High-Dimensional Matched Subspace Detection When Data are Missing

We consider the problem of deciding whether a highly incomplete signal lies within a given subspace. This problem, Matched Subspace Detection, is a classical, well-studied problem when the signal is completely observed. High- dimensional testing problems in which it may be prohibitive or impossible to obtain a complete observation motivate this work. The signal is represented as a vector in R^n, but we only observe m << n of its elements. We show that reliable detection is possible, under mild incoherence conditions, as long as m is slightly greater than the dimension of the subspace in question

Simultaneous Multicolor Detection of Faint Galaxies in the Hubble Deep Field

We present a novel way to detect objects when multiband images are available. Typically, object detection is performed in one of the available bands or on a somewhat arbitrarily co-added image. Our technique provides an almost optimal way to use all the color information available. We build up a composite image of the N passbands where each pixel value corresponds to the probability that the given pixel is just sky. By knowing the probability distribution of sky pixels (a chi-square distribution with N degrees of freedom), the data can be used to derive the distribution of pixels dominated by object flux. From the two distributions an optimal segmentation threshold can be determined. Clipping the probability image at this threshold yields a mask, where pixels unlikely to be sky are tagged. After using a standard connected-pixel criterion, the regions of this mask define the detected objects. Applying this technique to the Hubble Deep Field data, we find that we can extend the detection limit of the data below that possible using linearly co-added images. We also discuss possible ways of enhancing object detection probabilities for certain well defined classes of objects by using various optimized linear combinations of the pixel fluxes (optimal subspace filtering).Comment: 8 pages, 5 figures (4 postscript, 1 JPEG). To be published in A