Axis Transform Research Articles

Constructing medial axis transform of extruded and revolved 3D objects with free-form boundaries

This paper presents an algorithm for generating the Medial Axis Transform (MAT) of 3D objects with free form boundaries that are obtained by extrusion along a line or revolution about an axis. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description (to within a defined tolerance) of the MAT. The algorithm uses the 2D MAT of the profile being extruded or revolved to identify the limiting entities (junction points, seams and points of extremal maximum curvature) of the 3D MAT. It is shown that the MAT points of the profile face are sufficient to determine the topology and geometry of the MAT of this class of solids. The algorithm works for multiply-connected objects as well. Results of implementation are presented and use of the algorithm to handle general solids is discussed.

Constructing Medial Axis Transform for Free-form Trimmed Surfaces

Indian Institute of Science, bgm@mecheng.iisc.ernet.in ABSTRACT This paper presents an algorithm for constructing Medial Axis Transform for free-form surfaces. The algorithm uses a combination of tracing and bisector based techniques. This algorithm finds bisector curves by a tracing method and then trims these bisector curves for finding the branch points and constructing the MAT. Results on typical free-form domains are presented. Keywords: Medial Axis Transform, Free-form surface, geodesic. 1. INTRODUCTION One of the fundamental issues in CAD has been to represent objects in a way that not only allows the user to create and modify their shape and other attributes but also enables reasoning about the shape for various tasks in the product design cycle. Objects are normally described by their boundaries, but they can also be described by shifting the description to the interior. This is done by representing the skeleton/symmetric axis of the shape. Various skeleton/symmetric axis representation of shapes like Medial Axis Transform (MAT), Voronoi diagrams, box skeletons, mid surfaces and 2.5D skeletons are possible. Of these MAT has shown to be useful in various applications such as finite element mesh generation, pattern analysis and image analysis, path generation for pocket machining, robot motion planning and more Work on MAT up to now has been restricted to planar 2D and 3D objects. Very little work has been reported on construction of the MAT for 2D free form surface domains. An algorithm to develop MAT on free form surfaces (exact representation) is presented in this paper. 1.1 Medial Axis Transform (MAT) The medial axis transform was first introduced by Blum [1] to describe biological shapes. It can be viewed as the locus of the center of a maximal disk as it rolls inside an object. 1.2 Definition of 2D MAT for Planar Domains The medial axis (MA) of the set D, denoted M(D), is defined as the locus of points inside D which lie at the centers of all closed disks which are maximal in D, together with the limit points of this locus. A closed disk is said to be maximal in a subset D of the 2D space if it is contained in D but is not a proper subset of any other disk contained in D. The radius function of the MA of D is a continuous, real-valued function defined on M(D) whose value at each point on the MA is equal to the radius of the associated maximal disk. The medial axis transform (MAT) of D is the MA along with its associated radius function [8]. Fig. 1. illustrates the MA segments corresponding to a simple convex domain.

Techniques for the calculation of medial surfaces of solids

The Medial Axis Transform (MAT) surface, or simply Medial Surface (MS), offers the possibility of mathematically-based volumetric reasoning about geometric models using global shape and proximity information. The MAT surface is an important tool which can be used for several applications, such as designing manufacturing applications, robot path planning and offsetting, but has the disadvantage that it is computationally intensive. This paper describes two techniques which have been developed to improve the efficiency of the calculation process of the MAT of planar polyhedral objects. The first is a thorough analysis of the cases which can arise when calculating the vertex positions of the MAT of an object as combinations of points, lines and planes. The second technique, presented here, is a divide-and-conquer method, which makes use of some properties of the MAT structure to subdivide the problem. The power of the algorithms is illustrated by some examples, which show the MAT of solid objects with complicated shape and structure.

Any open bounded subset of [formula omitted] has the same homotopy type as its medial axis

Medial axis Transform is sometimes used as an intermediate representation in algorithms for meshing or recognition of shapes from digitized data. This raises the question whether the Medial Axis captures fundamental topological invariants of the object. The (positive) answer has been known already in the case of smooth objects. The main result presented here is the homotopy equivalence of any bounded open subset ofRnwith its medial axis.

