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Discovering Structural Regularity in 3D Geometry
, 2008
"... We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point or meshbased models. Our method assumes no p ..."
Abstract

Cited by 78 (9 self)
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We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point or meshbased models. Our method assumes no prior knowledge of the geometry or spatial location of the individual elements that define the pattern. Structure discovery is made possible by a careful analysis of pairwise similarity transformations that reveals prominent lattice structures in a suitable model of transformation space. We introduce an optimization method for detecting such uniform grids specifically designed to deal with outliers and missing elements. This yields a robust algorithm that successfully discovers complex regular structures amidst clutter, noise, and missing geometry. The accuracy of the extracted generating transformations is further improved using a novel simultaneous registration method in the spatial domain. We demonstrate the effectiveness of our algorithm on a variety of examples and show applications to compression, model repair, and geometry synthesis.
Symmetry factored embedding and distance
 ACM Trans. Graph. (Proc. SIGGRAPH
, 2010
"... We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is “factored out, ” and the SFD is the Euclidean distance in that space. These constructions ..."
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Cited by 20 (4 self)
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We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is “factored out, ” and the SFD is the Euclidean distance in that space. These constructions characterize the space of symmetric correspondences between points – i.e., orbits. A key observation is that a set of points in the same orbit appears as a clique in a correspondence graph induced by pairwise similarities. As a result, the problem of finding approximate and partial symmetries in a point set reduces to the problem of measuring connectedness in the correspondence graph, a wellstudied problem for which spectral methods provide a robust solution. We provide methods for computing the SFE and SFD for extrinsic global symmetries and then extend them to consider partial extrinsic and intrinsic cases. During experiments with difficult examples, we find that the proposed methods can characterize symmetries in inputs with noise, missing data, nonrigid deformations, and complex symmetries, without a priori knowledge of the symmetry group. As such, we believe that it provides a useful tool for automatic shape analysis in applications such as segmentation and stationary point detection. 1
Detecting approximate symmetries of discrete point subsets
, 2008
"... Detecting approximate symmetries of parts of a model is important when attempting to determine the geometrical design intent of approximate boundaryrepresentation (Brep) solid models produced e.g. by reverse engineering systems. For example, such detected symmetries may be enforced exactly on the ..."
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Cited by 3 (2 self)
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Detecting approximate symmetries of parts of a model is important when attempting to determine the geometrical design intent of approximate boundaryrepresentation (Brep) solid models produced e.g. by reverse engineering systems. For example, such detected symmetries may be enforced exactly on the model to improve its shape, to simplify its analysis, or to constrain it during editing. We give an algorithm to detect local approximate symmetries in a discrete point set derived from a Brep model: the output comprises the model’s potential local symmetries at various automatically detected tolerance levels. Nontrivial symmetries of subsets of the point set are found as unambiguous permutation cycles, i.e. vertices of an approximately regular polygon or an antiprism, which are sufficiently separate from other points in the point set. The symmetries are detected using a rigorous, tolerancecontrolled, incremental approach, which expands symmetry seed sets by one point at a time. Our symmetry cycle detection approach only depends on interpoint distances. The algorithm takes time O(n 4) where n is the number of input points. Results produced by our algorithm are demonstrated using a variety of examples.
Detecting Design Intent in Approximate CAD Models Using Symmetry
"... Finding design intent embodied as highlevel geometric relations between a CAD model’s subparts facilitates various tasks such as model editing and analysis. This is especially important for boundaryrepresentation models arising from, e.g., reverse engineering or CAD data transfer. These lack expl ..."
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Cited by 3 (0 self)
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Finding design intent embodied as highlevel geometric relations between a CAD model’s subparts facilitates various tasks such as model editing and analysis. This is especially important for boundaryrepresentation models arising from, e.g., reverse engineering or CAD data transfer. These lack explicit information about design intent, and often the intended geometric relations are only approximately present. The novel solution to this problem presented is based on detecting approximate local incomplete symmetries, in a hierarchical decomposition of the model into simpler, more symmetric subparts. Design intent is detected as congruencies, symmetries and symmetric arrangements of the leafparts in this decomposition. All elementary 3D symmetry types and common symmetric arrangements are considered. They may be present only locally in subsets of the leafparts, and may also be incomplete, i.e. not all elements required for a symmetry need be present. Adaptive tolerance intervals are detected automatically for matching interpoint distances, enabling efficient, robust and consistent detection of approximate symmetries. Doing so avoids finding many spurious relations, reliably resolves ambiguities between relations, and reduces inconsistencies. Experiments show that detected relations reveal significant design intent.
www.elsevier.com/locate/cad Detecting approximate symmetries of discrete point subsets ✩
, 2006
"... Detecting approximate symmetries of parts of a model is important when attempting to determine the geometrical design intent of approximate boundaryrepresentation (Brep) solid models produced e.g. by reverse engineering systems. For example, such detected symmetries may be enforced exactly on the ..."
Abstract
 Add to MetaCart
Detecting approximate symmetries of parts of a model is important when attempting to determine the geometrical design intent of approximate boundaryrepresentation (Brep) solid models produced e.g. by reverse engineering systems. For example, such detected symmetries may be enforced exactly on the model to improve its shape, to simplify its analysis, or to constrain it during editing. We give an algorithm to detect local approximate symmetries in a discrete point set derived from a Brep model: the output comprises the model’s potential local symmetries at various automatically detected tolerance levels. Nontrivial symmetries of subsets of the point set are found as unambiguous permutation cycles, i.e. vertices of an approximately regular polygon or an antiprism, which are sufficiently separate from other points in the point set. The symmetries are detected using a rigorous, tolerancecontrolled, incremental approach, which expands symmetry seed sets by one point at a time. Our symmetry cycle detection approach only depends on interpoint distances. The algorithm takes time O(n 4) where n is the number of input points. Results produced by our algorithm are demonstrated using a variety of examples.