Steven L. Tanimoto

Oct-trees and their use in representing three-dimensional objects

Abstract Many of the programming techniques used in solving two-dimensional problems can be extended to three dimensions. Here oct-trees are developed as a three-dimensional analog of quad-trees. Oct-trees can be used in geometric modeling and space planning. A fast algorithm is given for 90° rotation of oct-tree representations of objects. A space-efficient algorithm is given for translation in space. A PASCAL program for experimenting with oct-trees is described.

Optimal packing and covering in the plane are NP-complete☆

This paper was motivated by a practical problem related to databases for image processing: given a set of points in the plane, find an efficient covering of that set using identical fured-size rectangles with sides parallel to the coordinate system [ 111. Also, the problem of packing as many square modules as possible into an irregularly-shaped region on a silicon chip was an additional motivation. In one dimension many packing and covering problems for sets of arbitrary objects are NP-complete [ 11, but when restricted to using identical objects they become trivial [S]. We prove that even severely restricted instances of packing and covering problems remain NP-hard in two or more dimensions. We shall recast these as combinatorial problems through the device of the intersection graph. In one dimension if the objects are intervals then their intersection graphs are intervalgraphs ([2,7,10]). Since any graph is the intersection graph of convex objects [ 131 in three or more dimensions the computational results for arbitrary graphs apply to intersection graph problems in those dimensions. There are comparatively few computational results for intersection graphs in two dimensions (see [3,6] ) although they have been studied [S]. Our results help to fii the gap by showing that some very constrained

VIVA: A visual language for image processing

Visual languages have been developed to help new programmers express algorithms easily. They also help to make experienced programmers more productive by simplifying the organization of a program through the use of visual representations. However, visual languages have not reached their full potential because of several problems including the following: difficulty of producing visual representations for the more abstract computing constructs; the lack of adequate computing power to update the visual representations in response to user actions; the immaturity of the subfield of visual programming and need for additional breakthroughs and standardization of existing mechanisms. Visualization of Vision Algorithms (VIVA) is a proposed visual language for image processing. Its main purpose is to serve as an effective teaching tool for students of image processing. Its design also takes account of several secondary goals, including the completion of a software platform for research in human/image interaction, the creation of a vehicle for studying algorithms and architectures for parallel image processing, and the establishment of a presentation medium for image-processing algorithms.

Template matching in pyramids

Abstract Correlation of a small “template” array with a large image array is an operation commonly used in image analysis. Edge detection, line and corner finding are some applications of the technique. Two problems with template matching are its computational cost and its sensitivity to noise. Pyramids (image hierarchies incorporating variable resolution) allow template matching to be performed in a new manner. Hierarchical template matching allows both a savings in computation time (by a problem-dependent amount) and a considerable degree of insensitivity to noise. These techniques are introduced and analyzed through a series of simple and empirical examples. An important feature of template matching locations that would be difficult to enforce and evene to express, in the traditional nonhierarchical framework. The hierarchical approach admits a large variety of image-processing operations.

A pyramidal approach to parallel processing

This paper presents the architecture of a parallel computer called a pyramid machine. The system consists of a pyramidal array of processing elements, each of which executes the instructions broadcast by a controller. Each processing element except those on the outside of the array is directly connected to thirteen neighboring elements: eight on the same level, four on the next finer level and one on the next coarser level. The architecture combines features of tree machines and features of mesh-connected parallel computers. As a result it is able to rapidly perform computations of local and global processing. The main areas of application are image processing, graphics and spatial problem solving. The motivation, basic structure, and applications of the system are discussed.

Quad-Trees, Oct-Trees, and K-Trees: A Generalized Approach to Recursive Decomposition of Euclidean Space

K-trees are developed as a K-dimensional analog of quad-trees and oct-trees. K-trees can be used for modeling K-dimensional data. A fast algorithm is given for finding the boundary size of a K-dimensional object represented by a K-tree. For K considered as con-stant; the algorithm provides a method for computing the perimeter of a quad-tree encoded image or the surface area of an oct-tree encoded object in worst case time proportional to the number of nodes in the tree. This improves upon the expected-case linear-time method of Samet [10] for the perimeter problem. Our method has been implemented in Pascal, and a computational example is given.

Image transmission with gross information first

When interactive raster graphics is done between a raster display terminal and a main computer over a low-bandwidth transmission line, the rate of interaction can be frustrating for a user. The problem is greatly eased by transmitting rough approximations of images before the details, thereby providing immediate feedback to the user. There are various schemes for doing this. Each requires some overhead in extra computation, and most require increased time to complete image transmission. Three methods based on pyramid data structures are presented.

A hierarchical cellular logic for pyramid computers

Hierarchical structure occurs in biological vision systems and there is good reason to incorporate it into a model of computation for processing binary images. A mathematical formalism is presented which can describe a wide variety of operations useful in image processing and graphics. The formalism allows for two kinds of simple transformations on the values (called pyramids) of a set of cells called a hierarchical domain: the first are binary operations on boolean values, and the second are neighborhood-matching operations. The implied model of computation is more structured than previously discussed pyramidal models, and is more readily realized in parallel hardware, while it remains sufficiently rich to provide efficient solutions to a wide variety of problems. The model has a simplicity which is due to the restricted nature of the operations and the implied synchronization across the hierarchical domain. A corresponding algebraic simplicity in the logic makes possible the concise representation of many cellular-data operations.

Multiplayer activities that develop mathematical coordination

Four computer applications are presented that encourage students to develop “mathematical coordination” --- the ability to manipulate numerical variables in cooperation with other students so as to achieve a definite goal. The programs enable a form of computer-supported cooperative learning (CSCL). In this paper we describe the rationale and design of the programs, the results of an informal evaluation, and possible future work. The games were developed using a special software and hardware environment that facilitates the rapid prototyping of computerbased cooperative-learning materials. This research is part of an ongoing project entitled “Mathematics Experiences Through Image Processing” whose objective is to develop and test educational materials that introduce K-12 students to mathematical ideas within the context of digital image processing activities.

A perspective on the evolution of live programming

Liveness in programming environments generally refers to the ability to modify a running program. Liveness is one form of a more general class of behaviors by a programming environment that provide information to programmers about what they are constructing. This paper gives a brief historical perspective on liveness and proposes an extension of a hierarchy given in 1990, to now account for even more powerful execution-oriented tools for programmers. In addition, while liveness concerns the timeliness of execution feedback, considering a broader array of forms of feedback is helpful both in better understanding liveness and in designing ever more powerful development tools.