Convert grayscale image to point cloud (similar to dither) - matlab

Question

I'm currently trying to implement a method to generate TSP art, and for that I need a list of points (x,y), the local density of which is proportional to the gray scale pixel value of a given image.
My first thought was: well that works pretty much like Inverse Transform Sampling for statistics (you want to draw a sample that matches a given probability density function but you can only create a sample that is uniformly distributed).
I implemented this and it works fairly well, as evident by executing this code:
%% Load image, adjust it for our needs
im=imread('http://goo.gl/DDwV3t'); %load random headshot from google
im=imadjust(im,stretchlim(im,[.01,.65]),[]);
im=im2double(rgb2gray(im));
im=im(10:end-5,50:end-5);
figure;imshow(im);title('original');
im=1-im; %we want black dots on white background
im=flipud(im); %and we want it the right way up
%% process per row
imrow = cumsum(im,2);
imrow=imrow*size(imrow,1)./repmat(max(imrow,[],2),1,size(imrow,2));
y=1:size(imrow,2);
ximrow_i = zeros(size(imrow));
for i = 1:size(imrow,1)
mask =logical([diff(imrow(i,:))>=0.01,0]); %needed for interp
ximrow_i(i,:) = interp1(imrow(i,mask),y(mask),y);
end
y=1:size(ximrow_i,1);
y=repmat(y',1,size(ximrow_i,2));
y1=y(1:5:end,1:5:end); %downscale a bit
ximcol_i1=ximrow_i(1:5:end,1:5:end); %downscale a bit
figure('Color','w');plot(ximcol_i1(:),y1(:),'k.');title('Inverse Transform Sampling on rows');
axis equal;axis off;
%% process per column
imcol=cumsum(im,1);
imcol=imcol*size(imcol,2)./repmat(max(imcol,[],1),size(imcol,1),1);
y=1:size(imcol,1);
yimcol_i=zeros(size(imcol));
for i = 1:size(imcol,2)
mask =logical([diff(imcol(:,i))>=0.01;0]);
yimcol_i(:,i) = interp1(imcol(mask,i),y(mask),y);
end
y=1:size(imcol,2);
y=repmat(y,size(imcol,1),1);
y1=y(1:5:end,1:5:end);
yimcol_i1=yimcol_i(1:5:end,1:5:end);
figure('Color','w');plot(y1(:),yimcol_i1(:),'k.');title('Inverse Transform Sampling on cols');
axis equal;axis off;
It has the shortcoming that I can only use this per-row or per-column, but not both. The Inverse Transform Sampling method does not work for multivariate PDFs in general, and I'm fairly sure I wont be able to get it to work in this case.
Is there a simple method to achieve my goal that I haven't seen yet?
I am aware that an algorithm called Voronoi Stippler has been used to create the desired result and I will investigate that, but for the moment I liked the simplicity of Inverse Transform Sampling and would like to know if I can extend that method to match my needs.

Answer 1

It turns out this is fairly simple and can be done by Rejection Sampling.
For the special case where the instrumental distribution is U(0,1) it works like this (if I understood it correctly):
im=imread('http://goo.gl/DDwV3t'); %load random headshot from google
im=imadjust(im,stretchlim(im,[.01,.65]),[]);
im=im2double(rgb2gray(im));
im=im(10:end-5,50:end-5);
im=1-flipud(im);
d = im > .9*rand(size(im));
d=d&(rand(size(d))>.95); %randomly sieve out some more points
[i,j]=ind2sub(size(d),find(d));
figure('Color','w');plot(j,i,'k.');title('Rejection Sampling');
axis equal;axis off;
The sampling is done in one line:
d = im > .9*rand(size(im));
Since I ended up with too many points I randomly sampled the result thus reducing the number of points by approximately the factor 20.
This is pretty much the result I originally desired.