Günther F. Schrack

Encoding and decoding the Hilbert order

Explicit formulas are presented to generate the coordinates of point on the two-dimensional Hilbert curve from its location code and vice versa. Execution-time assessments suggest that the proposed algorithms are faster than the ones published previously.

Optimized relative step size random searches

New results concerning the family of random searches as proposed by Rastrigin are presented. In particular, the random search with reversals and two optimized relative step size random searches are investigated. Random searches with reversals are found to be substantially better than their counterparts. A new principle of updating the step size for this family of searches is proposed.

A new ordering strategy applied to spatial data processing

Ordering a set of multi-dimensional data is an essential task in modern spatial data processing systems such as raster geographical information systems. This can effectively be carried out by a transformation called spatial ordering. It has been reported that the H ilbert order exhibits desirable performance. However, its encoding or decoding process in the n -dimensional space (n 3) is complicated and would place a considerable computational burden on the application problems. In this paper, we propose a new spatial order with efficient algorithms. The performance evaluation suggests that this new order is competitive with the Hilbert order.

An algorithm for encoding and decoding the 3-D Hilbert order

Spatial ordering is a transformation relating an n-dimensional voxel to a set of natural numbers. Interest in the Hilbert order (H-order) has been high because of its desirable performance when applied to image processing. In this correspondence, an encoding/decoding algorithm for the three-dimensional (3-D) Hilbert order is presented.

Integer dilation and contraction for quadtrees and octrees

Integer dilation and contraction are functions used in conjunction with quadtree and octree mapping systems. Dilation is the process of inserting a number of zeros before each bit in a word and contraction is the process of removing those zeros. Present methods of dilation and contraction involve lookup tables which consume considerable amounts of memory for mappings of large or high resolution display devices but are very fast under practical limits. A method is proposed which rivals the speed of the tabular methods but eliminates the tables, thereby eliminating the associated memory consumption. The proposed method is applicable to both dilation and contraction for both quadtrees and octrees.

On Spatial Orders and Location Codes

Spatial orders such as the Morton (Z) order, U-order, or X-order have applications in matrix manipulation, graphic rendering and data encryption. It is shown that these spatial orders are single examples of entire classes of spatial orders which can be defined in arbitrary numbers of dimensions and base values. Secondly, an algorithm is proposed which can be used to transform between these spatial orders and Cartesian coordinates. It is shown that the efficiency of the algorithm improves with a larger base value. By choosing a base value that corresponds to the available memory page size, the computational effort required to perform operations such as matrix multiplication can be optimized.

Multiple-seed 3D connectivity filling for inaccurate borders

Abstract Reconstructing a volume representation in the presence of inaccurate boundary data is becoming a fundamental issue in a variety of fields, such as modeling of natural objects, medical imaging, graphics, and robotics. Given boundary data, parts of which are incorrect, a low-level representation of the enclosed volume in terms of octrees is found. This representation is useful in providing accurate volume estimates related to the enclosing boundary. The connectivity filling technique proposed here is, basically, a region-growing technique which simultaneously moves from the border elements toward the outer and inner normals: in the first case it propagates the label white and in the second case the label black . A labeling component algorithm and a border evaluation technique are then used to resolve ambiguities that may arise when gaps or superpositions of border elements occur. An example has been especially designed to illustrate the problem and its proposed solution. Illustrations of the algorithms capability applied to two natural objects conclude the presentation.

Adaptive display of linear octrees

Abstract A new approach for displaying three-dimensional objects in an octree environment is presented. The algorithm accepts as input a linear octree with surface elements properly identified, uses the octal encoding and viewing parameters to eliminate occluded elements and determines the amount of supersampling needed according to user-supplied viewing specifications. The final visibility test is carried out by creating a finite number of buckets obtained via a modified radix-sort algorithm. The set of top elements of each of the buckets forms the visible part of the object.

A fast distance approximation algorithm for encoded quadtree locations

Several distance approximation functions to the Euclidean distance of two points in the two-dimensional Euclidean plane are reviewed. An improved distance function is proposed and compared with previous ones as to their maximum relative errors and their execution times. The improved distance function is then applied for distance calculations in the domain of linear quadtrees using dilated integer arithmetic. The maximum relative error is discussed and execution times are reported.<<ETX>>

CNS-HLS Mapping Using Fuzzy Sets

Existing specification systems, such as the hue/ lighness/saturation system, use triples of numbers to spec1fy color, which can be tedious to work with. The Color Naming system allows users to specify color in natural lanuage phrases or words and is a siperior way for humans to select colors. the problems addressed in this article is how to map the linguistic values of the CNS to the numerical values of the HLS system so that the CNS can be part of an integrated graphics system. The algorithm used is based on fuzzy set theory and is implemented for the Tektronix 4027 color display terminal.