Edge detection of piecewise smooth functions from undersampled Fourier data using variance signatures

Detecting edges in piecewise smooth images is important in a variety of applications. In some cases, such as magnetic resonance imaging, data are collected as Fourier samples, and are often oversampled at low frequencies but more sparsely acquired in the high frequency range. This undersampling causes Fourier-based edge detection algorithms to yield false positives in smooth regions. The problem is compounded when the data are noisy. In this paper, we demonstrate that even when the given data are noisy and undersampled, the edge response behavior obtained using certain Fourier-based edge detection methods can be accurately characterized. In particular, the responses of different high order methods vary little, both in regions away from the discontinuities and at true edges. By contrast, the variance is large in cells neighboring each true edge. By locating these "double peaks" in variance, it is possible to determine edges from noisy and undersampled Fourier data. Alternative methods of generating and refining variance data are also explored. Specifically, a method based on regularized image reconstruction that leverages the sparsity of edges and image smoothness between these edges is effective at eliminating false positives.