Deconvolution of Gaussian Random Fields using the Graph Fourier Transform

Real-life acquisition systems are fundamentally limited in their ability to reproduce point sources. For example, a point source object, a star say, observed with an optical telescope is blurred by the imperfect lenses composing the system. Mathematically, the point sources are convolved with a device-specific kernel. This kernel, which depends only on the characteristics of the acquisition system, can to a certain extent be designed so as to minimise the effect of the convolution. But in many applications careful design of the sensing device is not good enough, and a proper deconvolution step is needed. In this paper, we propose an efficient deconvolution algorithm for point source Gaussian random fields as sensed by phased arrays. The algorithm first obtains a continuous least-squares estimate of the random field’s second order moment. The procedure is then in two subsequent steps, decoupling sources localisation and intensity recovery. First, we sample the continuous estimate at a high enough resolution and use the covariance function to construct a weighted graph. We then define a signal on this graph by assigning to each of the sample locations in the field their corresponding intensity (variance of the field at this location). The Graph Fourier Transform (GFT) is then used in order to filter out the convolution artifacts within the estimate. Candidate locations of the sources can then be identified with local maxima of the filtered estimate. From these locations a deconvolution problem is solved by means of weighted linear regression and the intensities of the sources within the field recovered. Finally, a multi-scale approach based on the filtering of the leading eigenvalues of the covariance operator is discussed, and its benefits in terms of efficiency and accuracy are highlighted.