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# Algorithm for Wavefront Sensing Using an Extended Scene

- Wednesday, 24 September 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:

- 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. - Within each sub-image, choose an
*N*×*N*-pixel cell centered at the coordinates determined in step 1. For the*i*th 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*). - 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*). - 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). - Fit
*u*and ν slopes to Φ(*u*,ν) over a small*u*,ν subdomain. - 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. - 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. - 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.*

### This Brief includes a Technical Support Package (TSP).

** Algorithm for Wavefront Sensing Using an Extended Scene** (reference NPO-44770) is currently available for download from the TSP library.

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