对数变换主成分分析的图像识别

Logarithm Transformation Based Principal Component Analysis for Image Recognition

为了进一步提升现有鲁棒主成分分析(PCA)算法处理含有异常样本数据的性能,提出了对数变换的PCA算法。为降低异常样本对目标函数值的影响,根据对数函数的性质建立了对数变换PCA算法的目标函数,证明了所提目标函数值小于标准PCA算法的。之后,给出了求解所提目标函数的一种优化算法,即为所提算法。通过迭代计算对角矩阵和进行特征值分解进行优化,证明了所提算法可以近似收敛于所提目标函数的最优解。分析证明了所提算法具有旋转不变性。为了更好地比较各算法处理异常样本数据的能力,使用AR数据库中的原样本作为异常样本,在其他数据库中人为添加了异常样本。与标准PCA算法、鲁棒PCA算法、包括贪婪求解的基于l1范数的PCA算法、非贪婪求解的基于l1范数的PCA算法、基于l2,p范数的PCA算法和基于最大相关熵的PCA算法在AR、Extended Yale B、CMU PIE这3个人脸数据库和MNIST这1个手写字符数据库进行了实验对比,结果表明:所提算法均得到了最低的重构误差和最高的识别精度;所提算法具有良好的收敛性能,一般迭代5到6次即可收敛。

To further improve the capability of the existing robust principal component analysis(PCA)in dealing with data containing outliers,a principal component analysis method based on logarithm transformation is proposed.For decreasing the interference of outliers to the value of the objective function and based on the property of the logarithm function,the objective function of the proposed logarithm transformation based PCA is established.It is theoretically proved that the objective value of the proposed algorithm is smaller than that of the standard PCA algorithm.An optimization algorithm for solving the proposed objective function is given.The proposed objective function is optimized by computing a diagonal matrix and eigenvalue decomposition iteratively.The optimization algorithm can approximately converge to the optimal solution of the proposed objective function.The rotation invariance property of the proposed algorithm is also analyzed.We use 3 face datasets(AR,Extended Yale B,CMU PIE)and 1 handwritten digit dataset(MNIST)for experimentation.For better comparing the capability of existing algorithms in dealing with outliers,the original data samples in the AR datasets are taken as outliers.Some data samples are artificially added to the other datasets as outliers.Compared with standard PCA algorithm and the existing robust PCA algorithms,including greedy PCA algorithms based on l1 norm,nongreedy PCA algorithms based on l1 norm,PCA algorithms based on l2,p norm and PCA algorithms based on maximum correntropy,the proposed algorithm is endowed with the lowest reconstruction error and highest recognition accuracy.Analyzing the relationship between the iteration number and the value of the objective function,the proposed algorithm has good convergence,it usually converges after 5 or 6 iterations.