Rongrong Wang

Nonlinear Dimensionality Reduction via the ENH-LTSA Method for Hyperspectral Image Classification

The problems of neglecting spatial features in hyperspectral imagery (HSI) and the high complexity of Local Tangent Space Alignment (LTSA) still exist in the nonlinear dimensionality reduction with LTSA for classification. Therefore, this paper proposes an innovative ENH-LTSA (Enhanced-Local Tangent Space Alignment) method to solve the two problems. First, random projection is introduced to preliminarily reduce the dimension of HSI data. It aims to improve the speed of neighbor searching and the local tangent space construction. Then, the new method presents the similarity measure via the adaptive weighted summation kernel (AWSK) distance. The AWSK distance considers both spectral and spatial features in HSI data, and attempts to ameliorate the k-nearest neighbors (KNNs) of each pixel. Furthermore, the adaptive spatial window is proposed to automatically estimate the proper window size for the description of spatial features. After that, fast approximate KNNs graph construction via Recursive Lanczos Bisection is incorporated into the new method to reduce the complexity of KNNs searching. When finishing constructing each local tangent space, the new method uses a fast low-rank approximate singular value decomposition to speed up eigenvalue decomposition of the global alignment matrix that is constituted with local manifold coordinates. Five groups of experiments with two different HSI datasets are designed to completely analyze and testify the ENH-LTSA method. Experimental results show that ENH-LTSA outperforms LTSA, both in classification results and in computational speed.

Restricted Isometry Property of Random Subdictionaries

We study statistical restricted isometry, a property closely related to sparse signal recovery, of deterministic sensing matrices of size m × N. A matrix is said to have a statistical restricted isometry property (StRIP) of order k if most submatrices with k columns define a near-isometric map of Rk into Rm. As our main result, we establish sufficient conditions for the StRIP property of a matrix in terms of the mutual coherence and mean square coherence. We show that for many existing deterministic families of sampling matrices, m = O(k) rows suffice for k-StRIP, which is an improvement over the known estimates of either m = Θ(k log N) or m = Θ(k log k). We also give examples of matrix families that are shown to have the StRIP property using our sufficient conditions.

Measures of Scalability

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.

From Compressed Sensing to Compressed Bit-Streams: Practical Encoders, Tractable Decoders

Compressed sensing is now established as an effective method for dimension reduction when the underlying signals are sparse or compressible with respect to some suitable basis or frame. One important, yet under-addressed problem regarding the compressive acquisition of analog signals is how to perform quantization. This is directly related to the important issues of how “compressed” compressed sensing is (in terms of the total number of bits one ends up using after acquiring the signal) and ultimately whether compressed sensing can be used to obtain compressed representations of suitable signals. In this paper, we propose a concrete and practicable method for performing “analog-to-information conversion”. Following a compressive signal acquisition stage, the proposed method consists of a quantization stage, based on

Sigma Delta quantization with Harmonic frames and partial Fourier ensembles

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A Lifted l1/l2 Constraint for Sparse Blind Deconvolution

(sigma-delta) quantization, and a subsequent encoding (compression) stage that fits within the framework of compressed sensing seamlessly. We prove that, using this method, we can convert analog compressive samples to compressed digital bitstreams and decode using tractable algorithms based on convex optimization. We prove that the proposed analog-to-information converter (AIC) provides a nearly optimal encoding of sparse and compressible signals. Finally, we present numerical experiments illustrating the effectiveness of the proposed AIC.

Resolving scaling ambiguities with the ℓ1/ℓ2 norm in a blind deconvolution problem with feedback

Sigma Delta (

Tightness of stability bounds by null space property

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