Untangling the Blum Medial Axis Transform

For over 30 years, Blum's Medial Axis Transform (MAT) has proven to be an intriguing tool for analyzing and computing with form, but it is one that is notoriously difficult to apply in a robust and stable way. It is well documented how a tiny change to an object's boundary can cause a large change in its MAT. There has also been great difficulty in using the MAT to decompose an object into a hierarchy of parts reflecting the natural parts-hierarchy that we perceive. This paper argues that the underlying cause of these problems is that medial representations embody both the substance of each part of an object and the connections between adjacent parts. A small change in an object's boundary corresponds to a small change in its substance but may involve a large change in its connection information. The problems with Blum's MAT are generated because it does not explicitly represent this dichotomy of information. To use the Blum MAT to it's full potential, this paper presents a method for separating the substance and connection information of an object. This provides a natural parts-hierarchy while eliminating instabilities due to small boundary changes. The method also allows for graded, fuzzy classifications of object parts to match the ambiguity in human perception of many objects.

Spatial Language: Cognitive and Computational Perspectives

Preface K.R. Coventry, P. Olivier. Contributors. 1. Reasoning about Shape using the Tangential Axis Transform or the Shape's `Grain' G. Edwards. 2. A Conceptual Model for Representing Verbal Expressions used in Route Descriptions A. Gryl, et al. 3. Resolving Ambiguous Descriptions through Visual Information I. Duwe, et al. 4. An Anthropomorphic Agent for the Use of Spatial Language T. Joerding, I. Wachsmuth. 5. Gesture, Thought, and Spatial Language K. Emmorey, S. Casey. 6. Organization of Temporal Situations N. Franklin, T. Federico. 7. Grounding Meaning in Visual Knowledge. A Case Study: Dimensional Adjectives A. Goy. 8. Understanding How We Think about Space C. Manning, et al. 9. The Real Story of `Over'? K.R. Coventry, G. Mather. 10. Generating Spatial Descriptions from a Cognitive Point of View R. Porzel, et al. 11. Multiple Frames of Reference in Interpreting Complex Projective Terms C. Eschenbach, et al. 12. Goal-Directed Effects on Processing a Spatial Environment. Indications from Memory and Language H.A. Taylor, S.J. Naylor. 13. Memory for Text and Memory for Space. Two Concurrent Memory Systems? M. Wagener-Wender. Author Index. Subject Index.

Skeletons for Representation and Reasoning in Engineering Applications

Skeletal transforms (of which the Medial Axis Transform (MAT) is the most popular) have been proposed in the literature as a useful shape abstraction tool. This paper develops this proposition with a description of the areas where skeletons (and the MAT in particular) could serve as an enabling representation of part geometry. The paper then addresses two issues that need to be resolved for the application of skeletons in the area of engineering analysis, numerical simulation-based analysis of design in particular. These are the generation of skeletal transforms for complex 2D and 3D geometries, and quadrilateral mesh generation. Generation of the midsurface abstraction of 3D objects (which is a skeleton with MAT like properties) and MAT for 2D closed regions is then described. The paper then presents an algorithm for 2D quadrilateral mesh generation based on the MAT. The paper concludes with a discussion of the outstanding issues in the use of skeletons/MAT in design and engineering applications.

Fast computation of cross-sections of 3D objects from their Medial Axis Transforms

Though Medial Axis Transform (MAT) is well known for object representation, its use in different kinds of computations remains unexplored till date. One of the main reasons is the popularity of other volumetric representations such as Octree for better data compression and ordered storage and retrieval facility, etc. In this paper an algorithm has been proposed based on Medial Axis Transform which allows one to visualize the internal structure of a 3D object across a cross-sectional plane given by its direction cosine vector. This not only enhances the computation speed but also improves the quality of the results compared to those obtained from the computation with direct voxel representation.

Terrain Reconstruction from Contours by Skeleton Construction

Generating terrain models from contour input is still an important process. Most methods have been unsatisfactory, as they either do not preserve the form of minor ridges and valleys, or else they are poor at modeling slopes. A method is described here, based on curve extraction and generalization techniques, that is guaranteed to preserve the topological relationships between curve segments. The skeleton, or Medial Axis Transform, can be extracted from the Voronoi diagram of a well-sampled contour map and used to extract additional points that eliminate cases of “flat triangles” in a triangulation. Elevation estimates may be made at these points. Based on this approach it is possible to make reasonable estimates of slopes for terrain models, and to extract meaningful intermediate points for triangulated irregular networks (TINs).

