Speckle imaging and hidden phase

Abstract

The Knox-Thompson, or cross-spectrum, method provides two two-dimensional difference equations for the phase of the object spectrum. We demonstrate that, in general, the object spectrum phase can be decomposed into a regular, single-valued function determined by the divergence of the phase gradient, as well as a multivalued function determined by the circulation of the phase gradient; this second function has been called the hidden phase. The standard least-squares solution to the two-dimensional difference equations will always miss this hidden phase. We present a solution method that gives both the regular and the hidden parts of the object spectrum phase. Finally, we illustrate several examples of imaging through turbulence and postprocessing with the Knox-Thompson method, including the hidden phase.