SOI and ISOI

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Abstract

The Successive Orthogonal Images (SOI) and Incremental Successive Orthogonal Images (ISOI) codes generate uniform samples of spheres, ellipsoids and of rotation groups. Both the SOI and ISOI codes provide rotation samples having excellent uniformity (good covering of the 2-sphere and SO(3) are obtained, which can be formulated in terms of spherical dispersion and discrepancy). The ISOI code has two additional advantages: incremental quality (samples are added one by one maintaining the uniformity of the resulting distribution), and explicit neighborhood structure (the samples are organized in a grid fashion, allowing efficient nearest neighbor calculations). echo '

Thank you for your interest in the SOI/ISOI code.

SOI CODE

The SOI codes are described in the following paper:

J. C. Mitchell. Discrete Uniform Sampling of Rotation Groups Using Orthogonal Images. SIAM Journal of Scientific Computing, 30(1):525-547, 2007. View Abstract

sampling.tar.gz

This archive contains the code for uniform deterministic sampling on SO(3)

ISOI CODE

The ISOI codes are described in the following paper:

A. Yershova, S. Jain, S. M. LaValle, and J. C. Mitchell. Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration International Journal of Robotics Research, November 2009. View Abstract

S2_sequence.tar.gz	This code provides a uniform deterministic sequence of samples over S^2. It generates an ordered sequence of points from the multiresolution grid structure provided by HEALPix. The output is parametrized by (x,y,z) coordinates in 3D.
SO3_grid.tar.gz	This code provides a uniform multiresolution grids over SO(3). The method uses Hopf coordinates to generate grid cells. The base-resolution grid consists of 72 points. The output is parametrized using unit quaternions, represented by (x,y,z,w) in 4D.
SO3_sequence.tar.gz	This code provides a uniform deterministic sequence over SO(3). The base-resolution grid consists of 72 points. The method uses Hopf coordinates to generate grid cells. The output is parametrized using unit quaternions, represented by (x,y,z,w) in 4D. The advantages of the deterministic sequences provided by this software are: uniformity (good covering of the 2-sphere and SO(3) is obtained, this can be formulated in terms of spherical dispersion and discrepancy), incremental quality (samples are added one by one maintaining the uniformity of the resulting distribution), explicit neighborhood structure (the samples are organized in a grid fashion, allowing efficient nearest neighbor calculations). It is important to note that the resulting sequence is infinite, that is, infinitely many samples can be generated retaining all of the above properties. Deterministic sequences were tested in sampling-based motion planning algorithms and compared to the performance of random sequences. While the performance efficiency is usually comparable, deterministic sequences provide important resolution completeness guarantees to motion planning methods.

References

If you use these programs, please cite the following two references: