Documentation
Last update : 14/11/2006
CCA - Computational Convex Analysis toolbox description
Description
The CCA package contains numerical algorithms to compute several fundamental transforms of convex analysis for convex and nonconvex functions. Most of its algorithms take a function as input, either as evaluated on a grid or given as a black box, and return the evaluation of the transform on a grid.
The transforms currently implemented are:
-
-
-
lft
: The
Legendre-Fenchel transform
(also called Legendre-Fenchel conjugate, Fenchel conjugate, or convex conjugate):
f*(s) = sup [ < s, x > - f(x)].
x
The notation <., .>
denotes the standard scalar product. Several linear-time algorithms are implemented (functions with names lft_*).
-
me
: The
Moreau envelope
(also called Moreau-Yosida approximate):
2
M(s) = inf f(x) + || s - x ||.
x
The notation ||.||
denotes the Euclidean norm. Several linear-time algorithms are implemented (functions with names me_*).
-
bb
: The lower
convex envelope
(also called
convex hull
): It is the largest convex function minoring a given function. The implementation uses the Beneath-Beyond algorithm to achieve a linear-time worst-case complexity when the points are sorted along the x-axis. It is used by some fast transform algorithms.
Unit tests are available in the tests/ directory to test that the package is rightly setup and to provide additional examples. To run all unit tests use (in the directory the package was unpacked) exec tests/test.sci;
See Also
lft_llt
,
me_llt
,
me_llt2d
,
bb
,
me_nep
,
me_nep2d
,
me_pe
,
me_pe2d
,
lft_direct
,
me_direct
,
me_direct2d
,
me_brute2d
,
Author
Yves Lucet, University of British Columbia, BC, Canada
Bibliography
Y. Lucet, 2006, Fast Moreau Envelope Computation I: Numerical Algorithms, Numerical Algorithms, 43 (2006), 235-249
Y. Lucet, 2005, A linear Euclidean distance transform algorithm based on the Linear-time Legendre Transform,
Proceedings of the Second Canadian Conference on Computer and Robot Vision (CRV 2005), IEEE Computer Society Press, 2005.
Y. Lucet, 1997, Faster than the fast Legendre transform, the linear-time Legendre transform, Numerical Algorithms, 16(2):171-185. Code in Netlib.
Y. Lucet, 1996, A fast computational algorithm for the Legendre-Fenchel transform, Computational Optimization and Applications, 6(1):27-57.