Name

pl_me_llt_2d — 2D Moreau envelope, LLT algorithm

Calling Sequence

[M,g,Conjpartial,Conj] = pl_me_llt_2d(Xr,Xc,f,Sr,Sc)

Parameters

Xr

column vector of length n.

Xc

column vector of length m.

f

matrix of size nxm. The function f is sampled on a grid Xr x Xc so f(i,j)=fu(Xr(i),Xc(j)) for some function fu.

Sr

column vector of size m1.

Sc

column vector of size m2. The Moreau envelope is computed on a grid SrxSc.

M

matrix of size m1xm2 containing the Moreau envelope of the function f.

g

matrix of size nxm containing the function g (see formula below).

Conjpartial

matrix containing the partial conjugate of the function g i.e. the one dimensional conjugate applied to each row.

Conj

matrix of size m1xm2 containing the conjugate of the function g (see formula below).

Description

Compute numerically the discrete Moreau envelope of a set of spatial points (Xr(i1),Xc(i2),f(i1,i2)) at slopes (Sr(j1),Sc(j2)), i.e.

                                                  2                   2
    M(j1,j2) =  min [ f(i1,i2) + (Sr(j1) - Xr(i1)) + (Sc(j2) - Xc(i2))  ].
           i1,i2

It reduces computation to one dimension, and computes the Legendre conjugate through the formula

           2                                                      2
M(j) = s(j) - 2 g*(j) with g*(j) = max [ s(j) * x(i) - 1/2 * (x(i) + f(i)) ]
                                    i

thereby resulting in a theta(n*m + m1*m2) 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=pl_me_llt_2d(x1,x2,F,s1,s2);
        //1d computation for separable function
        Ms1=pl_me_direct(x1,f(lambda1,x1),s1);
        Ms2=pl_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

pl_me_brute_2d , pl_me_direct_2d , pl_me_nep_2d , pl_me_pe_2d , pl_me_llt

Authors

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

See pl_me_llt

Used Functions

Computation is reduced to one dimension, which is then handled by pl_lft_llt.