spatialTransformInverse edit page

the inverse of a transform that has no closed form one

A polynomial, a spline or a scattered field maps position to position perfectly well but cannot be solved backwards in closed form. Since the displacement is smooth by construction - it was fitted through a smooth basis - the inverse is reached by iterating

q <- p - u(q),   u(q) = eval(T,q) - q

which converges geometrically as long as the displacement gradient is below one, i.e. as long as the field does not fold. A field that does fold has no inverse to find and the iteration is refused rather than returning a wrong answer.

inv of this returns the original transform, so a round trip costs nothing.

Syntax

Tinv = inv(T)                          % how one is normally made
Tinv = spatialTransformInverse(T)
Tinv = spatialTransformInverse(T,'tol',1e-10,'iterMax',100)

Input

T spatialTransform, the forward map

Output

Tinv spatialTransformInverse

Class Properties

T the forward transform
tol convergence tolerance, relative to the coordinate scale
iterMax iteration budget

See also

spatialTransform spatialTransformField spatialTransform

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/spatialTransformInverse.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.