Markku Tamminen

A general approach to connected-component labeling for arbitrary image representations

An improved and general approach to connected-component labeling of images is presented. The algorithm presented in this paper processes images in predetermined order, which means that the processing order depends only on the image representation scheme and not on specific properties of the image. The algorithm handles a wide variety of image representation schemes (rasters, run lengths, quadrees, bintrees, etc.). How to adapt the standard UNION-FIND algorithm to permit reuse of temporary labels is shown. This is done using a technique called age balancing, in which, when two labels are merged, the older label becomes the father of the younger label. This technique can be made to coexist with the more conventional rule of weight balancing, in which the label with more descendants becomes the father of the label with fewer descendants. Various image scanning orders are examined and classified. It is also shown that when the algorithm is specialized to a pixel array scanned in raster order, the total processing time is linear in the number of pixels. The linear-time processing time follows from a special property of the UNION-FIND algorithm, which may be of independent interest. This property states that under certain restrictions on the input, UNION-FIND runs in time linear in the number of FIND and UNION operations. Under these restrictions, linear-time performance can be achieved without resorting to the more complicated Gabow-Tarjan algorithm for disjoint set union.

Efficient component labeling of images of arbitrary dimension represented by linear bintrees

An algorithm is presented to perform connected-component labeling of images of arbitrary dimension that are represented by a linear bintree. The bintree is a generalization of the quadtree data structure that enables dealing with images of arbitrary dimension. The linear bintree is a pointerless representation. The algorithm uses an active border which is represented by linked lists instead of arrays. This results in a significant reduction in the space requirements, thereby making it feasible to process three- and higher-dimensional images. Analysis of the execution time of the algorithm shows almost linear behavior with respect to the number of leaf nodes in the image, and empirical tests are in agreement. The algorithm can be modified easily to compute a (d-1)-dimensional boundary measure (e.g. perimeter in two dimensions and surface area in three dimensions) with linear performance. >

Computing Geometric Properties of Images Represented by Linear Quadtrees

The region quadtree is a hierarchical data structure that finds use in applications such as image processing, computer graphics, pattern recognition, robotics, and cartography. In order to save space, a number of pointerless quadtree representations (termed linear quadtrees) have been proposed. One representation maintains the nodes in a list ordered according to a preorder traversal of the quadtree. Using such an image representation and a graph definition of a quadtree, a general algorithm to compute geometric image properties such as the perimeter, the Euler number, and the connected components of an image is developed and analyzed. The algorithm differs from the conventional approaches to images represented by quadtrees in that it does not make use of neighbor finding methods that require the location of a nearest common ancestor. Instead, it makes use of a staircase-like data structure to represent the blocks that have been already processed. The worst-case execution time of the algorithm, when used to compute the perimeter, is proportional to the number of leaf nodes in the quadtree, which is optimal. For an image of size 2n × 2n, the perimeter algorithm requires only four arrays of 2n positions each for working storage. This makes it well suited to processing linear quadtrees residing in secondary storage. Implementation experience has confirmed its superiority to existing approaches to computing geometric properties for images represented by quadtrees.

The EXCELL Method for Efficient Geometric Access to Data

The extendible cell (EXCELL) method provides a data structure for efficient geometric access. It stores geometric data into computer storage blocksm corresponding to disjoint variable sized rectangular cells accessible by an address calculation type directory. We describe the method for point files and files of more complicated figures analyzing performance. We report algorithms for the nearest neighbour and point-in-polygon-network problems and describe applications to geographical data bases, hidden line elimination and geometric modeling.

Efficient octree conversion by connectivity labeling

We present an algorithm for converting from the boundary representation of a solid to the corresponding octree model. The algorithm utilizes an efficient new connected components labeling technique. A novelty of the method is the demonstration that all processing can be performed directly on linear quad and octree encodings. We illustrate the use of the algorithm by an application to geometric mine modeling and verify its performance by analysis and practical experiments.

Localized set operations for solid modeling

Set operation algorithms form an important component of solid modeling systems. Their efficiency can be enhanced by localizing the search for geometric intersections to the region of overlap using a spatial directory. We present an algorithm that employs a three-dimensional extendible cell (EXCELL) directory to the set operation problem, and demonstrate by practical experiments the efficiency and the local nature of the algorithm.

The extendible cell method for closest point problems

The extendible cell method is an application of order preserving extendible hashing to multidimensional point files. We derive some of its performance characteristics and show its expected case optimality for closest point problems.

A Hierarchical Global Wiring Algorithm for Custom Chip Design

We present a global wiring algorithm used in a top-down physical design environment, i.e., macros are laid out after global wiring is done, and wires are allowed to pass through macros (the wiring-through model). The floorplan of the chip is in the form of a slicing structure. The algorithm is based on a hierarchical scheme. The final result is obtained through a series of refinement as the problem is recursively decomposed into a set of small-sized problems and then solved efficiently. The worst-case run-time for an arbitrary slicing tree (totally skewed) is O(M /sup 2/ N). When the floorplan is represented by a balanced slicing tree, the run-time of the overall algorithm is O(MN), where M is the number of macros and N the number of nets. The algorithm has been implemented in the C language and is used for chip designs. Experiments on both real and randomly generated designs show that the hierarchical router performs equally well as a flat global router in terms of wire length and wireability handling, but much faster in run-time (at least 10 times for an example with 100 macros and 1000 nets, and the gap being even larger for bigger-size problems).

Encoding pixel trees

Abstract Linear representations of quad-trees and other pixel trees are studied for efficient encoding of both black-and-white and multicolor images. Based on a random-line model it is shown that asymptotically, as resolution is increased with respect to image detail, a multicolor image can be represented with less than 5 bits per intersected pixel. This can be compared to the 2–3 bits required by chain encoding schemes. It is also shown that binary pixel trees support linear encoding with the same or better efficiency than the traditional quad-tree. It is shown how geometric probability can help in the understanding of the behavior of image encoding schemes.

Ray-casting and block model conversion using a spatial index

Abstract A spatial index is a data structure designed to facilitate spatial search, exemplified by the point-in-polyhedron inclusion problem. The 3D extensible cell (EXCELL) index is presented together with algorithms for spatial search and for converting a complex polyhedron (boundary representation) into an octree-like block model. We illustrate the techniques by an application to geometric mine modelling and demonstrate the efficiency of the approach by practical experiments.