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We consider the problem of recovering a complex vector (up to a global unimodular constant) given noisy and incomplete outer product measurements. Such problems arise when implementing distributed clock synchronization schemes, radar autofocus methods, and phaseless signal recovery. This problem is known as vector synchronization and is a variant of the more common angular synchronization problem. In applications with windowed measurements and/or convolutional models - for example, phase retrieval from STFT magnitude data, the outer product measurement matrix is highly incomplete and has a block diagonal structure. We describe a vector synchronization technique which applies an eigenvector computation to blocks of this matrix followed by a block compatibility operation to piece together the final solution. We provide theoretical guarantees (in the noiseless case) and empirical simulations demonstrating the accuracy and efficiency of the method. 

The underlying physics of imaging processes and associated instrumentation limitations mean that blurring artifacts are unavoidable in many applications such as astronomy, microscopy, radar and medical imaging. In several such imaging modalities, convolutional models are used to describe the blurring process; the observed image or function is a convolution of the true underlying image and a point spread function (PSF) which characterizes the blurring artifact. In this work, we propose and analyze a technique - based on convolutional edge detectors and Gaussian curve fitting - to approximate unknown Gaussian PSFs when the underlying true function is piecewise-smooth. For certain simple families of such functions, we show that this approximation is exponentially accurate. We also provide preliminary two dimensional extensions of this technique. These findings - confirmed by numerical simulations - demonstrate the feasibility of recovering accurate approximations to the blurring function, which serves as an important prerequisite to solving deblurring problems.