General Sweep Mathematical Morphology

The sweep operation to generate a new object by sweeping an object along a space curve trajectory provides a natural design tool in solid modeling. The simplest sweep is linear extrusion defined by a two-dimensional (2D) area swept along a linear path normal to the plane of the area to create a volume. Another simple sweep is rotational sweep defined by rotating a 2D object about an axis. Though simple, these two sweeps are often seen in real applications. Sweeps that generate area or volume changes in size, shape, or orientation during the sweeping process, and follow an arbitrarily curved trajectory, are called general sweeps (Requicha, 1980). General sweeps of solids are useful in modeling the region swept out by a machine-tool cutting head or a robot following a path. General sweeps of 2D cross sections are known as generalized cylinders in computer vision, and are usually modeled as parameterized 2D cross sections swept at right angles along an arbitrary curve. Being the simplest of general sweeps, generalized cylinders are somewhat easy to compute. However, general sweeps of solids are difficult to compute since the trajectory and object shape may make the sweep object self-intersect (Foley et al., 1995). Mathematical morphology involves the geometric analysis of shapes and textures in images. Appropriately used, mathematical morphological operations tend to simplify image data presenting their essential shape characteristics and eliminating irrelevancies (Haralick et al., 1987; Serra, 1982; Shih and Mitchell, 1989, 1992). As the object recognition, feature extraction, and defect detection correlate directly with shape, it becomes apparent that mathematical morphology is the natural processing approach to deal with the machine vision recognition process and the visually guided robot problem. The mathematical morphological operations can be thought of working with two images. Conceptually, the image being processed is referred to as the active image and the other image being a kernel is referred to as the structuring element. Each structuring element has a designed shape, which can be thought of as a probe or filter of the active image. We can modify the active image by probing it with various structuring elements. The two fundamental mathematical morphological operations are dilation and erosion. Dilation combines two sets using vector addition of set elements. Dilation by disk structuring elements corresponds to isotropic expansion algorithm popular to binary image processing. Dilation by small square ( 3 × 3) is an eight-neighborhood operation that can be easily implemented by adjacently connected array architectures and is the one known by the name "fill," "expand," or "grow." Erosion is the morphological dual to dilation. It combines two sets using vector subtraction of set elements. Some equivalent terms of erosion are "shrink" and "reduce.". The traditional morphological operations perform vector additions or subtractions by a translation of structuring element to the object pixel. They are far from being capable of modeling the swept volumes of structuring elements moving with complex, simultaneous translation, scaling, and rotation in Euclidean space. In this chapter, we developed an approach that adopts sweep morphological operations to study the properties of swept volumes. We present the theoretical framework for representation, computation, and analysis of a new class of general sweep mathematical morphology and its practical applications. Geometric modeling is the foundation for CAD/CAM integration (Pennington et al., 1983). The goal of automated manufacturing inspection and robotic assembly is to generate a complete process automatically. The representation must not only possess the nominal geometric shapes, but also reason the geometric inaccuracies (or tolerances) into the locations and shapes of solid objects. Boundary representation and constructed solid geometry (CSG) representation are popularly used as an internal database (Requicha and Voelcker, 1982; Rossignac, 2002) for geometric modeling. Boundary representation consists of two kinds of information-topological information and geometric information, including vertex coordinates, surface equations, and connectivity between faces, edges, and vertices. There are several advantages in boundary representation: large domain, unambiguity, uniqueness, and explicit representation of faces, edges, and vertices. There are also several disadvantages: verbose data structure, difficulty in creating, difficulty in checking validity, and variational information unavailability. The CSG representation works by constructing a complex part by hierarchically combining simple primitives using Boolean set operations (Mott-Smith and Baer, 1972). There are several advantages of using CSG representation: large domain, unambiguity, easy-to-check validity, and easy creativity. There are also several disadvantages: nonuniqueness, difficulty in editing graphically, input data redundancy, and variational information unavailability (Voelcker and Hunt, 1981). The framework we propose for geometric modeling and representation is sweep mathematical morphology. The sweep operation to generate a volume by sweeping a primitive object along a space curve trajectory provides a natural design tool. The simplest sweep is linear extrusion defined by a 2D area swept along a linear path normal to the plane of the area to create a volume (Chen et al., 1999). Another sweep is rotational sweep defined by rotating a 2D object about an axis. General sweep is useful in modeling the region swept out by a machine-tool cutting head or a robot following a path (Blackmore et al., 1994). General sweeps of 2D cross sections are known as generalized cylinders in computer vision and are usually modeled as parameterized 2D cross sections swept at right angles along an arbitrary curve. Being the simplest of general sweeps, generalized cylinders are somewhat easy to compute. However, general sweeps of solids are difficult to compute since the trajectory and object shape may make the sweep object self-intersect (Foley et al., 1995). A generalized sweeping method for CSG modeling was developed by Shiroma et al. (1982, 1991) to generate swept morphological operations that tend to simplify image data representing their volume. It is shown that the complex solid shapes can be generated with a blending surface to join two disconnected solids, fillet volumes for rounding corners, and swept volumes formed by the movement of numeric control (NC) tools. Ragothama and Shapiro (1998) presented a B-Rep method for deformation in parametric solid modeling. This chapter is organized as follows. Section II presents the theoretical development of general sweep mathematical morphology along with its properties. Section III describes an application of sweep morphology, which represents the blending of swept surfaces with deformations. Section IV presents the usage of sweep morphology for image enhancement, Section V the edge linking, and Section VI the shortest path planning. Section VII describes modeling based on the sweep mathematical morphology. Section VIII describes the formal languages. Section IX proposes the representation scheme for 2D and three-dimensional (3D) objects. Section X introduces the adopted grammars. Section XI applies the parsing algorithm to determine whether a given object belongs to the language. The conclusions are drawn in Section XII.