A Scale-Space Medialness Transform Based on Boundary Concordance Voting

The Concordance-based Medial Axis Transform (CMAT) presented in this paper is a multiscale medial axis (MMA) algorithm that computes the medial response from grey-level boundary measures. This non-linear operator responds only to symmetric structures, overcoming the limitations of linear medial operators which create “side-lobe” responses for symmetric structures and respond to edge structures. In addition, the spatial localisation of the medial axis and the identification of object width is improved in the CMAT algorithm compared with linear algorithms. The robustness of linear medial operators to noise is preserved in our algorithm. The effectiveness of the CMAT is accredited to the concordance property described in this paper. We demonstrate the performance of this method with test figures used by other authors and medical images that are relatively complex in structure. In these complex images the benefit of the improved response of our non-linear operator is clearly visible.

Qualitative sketch optimization

AbstractThe “sketch” drawn by a human designer represents a shape class of wider variability than can be captured by traditional CAD models; these typically work with parametrizations based on a nearly finished shape. Traditional Qualitative Reasoning is also unable to model this degree of ambiguity in shape. Cognitively, shapes are often represented in terms of an axial model. In defining 2D contours, such an axial representation is called the Medial Axis Transform or MAT. By perturbing the parameters of the MAT—length, link angle, and the node radius—one can define a shape class. Unlike the contour-to-MAT transform, which is well-known to be unstable, the MAT-to-contour process is an integrative process and is very stable. The variation in these parameters can be controlled by defining a suitable discretization over the parameter space. This leads to a broad class of similar shapes from which an optimized shape can be obtained for a given set of criteria. The optimizing criteria may involve the boundary description for each shape; the axial model is only used for generating the shape class. This Qualitative MAT model has been tested in several design optimization contexts, using Genetic Algorithms, and we show results for Automobile contours, IC engine parts, building profiles, etc.

Box-skeletons of discrete solids

The Medial Axis Transform (MAT) was defined by Blum in the 1960s as an alternate description of the shape of an object. Since then, its potential applicability in a wide range of engineering domains has been acknowledged. However, this potential has never quite been realized, except recently in two dimensions. One reason is the difficulty in defining algorithms for finding the MAT, especially in three dimensions. Another reason is the lack of incentive for modelling designs directly in MATs.Given this impasse, some lateral thinking appears to be in order. Perhaps the MAT per se is not the only skeleton which can be used. Are there other, more easily derived skeletons, which share those properties of the MAT which are of interest in engineering design?In this work, we identify a set of properties of the MAT which, we argue, are of primary interest. Briefly, these properties are dimensional reduction (in the sense of having no interior), homotopic equivalence, and invertibility. For the restricted class of discrete objects, we define an algorithm for identifying a point set, called a skeleton, which shares these properties with the MAT. Furthermore, this skeleton is to the box-norm (L∞ norm) what the MAT is to the Euclidean norm, and hence the deviation of this skeleton from the MAT is bounded.The algorithm will be developed for both 2D and 3D cases. Proofs of correctness of the algorithm shall be indicated. The use of this skeleton in automated numerical analysis of injection moulded parts shall be demonstrated on industrialsized parts. The use of the 3D skeleton in aiding automatic mesh generation for finite element analysis is also of interest, and shall be discussed.

