Industrial geometry: recent advances and applications in CAD

Abstract

Industrial Geometry aims at unifying existing and developing new methods and algorithms for a variety of application areas with a strong geometric component. These include CAD, CAM, Geometric Modelling, Robotics, Computer Vision and Image Processing, Computer Graphics and Scientific Visualization. In this paper, Industrial Geometry is illustrated via the fruitful interplay of the areas indicated above in the context of novel solutions of CAD related, geometric optimization problems involving distance functions: approximation with general B-spline curves and surfaces or with subdivision surfaces, approximation with special surfaces for applications in architecture or manufacturing, approximate conversion from implicit to parametric (NURBS) representation, and registration problems for industrial inspection and 3D model generation from measurement data. Moreover, we describe a ‘feature sensitive’ metric on surfaces, whose definition relies on the concept of an image manifold, introduced into Computer Vision and Image Processing by Kimmel, Malladi and Sochen. This metric is sensitive to features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. We illustrate its applications at hand of feature sensitive curve design on surfaces and local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.

Introduction

During the past decades, geometric methods have played an increasingly important role in a variety of areas dealing with computing for industrial applications; these include Computer-Aided Design and Manufacturing, Geometric Modeling, Computational Geometry, Robotics, Computer Vision, Pattern Recognition and Image Processing, Computer Graphics and Scientific Visualization.

These areas originated from different requirements in specific applications and thus they have seen rather disjunct developments. In fact, very similar problems have been treated by different communities. These communities still have different favorite solutions to nearly the same problems. Let us illustrate this at hand of curve approximation. According to industry standards, the CAD approach uses B-spline curves and a method for data fitting which iterates between parameter estimation and linear least squares approximation [11], [34]. Computer Vision and Image Processing developed another method, active contours [4], [12], which have originally been formulated as parametric curves. Nowadays, the advantages of (discretized) implicit representations and the formulation of the curve evolution via partial differential equations in the level set method [19], [30] are highly appreciated, in particular, for difficult curve approximation problems which arise in image segmentation. Curve approximation also appears in higher dimensional spaces. For example, in the space of rigid body motions, it leads to motion design for Robotics [17] or Computer Animation.

In recent years, these different areas of research have started to become increasingly interconnected, and have even begun to merge. A driving force in this process is the increasing complexity of applications, where one field of research alone would be insufficient to achieve useful results. Novel technologies for acquisition and processing of data lead to new and increasingly challenging problems, whose solutions require the combination of techniques from different branches of applied geometry. The thereby emerging research area, which aims at unifying existing and developing new methods and algorithms for a variety of application areas with a strong geometric component, shall be called Industrial Geometry.

Let us continue the example from curve approximation addressed above. The viewpoint of Industrial Geometry would be to investigate the various algorithms from a common perspective. Since all the available algorithms are solving non-linear geometric optimization problems, it is appropriate to study and compare the known approaches from the optimization perspective. In the present paper, we will point to recent results in this direction.

It is impossible to outline all major current research streams in Industrial Geometry in this paper. Therefore, we will focus just on a few topics. We will briefly look at the level set method [19], [31] and on hybrid data structures for geometric computing [15]. The major part of this paper is devoted to geometric optimization problems which involve distance functions. Here we will present a survey with some new results on a recently developed class of optimization algorithms, which can be called squared distance minimization. The benefits of the optimization viewpoint rather than the perspective of a specific application will become obvious. With nearly the same algorithms we can solve a wide class of curve and surface approximation problems and a number of registration (surface matching) problems.

The methods we are using for the topics indicated so far have a relation to Computer Vision and Image Processing. As a further example for the fruitful use of techniques which originate in these fields, we discuss a new metric on surfaces. It is sensitive to features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. This new metric can be easily understood with the concept of an image manifold [14], and it has a number of interesting applications [25]. For example, we can design curves on surfaces whose shape is adapted to the features of the surface. Moreover, we briefly address local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects. Image processing frequently uses mathematical morphology for basic topological and geometric operations [10], [30]; this work describes similar operations on surfaces, which—if desired—can be made sensitive to the features.