# Efficient subpixel image registration by cross-correlation.

Registers two images (2-D rigid translation) within a fraction of a pixel specified by the user. Instead of computing a zero-padded FFT (fast Fourier transform), this code uses selective upsampling by a matrix-multiply DFT (discrete FT) to dramatically reduce computation time and memory without sacrificing accuracy. With this procedure all the image points are used to compute the upsampled cross-correlation in a very small neighborhood around its peak. This algorithm is referred to as the single-step DFT algorithm in [1].

[1] Manuel Guizar-Sicairos, Samuel T. Thurman, and James R. Fienup, "Efficient subpixel image registration algorithms," Opt. Lett. 33, 156-158 (2008).

## Contents

## Syntax

The code receives the FFT of the reference and the shifted images, and an (integer) upsampling factor. The code expects FFTs with DC in (1,1) so do not use fftshift.

output = dftregistration(fft2(f),fft2(g),usfac);

The images are registered to within 1/usfac of a pixel.

output(1) is the normalized root-mean-squared error (NRMSE) [1] between f and g.

output(2) is the global phase difference between the two images (should be zero if images are real-valued and non-negative).

output(3) and output(4) are the row and column shifts between f and g respectively.

[output Greg] = dftregistration(fft2(f),fft2(g),usfac);

Greg is an optional output, it returns the Fourier transform of the registered version of g, where the global phase difference [output(2)] is also compensated.

## Obtain a reference and shifted images

To illustrate the use of the algorithm, lets obtain a reference and a shifted image. First we read the reference image f(x,y)

```
f = im2double(imread('cameraman.tif'));
```

Define g(x,y) as a version of f(x,y) shifted by fractional values of a pixel and multiplied by a global phase.

deltar = 3.48574; deltac = 8.73837; phase = 2; [nr,nc]=size(f); Nr = ifftshift([-fix(nr/2):ceil(nr/2)-1]); Nc = ifftshift([-fix(nc/2):ceil(nc/2)-1]); [Nc,Nr] = meshgrid(Nc,Nr); g = ifft2(fft2(f).*exp(i*2*pi*(deltar*Nr/nr+deltac*Nc/nc))).*exp(-i*phase); figure(1); subplot(1,2,1); imshow(abs(f)); title('Reference image, f(x,y)') subplot(1,2,2); imshow(abs(g)); title('Shifted image, g(x,y)')

We have shifted the image by 8.73837 and 3.48574 pixels in the x and y direction, respectively, and added a phase of 2 radians to g(x,y). The shift was implemented by applying a linear phase on its FT, thus we have assumed that the images wrap around (features leaving one side of the window reappear on the opposite side) and that the image is band-limited (interpolated by a sinc function). Cross-correlation image registration by DFTs (both the matrix-multiply DFT and the zero-padded FFT) share these assumptions.

This registration technique is well suited to compare images that are captured in Fourier domain (i.e. to evaluate an image reconstruction by holography or phase retrieval) which are strictly band-limited and exhibit the wrap-around effect.

Even though the registration code assumes band-limited images that wrap around, we have obtained very good results when applying it to band-limited microscope images, and aliased imagery. That is when shifting the image brings in new content instead of wrapping it around or when the images are not band-limited.

## Sample Image Registration

dftregistration.m receives the FT of f and g and the upsampling factor. The code expects DC of the FTs at (1,1) so don't use fftshift.

We now use the image registration code to register f and g within 0.01 pixels by specifying an upsampling parameter of 100

[output Greg] = dftregistration(fft2(f),fft2(g),100); display(output),

output = 0.0008 2.0000 3.4900 8.7400

The pixel shift error (difference between the true and obtained shifts) is 0.0016 and 0.0043 in the x and y directions respectively. Well within the expected accuracy of 0.01. Notice that using the conventional zero-padded FFT approach with the same accuracy, would require computation of a 25,600x25,600 FFT, which would require more than 19 Gbytes of RAM and a very comfortable chair.

The following plot shows the reference image and the registered image.

figure(1); subplot(1,2,1); imshow(abs(f)); title('Reference image, f(x,y)') subplot(1,2,2); imshow(abs(ifft2(Greg))); title('Registered image, gr(x,y)')

## Disclaimer

I have made every effort to evaluate the proper working of this code under many different conditions. However, it is the responsibility of the user to ensure that this registration code is adequate for their application.

Feel free to e-mail me with questions or comments.