Non-commutative morphology: Shapes, filters, and convolutions

Lysenko, Mikola; Shapiro, Vadim; Nelaturi, Saigopal

doi:10.1016/j.cagd.2011.06.008

Cited by 6 publications

(9 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The inner product in (27) can be viewed as a 6D noncommutative convolution over SE(3) [24]. For numerical tractability, we decompose the motion into rotational and translational parts, and view the latter as a 3D commutative convolution:…”

Section: Discrete Correlationsmentioning

confidence: 99%

“…, respectively, their Fourier correlation is obtained by substituting (24) in (45), or directly applying CFT to (44) aŝ…”

Section: Fourier Correlationsmentioning

confidence: 99%

“…Roerdink [35] generalized the concept of Minkowski sums/differences to Minkowski products/quotients over general groups, whose noncommutative convolutional formulation was presented by Lysenko et al [24]. Nelaturi and Shapiro [26] applied the concept to SE(3) (in this context referred to as 'configuration products/quotients') and showed its applicability to direct and inverse C−space problems ranging from computing general sweeps to solving for maximal shapes and motions subject to containment constraints.…”

Section: Morphological Operationsmentioning

confidence: 99%

“…Analytic modeling relies on describing shape and configuration pointsets as sublevel sets of functions and formulating fundamental operations (e.g., pertaining to detecting collisions, similarity, complementarity, or symmetry) in terms of correlations between those functions. For example, Minkowski operations [35] that are central to mathematical morphology are formalized as convolutions of constituent functions and computed efficiently in the Fourier domain [24]. Minkowski operations have been used extensively to formulate important problems in robot path planning [22], mechanism workspace design [26], virtual reality (graphics/haptics) [5,4], protein docking [1], packaging and nesting [9], and more.…”

Section: Introductionmentioning

confidence: 99%

“…Minkowski operations have been used extensively to formulate important problems in robot path planning [22], mechanism workspace design [26], virtual reality (graphics/haptics) [5,4], protein docking [1], packaging and nesting [9], and more. Unfortunately, their combinatorial computation even for 3D polyhedral objects quickly becomes impractical with increasing number of polygons [24]. This can be alleviated using classical FFTs [11] for numerical implementation of convolutions, which take advantage of uniform spatial sampling.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Analytic methods for geometric modeling via spherical decomposition

Behandish

Ilieş

2016

Computer-Aided Design

View full text Add to dashboard Cite

Analytic methods are emerging in solid and configuration modeling, while providing new insights into a variety of shape and motion related problems by exploiting tools from group morphology, convolution algebras, and harmonic analysis. However, most convolution-based methods have used uniform grid-based sampling to take advantage of the fast Fourier transform (FFT) algorithm. We propose a new paradigm for more efficient computation of analytic correlations that relies on a grid-free discretization of arbitrary shapes as countable unions of balls, in turn described as sublevel sets of summations of smooth radial kernels at adaptively sampled 'knots'. Using a simple geometric lifting trick, we interpret this combination as a convolution of an impulsive skeletal density and primitive kernels with conical support, which faithfully embeds into the convolution formulation of interactions across different objects. Our approach enables fusion of search-efficient combinatorial data structures prevalent in time-critical collision and proximity queries with analytic methods popular in path planning and protein docking, and outperforms uniform grid-based FFT methods by leveraging nonequispaced FFTs. We provide example applications in formulating holonomic collision constraints, shape complementarity metrics, and morphological operations, unified within a single analytic framework.

show abstract

Section: Discrete Correlationsmentioning

confidence: 99%

“…, respectively, their Fourier correlation is obtained by substituting (24) in (45), or directly applying CFT to (44) aŝ…”

Section: Fourier Correlationsmentioning

confidence: 99%