Spatial domain morphological filtering for interpolation of the Fourier domain

We establish here a method to partially recover the missing frequencies in data acquired through sub-sampling in the Fourier domain. Non-linear open and close operations are applied recursively to the raw spatial image estimated from its sparse Fourier samples. The mean of the open and closed images updates the estimated image after applying each structuring element (SE). The updated Fourier spectra are required at each stage to remain consistent with the known sample values. The SE size is incremented until the values estimated for the missing Fourier coefficients converge. The average image obtained from applying multiple sets of randomly selected SEs forms the final result. A denoising scheme is finally used to supplement the interpolated result. The denoising estimates the alias artefacts that arise from the initial sub-sampling by assuming that a narrow region around the borders of the reconstructed image has constant grey level. Convergence occurs in around 30 iterations that usually requires, in total a little less than one minute on a 64GB RAM, i7 machine. Since the computational complexity of the method is low, it can serve as a reliable initialization for more sophisticated iterative reconstruction schemes. Results for several simulated and real sub-sampled image data show an improvement of about 0.1 in Structural SIMilarity index (SSIM, whose maximum possible value is 1) in comparison to the raw inverse FFT image estimate. This method is more likely to underestimate the missing frequency values.