Multiscale Sampling Geometries and Methods for Deterministic and Stochastic Reconstructions of Magnitude and Phase Spectra of Satellite Imagery

This paper presents new methods for phase and magnitude interpolation and demonstrates their usefulness in reconstructing images from a limited number of frequency samples. A collection of multiscale frequency domain sampling geometries are developed based on the partition of the spectrum into low-, medium-, and high-frequency blocks. A nonstationary statistical approach is introduced that is based on adaptively selecting the best stochastic model in each frequency block. To develop effective models, the magnitude spectrum is preprocessed using a logarithmic transformation. Phase interpolation requires preprocessing by an appropriate phase unwrapping method. The new stochastic interpolation method is compared against cubic spline, bilinear, and nearest neighbor interpolation methods. Image reconstruction results are presented for sampling rates that retain 6.01% to 28.91% of the 2-D fast Fourier transform (FFT) samples. Image interpolation methods are compared based on the peak signal-to-noise ratio and the mean structural similarity index for satellite images of rural, natural, and urban images. The results indicate that the stochastic (Kriging) interpolation approach provides the best rural image reconstructions using just 6.01% of the 2-D FFT samples. Bilinear interpolation also gave excellent reconstructions for natural and urban images. For natural and urban images, stochastic interpolation gave the best magnitude-only interpolation results.