The main objective of this thesis is to apply ideas and techniques from modern control theory, especially from robust state estimation and model validation, to various important problems in computer vision. Robust model validation is used in texture recognition where new approaches for classifying texture samples and segmenting textured images are developed. Also, a new model validation approach to motion primitive recognition is demonstrated by considering the motion segmentation problem for a mobile wheeled robot. A new approach to image inpainting based on robust state estimation is proposed where the implementation presented here concerns with recovering corrupted frames in video sequences. Another application addressed in this thesis based on robust state estimation is video-based tracking. A new tracking system is proposed to follow connected regions in video frames representing the objects in consideration. The system accommodates tracking multiple objects and is designed to be robust towards occlusions. To demonstrate the performance of the proposed solutions, examples are provided where the developed methods are applied to various gray-scale images, colored images, gray-scale videos and colored videos. In addition, a new algorithm is introduced for motion estimation via inverse polynomial interpolation. Motion estimation plays a primary role within the video-based tracking system proposed in this thesis. The proposed motion estimation algorithm is also applied to medical image sequences. Motion estimation results presented in this thesis include pairs of images from a echocardiography video and a robot-assisted surgery video.

The main objective of this thesis is to apply ideas and techniques from modern control theory, especially from robust state estimation and model validation, to various important problems in computer vision. Robust model validation is used in texture recognition where new approaches for classifying texture samples and segmenting textured images are developed. Also, a new model validation approach to motion primitive recognition is demonstrated by considering the motion segmentation problem for a mobile wheeled robot. A new approach to image inpainting based on robust state estimation is proposed where the implementation presented here concerns with recovering corrupted frames in video sequences. Another application addressed in this thesis based on robust state estimation is video-based tracking. A new tracking system is proposed to follow connected regions in video frames representing the objects in consideration. The system accommodates tracking multiple objects and is designed to be robust towards occlusions. To demonstrate the performance of the proposed solutions, examples are provided where the developed methods are applied to various gray-scale images, colored images, gray-scale videos and colored videos. In addition, a new algorithm is introduced for motion estimation via inverse polynomial interpolation. Motion estimation plays a primary role within the video-based tracking system proposed in this thesis. The proposed motion estimation algorithm is also applied to medical image sequences. Motion estimation results presented in this thesis include pairs of images from a echocardiography video and a robot-assisted surgery video.

Master of Science / Department of Statistics / Weixin Yao / In the fitting of mixtures of linear regression models, the normal assumption has been traditionally used for the error term and then the regression parameters are estimated by the maximum likelihood estimate (MLE) using the EM algorithm. Under the normal assumption, the M step of the EM algorithm uses a weighted least squares estimate (LSE) for the regression parameters. It is well known that the LSE is sensitive to outliers or heavy tailed error distributions. In this report, we propose a robust mixture of linear regression model, which replaces the least square criterion with some robust criteria in the M step of the EM algorithm. In addition, we will use a simulation study to demonstrate how sensitive the traditional mixture regression estimation method is to outliers or heavy tailed error distributions and compare it with our proposed robust mixture regression estimation method. Based on our empirical studies, our proposed robust estimation method works comparably to the traditional estimation method when there are no outliers and the error is normally distributed but is much better if there are outliers or the error has heavy tails (such as t-distribution). A real data set application is also provided to illustrate the effectiveness of our proposed methodology.

Master of Science / Department of Statistics / James J. Higgins / Completely random and randomized block designs involving n factors at each of two levels are used to screen for the effects of a large number of factors. With such designs it may not be possible either because of costs or because of time to run each treatment combination more than once. In some cases, only a fraction of all the treatments may be run. With a large number of factors and limited observations, even one outlier can adversely affect the results. Robust regression methods are designed to down-weight the adverse affects of outliers. However, to our knowledge practitioners do not routinely apply robust regression methods in the context of fractional replication of 2^n factorial treatment structures. The purpose of this report is examine how robust regression methods perform in this context.

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Master of Science / Department of Statistics / Weixin Yao / In practice, when applying a statistical method it often occurs that some observations deviate from the usual model assumptions. Least-squares (LS) estimators are very sensitive to outliers. Even one single atypical value may have a large effect on the regression parameter estimates. The goal of robust regression is to develop methods that are resistant to the possibility that one or several unknown outliers may occur anywhere in the data. In this paper, we review various robust regression methods including: M-estimate, LMS estimate, LTS estimate, S-estimate, [tau]-estimate, MM-estimate, GM-estimate, and REWLS estimate. Finally, we compare these robust estimates based on their robustness and efficiency through a simulation study. A real data set application is also provided to compare the robust estimates with traditional least squares estimator.

Diffusion tensor imaging (DTI) is a relatively new technology of magnetic resonance imaging, which enables us to observe the insight structure of the human body in vivo and non-invasively. It displays water molecule movement by a 3×3 diffusion tensor at each voxel. Tensor field processing, visualisation and tractography are all based on the diffusion tensors. The accuracy of estimating diffusion tensor is essential in DTI. This research focuses on exploring the potential improvements at the tensor estimation of DTI. We analyse the noise arising in the measurement of diffusion signals. We present robust methods, least median squares (LMS) and least trimmed squares (LTS) regressions, with forward search algorithm that reduce or eliminate outliers to the desired level. An investigation of the criterion to detect outliers is provided in theory and practice. We compare the results with the generalised non-robust models in simulation studies and applicants and also validated various regressions in terms of FA, MD and orientations. We show that the robust methods can handle the data with up to 50% corruption. The robust regressions have better estimations than generalised models in the presence of outliers. We also consider the multiple tensors problems. We review the recent techniques of multiple tensor problems. Then we provide a new model considering neighbours' information, the Bayesian single and double tensor models using neighbouring tensors as priors, which can identify the double tensors effectively. We design a framework to estimate the diffusion tensor field with detecting whether it is a single tensor model or multiple tensor model. An output of this framework is the Bayesian neighbour (BN) algorithm that improves the accuracy at the intersection of multiple fibres. We examine the dependence of the estimators on the FA and MD and angle between two principal diffusion orientations and the goodness of fit. The Bayesian models are applied to the real data with validation. We show that the double tensors model is more accurate on distinct fibre orientations, more anisotropic or similar mean diffusivity tensors. The final contribution of this research is in covariance tensor estimation. We define the median covariance matrix in terms of Euclidean and various non-Euclidean metrics taking its symmetric semi-positive definiteness into account. We compare with estimation methods, Euclidean, power Euclidean, square root Euclidean, log-Euclidean, Riemannian Euclidean and Procrustes median tensors. We provide an analysis of the different metric between different median covariance tensors. We also provide the weighting functions and define the weighted non-Euclidean covariance tensors. We finish with manifold-valued data applications that improve the illustration of DTI images in tensor field processing with defined non-weighted and weighted median tensors. The validation of non-Euclidean methods is studied in the tensor field processing. We show that the root square median estimator is preferable in general, which can effectively exclude outliers and clearly shows the important structures of the brain. The power Euclidean median estimator is recommended when producing FA map.

by Lai Siu Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1989. / Bibliography: leaves 50-52.