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摘要: 张量主成分分析(Tensor principal component analysis, TPCA)在彩色图像低维表征领域得到广泛深入研究, 采用${\rm{F}}$范数平方作为低维投影的距离度量方式, 表征含离群数据和噪声图像的鲁棒性较弱. ${L}_{1}$范数能够抑制噪声的影响, 但所获的低维投影数据缺乏重构误差约束, 其局部表征能力也较弱. 针对上述问题, 利用${\rm{F}}$范数作为目标函数的距离度量方式, 提出一种基于$\rm{F}$范数的分块张量主成分分析算法(Block TPCA with $\rm{F}$-norm, BlockTPCA-F), 提高张量低维表征的鲁棒性. 考虑到同时约束投影距离与重构误差, 提出一种基于比例$\rm{F}$范数的分块张量主成分分析算法(Block TPCA with proportional F-norm, BlockTPCA-PF), 其最大化投影距离与最小化重构误差均得到了优化. 然后, 给出其贪婪的求解算法, 并对其收敛性进行理论证明. 最后, 对包含不同噪声块和具有实际遮挡的彩色人脸数据集进行实验, 结果表明, 所提算法在平均重构误差、图像重构与分类率等方面均得到明显提升, 在张量低维表征中具有较强的鲁棒性.Abstract: Tensor principal component analysis (TPCA) has been widely and deeply studied in the field of low-dimensional representation of color images. Using the square F-norm as the distance metric of low-dimensional projection, the robustness of representing the images with outliers and noise is weak. ${L}_{1}$ can suppress the influence of noise, but the obtained low-dimensional projection data lacks the constraint of reconstruction errors, and the local representation ability is also weak. To solve the above problems, using F-norm as the distance metric of the objective function, a block TPCA with F-norm (BlockTPCA-F) algorithm is proposed to improve the robustness of tensor representation with low dimension. Considering the constraints of projection distances and reconstruction errors at the same time, a block TPCA with proportional F-norm (BlockTPCA-PF) algorithm is presented. The maximum projection distance and the minimum reconstruction error in the objective function are optimized. Then, the greedy algorithms for BlockTPCA-F and BlockTPCA-PF are given respectively, and the convergence is proved theoretically. Finally, experiments are carried out on color face datasets with different artificial noise blocks or actual occluded faces. The results show that the proposed algorithms have been significantly improved in the average reconstruction error, the image reconstruction and the classification rate, and they have strong robustness in tensor low-dimensional representation.
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图 3 ${{\cal{X}}_m}$, ${{\cal{Y}}_m}$与${{\cal{E}}_m}$之间的关系
Fig. 3 The relation between ${{\cal{X}}_m}$, ${{\cal{Y}}_m}$ and ${{\cal{E}}_m}$
表 1 20%噪声下最优平均分类率
Table 1 Optimal average classification rate under 20% noise
NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF AB 10 0.9048 0.8974 0.9079 0.9153 0.9153 0.9143 20 0.9132 0.9111 0.9132 0.9101 0.9164 0.9175 30 0.9090 0.9069 0.9090 0.9069 0.9090 0.9101 40 0.9058 0.9048 0.9058 0.9026 0.9090 0.9090 50 0.9048 0.9058 0.9058 0.9005 0.9079 0.9058 GT 10 0.6940 0.7020 0.7055 0.7055 0.6900 0.6915 20 0.7015 0.6935 0.6935 0.6950 0.7070 0.7090 30 0.7005 0.6875 0.6880 0.6905 0.7020 0.7035 40 0.6900 0.6860 0.6850 0.6845 0.7010 0.7035 50 0.6855 0.6820 0.6850 0.6840 0.6970 0.7000 
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表 3 60%噪声下最优平均分类率
Table 3 Optimal average classification rate under 60% noise
NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF AB 10 0.7958 0.7958 0.7937 0.7915 0.8148 0.8116 20 0.7810 0.7810 0.7788 0.7779 0.7926 0.7947 30 0.7810 0.7746 0.7820 0.7757 0.7788 0.7799 40 0.7841 0.7746 0.7767 0.7799 0.7778 0.7799 50 0.7778 0.7757 0.7778 0.7799 0.7757 0.7757 GT 10 0.5354 0.5550 0.5690 0.5700 0.5690 0.5680 20 0.5344 0.5450 0.5665 0.5680 0.5580 0.5580 30 0.5238 0.5455 0.5590 0.5590 0.5510 0.5520 40 0.5101 0.5435 0.5470 0.5470 0.5500 0.5510 50 0.5048 0.5405 0.5450 0.5455 0.5470 0.5485
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表 4 AR人脸数据集最优平均分类率
Table 4 Optimal average classification rate of AR face dataset
NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF AR 10 0.7692 0.7653 0.7692 0.7731 0.8077 0.8077 20 0.7654 0.7653 0.7654 0.7616 0.8001 0.8001 30 0.7654 0.7615 0.7653 0.7654 0.8039 0.8039 40 0.7654 0.7692 0.7692 0.7654 0.8038 0.8038 50 0.7692 0.7615 0.7654 0.7692 0.8039 0.8039
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表 2 40%噪声下最优平均分类率
Table 2 Optimal average classification rate under 40% noise
NPC MPCA TPCA-$L _{1} $-G TPCA-$L _{1} $-NG TPCA-F BlockTPCA-F BlockTPCA-PF AB 10 0.8804 0.8794 0.8847 0.8772 0.8889 0.8889 20 0.8783 0.8772 0.8751 0.8709 0.8889 0.8889 30 0.8624 0.8571 0.8593 0.8635 0.8751 0.8730 40 0.8519 0.8497 0.8508 0.8614 0.8603 0.8571 50 0.8497 0.8476 0.8466 0.8519 0.8529 0.8497 GT 10 0.6243 0.6645 0.6650 0.6630 0.6690 0.6690 20 0.5788 0.6335 0.6300 0.6295 0.6590 0.6590 30 0.5556 0.6115 0.6115 0.6115 0.6455 0.6455 40 0.5471 0.6050 0.6070 0.6070 0.6320 0.6320 50 0.5439 0.6035 0.6070 0.6065 0.6220 0.6220
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