Boundary extraction algorithm for cutting area detection

Abstract

In sculptured surface machining, many machining operations require cutting areas to be expressed in terms of a boundary representation. To extract cutting areas from a sculptured surface, this paper employs a regular grid model (Z-map model) and presents a procedure extracting areas from a Z-map model. The extracted areas may correspond to various machining features, depending on what the Z-map stores, such as curvature (fillet, uncut area), slope (wall, floor) and Z-values (contour). The core of the procedure is a ‘boundary extraction algorithm’ extracting boundaries and identifying the inclusion relationships among the boundaries from a binary image. The time complexity of the boundary extraction algorithm is O(n), where n is the number of runs. In terms of its time complexity and simplicity, the proposed algorithm has advantages over the prior ones. Empirical tests show the performance of the proposed algorithm.

Introduction

NC-data generation for sculptured surface machining is usually handled by NC-programmers with commercial CAM systems. Generally, NC-data generation involves three steps: (1) process planning by NC-programmers, splitting the sculptured surface into several cutting areas according to the machining features and assigning cutting parameters (tool, tool-path topology) to each cutting area; (2) modeling boundary curves to represent the cutting areas in terms of a boundary representation and (3) NC-data generation from a CAM system with the boundary curves and the cutting parameters. Many researchers have studied CAPP (computer-automated process planning) to replace the first and second steps of the NC-data generation; however, there remain many obstacles to be overcome for the implementation of a practical CAPP system [3]. Especially, the second step, modeling boundary curves, has rarely been brought into focus even though it is essential for the implementation of a CAPP system. The fact that no proper solution has been found for the boundary curve modeling serves as a motivation of this paper.

Direct extraction of cutting areas from a sculptured surface is extremely difficult, because of two major difficulties: (1) numerical search operations, computationally expensive and unstable, are required to extract feature curves (wall, floor, fillet, etc.) from a parametric surface and (2) the feature curves have to be merged for the boundary curves by finding sets of connected feature curves, because a sculptured surface may consist of thousands of parametric surfaces.

To avoid the mentioned difficulties, we employ a regular grid model called a Z-map [3] model. This paper presents a procedure extracting cutting areas from a Z-map model, a special form of discrete non-parametric surface model. A Z-map is a 2D-array of real numbers in which Z-values of the surface, sampled at the regular grid-points, are stored. The detailed procedure obtaining a Z-map from a sculptured surface can be found in Choi and Jerard [3]. Owing to its simplicity, a Z-map model is used widely in three-axis NC machining applications [1], [2], [3], [4], [5].

We describe the suggested procedure informally with the input and output of the procedure (Fig. 1). The input includes a Z-map and a value range defining a machining feature, cutting areas. With the input, boundary curves enclosing the cutting areas and the inclusion relationships among the boundary curves are extracted. The machining feature, we can find, depends on what the Z-map stores, such as curvatures, slopes and Z-values. Curvatures and slopes can be calculated by pre-processing each of the grid-points in a Z-map storing Z-values. Consequently, we can obtain cutting areas corresponding to a specific slope range (wall, floor) or curvature range (fillet, uncut area), which may be considered as machining features. If the Z-map represents a CL-surface [1], [3], Z-constant contouring tool-path can be obtained by specifying the height with the value-range (V_Low=height, V_High=infinite).

Most of the prior results [6], [7], [8] are found in computer vision and several algorithms have been suggested for the boundary extraction problem from a binary image. Although those algorithms are not designed to extract cutting areas from a Z-map, they can be applied to the problem, because a Z-map can be easily transformed to a binary image with a value-range. Fig. 2 shows an example of a binary image obtained from a Z-map. If a grid-point belongs to the value-range (V_Low, V_High), then the grid-point is classified as an object-point (black), otherwise a background-point (white).

Freeman [6] proposed the concept of a chain code which is a data structure representing the boundaries by means of eight direction vectors (Fig. 3a). The basic idea of the conventional chain-coding algorithm is to trace the boundary pixels counterclockwise or clockwise and to sequentially generate codes considering the neighborhood relationship between adjacent boundary pixels. Kim et al. [7] proposed a two-step algorithm to compute chain codes from a particular run-length coded representation (Fig. 3b) that preserves connectivity information between runs. They also suggested an algorithm to detect the inclusion relationships between external and internal boundaries. More recently, Zingaretti et al. [8] presented an algorithm to extract boundaries from multi-valued images.

We propose a procedure to extract cutting areas from a Z-map for sculptured surface machining, and the core of the procedure is a ‘boundary extraction algorithm’ to extract boundaries and identify the inclusion relationships among the boundaries from a run-length coded binary image. The main features of the proposed algorithm are as follows: (1) a connectivity-net is devised to store the connectivity information between runs; (2) a recursive function is designed for the efficient construction of the connectivity-net; and (3) a hierarchical boundary extraction scheme is employed to identify inclusion relationships among the boundaries from the connectivity-net. The time complexity of the boundary extraction algorithm is O(n), where n is the number of runs.

The overall structure of the ‘boundary extraction algorithm’ is given in Section 2, followed by the connectivity-net construction in Section 3. Empirical performance tests are given in Section 4. Finally, concluding remarks are given in Section 5.