Iterative closest point

The minimization process use ICP algorithm (Iterative Closest Point) described by Greespan [8] and Besl et al. [9]. Defining D the set of data points of the surface Sj and M the set of points of the model or surface S2, this method establish a matching of D and Mpoints. Thus for each point of D there is a point (the nearest) of the model M. By the correspondence established above, the transformation that minimizes the distance criterion is calculated and applied to the points of the set D and the overall error is calculated using least squares method. 

CE-MRA contrast enhancement magnetic resonance ICP iterative closest point... 

Iterative closest point When dealing with two sets of point clouds with no matched correspondences, an iterative closest point can be used to rigidly... 

To eliminate the marker used for robot detection, a Metric-Based Iterative Closest Point (MbICP) algorithm is used in Zhuang, Gu, Wang, and Yu (2010). It derives the real-time optimal relative observations between any two robots equipped with laser range finders. In the same work, an adaptive localization period selection algorithm is proposed, based on autonomous motion state estimation. 

This process is repeated until the intersection is nonempty. It would also be possible to compute the two closest points between the two convex hulls and choose the point lying half way between the two convex hulls or we could increase the size of all convex hulls by a certain percentage before computing the intersected hull in order to avoid an empty intersection. This could also be done iteratively. First we see if the intersection is indeed empty. If the intersection is empty, we increase all convex hulls by a certain percentage. If it is still empty, we increase it even further. This process continues until the intersection is nonempty. For our experiments we have used the latter method, as this method produced the best results. 

Figure 5.12 Trajectory of multiplier iterations when finding the closest points on two ellipses.

Included in the methods discussed below are Newton-based methods (Section 10.3.1), the geometry optimization by direct inversion of the iterative subspace, or GDIIS, method (Section 10.3.2), QM/MM optimization techniques (Section 10.3.3), and algorithms designed to find surface intersections and points of closest approach (Section... 

Figure 5.12 General pnx edure of Jf-means. The colored dots represent the centroids of clusters. To cluster the points into predefined three clusters 1. The process is initialized hy randomly selecting three objects as cluster centroids. 2. After the first iteration, points are assigned to the closest centroid, and cluster centroids are recalculated. 3. The iterative procedures of point reassignment and centroid update are repeated. 4. Centroids are stable the termination condition is met. (See color insert)...

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

Because the maximum differences between r = 5 and r = oo for u, v, T, and p were 0.8%, 0.4%, 0.4%, and 0.8% respectively, the computational domain was chosen in the range 1 < r < 5 and 0 < 9 < n. The conditions at infinity were used at r = 5. The grid was equally distributed in both the r and 0 directions. The temperature and velocity were evaluated at the cell nodes, while the pressures were evaluated at the center of the cell. Equations (81)-(83) were discretized by using the three-point finite difference formula with second-order error. The pressure on the node of the cell was calculated by the interpolation of pressure at the center of the four closest cells. The discretized equation was solved along with the boundary conditions in an iterative procedure. 

In this optimization problem, the focus is to select that point on the limit state equation that is closest to the origin, in the standard normal space. In Fig. 4, T represents the standard normal transformation function from the original space (a ) to the standard normal space ( ). This optimization is solved using the Rackwitz-Fiessler (Fiessler et al. 1979) algorithm, an iterative procedure, as follows ... 

Figure 3.1 Illustration of the Monte Carlo packing-generation method for a system of N = 6 bidisperse frictionless disks (half small and half large with diameter ratio 1.4) in 2D. (a) The x- and y-coordinates for N = 6 random points are first generated in a square cell with periodic boundary conditions and the particles are grown uniformly until the closest pair of disks is in contact, (b) An attempt is made to move each particle randomly from the original (dashed outline) to the new position (shaded disk), and the move is accepted if it does not give rise to particle overlap, (c) The disks are expanded uniformly from the original (dashed outline) to the new size (shaded disk) until the two closest disks touch. This process is repeated for Nj iterations to obtain a single static packing.

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