Non-commutative morphology: Shapes, filters, and convolutions

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.

► Group morphology = typed non-commutative mathematical morphology.
► We introduce Minkowski product, which generalizes Minkowski sum.
► We represent the Minkowski product by sublevel sets of group convolutions.
► One can compute group convolutions by factorization or generalized Fourier transform.
► Applications: configuration space obstacles, sweeping, workspaces, symmetry.