Irene Gargantini

An effective way to represent quadtrees

A quadtree may be represented without pointers by encoding each black node with a quaternary integer whose digits reflect successive quadrant subdivisions. We refer to the sorted array of black nodes as the “linear quadtree” and show that it introduces a saving of at least 66 percent of the computer storage required by regular quadtrees. Some algorithms using linear quadtrees are presented, namely, (<italic>i</italic>) encoding a pixel from a 2<supscrpt><italic>n</italic></supscrpt> × 2<supscrpt><italic>>n</italic></supscrpt> array (or screen) into its quaternary code; (<italic>ii</italic>) finding adjacent nodes; (<italic>iii</italic>) determining the color of a node; (<italic>iv</italic>) superposing two images. It is shown that algorithms (<italic>i</italic>)-(<italic>iii</italic>) can be executed in logarithmic time, while superposition can be carried out in linear time with respect to the total number of black nodes. The paper also shows that the dynamic capability of a quadtree can be effectively simulated.

Circular arithmetic and the determination of polynomial zeros

SummarySuppose all zeros of a polynomialp but one are known to lie in specified circular regions, and the value of the logarithmic derivativep′p−1 is known at a point. What can be said about the location of the remaining zero? This question is answered in the present paper, as well as its generalization where several zeros are missing and the values of some derivatives of the logarithmic derivative are known. A connection with a classical result due to Laguerre is established, and an application to the problem of locating zeros of certain transcendental functions is given. The results are used to construct (i) a version of Newtons method with error bounds, (ii) a cubically convergent algorithm for the simultaneous approximation of all zeros of a polynomial. The algorithms and their theoretical foundation make use of circular arithmetic, an extension, based on the theory of Moebius transformations, of interval arithmetic from the real line to the extended complex plane.

Linear octtrees for fast processing of three-dimensional objects

Abstract A new, effective way of storing octtrees for three-dimensional representation of objects is given. The 10 fields normally required to identify a node of an octtree are reduced to only one. Algorithms are presented for (i) mapping cubic pixels from and to space array (with subscripts I, J, K ), (ii) finding the stereographic projections on the IJ, IK , and JK planes, (iii) performing union (intersection) of two objects centered on the same array, and (iv) finding the pixel adjacent to a given one in a specified direction. The newly proposed data structure is a (dynamically built) array of sorted octal codes which reflects the successive octant subdivisions; it represents a dramatic improvement with respect to octtrees when space complexity is considered. Also, the formulation of the procedures mentioned above takes advantage of this “natural” structure and results in very simple algorithms, easy to code and optimize. Some of the proposed procedures could also be implemented in parallel mode.

Parallel laguerre iterations: The complex case

SummaryConsider a polynomialP(z) of degreeN whose zeros are known to lie insideN closed disks, each disk containing one and only one root. In this paper we show that if the given disks are “sufficiently well separated”, then the first derivative ofP(z) never vanishes inside the initial inclusion regions. The formulation of the square-root iteration in terms of circular regions is then possible and leads to an iterative scheme with degree four convergence. The corresponding algorithm makes use of circular arithmetic and in particular of the definition of square root of a disk. A criterion for the selection of the appropriate square-root set is also given. The procedure can be used to simultaneously refine all (complex or real) roots ofP(z) together with their error bounds.

Translation, rotation and superposition of linear quadtrees

Abstract In Gargantini (1982a) it has been shown that storing black nodes of a quadtree is sufficient to retrieve any basic property associated with quadtrees. To achieve this, each black node must be represented as a quaternary integer whose digits (from left to right) describe the path from the root to that node. The sorted sequence of quaternary integers representing a given region is called the linear quadtree associated with that region. Such a structure has been shown to save more than two-thirds of the memory locations used by regular quadtrees. In this paper we present procedures for translating and rotating a region and consider the superposition of binary images with different characteristics (such as different resolution parameter, different pixel size and/or different center). Translation, rotation, and superposition are shown to be O(N log N) operations; for translation N is the number of black pixels; for rotation N is the number of black nodes; for superposition N is the sum of black nodes or black pixels of the two images, depending on whether or not the two regions are centered on the same raster.

