Compressive sensing with local geometric features

Abstract

We propose a framework for compressive sensing of images with local geometric features. Specifically, let x ∈ R N be an N -pixel image, where each pixel p has value xp. The image is acquired by computing the measurement vector Ax, where A is an m × N measurement matrix for some m N . The goal is then to design the matrix A and recovery algorithm which, given Ax, returns an approximation to x.In this paper we investigate this problem for the case where x consists of a small number (k) of "local geometric objects" (e.g., stars in an image of a sky), plus noise. We construct a matrix A and recovery algorithm with the following features: (i) the number of measurements m is O(k log k N ), which undercuts currently known schemes that achieve m = O(k log(N/k)) (ii) the matrix A is ultra-sparse, which is important for hardware considerations (iii) the recovery algorithm is fast and runs in time sub-linear in N . We also present a comprehensive study of an application of our algorithm to a problem in satellite navigation.

How to find real-world applications of compressive sensing

“…al. 18 Missile detection considered later in this paper bears similarities to star point source detections algorithms. This also bears some resemblance to the application of compressive sampling to radio interferometry 17 where the use of partial Fourier reconstruction is described.…”

Compressive sensing using locality-preserving matrices

“…That is, all pixels with the same coordinates modulo √ m are added together. This is the construction used in [GIPR11] and [HPYI12]. Note that the resulting mapping from the pixels onto the sensor array is discontinuous, as e.g., the neighboring pixels (0, √ m − 1) and (0, √ m) are mapped to distant sensors.…”

“…The geometric approach has been shown to be useful for sparse recovery and processing of point sources, such as stars [GIPR11], muzzle flashes [HPYI12] or tracked objects [TAN10]. However, the theoretical guarantees for this method are not fully satisfactory.…”

“…In particular, it is not known whether the construction satisfies the p / q approximation guarantee of Equation 1. Instead, the paper [GIPR11] showed a recovery guarantee for a class of images that possess additional geometric structure, namely that contain a small number of distinguishable objects (e.g., stars) plus some noise. Moreover, the proof applied only to a variation of the geometric construction where the image was partitioned into pieces of constant size which were then pseudorandomly permuted.…”

Shift-encoded optically multiplexed imaging

“…While it is feasible to detect objects in a multiplexed image, the angular position of a detected object is uncertain since it could have appeared in any of the image channels. Static encoding schemes, such as changing the point spread function of each channel, 8 combining measurements from two or more multiplexing sensors with different multiplexing channel parameters, 9 or relying on differential channel overlap and rotation, 3 enable disambiguation and tracking of localized objects. Dynamic time-encoded schemes, in which the encoding varies across frames, enable imaging of scenes without any a priori knowledge of the spatial structure.…”