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Mean shift (MS) and subspace constrained mean shift (SCMS) algorithms are non-parametric, iterative methods to find a representation of a high dimensional data set on a principal curve or surface embedded in a high dimensional space. The representation of high dimensional data on a principal curve or surface, the class of mean shift type algorithms and their properties, and applications of these algorithms are the main focus of this dissertation. Although MS and SCMS algorithms have been used in many applications, a rigorous study of their convergence is still missing. This dissertation aims to fill some of the gaps between theory and practice by investigating some convergence properties of these algorithms. In particular, we propose a sufficient condition for a kernel density estimate with a Gaussian kernel to have isolated stationary points to guarantee the convergence of the MS algorithm. We also show that the SCMS algorithm inherits some of the important convergence properties of the MS algorithm. In particular, the monotonicity and convergence of the density estimate values along the sequence of output values of the algorithm are shown. We also show that the distance between consecutive points of the output sequence converges to zero, as does the projection of the gradient vector onto the subspace spanned by the D-d eigenvectors corresponding to the D-d largest eigenvalues of the local inverse covariance matrix. Furthermore, three new variations of the SCMS algorithm are proposed and the running times and performance of the resulting algorithms are compared with original SCMS algorithm. We also propose an adaptive version of the SCMS algorithm to consider the effect of new incoming samples without running the algorithm on the whole data set. As well, we develop some new potential applications of the MS and SCMS algorithm. These applications involve finding straight lines in digital images; pre-processing data before applying locally linear embedding (LLE) and ISOMAP for dimensionality reduction; noisy source vector quantization where the clean data need to be estimated before the quanization step; improving the performance of kernel regression in certain situations; and skeletonization of digitally stored handwritten characters. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-30 18:01:12.959

Robust, non-intrusive human eye detection problem has been a fundamental and challenging problem for computer vision area. Not only it is a problem of its own, it can be used to ease the problem of finding the locations of other facial features for recognition tasks and human-computer interaction purposes as well. Many previous works have the capability of determining the locations of the human eyes but the main task in this thesis is not only a vision system with eye detection capability / Our aim is to design a real-time, robust, scale-invariant eye tracker system with human eye movement indication property using the movements of eye pupil. Our eye tracker algorithm is implemented using the Continuously Adaptive Mean Shift (CAMSHIFT) algorithm proposed by Bradski and the EigenFace method proposed by Turk &amp / Pentland. Previous works for scale invariant object detection using Eigenface method are mostly dependent on limited number of user predefined scales which causes speed problems / so in order to avoid this problem an adaptive eigenface method using the information extracted from CAMSHIFT algorithm is implemented to have a fast and scale invariant eye tracking. First of all / human face in the input image captured by the camera is detected using the CAMSHIFT algorithm which tracks the outline of an irregular shaped object that may change size and shape during the tracking process based on the color of the object. Face area is passed through a number of preprocessing steps such as color space conversion and thresholding to obtain better results during the eye search process. After these preprocessing steps, search areas for left and right eyes are determined using the geometrical properties of the human face and in order to locate each eye indivually the training images are resized by the width information supplied by the CAMSHIFT algortihm. Search regions for left and right eyes are individually passed to the eye detection algortihm to determine the exact locations of each eye. After the detection of eyes, eye areas are individually passed to the pupil detection and eye area detection algorithms which are based on the Active Contours method to indicate the pupil and eye area. Finally, by comparing the geometrical locations of pupil with the eye area, human gaze information is extracted. As a result of this thesis a software named &ldquo / TrackEye&rdquo / with an user interface having indicators for the location of eye areas and pupils, various output screens for human computer interaction and controls for allowing to test the effects of color space conversions and thresholding types during object tracking has been built.