Viewing Transformations of Voxel-Based Objects Via Linear Octrees

This article gives three viewing-transformation algorithms for displaying on a screen 3D pictures represented by linear octrees. All the procedures take advantage of the recursive labeling used to identify the successive decomposition of an object into octants. The first algorithm performs transformations directly on the linear octree, while the second and third algorithms determine the 3D border of the given object first and then project onto the screen the surface voxels thus found. All the algorithms perform the viewing transformations in O(RN) time, where R is the resolution of the picture and N is the number of elements in the linear octree. One of the algorithms provides views of the object at different layers of gray level, while another allows internal views.

Filling by quadrants or octants

Abstract Filling by quadrants or by octants is shown to be executable in time proportional to the lenght of the border multiplied by n, the logarithm of the diameter of the image. The underlying data structure is the linear quadtree in two dimensions or the linear octtree in three dimensions. The input is the border to be filled while the output is the linear quad or octtree representing the filled region(s). The latter can be a set of connected or disjoint black blocks. The basic idea behind the algorithm is to allow the region to grow “inwards” while restraining its growth “outwards” by the use of the block-bits technique introduced by the authors in a previous paper. The new features introduced by this paper are: (i) the low worst-case time complexity, as compared with previous algorithms, (ii) the fact that the basic space requirements consist of the input, output and 4n or 8n pointers, and (iii) its 3D implementation. The last capability has been developed for medical imaging purposes and 3D modelling.

Determination of the 3D border by repeated elimination of internal surfaces

A novel approach to the 3D border determination is presented: it starts by representing the 3D object in linear octtree form, proceeds by eliminating internal boundaries between nodes of the same size while deleting internal nodes and terminates when only border voxels remain. The algorithm basically performs a mapping of the 3D object into its own border, with both input and output being represented as linear octtrees. The algorithm is shown to be executable inO(kn(N+M)) time, wherek andN are the maximum node grouping and number of nodes (respectively) of the initial linear octtree,n is the resolution of the bilevel image andM is the number of border voxels. The range of applicability of the proposed algorithm is quite wide: it can determine the external border of a simply connected region as well as the external and internal borders of a set of multiply connected objects, all at the same time.ZusammenfassungEine neue Methode zur 3D-Objektgrenzen-Bestimmung wird präsentiert: sie beginnt mit der 3D-Objektdarstellung in „linear octtree form”, gefolgt von einer Eliminierung interner Grenzen zwischen Knoten („nodes”) gleicher Größe und einer Löschung interner Knoten. Das Verfahren endet mit der Feststellung, daß nur mehr „border voxels” vorliegen. Der Algorithmus führt grundsätzlich eine Abbildung eines 3D-Objektes in seinen eigenen Grenzen durch, wobei es sich sowohl bei „input” als auch bei „output” um „linear octtrees” handelt. Es wird gezeigt, daß die Exekutionszeit des Algorithmus von der Ordnungkn(N+M) ist, worink die maximale Knotenzusammenfassung undN die Anzahl der Knoten des ursprünglichen „octtrees” bedeuten.n ist die Genauigkeit des Binärbildes („bilevel image”) undM ist die Anzahl der „border voxels”. Der Anwendungsbereich des vorgeschlagenen Algorithmus ist ziemlich groß: er bestimmt die äußeren Grenzen eines einfach zusammenhängenden Gebietes genauso wie die äußeren und inneren Grenzen eines Satzes mehrfach zusammenhängender Objekte.

Linear quadtrees: a blocking technique for contour filling

Abstract Given a linear quadtree forming a regions contour, an algorithm is presented to determine all the pixels 4-connected to the borders elements. The procedure, based on a connectivity technique, associates a two-valued state (“blocked” or “unblocked”) with each node and fills increasingly larger quadrants with black nodes whose state is known to be unblocked. Advantages of the proposed procedure over existing ones are: (i) multiply connected regions can be reconstructed; (ii) the border can be given as a set of either 4- or 8-connected pixels.

Detection of connectivity for regions represented by linear quadtrees

An algorithm is given for the determination of all connected components of a given region. The data structure used to represent the region (or image) is the linear quadtree, a recently introduced structure which basically consists of a sorted array of quaternary integers. A linear quadtree is built in a way similar to that of a regular quadtree but contains, as elements, only black nodes. The proposed algorithm has computational complexity proportional to 12nN, where n is the resolution parameter of the image, and N is the number of black nodes. Space complexity is proportional to 6N. The algorithm is shown to compare favourably, both in terms of space and time, with the existing method due to H. Samet.