text: In this paper, we propose an auto-encoder based generative neural network model whose encoder compresses the inputs into vectors in the tangent space of a special Lie group manifold: upper triangular positive definite affine transform matrices (UTDATs). UTDATs are representations of Gaussian distributions and can straightforwardly generate Gaussian distributed samples. Therefore, the encoder is trained together with a decoder (generator) which takes Gaussian distributed latent vectors as input. Compared with related generative models such as variational auto-encoder, the proposed model incorporates the information on geometric properties of Gaussian distributions. As a special case, we derive an exponential mapping layer for diagonal Gaussian UTDATs which eliminates matrix exponential operator compared with general exponential mapping in Lie group theory. Moreover, we derive an intrinsic loss for UTDAT Lie group which can be calculated as l-2 loss in the tangent space. Furthermore, inspired by the Lie group theory, we propose to use the Lie algebra vectors rather than the raw parameters (e.g. mean) of Gaussian distributions as compressed representations of original inputs. Experimental results verity the effectiveness of the proposed new generative model and the benefits gained from the Lie group structural information of UTDATs.
  @cite_10: We present a new algorithm to detect pedestrian in still images utilizing covariance matrices as object descriptors. Since the descriptors do not form a vector space, well known machine learning techniques are not well suited to learn the classifiers. The space of d-dimensional nonsingular covariance matrices can be represented as a connected Riemannian manifold. The main contribution of the paper is a novel approach for classifying points lying on a connected Riemannian manifold using the geometry of the space. The algorithm is tested on INRIA and DaimlerChrysler pedestrian datasets where superior detection rates are observed over the previous approaches.
  @cite_2: This paper introduces a feature descriptor called shape of Gaussian (SOG), which is based on a general feature descriptor design framework called shape of signal probability density function (SOSPDF). SOSPDF takes the shape of a signal's probability density function (pdf) as its feature. Under such a view, both histogram and region covariance often used in computer vision are SOSPDF features. Histogram describes SOSPDF by a discrete approximation way. Region covariance describes SOSPDF as an incomplete parameterized multivariate Gaussian distribution. Our proposed SOG descriptor is a full parameterized Gaussian, so it has all the advantages of region covariance and is more effective. Furthermore, we identify that SOGs form a Lie group. Based on Lie group theory, we propose a distance metric for SOG. We test SOG features in tracking problem. Experiments show better tracking results compared with region covariance. Moreover, experiment results indicate that SOG features attempt to harvest more useful information and are less sensitive against noise.
summary: Geometric theories have been applied to analyze image feature space. In @cite_10 , covariance matrices are used as image feature representations for object detection. Because covariance matrices are symmetric positive definite (SPD) matrices, which form a Riemannian manifold, a corresponding boosting algorithm is designed for SPD inputs. In @cite_2 , Gaussian distributions are used to model image features and the input space is analyzed using Lie group theory.