How to create random points alongside a complex polyline? - matlab

Question

I would like to populate random points on a 2D plot, in such a way that the points fall in proximity of a "C" shaped polyline.
I managed to accomplish this for a rather simple square shaped "C":
This is how I did it:
% Marker color
c = 'k'; % Black
% Red "C" polyline
xl = [8,2,2,8];
yl = [8,8,2,2];
plot(xl,yl,'r','LineWidth',2);
hold on;
% Axis settings
axis equal;
axis([0,10,0,10]);
set(gca,'xtick',[],'ytick',[]);
step = 0.05; % Affects point quantity
coeff = 0.9; % Affects point density
% Top Horizontal segment
x = 2:step:9.5;
y = 8 + coeff*randn(size(x));
scatter(x,y,'filled','MarkerFaceColor',c);
% Vertical segment
y = 1.5:step:8.5;
x = 2 + coeff*randn(size(y));
scatter(x,y,'filled','MarkerFaceColor',c);
% Bottom Horizontal segment
x = 2:step:9.5;
y = 2 + coeff*randn(size(x));
scatter(x,y,'filled','MarkerFaceColor',c);
hold off;
As you can see in the code, for each segment of the polyline I generate the scatter point coordinates artificially using randn.
For the previous example, splitting the polyline into segments and generating the points manually is fine. However, what if I wanted to experiment with a more sophisticated "C" shape like this one:
Note that with my current approach, when the geometric complexity of the polyline increases so does the coding effort.
Before going any further, is there a better approach for this problem?

Answer 1

A simpler approach, which generalizes to any polyline, is to run a loop over the segments. For each segment, r is its length, and m is the number of points to be placed along that segment (it closely corresponds to the prescribed step size, with slight deviation in case the step size does not evenly divide the length). Note that both x and y are subject to random perturbation.
for n = 1:numel(xl)-1
r = norm([xl(n)-xl(n+1), yl(n)-yl(n+1)]);
m = round(r/step) + 1;
x = linspace(xl(n), xl(n+1), m) + coeff*randn(1,m);
y = linspace(yl(n), yl(n+1), m) + coeff*randn(1,m);
scatter(x,y,'filled','MarkerFaceColor',c);
end
Output:
A more complex example, using coeff = 0.4; and xl = [8,4,2,2,6,8];
yl = [8,6,8,2,4,2];
If you think this point cloud is too thin near the endpoints, you can artifically lengthen the first and last segments before running the loop. But I don't see the need: it makes sense that the fuzzied curve is thinning out at the extremities.

Answer 2

With your original approach, two places with the same distance to a line can sampled with a different probability, especially at the corners where two lines meet. I tried to fix this rephrasing the random experiment. The random experiment my code does is: "Pick a random point. Accept it with a probability of normpdf(d)<rand where d is the distance to the next line". This is a rejection sampling strategy.
xl = [8,4,2,2,6,8];
yl = [8,6,8,2,4,2];
resolution=50;
points_to_sample=200;
step=.5;
sigma=.4; %lower value to get points closer to the line.
xmax=(max(xl)+2);
ymax=(max(yl)+2);
dist=zeros(xmax*resolution+1,ymax*resolution+1);
x=[];
y=[];
for n = 1:numel(xl)-1
r = norm([xl(n)-xl(n+1), yl(n)-yl(n+1)]);
m = round(r/step) + 1;
x = [x,round(linspace(xl(n)*resolution+1, xl(n+1)*resolution+1, m*resolution))];
y = [y,round(linspace(yl(n)*resolution+1, yl(n+1)*resolution+1, m*resolution))];
end
%dist contains the lines:
dist(sub2ind(size(dist),x,y))=1;
%dist contains the normalized distance of each rastered pixel to the line.
dist=bwdist(dist)/resolution;
pseudo_pdf=normpdf(dist,0,sigma);
%scale up to have acceptance rate of 1 for most likely pixels.
pseudo_pdf=pseudo_pdf/max(pseudo_pdf(:));
sampled_points=zeros(0,2);
while size(sampled_points,1)<points_to_sample
%sample a random point
sx=rand*xmax;
sy=rand*ymax;
%accept it if criteria based on normal distribution matches.
if pseudo_pdf(round(sx*resolution)+1,round(sy*resolution)+1)>rand
sampled_points(end+1,:)=[sx,sy];
end
end
plot(xl,yl,'r','LineWidth',2);
hold on
scatter(sampled_points(:,1),sampled_points(:,2),'filled');