the heat kernel of the circle, a wrapped Gaussian: what smooths a density of angles

Syntax

psi = S1GaussKernel()

Description

The periodic heat kernel, obtained by wrapping a Gaussian of standard deviation sigma around the circle. The mean over the circle is one, and mode k is multiplied by exp(-k**2*sigma**2/2). Larger sigma gives a broader smoothing kernel.

Evaluate psi at x, an angle in radians. Use psi.halfwidth() for the angular width in radians and psi.bandwidth for the maximum degree of its harmonic representation.

Keywords

keyword default description
sigma   Gaussian standard deviation in radians; five degrees when neither sigma nor halfwidth is supplied
halfwidth   angular halfwidth in radians, taking precedence over sigma

Output

psi S1GaussKernel

See also

S1Kernel, calcDensity, calcKernel

Members

bandwidth the modes beyond 6 / sigma are below e^{-18}
halfwidth the halfwidth in closed form, sigma sqrt(2 log 2)
source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/S1GaussKernel_index_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX 
@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.