Computer Aided Geometric Design 28 (2011) 497–522 Contents lists available at ScienceDirect Computer Aided Geometric Design www.elsevier.com/locate/cagd Non-commutative morphology: Shapes, filters, and convolutions Mikola Lysenko ∗ , Vadim Shapiro, Saigopal Nelaturi Spatial Automation Lab, Department of Mechanical Engineering, University of Wisconsin–Madison, 1513 University Avenue, Madison, WI 53706-1572, United States article info abstract Article history: Available online 28 July 2011 Keywords: Minkowski sum Group morphology Mathematical morphology Configuration space Group morphology is a generalization of mathematical morphology which makes an explicit distinction between shapes and filters. Shapes are modeled as point sets, for example binary images or 3D solid objects, while filters are collections of transformations (such as translations, rotations or scalings). The action of a filter on a shape generalizes the basic morphological operations of dilation and erosion. This shift in perspective allows us to compose filters independent of shapes, and leads to a non-commutative generalization of the Minkowski sum and difference which we call the Minkowski product and quotient respectively. We show that these operators are useful for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. To compute these new operators, we propose the use of group convolution algebras, which extend classical convolution and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function. Published by Elsevier B.V. 1. Introduction 1.1. Motivation Representation and analysis of shapes using mathematical morphology, pioneered by Matheron and Serra (1983), has found numerous applications in geometric computing, from image processing to solid modeling. In mathematical morphol- ogy, shapes are modeled by subsets of the Euclidean plane and are analyzed using morphological operators constructed from dilation, erosion and set operations. Alternatively, one can choose Minkowski addition and subtraction as basic operations in mathematical morphology, and this provides the connection to robotics and solid modeling applications where Minkowski operations have been used to formulate a variety of computations and applications – including intersection computation (Schwartz, 1981), motion planning (Lozano-Perez, 1983), blending (Rossignac and Requicha, 1986), packaging (Middleditch, 1988), design (Boissonnat et al., 1997; Caine, 1994), tolerances (Suresh and Voelcker, 1994) and numerous other applications involving relative motions/configurations. Yet for all their ubiquity and versatility, mathematical morphology and Minkowski operations have not been widely adopted in solid modeling systems and applications. There are several factors which contribute to this situation. First, it is not obvious how to extend the standard morphological operators to more general motions and transforma- tions in solid modeling. The tools of configuration space modeling (Lozano-Perez, 1983; Nelaturi and Shapiro, 2009; Roerdink, 1990) partially address these issues for rigid motions, but they do not deal with many other types of trans- formations, such as scaling or shearing. These deficiencies are further compounded by a second problem in that few * Corresponding author. E-mail addresses: mikolalysenko@gmail.com (M. Lysenko), vshapiro@engr.wisc.edu (V. Shapiro), saigopan@cae.wisc.edu (S. Nelaturi). 0167-8396/$ – see front matter Published by Elsevier B.V. doi:10.1016/j.cagd.2011.06.008