3-D imaging correlography and coherent image reconstruction

By illuminating an object with a laser and collecting far-field speckle intensity patterns, at a regularly spaced sequence of wavelengths, one obtains the squared magnitude of the 3-D Fourier transform of the object. Performing 3-D phase retrieval to reconstruct a 3-D image (consisting of complex-valued voxels) is relatively difficult unless one has a tight support constraint. An alternative is to perform averaging of the autocovariance of the far-field speckle intensities, over an ensemble of speckle realizations, to estimate the squared magnitude of the Fourier transform of the underlying (incoherent) reflectivity of the object, by the correlography method. This also gives us an incoherent-image-autocorrelation estimate, from which we can derive an initial support constraint. Since the image, being incoherent, is real-valued and nonnegative, performing phase retrieval on this data is easier and more robust. Unfortunately the resolution for correlography is only moderate since the SNR is low at the higher spatial frequencies. However, one can then use a thresholded version of that reconstructed incoherent image as a tight support constraint for performing phase retrieval on the original speckle intensity patterns to reconstruct a fine-resolution, coherent image. The fact that the objects are opaque plays an important role in the robustness of this approach. We will show successful reconstruction results from real data collected in the laboratory as part of the PROCLAIM (Phase Retrieval with an Opacity Constraint for LAser IMaging) effort.