Keith Knox (Boeing LTS), C. E. Mannix, Jr. (The Boeing Co.)
Keywords: Imaging
Abstract:
Short-exposure images of stars and other unresolved objects appear as speckle patterns that randomly move across the image. These random patterns and motion are induced by an accumulated random phase disturbance caused by the light traveling through the time-varying atmosphere. Larger phase disturbances are responsible for the random motion of the speckle patterns, while smaller phase disturbances define details of the individual speckles.
A smoother appearance of the speckle images is possible by estimating the amount of random motion between speckle images and re-centering the individual images. This is particularly useful when several frames are averaged together to increase the signal-to-noise ratio. By re-centering the individual images before averaging, blur from the random motion of the speckle patterns is reduced and fainter details in the averaged image can be detected.
The problem then becomes how to estimate the amount of frame-to-frame random motion. One method is to perform the cross-correlation between the first frame and all other frames in an image sequence. The shift in the peak in this correlation function indicates the difference in positions of the images in the two frames. Re-centering all but the first of the frames removes a large amount of the random motion. The problem is that the shift in the cross-correlation peak is measured in an integer number of pixels and this lack of precision leaves a significant amount of the random motion intact.
One innovation to be described in this talk is the fact that there is nothing unique about the first frame in the sequence. Any frame in the sequence could be used as the reference from which the differences in location of all the other images are measured. In fact, one can measure the differences in position for all independent pairs of frames in the image sequence. This provides a combinatorially large number of measurements compared to the number of image frames, or shifts. Since the number of measurements is much larger than the number of variables, this problem is significantly over-determined, meaning that a least square solution is possible.
The major innovation in this method is a closed-form solution to the least squares problem of estimating the random shifts. The least squares solution, using the pseudoinverse, involves a matrix inversion of an N-1 x N-1 square matrix, where N is the number of image frames. For a large number of frames, numerically calculating this inverse can become computationally prohibitive. We have discovered a closed-form solution to this pseudoinverse solution that does not require the calculation of the inverse matrix. In fact, it does not even require holding the least squares matrix, or even the complete cross-correlation vector, in memory. Instead, the exact least squares solution can be incrementally determined as each cross-correlation is measured.
This method will be illustrated with images of an unresolved object for which the new re-centering method has eliminated almost all of the random shifts in the image sequence.
Date of Conference: September 10-14, 2006
Track: Imaging