Yingkang Hu’s research while affiliated with Georgia Southern University and other places

The power and flexibility of polynomial surfaces are unleashed when their degrees are no longer restricted to four or lower, as they are used in early CT phantoms. They have proved useful and appropriate for geometric simulation of human and animal anatomy. In this paper a general algorithm is presented for the x-ray transform of any polynomial surface, as long as its ray-surface intersection equation is implemented on the computer. A versatile and powerful polynomial utility C++ class is created to simplify the implementation. Three groups of surfaces are implemented and applied to build a heart phantom closely simulating the Visible Man's heart. The x-ray transform algorithm is tested and verified by the successful reconstruction of the heart phantom.

Analytic simulation in computed tomography(CT) generates projection data for evaluating and improving CT image reconstruction algorithms and has played an important role in the research and development of x-ray CT. The simulation is desired to be as realistic as possible while the computation needs to be efficient and accurate. Early primitive equation-based phantoms such as Shepp-Logan and FORBILD use only boxes, cylinders, and quadrics to simulate body parts. The superquadrics have been used in Computer Graphics since 1980's, and in CT since 1990's. While their more complex shapes make them more realistic in simulation, the difficulties in solving their equations increase dramatically, which restricts their use, especially in ray-tracing and x-ray transform. Zhu et al. developed an algorithm for ray-intersecting the superellipsoid and used it to build a thorax phantom. No algorithms for ray-intersecting the supertoroid, however, are known up to now to the best of our knowledge. In this paper we propose such an algorithm and use it in computation of x-ray transform. The algorithm was tested by a phantom consisting of the top portion of vertebrae and a few ribs. Cone-beam data were produced from the phantom, and then used to reconstruct the phantom.

Analytic simulation in computed tomography(CT) generates projection data for evaluating and improving CT image reconstruction algorithms and has played an important role in the research and development of x-ray CT. The simulation is desired to be as realistic as possible while the computation needs to be efficient and accurate. Early primitive equation-based phantoms such as Shepp-Logan and FORBILD use only boxes, cylinders, and quadrics to simulate body parts. The superquadrics have been used in Computer Graphics since 1980's, and in CT since 1990's. While their more complex shapes make them more realistic in simulation, the difficulties in solving their equations increase dramatically, which restricts their use, especially in ray-tracing and x-ray transform. Zhu et al. developed an algorithm for ray-intersecting the superellipsoid and used it to build a thorax phantom. No algorithms for ray-intersecting the supertoroid, however, are known up to now to the best of our knowledge. In this paper we propose such an algorithm and use it in computation of x-ray transform. The algorithm was tested by a phantom consisting of the top portion of vertebrae and a few ribs. Cone-beam data were produced from the phantom, and then used to reconstruct the phantom.

In this short paper, a unified framework for performing density-weighted fuzzy c-means (FCM) clustering of feature and relational datasets is presented. The proposed approach consists of reducing the original dataset to a smaller one, assigning each selected datum a weight reflecting the number of nearby data, clustering the weighted reduced dataset using a weighted version of the feature or relational data FCM algorithm, and if desired, extending the reduced data results back to the original dataset. Several methods are given for each of the tasks of data subset selection, weight assignment, and extension of the weighted clustering results. The newly proposed weighted version of the non-Euclidean relational FCM algorithm is proved to produce the identical results as its feature data analog for a certain type of relational data. Artificial and real data examples are used to demonstrate and contrast various instances of this general approach.

The visual assessment of tendency (VAT) technique, developed by J.C. Bezdek, R.J. Hathaway and J.M. Huband, uses a visual approach to find the number of clusters in data. In this paper, we develop a new algorithm that processes the numeric output of VAT programs, other than gray level images as in VAT, and produces the tendency curves. Possible cluster borders will be seen as high-low patterns on the curves, which can be caught not only by human eyes but also by the computer. Our numerical results are very promising. The program caught cluster structures even in cases where the visual outputs of VAT are virtually useless.

We improve the visual assessment of tendency (VAT) technique, which, developed by J.C. Bezdek, R.J. Hathaway and J.M. Huband, uses a visual approach to find the number of clusters in data. Instead of using square gray level images of dissimilarity matrices as in VAT, we further process the matrices and produce the tendency curves. Possible cluster structure will be shown as peak-valley patterns on the curves, which can be caught not only by human eyes but also by the computer. Our numerical experiments showed that the computer can catch cluster structures from the tendency curves even in cases where the visual outputs of VAT are virtually useless.

The efficiency of optimization in fuzzy c-means clustering is investigated. Numerous, powerful, general-purpose simultaneous optimization (SO) methods, and hybrid methods combining these and the most widely used alternating optimization (AO) method, are extensively tested for speed comparison. AO is clearly the best and simplest of the methods we tested when used on data sets of small or moderate sizes, especially those containing well-separated clusters. This justifies the extremely wide use of AO. On large-scale problems, some methods we tested are significantly faster than AO.

Fuzzy c-means (FCM) is a useful clustering technique. Modifications of FCM using L₁ norm distances increase robustness to outliers. Object and relational data versions of FCM clustering are defined for the more general case where the L_p norm (p&ges;1) or semi-norm (0<p<1) is used as the measure of dissimilarity. We give simple (though computationally intensive) alternating optimization schemes for all object data cases of p>0 in order to facilitate the empirical examination of the object data models. Both object and relational approaches are included in a numerical study

It is well known that the adaptive algorithm is simple and easy to program but the results are not fully competitive with other nonlinear methods such as free knot spline approximation. We modify the algorithm to take full advantages of nonlinear approximation. The new algorithms have the same approximation order as other nonlinear methods, which is proved by characterizing their approximation spaces. One of our algorithms is implemented on the computer, with numerical results illustrated by figures and tables.