Offsetting operations in solid modelling
Computer Aided Geometric Design
Application of vector sum operator
Computer-Aided Design
A Representation Theory for Morphological Image and Signal Processing
IEEE Transactions on Pattern Analysis and Machine Intelligence
A coordinate-free approach to geometric programming
Theory and practice of geometric modeling
Minkowski operations for satellite antenna layout
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Representations for Rigid Solids: Theory, Methods, and Systems
ACM Computing Surveys (CSUR)
A Reflective Symmetry Descriptor
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part II
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
A planar-reflective symmetry transform for 3D shapes
ACM SIGGRAPH 2006 Papers
Partial and approximate symmetry detection for 3D geometry
ACM SIGGRAPH 2006 Papers
Accurate Minkowski sum approximation of polyhedral models
Graphical Models - Special issue on PG2004
Spatial Planning: A Configuration Space Approach
IEEE Transactions on Computers
Discovering structural regularity in 3D geometry
ACM SIGGRAPH 2008 papers
A kinetic framework for computational geometry
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Covering Minkowski sum boundary using points with applications
Computer Aided Geometric Design
Configuration products in geometric modeling
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Group morphology with convolution algebras
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
A sweep and translate algorithm for computing voxelized 3D Minkowski sums on the GPU
Computer-Aided Design
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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.