陈凯旋,吴小俊(江南大学物联网工程学院, 无锡 214122)
目的 在基于图像集的分类任务中，用SPD （symmetric positive definite）矩阵描述图像集，并考虑所得到的黎曼流形，已被证明对许多分类任务有较好的效果。但是，已有的经典分类算法大多应用于欧氏空间，无法直接应用于黎曼空间。为了将欧氏空间的分类方法应用于解决图像集的分类，综合考虑SPD流形的LEM （Log-Euclidean metric）度量和欧氏空间分类算法的特性，实现基于图像集的分类任务。方法 通过矩阵的对数映射将SPD流形上的样本点映射到切空间中，切空间中的样本点与图像集是一一对应的关系，此时，再将切空间中的样本点作为欧氏空间中稀疏表示分类算法的输入以实现图像集的分类任务。但是切空间样本的形式为对称矩阵，且维度较大，包含一定冗余信息，为了提高算法的性能和运行效率，使用NYSTRÖM METHOD和（2D）2PCA （two-directional two-dimensional PCA）两种方法来获得包含图像集的主要信息且维度更低的数据表示形式。结果 在实验中，对人脸、物体和病毒细胞3种不同的对象进行分类，并且与一些用于图像集分类的经典算法进行对比。实现结果表明，本文算法不仅具有较高的识别率，而且标准差也相对较小。在人脸数据集上，本文算法的识别率可以达到78.26%，比其他算法高出10%左右，同时，具有最小的标准差2.71。在病毒数细胞据集上，本文算法的识别率可以达到58.67%，在所有的方法中识别率最高。在物体识别的任务中，本文算法的识别率可以达到96.25%，标准差为2.12。结论 实验结果表明，与一些经典的基于图像集的分类算法对比，本文算法的识别率有较大的提高且具有较小的标准差，对多种数据集有较强的泛化能力，这充分说明了本文算法可以广泛应用于解决基于图像集的分类任务。但是，本文是通过（2D）2PCA和NYSTRÖM METHOD对切空间中样本进行降维来获得更低维度的样本，以提高算法的运行速度和性能。如何直接构建维度更低，且具有判别性的SPD流形将是下一步的研究重点。
Sparse representation in tangent space for image set classification
Chen Kaixuan,Wu Xiaojun(School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China)
Objective In image set classification, symmetric positive definite (SPD) matrices are usually utilized to model image sets. The resulting Riemannian manifold yields a high discriminative power in many visual recognition tasks. However, existing classic classification algorithms are mostly applied in the Euclidean space and cannot work directly on SPD matrices. To apply the classification algorithm of Euclidean space to image set classification, this work comprehensively reviews the unique Log-Euclidean metric (LEM) of the SPD manifold and the properties of the existing classical classification algorithm, and the classification task based on the image sets is achieved.Method Given that the SPD matrices lie on Riemannian space, we map the samples on the SPD manifold to the tangent space through logarithm mapping, and each sample in the tangent space corresponds to an image set. The form of the samples in the tangent space is a symmetrical matrix, and its dimensionality conforms with the samples on the SPD manifold. The symmetric matrix in the tangent space contains redundant information and has a large dimension. To improve the performance and efficiency of the algorithm, we need to reduce the dimensionality of the data in the tangent space. In our technique, we use the Nyström method and (2D)2PCA to obtain low-dimensional data that contain the main information of the image sets. 1) The Nyström method can approximate the infinite-dimensional samples in the reproducing kernel Hilbert space (RKHS). The dimensionality of the samples mapped into the RKHS by kernel mapping is infinite, and the Riemannian kernel is obtained by the inner product of the samples in the tangent space using the LEM of the SPD manifold. For a set of M training samples, the Riemannian kernel matrix K=[k(xi,xj)]M×M can be written as K≌ZTZ=VΣ1/2 Σ1/2 VT. Here, Zd×M=Σ1/2VT, Σ and V are the top d eigenvalues and the eigenvectors of K, where d is the rank of the kernel matrix K. The projection matrix can be denoted as Σ-1/2VT, and the d-dimensional vector approximation of the random sample y in the RKHS can be written as Σ-1/2VT(k(y,x1),…,k(y,xM))T. 2) (2D)2PCA (two-directional two-dimensional PCA) is a well-known dimensionality reduction (DR) technique for two-dimensional data in machine learning and pattern recognition. (2D)2PCA overcomes the limitation of PCA of working only on one-dimensional data and of 2DPCA being used for the row and column DR of two-dimensional data to obtain two direction projection matrices. In our experiments, the row direction projection matrix WR is consistent with column direction projection matrix WC:W=WR=WC, in which WD×d is a projection matrix for the row and column directions, because the form of the samples in the tangent space is a symmetrical matrix. The sample x∈RD×D can be reduced as x'=WTxW, where x'∈Rd×d and an efficient low-dimensional representation of the sample in the tangent space is achieved. To this end, the SPD matrices are transformed into low-dimensional descriptors with respect to the corresponding image sets. The classical sparse representation classification algorithm, Fisher discrimination dictionary learning in Euclidean space, which has good recognition rates for single images, can be utilized to classify the points of low-dimensional descriptors.Result Our approach is used for several tasks, including face identification, object classification, and virus cell recognition, and we experiment on the YouTube celebrities (YTC), ETH-80, and Virus datasets. Results show that our algorithm has not only a higher recognition rate but also a relatively smaller standard deviation than several classical algorithms for image set classification, such as covariance discriminant learning (CDL) and projection metric learning. In the experiment on the ETH-80 dataset, our approach achieves a recognition rate of 96.25%, which indicates a considerable improvement over those of discriminative canonical correlations (DCC), CDL, and other classic methods. The standard deviation is only 2.12, which is smaller than that of the other methods; this finding indicates that our method has the best robustness on the ETH-80 dataset. For the YTC dataset, the recognition rate of our method is 78.26%, which is 10% higher than that of the other methods. This result shows the evident advantage of our method for the YTC dataset. The standard deviation is smallest, which shows that our method also has the best robustness for the YTC dataset. However, for the Virus dataset, our method has the highest recognition rate 58.67% but a large standard deviation. This finding indicates that the robustness of our method for this dataset is insufficient.Conclusion In this work, we consider the geometrical properties of the SPD manifolds and the related Riemannian metric and combine them with the classical classification algorithm of the Euclidean space to implement image set classification and achieve good results. Experimental results show that our proposed method achieves high accuracies and generally small standard deviation. Therefore, our method can be widely applied in image set classification. Our future work will focus on constructing a lower-dimensional and more discriminative SPD manifold than that in this research.