Statistical Compressed Sensing of Gaussian Mixture Models
Statistical Compressed Sensing of Gaussian Mixture Models
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DOI：
10.1109/tsp.2011.2168521
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发表时间：
2011-01
影响因子：
5.4
通讯作者：
Guoshen Yu;G. Sapiro
中科院分区：
文献类型：
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作者：
Guoshen Yu;G. Sapiro
介绍了一种新的压缩感知框架，即统计压缩感知（SCS），其目的是有效地采样遵循统计分布的信号集合，并平均实现准确的重建。深入研究了基于高斯模型的SCS。对于遵循单个高斯模型的信号，具有O（k）测量值的高斯或伯努利传感矩阵，远小于O（k）测量值。（k log（N/k）），其中N是信号维度，并且具有经由线性滤波实现的最优解码器，比在常规CS中应用的追踪解码器明显更快，SCS的误差以压倒性的概率被示出为由常数乘以最佳k项近似误差的严格上界。故障概率也显着小于传统的稀疏导向CS。更强而更简单的结果进一步表明，对于任何传感矩阵，高斯SCS的误差上限是常数乘以概率为1的最佳k项近似，并且可以有效地计算边界常数。对于高斯混合模型（Gestival），假设多个高斯分布，每个信号都遵循其中一个未知的索引，分段线性估计器被引入到解码SCS。模型选择的准确性是分段线性解码器的核心，根据高斯分布的属性和感测测量的数量进行分析。提出了一种基于高斯混合模型的SCS的最大-最大（Max-Max）算法，该算法迭代估计高斯模型参数、信号模型选择和信号解码。在真实的图像传感应用中，基于GMM的SCS显示，导致改进的结果相比，传统的CS，在一个相当低的计算成本。
A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS based on sparse models, where N is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximization-maximization (Max-Max) algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost.