2D Moreau envelope, PE algorithm
M = pl_me_pe_2d(x0,xn,y0,yn,f)
real number. Lower bound of the x-interval.
real number. Upper bound of the x-interval.
real number. Lower bound of the y-interval.
real number. Upper bound of the y-interval.
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.
matrix of size nxm containing the Moreau envelope of the function f.
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 pl_me_pe) resulting in a theta(n*m) linear-time algorithm.
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_pe_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); | ![]() | ![]() |
Yves Lucet
, University of British Columbia, BC, Canada
Computation is reduced to one dimension, which is then
handled by me_pe_d
(me_pe_d is the same function as pl_me_pe except the grid is assumed to be 1,..,n times 1,...,m).