Abstracts for Solid Modeling Articles in March 1996 TVCG

Abstracts for Solid Modeling Articles in March 1996 TVCG

Machine identification of buds in images of plant shoots

Tissue cultures find increasingly widespread applications for cloning of many plants. Commercial propagation by tissue cultures is limited to ornamental plants, because the cost of skilled labor required cannot compete with conventional propagation methods. To cut down the cost, some automation is essential. A cost-effective approach is to chop the plantlets into segments on a conveying production line while using machine vision for identifying and locating the number and positions of propagation organs in images of the plantlet segments. Plantlet segments without propagation organs will be rejected, while segments with viable buds will be selected for subculturing. To this end, a machine-vision-controlled automatic subculturing system for potato tissue cultures is proposed as a simpler and more cost-effective solution than the popular trend of imitating the manual sub-culturing task by a robot. A simple and relatively fast image-processing algorithm particularly suitable for classification of potato tissue cultures, was developed. In lieu of the general Medial Axis Transform approach, this specialized algorithm takes advantage of the inherent difference between the geometrical shape and gray scale levels of the stems and the leaves as well as of the rather simple connectivity rules of attachment between them. The results indicate that machine inspection and classification of tissue culture plantlets is possible, but considerably more work needs to be done before this technique is fully developed for automating tissue culture processes.

Motion Planning Using Medial Axis

Abstract This paper addresses the problem of planning the motion of a polygonal object through a set of planar obstacles. We propose a two-disk motion planning strategy to navigate the object within the free space between the obstacles from an initial location to a final location. This method makes use of the Medial Axis Transform (MAT) of the free space which can be generated efficiently using the method developed in [11]. We determine two minima] overlapping disks that fully enclose the moving object, and then constrain the centers of the two disks to move continuously aJong a path on the medial axis. In this paper we direct our efforts to the problem of finding the two enclosing disks for a moving object which is considered as a polygon. The problem has been considered as being optimally cutting a polygon into two smaller polygons such that each of smaller polygons can be covered by a minimal disk. We have proved that if the cut is optimal. the resultant minimal disks for two smaller polygons have equal diameters. Based on the geometry of a convex polygon, we formulated the problem as an optimization problem to determine a local optimum for a given edge pair. For a concave polygon, we propose a method to create a hypothetical convex polygon for approximating a given concave polygon, and then perform optimization using the method described above. We have proved that the maximal radii of the disks covering the hypothetical convex polygon generated from a concave polygon is 1.25 times of the radii of the disks that covers the original concave polygon. This shows that the approximation of a concave polygon using a minimal convex polygon does not yield too conservative solution. Simulations are presented for a variety of polygons. The method is being used for disassembly motion planning of a subassembly within its parent subassembly.

Optimal Parallel Algorithms for Region Labeling and Medial Axis Transform of Binary Images

Given a $\sqrt{n} \times \sqrt{n} $ binary image, region labeling labels each 1 of the image so that two 1’s have the same label if and only if they are in the same region (i.e., connected) and medial axis transform finds for each 1 of the image the largest square subimage having it as top-left corner and consisting only of 1’s. Both can be solved in $\theta ( n )$ sequential time. $O( \log n )$ time, $n/\log n$ processor parallel algorithms for both problems in the EREW PRAM are presented.

A quadtree medial axis transform

As printed Quadtree skeletons are exact representations of the image and are used because they are observed to yield space efficiently and a decreased sensitivity to shifts in contrast with the quadtree. The QMAT can be used as the underlying representation when solving most problems that can be solved by using a quadtree. An algorithm is presented for the computation of the QMAT of a given quadtree by only examining each BLACK node's adjacent and abutting neighbors. Corrected Abstract (published as corrigendum in CACM 27, 2 (February 1984) p. 151) The skeletal and medial axis transform concepts used in traditional image processing representations are adapted to the quadtree representation. The result is the definition of of a new data structure termed the Quadtree Medial Axis Transform (QMAT). A QMAT results in a partition of the image into a set of nondisjoint squares having sides whose lengths are sums of powers of 2 rather than, as is the case with quadtrees, a set of disjoint squares having sides of lengths which are powers of 2. The motivation is not to study skeletons for the usual purpose of obtainings approximations of the image. Instead, quadtree skeletons are exact representations of the image and are used because they are observed to yield space efficiency and a decreased sensitvity to shifts in contrast with the quadtree. The QMAT can be used as the underlying representation when solving most problems that can be solved by using a quadtree. An algorithm is presented for the computation of the QMAT of a given quadtree by only examining each BLACK node's adjacent and abutting neighbors. Analysis of the algorithm reveals an average execution time proportional to the complexity of the image, i.e., the number of BLACK blocks.