Convex Minimization on a Grid and Applications

Mirolo, Claudio

doi:10.1006/jagm.1997.0908

Cited by 7 publications

(12 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our approach builds upon the work done in [3], but with a few important extensions. The principal refinements include: the local characterization of the polyhedral graph of the convex function; the adoption of a new polygonal cell-subdivision with cell-shift operations, instead of an isothetic grid, as a discrete minimization structure; the incremental techniques aimed at controlling the focus of the minimization relative to a previous solution.…”

Section: Grounds Of the Approachmentioning

confidence: 99%

“…Moreover, the combinatorially complex algorithms dealing with polygon-drum and drum-drum pairs, while essential with an isothetic grid, can now be replaced by simpler tools. Thus, after briefly summarizing, in the next subsection, the key results from [3], we will then proceed by analyzing some useful relationships between the configurations of pairs of planar sections and the linearity properties of the convex function. Indeed, a deeper understanding of such relationships is at the root of the major improvements of the minimization process.…”

Section: Grounds Of the Approachmentioning

confidence: 99%

“…Thus the next question is how to transform a search problem on a continuous domain into a search problem on a discrete domain. Following [3], such a discrete domain can be defined as the set of rectangles of an isothetic grid. Then, under reasonable assumptions, the grid rectangle containing the point of minimum can be found in O(log k) minimization steps in the average, where k is the size of the grid.…”

Section: A Collision Translations and Convex Minimizationmentioning

confidence: 99%

“…A first step along the line of research investigating the power of our novel approach was the algorithm analyzed in [3], in regard to which the present work introduces a few major technical improvements as well as provides a richer collection of data as a basis for the assessment. More specifically, the main original contributions of this paper are:…”

Section: Introductionmentioning

confidence: 99%

“…The deeper connection of the minimization process with the shape of the convex function, the tools developed to solve the related subproblems and the significant refinement (ii) make the new algorithm different in several important respects from that outlined in [3].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Incremental Convex Minimization for Computing Collision Translations of Convex Polyhedra

2007

View full text Add to dashboard Cite

Abstract-The subject of this paper is an asymptotically fast and incremental algorithm for computing collision translations of convex polyhedra, where the problem at hand is reduced to determining collision translations of pairs of planar sections and minimizing a bivariate convex function. There are two main reasons, in our view, why the algorithm is worth consideration. On the one hand, the addressed proximity measure, namely collision translation, is not as widely studied as distance. On the other, its peculiar computation strategy may be interesting in itself, being well suited to work without initialization and also endowed with an inherently embedded mechanism to exploit spatial coherence. After outlining the main ideas of this novel approach and providing an estimation of the computational costs, we summarize a broad set of numerical experiments meant to explore extensively the behavior of the algorithm, both without and with initialization. Finally, in order to assess the efficacy and the potential of the approach under analysis, the attained performances are contrasted with those of other popular algorithms designed to compute distances between polyhedra. A thorough comparison of the reported query times and, more significantly, of the corresponding trends shows that the behavior of the collision translation algorithm is quite interesting, especially when used without initialization or under variable coherence, which should encourage further work on this approach.

show abstract

Section: Grounds Of the Approachmentioning

confidence: 99%

Section: Grounds Of the Approachmentioning

confidence: 99%

Section: A Collision Translations and Convex Minimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations