Corrections to “Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds” [Dec 10 6140-6155]

Electr. & Comput. Eng. Dept., Duke Univ., Durham, NC, USA
IEEE Transactions on Signal Processing (Impact Factor: 2.79). 03/2011; 59(3):1329. DOI: 10.1109/TSP.2010.2070796
Source: DBLP


Nonparametric Bayesian methods are employed to constitute a mixture of low-rank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a low-dimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds, based on compressive sensing (CS) projection measurements. The statistical CS inversion is performed analytically. We derive the required number of CS random measurements needed for successful reconstruction, based on easily-computed quantities, drawing on block-sparsity properties. The proposed methodology is validated on several synthetic and real datasets.

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Available from: David B Dunson,
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    • "There are multiple reasons for adopting a GMM representation, which can be seen as a union of (linear or affine) subspaces, where each subspace is associated with the translation of the image of the (possibly low-rank) covariance matrix of each Gaussian component within the GMM. In fact, low-rank GMM priors have been shown to approximate signals in compact manifolds [11] and have been shown to provide state-of-the-art results in practical problems in image processing [12], dictionary learning [11], image classification [13] and video compression [14]. Of particular relevance, the adoption of GMM priors also offers an opportunity to analyze phase transitions in the classification "

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    • "This framework can be generalized to other applications and algorithms. For instance, Gaussian mixture model based dictionary learning approaches [49]–[53] that learn a union of subspaces can also benefit from side information. "
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    ABSTRACT: A blind compressive sensing algorithm is proposed to reconstruct hyperspectral images from spectrally-compressed measurements. The wavelength-dependent data are coded and then superposed, mapping the three-dimensional hyperspectral datacube to a two-dimensional image. The inversion algorithm learns a dictionary in situ from the measurements via globallocal shrinkage priors. By using RGB images as side information of the compressive sensing system, the proposed approach is extended to learn a coupled dictionary from the joint dataset of the compressed measurements and the corresponding RGB images, to improve reconstruction quality. A prototype camera is built using a liquid-crystal-on-silicon modulator. Experimental reconstructions of hyperspectral datacubes from both simulated and real compressed measurements demonstrate the efficacy of the proposed inversion algorithm, the feasibility of the camera and the benefit of side information.
    IEEE Journal of Selected Topics in Signal Processing 09/2015; 9(6). DOI:10.1109/JSTSP.2015.2411575 · 2.37 Impact Factor
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    • "which is an analytical solution with [44] "
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    ABSTRACT: We develop a new compressive sensing (CS) inversion algorithm by utilizing the Gaussian mixture model (GMM). While the compressive sensing is performed globally on the entire image as implemented in our lensless camera, a low-rank GMM is imposed on the local image patches. This low-rank GMM is derived via eigenvalue thresholding of the GMM trained on the projection of the measurement data, thus learned {\em in situ}. The GMM and the projection of the measurement data are updated iteratively during the reconstruction. Our GMM algorithm degrades to the piecewise linear estimator (PLE) if each patch is represented by a single Gaussian model. Inspired by this, a low-rank PLE algorithm is also developed for CS inversion, constituting an additional contribution of this paper. Extensive results on both simulation data and real data captured by the lensless camera demonstrate the efficacy of the proposed algorithm. Furthermore, we compare the CS reconstruction results using our algorithm with the JPEG compression. Simulation results demonstrate that when limited bandwidth is available (a small number of measurements), our algorithm can achieve comparable results as JPEG.
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