We consider the general problem of matching a subspace to a signal in RN that has been observed indirectly (compressed) through a random projection. We are interested in the case where the collection of K-dimensional subspaces is continuously parameterized, i.e. naturally indexed by an interval from the real line, or more generally a region of RD. Our main results show that if the dimension of the random projection is on the order of K times a geometrical constant that describes the complexity of the collection, then the match obtained from the compressed observation is nearly as good as one obtained from a full observation of the signal. We give multiple concrete examples of collections of subspaces for which this geometrical constant can be estimated, and discuss the relevance of the results to the general problems of template matching and source localization. 1

• Integrates linear acquisition with dimensionality reduction

The performance of existing approaches to the recovery of frequency-sparse signals from compressed measurements is limited by the coherence of required sparsity dictionaries and the discretization of frequency parameter space. In this pa-per, we adopt a parametric joint recovery-estimation method based on model selection in spectral compressive sensing. Numerical experiments show that our approach outperforms most state-of-the-art spectral CS recovery approaches in fi-delity, tolerance to noise and computation efficiency. Index Terms — Compressive sensing, frequency-sparse signal, model selection, parametric estimation, maximum likelihood estimator 1.

Light rays incident on a transparent object of uniform refractive index undergo deflections, which uniquely characterize the surface geometry of the object. Associated with each point on the surface is a deflection map which describes the pattern of deflections in various directions and it tends to be sparse when the object surface is smooth. This article presents a novel method to efficiently acquire and reconstruct sparse deflection maps using the framework of Compressed Sensing (CS). To this end, we use a particular implementation of schlieren deflectometer, which provides linear measurements of the underlying maps via optical comparison with programmable spatial light modulation patterns. To optimize the number of measurements needed to recover the map, we base the design of modulation patterns on the principle of spread spectrum CS. We formulate the map reconstruction task as a linear inverse problem and provide a complete characterization of the proposed method, both on simulated data and experimental deflecto-metric data. The reconstruction techniques are designed to incorporate various types of prior knowledge about the deflection spectrum. Our results show the capability and advantages of using a CS based approach for deflectometric imaging. Further, we present a method to characterize deflection spectra that captures its essence in a few parameters. We demonstrate that these parameters can be extracted directly from a few compressive measurements, without needing any costly reconstruction procedures, thereby saving a lot of computations. Then, a connection between the evolution of these parameters as a function of spatial locations and the optical characteristics of the objects under study is made. The experimental results with simple plano-convex lenses and multifocal intra-ocular lenses show how a quick characterization of the objects can be obtained using compressed sensing.

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