Research On Lie Group Based Image Generalized Gaussian Feature Structure Analysis

Mapping high-dimensional image data into low-dimensional feature space, then mea-sure distance, compute mean and classification in the mapped feature space is the mostpopular framework to solve computer vision and pattern recognition problems. Statisticalfeatures is most important in image feature extraction. Such kind of image feature estimatethe probability density of the distribution of low-level image signals, and take the estimat-ed parameters to describe images. Histogram is a kind of feature description scheme oftenused in computer vision. Different histogram based features have been used to solve differ-ent problems. For example, there are grayscale histograms and color histogram for trackinghistogram of oriented gradients for object detection. Although histograms are successfullyused in solving these problems, they have several disadvantages: unable to describe high di-mensional signal, sensitive to bin size et al. Therefore, continuous distribution features areproposed for image representation, such as region covariance, spatiogram, gaussian mixturemodels and hierarchical gaussian et al. Because gaussian functions plays a very importantrole in these features, we call them Gaussian Type Features.Feature distance measure, mean computation and classification are highly related tothe topological structure of feature space. However, structures of feature spaces have notbeen analysed before. Recently, region covariance is identified as Riemannian manifold.Results on object tracking and detection show that taking the manifold properties of regioncovariance features into consideration can improve the performance signanficantly. How-ever, only covariance matrices form a Riemannian, other Gaussian-type features do not.Since Gaussian-type image features are highly related to Gaussian function space, in orderto analyse Gaussian-type features, we need to analyse Gaussian function space first.What kind of space is Gaussian function space is? In this thesis, we identify it by anal-yse the connections bwtween Gaussian functions and affine transformations. An invertibleaffine transform of an multivariate Gaussian distributed random vector is also multivariateGaussian distributed. If we start from the zero-mean unit covariance multivariate Gaussian random vector, we can find an unique Gaussian function for any invertible affine transforma-tions. Moreover, if we restrict the affine transformations to upper triangular positive definiteaffine transformations, we can find an unique such kind of transformation for any Gaus-sian functions. So there exist a bijection between Gaussian functions and upper triangularpositive definite affine transformations. That is to say, we can analyse upper triangular pos-itive definite affine transformations instead of Gaussian functions. It’s known that invertibleaffine transformations form a Lie group. Since upper triangular affine transformations areclosed under the group operations of affine Lie group, they form a sub-group of affine Liegroup. Because any sub-group of a Lie group is also a Lie group, upper triangular positivedefinite affine transformations form a Lie group. Therefore, Gaussian function space is aLie group.Based on the core contribution of this thesis: Gaussian function spaces is a Liegroup, we analyse the structure of image Gaussian-type feature: we propose image regionGaussian feature and apply it to object detection problem; based on the theories of measuredistance of two Lie group elements and Lie group tangent space projection, we designa Lie group boosting algorithm for object detector training; based on the theory of Liegroup element distance measure, we propose a spatiogram similarity measure and applyit to object tracking problem, experimental results show better performance than otherspatiogram similarity measures; by integrating Lie group element distance measure theorywith earth mover’s distance, we propose Lie group earth mover’s distance for Gaussianmixture model comparison and apply it to image retrieval problem, experimental resultsshow that the proposed distance measure obtain the best performance compared withother distance measure; based on tangent space projection theory of Lie group and theproperties of a special kind of Gaussian mixture models: maximum a posteriori estimatedGaussian mixture models, we propose a new image feature descriptors: Lie algebrizedGaussian features and apply it scene classification problem, experimental results showbetter formance than the state-of-the-art methods.