Harvey H. Atkinson

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.

Improvements to a recent 3d-border algorithm

Abstract A linear oct-tree can be mapped into its three-dimensional border-representation by the repeated elimination of the internal boundaries existing between any two nodes. This paper shows that such a mapping can be efficiently implemented by a recursive algorithm in a single pass through the input data. Among the advantages of the procedure presented here with respect to the algorithm previously introduced by the authors are the elimination of costly dynamic insertions and deletions, the reduction of memory requirements and the search of smaller arrays.

Counting regions, holes, and their nesting level in time proportional to the border

Given a 2-dimensional black-and-white digital picture it is sometimes of interest to determine all the black components (together with the number of white holes in each one) and evaluate their level of nesting. Algorithms for solving this problem appear in the literature, but without time-complexity analyses. The algorithm proposed in this paper separates all maximally connected subsets and counts holes, regions, and nesting levels in time proportional to the number of components multiplied by the number of border pixels. The efficiency of this algorithm is due to the use of linear quadtrees to represent pictures. This paper also contains an upper bound for the size of a linear quadtree in terms of the number of border pixels of the picture it represents, and a proof that the bound is sharp to within a factor of two. The bound is then used to analyze the above-mentioned algorithm and could also be used to allocate static memory for storing linear quadtrees. This work was originally motivated by the purpose of determining the age of a fish scale by counting its growth zones, which have the structure of (roughly) concentric annular regions: in a digitized, segmented picture thee regions appear most often as connected subsets of black pixels on a white background.

Ray Tracing an Octree: Numerical Evaluation of the First Intersection

An algorithm is presented which finds the first intersection of a directed semi‐infinite straight‐line (called ray) with an octree, without resorting to the evaluation of neighbouring nodes. Given a pointer‐based region octree, intersections of the ray with a nodes bisecting planes are first evaluated to determine in which sub‐octants the ray‐node intersections may lie; a local ordering then determines the sequence in which these sub‐octants should be examined so that the intersection closest to the rays origin can be selected.

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.

Three-dimensional modeling by combining artificial with real data

Issues arising in the graphics simulation of bone grafting and in the use of commercial libraries to create custom-designed models revolve on the ability to combine data defined in Volume Representation--as that generated by Computed Tomography scans--and user-defined data introduced in Boundary Representation--as vertices of polygons, for instance. To address these issues the authors propose a hybrid filling technique, which also processes conflicting adjacency information created by subsampling or digitization errors. A graphics experiment has been designed to illustrate the problems arising in the presence of objects defined partially in Volume and partially in Boundary Representation, and a method to handle them.