Find distance and total number of overlapping points in a figure

I want to find the distance and total number of overlapping points as shown in the encircled ellipses in the attached figure. Also is there anyway I can set a distance formula so that the points lie within that distance would be counted? And how to find how many of those points are there which fall in that specific distance range? Thnaks

[Adam Danz]

I just noticed your tags 'image analysis'. The data in your demo are not image data. If the demo data does not represent the actual data, you'll need to clairify that. Otherwise, consider updating those tags.

Answer 1

[Adam Danz]

The points are overlapping because of the marker size and the data resolution set by the axis limits. If you zoom in or use smaller marker size, those points will not overlap. You need to set a minimal distance threshold that defines what "overlapping" or "near" means.

You can use pdist or apply the distance formula to all pairs of points to get the distance between all pairs of points and then determine which paired-distances fall below your threshold.

Demo

This demo creates a set of (x,y) coordinates and uses pdist to compute the paired distances. It then plots a distribution of distances that you can use to choose a threshold that defines a distance considered to be 'overlapping'. Lastly, it marks the paired coordinates whose distances are below the threshold.

• xy is an nx2 matrix of [x,y] coordinates
• threshold is a value you choose to define 'near'
• nearIdx is a vector of indices showing which values in xy are 'near'

% Create (x,y) coordinates
rng('default')
xy = rand(30,2);

% Plot raw data
figure()
tiledlayout(2,2)
ax1 = nexttile();
plot(ax1, xy(:,1), xy(:,2), 'bo', 'MarkerSize', 10)
title(ax1, 'Raw data')

% Compute paired distances
pairedDist = squareform(pdist(xy)) + triu(inf(size(xy,[1,1])));

% Plot distribution of distances to choose a threshold
ax2 = nexttile();
histogram(ax2, pairedDist(:),50)
title(ax2, 'Paired distances')

threshold = 0.10;
xline(ax2, threshold, 'r-', 'Theshold')

% Find points that are below threshold distance
[xIdx,yIdx] = find(pairedDist < threshold);
nearIdx = [xIdx; yIdx];

% Plot the paired points that are 'near' 
hold(ax1, 'on')
plot(ax1, xy(nearIdx,1), xy(nearIdx,2), 'rx')

% Set equal aspect ratio to visually inspect distances
axis(ax1, 'equal')

[Sajjad Ahmad Khan]

@Adam Danz Thank you so much for your help! I found it useful and I also changed the tags. I tried the code on real dataset (see the attached file). I set the threshold level of 1, however I am getting some of localizations far from each others (see the rectangle regions). The regions enclosed in ellipses seems to lie within the threshold level of 1 but the localizations detected in rectangular regions don't seem to be near. Can you comment on that? Thanks!

[Adam Danz]

It doesn't look right. Each pair of neighboring points should get a red x and for some pairs I only see 1 red x. There are also clusters that look pretty close but I can't really estimate their distances given the unequal aspect ratio of axes. Lastly, it could be that some points are exactly on top of each other or so close to each other that they appear as 1 marker.

Important: Since your x and y axis scales are completely different, it will be hard to visually inspect for neighboring points. If you execute axis equal the aspect ratio will be equal. Otherwise visual inspection is nearly impossible.

Is your implementation exactly the same as mine? In other words, did you copy-paste my code and merely replace xy with your coordinates? If not, look for differences between your implementation and my code. If you're certain that the implementation is exactly the same.

[Sajjad Ahmad Khan]

@Adam Danz Yes I implemented the same code as shared by you. Now I set axis equal and got this plot shown in the figure. I also have a zoomed-in plot which I am sharing in a comment below this comment.

[Sajjad Ahmad Khan]

@Adam Danz Here is the zoom-in plot. Here ellipses represent the two circles that overlap due to threshold limit but there are rectangular regions which only have one circle. I am sharing the XY data in a separate comment for your consideration. Thanks!

[Adam Danz]

You have to explore your data a bit more carefully.

> but there are rectangular regions which only have one circle

To investigate this, I loaded the attached data and searched for data points within the highlighted rectange which has the following coordinates.

data = load('XYlocalizations.mat');
xy = data.XY; 
idx = xy(:,1)<202 & xy(:,1)>200 & xy(:,2)<155.5 & xy(:,2)>155;
sum(idx)

ans = 2

Note that there are two points there. Let's see if they are exact duplicates,

xy(idx,:)

ans = 2×2

201.6977 155.1422 201.6977 155.1422

Yep, you've got two coordinates in exactly the same location.

Now let's look at how many 2D points are exact matches,

pairedDist = squareform(pdist(xy)) + triu(inf(size(xy,[1,1])));
sum(pairedDist(:)==0)

ans = 57

You've got 57 points that have duplicate (x,y) coordinates.

Finally, let's label coordinates that contain duplicates using a black square

% Continuing from the code within my answer...
axis(ax1, 'equal')
xlim(ax1, [174.09, 212.12])
ylim(ax1, [135.43, 165.75])

threshold0 = 0; % for near-duplicates, use something like 0.001
[x0Idx,y0Idx] = find(pairedDist <= threshold0);
duplicateIdx = [x0Idx; y0Idx];
plot(ax1, xy(duplicateIdx,1), xy(duplicateIdx,2), 'ks','Markersize',15, 'LineWidth', 1)

Since my answer solved your question, please consider accepting the answer (blue accept-this-answer button) so the question is marked as solved.

[Sajjad Ahmad Khan]

@Adam Danz thank you so much! I accepted the answer.