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The combinatorics that appears here is familiar in knot theory :. The fuzzy spheres appear in D-brane geometry :. The fuzzy 2-sphere was introduced in:. Discussion of the spectral Riemannian geometry of the fuzzy 2-sphere :. In the context of the IKKT matrix model :. Sanyaje Ramgoolam , Bill Spence , S. Thomas, Section 3. Constantinos Papageorgakis , p. The fuzzy 3-sphere was first discussed in the context of D0-brane -systems in. Harald Grosse , Ctirad Klimcik , P. Sanjaye Ramgoolam , Section 5 of: On spherical harmonics for fuzzy spheres in diverse dimensions , Nucl.
The output of the algorithm is a set of detected shapes with assigned points and all remaining points not covered by these shapes. Each input point can be assigned to at most one detected shape. Steps are repeated for a prescribed number of iterations and the shape with the highest number of inliers, referred to as the largest shape, is kept. In our context, the error between a point and a shape is defined by its distance and normal deviation to the shape. A random subset corresponds to the minimum number of points with normals required to uniquely define a primitive.
For very large point sets, the basic RANSAC method is not practical when testing all possible shape candidates against the input data in order to find the largest shape. Shape candidates are constructed until the probability to miss the largest candidate is lower than a user-specified threshold. The largest shape is repeatedly extracted until no more shapes, restricted to cover a minimum number of points, can be extracted.
An additional gain in efficiency is achieved through exploiting the normal attributes during initial shape construction and enumeration of inliers. The support of a shape refers to the footprint of the points covered by the primitive. To avoid generating shapes with the fragmented support, we enforce a connectivity constraint by considering only one connected component, referred to as cluster , selected as the one covering the largest number of inliers see Section Parameters for more details.
Property maps provide a means to interface with the user-specific data structures. The following example reads a point set from a file and detects only planar shapes. The default parameters are used for detection. It can be used, for example, to terminate the algorithm based on a timeout. In the following example, the algorithm stops if it takes more than half a second and prints out the progress made.
Shape detection is performed on five shape types plane, cylinder, sphere, cone, and torus. The input point set is sampled on a surface mostly composed of piecewise planar and cylindrical parts, in addition to free-form parts. Basic information of the detected shapes is written to the standard output: if the shape is either a plane or a cylinder, specific parameters are recovered, otherwise the general method info is used to get the shape parameters in a string object.
Note that specific parameters can be recovered for any of the provided shapes. This example illustrates how to access the points assigned to each shape and compute the mean error. A timer measures the running performance. This class must provide the following functions: construct a shape from a small set of given points, compute the squared distance from a query point to the shape, and compute the normal deviation between a query point with the normal and the normal to the shape at the closest point from the query.
The used shape parameters are added as members to the derived class. Note that the RANSAC approach is efficient for shapes that are uniquely defined by a small number of points, denoted by the number of required samples. The algorithm aims at detecting the largest shape via many random samples, and the combinatorial complexity of possible samples increases rapidly with the number of required samples. The last two functions are used to determine the number of inlier points to the shape.
They compute respectively the squared distance from a set of points to the shape, and the dot product between the point normals and the normals at the shape for the closest points on the shape. By default, the connected component is detected via the neighbor graph as mentioned above. Another optional method can be implemented to provide a helper function providing the shape parameters written to a string:.
The property maps are used to map the indices to the corresponding points and normals. A selective error tolerance parameter leads to higher running time and fewer shapes, as many shape candidates are generated to find the largest shape. We plot the detection performance against the epsilon error tolerance parameter for detecting planes in a complex scene with 5M points see Figure Figure The probability parameter controls the endurance when searching for the largest candidate at each iteration.
It barely impacts the number of detected shapes, has a moderate impact on the size of the detected shapes, and increases the running time. We plot the performance against the probability parameter see Figure Figure This shape detection component is based on the region growing algorithm applied to a set of user-specified items.
Shapes are detected by growing regions from seed items, where each region is created as follows:. Using this generic framework, users can grow any type of regions on a set of arbitrary items with their own propagation and seeding conditions see an example. The concept NeighborQuery provides the means for accessing neighbors of an item.
To create a model that respects this concept, the user has to provide an overload of the operator:. The concept RegionType provides the means for validating regions. In fact, a model of this concept maintains a description of the region type that is used in region growing. To create a model that respect this concept, three functions have to be defined:.
The SeedMap property map, provided as an optional parameter to the main class, enables to define the seeding order of items that is which items are used first to grow regions from. Such items are referred to as seed items. The SeedMap maps the index of an item to its order number in the overall region growing processing queue. It also shows how to skip unnecessary items and change their default seeding order. We choose a simple custom object item. We define four such objects, where for each object, we manually store indices of its neighbors.
The operator NeighborQuery::operator does nothing but accessing these neighbors. The RegionType class defines the three necessary functions:. We also define a SeedMap , such that the second object is handled first, while the first object follows. Moreover, the last object is always skipped. Notice that in this example, the container with objects is std::vector , which is random access.
However, the generic region growing algorithm can be applied to any other type of container, e. The result of using these classes with the region growing main class is that the first two objects form the first region, the third object forms the second region, and the last object is skipped.
If one wants to detect lines see 2D Example.
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