Edge detection, image segmentation and their applications in microarray image analysis

We consider the problem of detecting jump location curves of regression surfaces. In the literature, most existing methods detect jumps in regression surfaces based on estimation of either the first-order or the second-order derivatives of the regression surface. Methods based on the first-order derivatives are usually better in removing the noise effect, whereas methods based on the second-order derivatives are often superior in localization of the detected jumps. In this thesis research, we suggest a new procedure for jump detection in regression surfaces, which combines the major strengths of the above two types of methods. Our method is based on estimation of both the first-order and second-order derivatives of the true regression surface. Theoretical justifications and numerical studies show that it works well in applications.; Jump detection in regression surfaces has many applications. One important application is image segmentation for analyzing gene microarray images. Gene microarray data are widely used in applications, including pharmaceutical and clinical research. By comparing gene expressions in normal and abnormal cells, microarrays can be used for identifying genes involved in particular diseases, and then these genes can be targeted by therapeutic drugs. Many gene expression data are produced from spotted microarray images. A microarray image consists of thousands of spots, with individual DNA sequences first printed at each spot and then equal amount of probe samples from treatment and control cells mixed and hybridized with the printed DNA sequences. To obtain gene expression data, the image needs to be segmented first to separate foregrounds from backgrounds for individual spots, and then averages of foreground pixels are used for computing the gene expression data. So image segmentation of microarray images is related directly to the reliability of gene expression data. Several image segmentation procedures have been suggested and included in some software packages handling gene microarray data. In this thesis research, a new image segmentation methodology is proposed based on local polynomial kernel smoothing. Theoretical arguments and numerical studies show that it has some good statistical properties and would perform well in applications.