Algorithm for Wavefront Sensing Using an Extended Scene

NASA’s Jet Propulsion Laboratory

Wednesday, October 01 2008

The restriction to a point source has been removed.

A recently conceived algorithm for processing image data acquired by a Shack-Hartmann (SH) wavefront sensor is not subject to the restriction, previously applicable in SH wavefront sensing, that the image be formed from a distant star or other equivalent of a point light source. That is to say, the image could be of an extended scene. (One still has the option of using a point source.) The algorithm can be implemented in commercially available software on ordinary computers.

The steps of the algorithm are the following:

1. Suppose that the image comprises M sub-images. Determine the x,y Cartesian coordinates of the centers of these sub-images and store them in a 2×M matrix.
2. Within each sub-image, choose an N×N-pixel cell centered at the coordinates determined in step 1. For the ith sub-image, let this cell be denoted as s_i(x,y). Let the cell of another sub-image (preferably near the center of the whole extended-scene image) be designated a reference cell, denoted r(x,y).
3. Calculate the fast Fourier transforms of the sub-sub-images in the central N’×N’ portions (where N’ < N and both are preferably powers of 2) of r(x,y) and s_i(x,y).
4. Multiply the two transforms to obtain a cross-correlation function C_i(u,ν), in the Fourier domain. Then let the phase of C_i(u,ν) constitute a phase function, Φ(u,ν).
5. Fit u and ν slopes to Φ(u,ν) over a small u,ν subdomain.
6. Compute the fast Fourier transform, S_i(u,ν) of the full N×N cell s_i(x,y). Multiply this transform by the u and ν phase slopes obtained in step 4. Then compute the inverse fast Fourier transform of the product.
7. Repeat steps 4 through 6 in an iteration loop, cumulating the u and ν slopes, until a maximum iteration number is reached or the change in image shift becomes smaller than a predetermined tolerance.
8. Repeat steps 4 through 7 for the cells of all other sub-images.

This work was done by Erkin Sidick, Joseph Green, Catherine Ohara, and David Redding of Caltech for NASA’s Jet Propulsion Laboratory.

The software used in this innovation is available for commercial licensing. Please contact Karina Edmonds of the California Institute of Technology at (626) 395-2322. Refer to NPO-44770.