Scilab Function
Last update : 3/11/2006

me_pe - Moreau envelope, PE algorithm

Calling Sequence

M = me_pe(x0,xn,f)

Parameters

Description

Compute numerically the discrete Moreau envelope of a set of planar points (X(i),f(i)) at slopes S(j)=X(j), i.e.

	   				      2
	   M(j) = min f(i) + || s(j) - x(i) ||.
		   i
It reduces computation to a 1,..,n grid with
	       2               2                    2
	M(j)= h * min [ ||j-i|| + f(x_0 + i * h) / h  ]
	           i

The algorithms runs in linear time Theta(n) with n=length(X)=length(f) by computing the lower parabolic envelope (PE).

Examples

	X=[-5:0.5:5]';
	Y=X.^2;
	M=me_pe(-5,5,Y)

See Also

me_pe2d ,   me_direct ,   me_llt ,   me_nep ,  

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
  • P. F. Felzenszwalb and D. P. Huttenlocher, 2004, Distance transforms of sampled functions, Tech. Rep. TR2004-1963, Cornell Computing and Information Science.