Tight Boundary around a Set of Points

I want to draw a tight boundary around a set of points. The code is used below to draw the black boundary around the blue points. The red areas show areas that are not tight enough for my application. Does anyone have a suggestion on how to get a tighter boundary? The data is attached.

   clc; clear; close all;
   load('xy.mat')
   xy = [x y];
   plot(xy(:,1), xy(:,2),'.')
   grid on;
   hold all;

   xyNormalized = xy;
   for iColumn = 1:2
      minValue = min(xy(:,iColumn));
      maxValue = max(xy(:,iColumn));
      xyNormalized(:,iColumn) = (xy(:,iColumn)-minValue)/(maxValue - minValue);
   end

   k = boundary(xyNormalized,1);
   xyBoundary = xy(k,:);
   h = plot(xyBoundary(:,1), xyBoundary(:,2),'k');

Answer 1

[Thorsten]

[ux, ia, ib] = uniquetol(x, 0.01);
for i= 1:max(ib), M(i,:) = [min(y(ib == i)) max(y(ib == i))]; end
plot(x, y, '.')
hold on
plot(ux, M(:,1), 'r-')
plot(ux, M(:,2), 'r-')

This is how it looks like

[Jason Nicholson]

Good answer. It works. I did have to tighten the tolerance to 0.001.

clc; clear; close all;

load('xy.mat')
xy = [x y];
plot(xy(:,1), xy(:,2),'.')
grid on;
hold all;

[ux, ia, ib] = uniquetol(x, 0.001);
for i= 1:max(ib) 
    M(i,:) = [min(y(ib == i)) max(y(ib == i))];
end

plot(ux, M(:,1), 'r-')
plot(ux, M(:,2), 'r-')

[Thorsten]

You're welcome. Please accept my answer if it works for you.

Answer 2

[FirefoxMetzger]

This draws a boundary along the highest elements of each class, the lowest elements of each class and has the left and right outer classes as boundary:

    load('xy.mat')
    xy = [x y];
    plot(xy(:,1), xy(:,2),'.')
    grid on;
    hold all;

    %split data into classes and find max/min for each class
    class_label = unique(x);
    upper_boundary = zeros(size(class_label));
    lower_boundary = zeros(size(class_label));
    for idx = 1:numel(class_label)
        class = y(x == class_label(idx));
        upper_boundary(idx) = max(class);
        lower_boundary(idx) = min(class);
    end

    % plot a nice boundary in red
    left_boundary = y(x == class_label(1));
    right_boundary = y(x == class_label(end));

    plot(class_label(1)*ones(size(left_boundary)),left_boundary,'r')
    plot(class_label,upper_boundary,'r')
    plot(class_label(end)*ones(size(right_boundary)),right_boundary,'r')
    plot(class_label,lower_boundary,'r')

[Jason Nicholson]

Good answer. This answer works well.

[Sinvaldo Moreno]

I believe the fast and easy way is get the boundaries:

 % code
load('xy.mat');
 xy = [x y]
[k, v] = boundary(xy(:,1),xy(:,2));
plot(xy(:,1), xy(:,2),'.');
hold on;
plot(xy(k,1),xy(k,2));

[Jason Nicholson]

Sinvaldo. This is not correct. I want a tight boundary.

The boundary command has an input s called the "shrink factor." A shrink factor of 0 corresponds to the convex hull of the points. A shrink factor of 1 corresponds to the tightest signel region boundary the points. By default, the shrink factor is 0.5 when it is not specified in the boundary command. For example, your code yields the following graph. Look at the answer that I accepted and you will see that the boundary is tighter than the boundary in the graph below.

Even if the shrink factor is set to 1 like in the code and picture below, I still don't think the boundary is tight enough for my purposes. Thanks for the answer but it just isn't what I am looking for.

[k, v] = boundary(xy(:,1),xy(:,2),1);

Answer 3

[Image Analyst]

You might try the envelope function in the Signal Processing Toolbox:

[Jason Nicholson]

Interesting idea. However, the envelope function needs the input to output to be single valued. In math this is called a function, f(x). i.e. There cannot be multiple y values for a single x value.