chi2_shift¶

image_registration.chi2_shifts.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)[source]¶

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 \(\chi^2\) to measure the errors on the fit

Equation 1 gives the \(\chi^2\) value as a function of shift, where Y is the model as a function of shift:

\[\chi^2(dx,dy) = \Sigma_{ij} \frac{(X_{ij}-Y_{ij}(dx,dy))^2}{\sigma_{ij}^2}\]

Equation 2-4:

\begin{align} \mathrm{Term~1:} & f(dx,dy) & = & \Sigma_{ij} \frac{X_{ij}^2}{\sigma_{ij}^2} \\ & f(dx,dy) & = & f(0,0) , \forall dx,dy \\ \mathrm{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) \\ \mathrm{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) \end{align}

The cross-correlation can be computed with fourier transforms, and is defined

\[CC_{m,n}(x,y) = \Sigma_{ij} x^*_{ij} y_{(n+i)(m+j)}\]

which can then be applied to our problem, noting that the cross-correlation has the same form as term 2 and 3 in \(\chi^2\) (term 1 is a constant, with no dependence on the shift)

\begin{align} \mathrm{Term~2:} & CC(X/\sigma^2,Y)[dx,dy] & = & \Sigma_{ij} \left(\frac{X_{ij}}{\sigma_{ij}^2}\right)^* Y_{ij}(dx,dy) \\ \mathrm{Term~3:} & CC(\sigma^{-2},Y^2)[dx,dy] & = & \Sigma_{ij} \left(\frac{1}{\sigma_{ij}^2}\right)^* Y^2_{ij}(dx,dy) \end{align}

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 \(\Delta \chi^2\) when determining the best-fit shift and error.

The resulting shifts are limited to an accuracy of +/-0.5 pixels in the upsampled image frame. That is not a Gaussian uncertainty but a quantized limit: the best solution will always be +/-0.5 pixels offset because we’re zooming in on an even grid, so the “best” position is required to be a discrete pixel center. If you’re looking at an image with exactly zero shift, it will have exactly +/- 1/usfac/2 systematic error in the resulting solution.

Parameters:

im1np.ndarray
im2np.ndarray: The images to register.
errnp.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_factorint or ‘auto’: upsampling factor; governs accuracy of fit (1/usfac is best accuracy) (can be “automatically” determined based on chi^2 error)
return_errorbool: Returns the “fit error” (1-sigma in x and y) based on the delta-chi2 values
return_chi2_arraybool: 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)
zeromeanbool: 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.
verbosebool: Print error message if upsampling factor is inadequate to measure errors
use_numpy_fftbool: Force use numpy’s fft over fftw? (only matters if you have fftw installed)
nthreadsbool: Number of threads to use for fft (only matters if you have fftw installed)
nfittedint: number of degrees of freedom in the fit (used for chi^2 computations). Should probably always be 2.
max_auto_sizeint: Maximum zoom image size to create when using auto-upsampling

Returns:

dx,dyfloat,float: Measures the amount im2 is offset from im1 (i.e., shift im2 by -1 * these #’s to match im1)
errx,erryfloat,float: optional, error in x and y directions
xvals,yvals,chi2n_upsampledndarray,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

>>> import image_registration
>>> 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')

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chi2_shift¶