Lorentzian Metric Tensor Learning In Tensor And Wavelet Feature Space And Its Applications

Feature extraction is a key problem in pattern recognition. And one of the most efficient techniques solving this problem is dimensionality reduction, which aims to find low dimensional representation of dataset useful for classification. Real applications are usually nonlinear. So nonlinear methods based on manifold learning have been developed to discover the structure of data. However, it is difficult to find analytic solutions. Linear algorithms based on manifold learning achieve the intention of researchers. Lorentzian Discriminant Projection is a supervised linear method based on manifold learning proposed recently. It forms a Lorentzian manifold using distance within and between class to detect local and global structure of dataset.Most algorithms for dimensionality reduction are vector-based. They consider an m×n image as a high dimensional vector in Rmn. But an image is intrinsically a 2D tensor. Such a vector representation fails to take into account the spatial locality of pixels in the image. In this paper, we consider an image as a 2D tensor and propose Tensor Lorentzian Discriminant Projection algorithm. It explicitly uses the relationship between pixels from column vectors and row vectors of the image matrix. Hence, structure information of image is used for computing. Face and texture recognition experimental results show that our method achieves better recognition accuracy, while being much more efficient.Although tensor model is supervisor to vector model, it is performed on original samples directly just like vector model. The original samples contain too much variation within class that confuses classification. Thus we develop a new method based on wavelet transform and Lorentzian geometry. It adopts wavelet transform on images to get the wavelet features that contain the global contour and detail information of image. We solve problems on wavelet domain where the differences between classes increase while that within class decrease. And then it reduces to Lorentzian metric tensor learning task in wavelet feature space to find essential characteristic of original samples. The experimental results on FRGC, PIE and ORL database demonstrate the effectiveness of our proposed method.