me_pe2d - 2D Moreau envelope, PE algorithm

Calling Sequence

M = me_pe2d(x0,xn,y0,yn,f)

Parameters

• x0 : real number. Lower bound of the x-interval.
• xn : real number. Upper bound of the x-interval.
• y0 : real number. Lower bound of the y-interval.
• yn : real number. Upper bound of the y-interval.
• f : column vector. The value of the function on the grid X=[(x0:hx:xn)'] x [(y0:hy:yn)']; where n=size(f,1), hx=(xn-x0)/(n-1), m=size(f,2), and hy=(yn-y0)/(m-1), so f(i,j)=fu(X(i,j)) for some function fu.
• M : matrix of size nxm containing the Moreau envelope of the function f.

Description

Compute numerically the discrete Moreau envelope of a set of spatial points (X(i1,i2),f(i1,i2)) at slopes (X(j1,j2)). It reduces computation to one dimension, and uses the one-dimensional parabolic envelope algorithm (see me_pe) resulting in a theta(n*m) linear-time algorithm.

Examples

        function f=f(lambda,x),f=lambda * x.^2,endfunction
        function g=g(lambda1,lambda2,x,y),g=f(lambda1,x)+f(lambda2,y),endfunction
        lambda1=1;lambda2=2;
        x1=(-10:10)';x2=(-5:5)';
        [X, Y]=ndgrid(x1,x2);F=g(lambda1,lambda2,X,Y);
        s1=(-4:4)';s2=(-5:6)';
        Xr=x1;Xc=x2;Sr=s1;Sc=s2;
        desired=me_pe2d(x1,x2,F,s1,s2);
        //1d computation for separable function
        Ms1=me_direct(x1,f(lambda1,x1),s1);
        Ms2=me_direct(x2,f(lambda2,x2),s2);
        t1 = Ms1 * ones(1,size(Ms2,1));
        t2 = ones(size(Ms1,1),1) * Ms2';
        correct=t1+t2;
        b = and(correct == desired);
  

See Also

me_brute2d,  me_direct2d,  me_llt2d,  me_nep2d,  me_pe,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

Used Function