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.