This project discusses two different but related research problems: (1) image segmentation for analyzing cDNA microarray images, and (2) jump-preserving surface estimation from noisy data. Image segmentation of microarray images is a critical stage in generating gene expression data, which are widely used in pharmaceutical and clinical research for identifying particular diseases. Several image segmentation procedures have been included in some software packages for handling gene microarray data. In this project, a new image segmentation methodology is proposed based on local linear kernel smoothing. It is expected that this method would possess some good theoretical properties. Preliminary numerical studies show that it outperforms some existing procedures. For the purposes of de-noising, change of image resolution, data compression, etc., images often need to be reconstructed. Since edges of an image carry critical information about objects, they should be preserved when the image is reconstructed. So edge-preserving image reconstruction is an important research problem. An image can be regarded as a surface of the image intensity function. In this project, the image reconstruction problem is studied from the perspective of surface estimation. A new jump-preserving surface estimation methodology is proposed based on local linear kernel smoothing. Compared to some existing procedures, this method is easy to use and simple to compute. It is also possible to develop profound statistical theory for this method based on existing theory about local linear kernel smoothing. Gene microarray data are widely used in pharmaceutical and clinical research. By comparing gene expression 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. Most gene expression data are produced by segmentation of microarray images. So image segmentation techniques are related directly to the quality of gene expression data. This project proposes a new image segmentation technique. Based on preliminary numerical studies, it could improve the current segmentation techniques, and consequently, improve the quality of gene expression data and have a positive impact on pharmaceutical and clinical studies involving gene microarrays. This project also suggests a new jump-preserving surface estimation methodology, which can be used directly for restoring true images from their noisy versions. Compared to some existing procedures, this method is easy to use and simple to compute. It should be helpful for several different sciences and industries using such techniques (e.g., medical sciences, meteorology, oceanography, military, space communication, etc.).
