose : bool
Print error message if upsampling factor is inadequate to measure errors
use_numpy_fft : bool
Force use numpy's fft over fftw? (only matters if you have fftw
installed)
nthreads : bool
Number of threads to use for fft (only matters if you have fftw
installed)
nfitted : int
number of degrees of freedom in the fit (used for chi^2 computations).
Should probably always be 2.
zoom_shape : [int,int]
Shape of iterative zoom image
rezoom_shape : [int,int]
Shape of the final output chi^2 map to use for determining the errors
rezoom_factor : int
Amount to zoom above the last zoom factor. Should be <=
rezoom_shape/zoom_shape
Other Parameters
----------------
return_error : bool
Returns the "fit error" (1-sigma in x and y) based on the delta-chi2
values
return_chi2_array : bool
Returns the x and y shifts and the chi2 as a function of those shifts
in addition to other returned parameters. i.e., the last return from
this function will be a tuple (x, y, chi2)
Returns
-------
dx,dy : float,float
Measures the amount im2 is offset from im1 (i.e., shift im2 by -1 *
these #'s to match im1)
errx,erry : float,float
optional, error in x and y directions
xvals,yvals,chi2n_upsampled : ndarray,ndarray,ndarray,
x,y positions (in original chi^2 coordinates) of the chi^2 values and
their corresponding chi^2 value
Examples
--------
Create a 2d array,
shift it in both directions,
then use chi2_shift_iterzoom to determine the shift
>>> np.random.seed(42) # so the doctest will pass
>>> image = np.random.randn(50,55)
>>> shifted = np.roll(np.roll(image,12,0),5,1)
>>> dx,dy,edx,edy = chi2_shift_iterzoom(image, shifted, upsample_factor='auto')
>>> shifted2 = image_registration.fft_tools.shift2d(image,3.665,-4.25)
>>> dx2,dy2,edx2,edy2 = chi2_shift_iterzoom(image, shifted2, upsample_factor='auto')
"""
chi2,term1,term2,term3 = chi2n_map(im1, im2, err, boundary=boundary,
nthreads=nthreads, zeromean=zeromean, use_numpy_fft=use_numpy_fft,
return_all=True, reduced=False)
# at this point, the chi2 map contains ALL of the information!
# below is sub-pixel zoom-in stuff
chi2zoom, zf, offsets = iterative_zoom.iterative_zoom(chi2,
mindiff=mindiff, zoomshape=zoom_shape, return_zoomed=True,
verbose=verbose, return_center=False, **kwargs)
if np.all(chi2zoom==0):
# if you've over-zoomed & broken things, you can zoom in by the same
# factor but with a bigger field of view
(yy,xx),chi2_rezoom = zoom.zoomnd(chi2, usfac=zf, offsets=offsets,
outshape=rezoom_shape, middle_convention=np.floor,
return_xouts=True, **kwargs)
else:
(yy,xx),chi2_rezoom = zoom.zoomnd(chi2, usfac=zf*rezoom_factor,
offsets=offsets, outshape=rezoom_shape,
middle_convention=np.floor, return_xouts=True,
**kwargs)
# x and y are swapped and negative
returns = [-off for off in offsets[::-1]]
if return_error:
errx_low,errx_high,erry_low,erry_high = chi2map_to_errors(chi2_rezoom, zf*rezoom_factor)
returns.append( (errx_low+errx_high)/2. )
returns.append( (erry_low+erry_high)/2. )
if return_chi2array:
yy = (chi2.shape[0]-1)/2 - yy
xx = (chi2.shape[1]-1)/2 - xx
returns.append((xx,yy,chi2_rezoom))
return returns