Image Classification On A Riemannian Manifold

Image classification is one of the most important problems for computer vision, pat-tern recognition and machine learning. Image classification algorithm is to extract one or more image features (such as color, density, texture, shape, spatial information, etc.), and then divide the feature space into several classes based on a decision rule. With the rapid development of computer multimedia and internet technology, image classification method becomes more and more important, and has an extensive application foreground. This cov-ers a wide variety of application areas such as high-level semantic understanding, large-scale image search, video intelligent monitoring, human-computer interaction, medical diagno-sis, virtual reality, etc.For the image classification algorithm, how to effectively construct and analyze image features is extremely important. Most previous approaches for image classification are based on global image features, and hence are sensitive to changes in environmental conditions. In contrast to global image features, a patch-based image representation has the potential to overcome these problems. And typical image classification methods explore image ap-pearances without considering the spatial causality among distinctive domains in an image. Most of existing feature description methods have viewed image feature as a vector with ignoring the topological structure of the feature space. And Riemannian manifold theory is commonly viewed as a power tool for analyzing the feature space. The space of these structured feature descriptors can be formulated as a connected Riemannian manifold (e.g. Grassmannian manifold, Lie group manifold, Stiefel manifold, etc.).Based on existing research, the image classification on a Riemannian manifold and its application are intensive studied. The main contents and innovative work of this article are shown as follows:(1) An ordered image patch based auto-regressive moving average (ARMA) modelTypical image classification methods explore image appearances without considering the spatial causality among distinctive domains in an image. To address the issue, we intro-duce an ordered-patch based image representation for integrating both the local appearance information and spatial relationships of the image. Then the sequence of these ordered patches is described by an ARMA model, which parameters can be further estimated. This part content is a research basis for studying the Riemannnian manifold based image classi-fication and its related image classification problems.(2) A Grassmannian manifold and ARMA model based image classificationWe introduce an ordered-patch based image representation and use the auto-regressive moving average (ARMA) model to characterize the representation. Firstly, each image is encoded as a sequence of ordered patches, integrating both the local appearance information and spatial relationships of the image. Secondly, the sequence of these ordered patches is described by an ARMA model, which can be further identified as a point on the image Grass-mannian manifold. Then image classification can be conducted on such a manifold under this manifold representation. Furthermore, an appropriate Grassmannian kernel for support vector machine (SVM) classification is developed based on a distance metric of the image Grassmannian manifold. Finally, experiments are conducted on several image datasets (such as MNIST, USPS, Yale, ORL, etc.) to demonstrate that the proposed algorithm outperforms other existing image classification methods.(3) A Lie group manifold and ARMA model based face image classificationIn order to efficiently deal with the complex nonlinear variations of face images, we present a Lie group manifold and ARMA model based face image classification method. Firstly, we present a ARMA model based face representation to capture both the appearance and spatial information of the face image. Secondly, the derived ARMA model can be pa-rameterized as a specially-structured upper triangular matrix, the space of which is proved to constitute a Lie group. A Lie group (LG) kernel is then designed to characterize the sim-ilarity between the ARMA models for any two face images and the kernel can be fed into classical kernel-based classifiers for different types of facial analysis. Finally, experimental evaluations on face recognition and head pose estimation are conducted on several chal-lenging datasets and the results show that the proposed classification algorithm outperforms other facial analysis methods.(4) Discriminative Analysis for symmetric positive definite (SPD) matrices on Lie Groups for improving image classificationFor addressing image classification problems, we study Discriminative Analysis for symmetric positive definite (SPD) matrices on Lie Groups, namely transforming a Lie Group (LG) into a dimension-reduced one by optimizing data separability. Particularly, we take the space of SPD matrices, e.g. covariance matrices, as a concrete example of Lie Groups, which has proved to be a powerful tool for high-order image feature representation. The discriminative transformation of a Lie Group is achieved by optimizing the within-class compactness as well as the between-class separability based on the popular graph embed-ding framework. A new kernel based on the geodesic distance between two samples in the dimension-reduced Lie group is then defined, and fed into classical kernel-based classifiers, e.g. Support Vector Machine, for various image classification tasks. Extensive experiments on five public datasets, i.e., Scene-15, Caltech101,UIUC-Sport, MIT-Indoor and VOC07, well demonstrate its effectiveness and state-of-the-art performances.