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摘要: 处理部分光照和遮挡等噪声的图像重构及分类问题因其极具挑战而备受关注，该问题的解决很大程度上取决于对误差的描述，常见的方法以向量形式存储误差矩阵且假定其服从于独立同分布，忽视了图像数据的内部结构信息.针对该问题，本文提出一种联合平滑矩阵多变量椭圆分布的稀疏表示算法（Sparse representation with smoothed matrix multivariate elliptical distribution，SMED）.该算法强调误差矩阵中各个像素间的依赖性并假定误差矩阵作为一个随机矩阵变量服从于矩阵多变量椭圆分布；之后，引入辅助变量光滑目标函数，使得模型易于获得全局最优解；最后，采用迭代加权最小二乘法优化求解模型.此外，文中对SMED算法的收敛性和复杂度进行了理论分析，并讨论了模型的参数敏感性.在AR、ExYaleB和PubFig三个公开数据集中的实验验证了所提算法具有鲁棒的鉴别力，且其综合性能明显优于经典算法.Abstract: As one of the most challerging problems, partial illumination or occlusion in image reconstruction and classification has recently received considerable attention. This problem heavily depends on the characterization of representation error, while most existing approaches ignore the internal structure information of image data for the error matrix needs to be stretched into a vector with each element being assumed independently corrupted. To improve the algorithms in this regard, a new sparse representation with smoothed matrix multivariate elliptical distribution (SMED) is proposed, which emphasizes the dependency between pixels of noise and assums that error matrix is a random matrix variate and follows a matrix multivariate elliptical distribution. Then, it introduces regularization terms to smooth the objective function, which makes the model easily to obtain a globally optimal solution. Finally, iteratively reweighted least square is adopted to solve the SMED model efficiently. Furthermore, comprehensive analyses are also performed including convergence behavior, computational complexity, together with parameter determination. Experimental results on the AR, ExYaleB and PubFig databases demonstrate that the proposed algorithm is a robust discriminative classifier with excellent performance, being superior to classical algorithms.1) 本文责任编委 黎铭
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图 1 多元分布比较图(常数取${C}_i=1$, $i=1, 2, 3$)
Fig. 1 The comparison chart of the multivariate distribution (where the constants are ${ C}_{i}=1$, $i=1, 2, 3$)
图 2 不同参数$\mu_{{c}}$和$\rho$值下的SMED$_{11}$算法收敛值曲线
Fig. 2 Convergence curves of SMED$_{11}$ algorithm on AR with different regularization parameters $\mu_{{c}}$ and $\rho$
图 3 不同光照影响及墨镜和围巾遮挡下的人脸重构性能对比
Fig. 3 The reconfiguration performance contrast of different algorithms under illuminition changes and sunglass or scraf occlusion
图 4 不同像素污损程度下的人脸识别率对比
Fig. 4 Comparison of recognition rates under the different pixel corruption levels
图 5 不同遮挡程度下的人脸识别率对比
Fig. 5 Comparison of recognition rates under the different occlusion levels
图 6 PubFig中1人的训练样本和测试样本示例图
Fig. 6 Sample images for one person from PubFig face database
表 1 AR人脸库真实影响下不同算法的识别率对比(%)
Table 1 Comparison of recognition rates on AR database (%)
算法 光照影响 墨镜遮挡 围巾遮挡 SMED$_{11}$ 95.83 74.66 67.64 SMED$_{1/21}$ 91.93 77.17 69.83 SMED$_{3/22}$ 93.16 72.67 60.33 SRC 90.50 68.83 55.83 CRC 95.33 69.67 63.50 RRC 88.17 71.16 69.83 FFS 83.00 65.50 50.83 HQ_A 87.83 70.00 44.50 HQ_M 93.51 75.83 48.93 
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表 2 AR数据库中各算法测试时间对比(s)
Table 2 Running time of competing algorithms on AR database (s)
分类器 测试时间 SMED$_{11}$ 0.1703 SMED$_{1/21}$ 0.1748 SMED$_{3/22}$ 0.1648 SRC 0.0917 CRC 0.0041 RRC 3.2165 FFS 0.1738 HQ_A 4.3898 HQ_M 13.2725
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表 3 ExYaleB、PubFig数据库中各算法测试时间对比(s)
Table 3 Running time of competing algorithms on ExYaleB, PubFig database (s)
分类器 测试时间 ExYaleB ExYaleB ExYaleB ExYaleB PubFig (白色遮挡) (黑色遮挡) (baboon遮挡) (鲜花遮挡) SMED$_{11}$ 0.2429 0.2508 0.2191 0.2376 1.6705 SMED$_{1/21}$ 0.3450 0.2452 0.2561 0.2445 1.7864 SMED$_{3/22}$ 0.2638 0.2435 0.2318 0.2186 1.6425 CRC 0.0037 0.0045 0.0031 0.0035 0.0152 RRC 0.2510 0.2370 0.2426 0.2375 2.5819 CESR 0.3244 0.1276 0.2966 0.2240 2.3499 HQ_M 12.2049 11.7532 12.3732 11.6457 6.8705
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