We present a new method for the modeling of freehand collected three-dimensional ultrasound data. The model is piecewise linear and based upon progressive tetrahedral domains created by a subdivision scheme which splits a tetrahedron on on its longest edge and guarantees a valid tetrahedrization. Least squares error is used to characterize the model and an effective iterative technique is used to compute the values of the model at the vertices of the tetrahedral grid. Since the subdivision strategy is adaptive, the complexity of the model conforms to the complexity of the data leading to an extremely efficient and highly compressed volume model. The model is evaluated in real time using piecewise linear interpolation, and gives a medical professional the chance to see images which would not be possible using conventional ultrasound techniques.

A cellular network is often modelled as a graph and the channel assignment problem is formulated as a coloring problem of the graph. Sen et al. (1998) introduced the notion of cellular graphs that models the hexagonal cell structure of a cellular network. Assuming a k-band buffering system where the interference does not extend beyond k cells away from the call originating cell, we provided two different formulations of the channel assignment problem: distance-k chromatic number problem and k-band chromatic bandwidth problem. The channel assignment algorithms presented in Sen et al. were non-optimal. In this paper we provide: (i) a new algorithm for the distance-k chromatic number problem that is optimal and (ii) a near optimal algorithm for the 2-band chromatic bandwidth problem that has a performance bound of 4/3. The complexity of the algorithms is O(p), where p is the number of cells.

We present a novel approach to solving the cracking problem. The cracking problem arises in many contexts in scientific visualization and computer graphics modeling where there is need for an approximation based upon domain decomposition that is fine in certain regions and coarse in others. This includes surface rendering approximation of images and multiresolution terrain visualization. In general, algorithms based upon adaptive refinement strategies must deal with this problem. The approach presented here is simple and general. It is based upon the use of a triangular Coons patch. Both the basic idea of using a triangular Coons patch in this context and the particular Coons patch that is used constitute the novel contributions of the paper.

A cellular network is often modelled as a graph and the channel assignment problem is formulated as a coloring problem of the graph. We introduce the notion of cellular graphs that models the hexagonal cell structures of a cellular network. Exploiting the regular structure of the cellular graphs we compute the upper and the lower bounds for a class of channel assignment problems. Assuming a k-band buffering system where the interference does not extend beyond k cells away from the call originating cell, we provide two different formulations of the channel assignment problem-distance-k chromatic number problem and k-band chromatic bandwidth problem. We give one algorithm for the first problem and two for the second, with all three algorithms assigning channels to the cells. The complexity of the algorithm for the first problem is O(p), where p is the number of cells. For the second problem, the complexity of the first algorithm is O(p) and the complexity of the second algorithm is O(k/sup 5/log k). All the algorithms are asymptotically optimal, in the sense that the order of the upper bound of the number of channels required is the same as the order of the lower bound.