How to perform shape factor analysis (circularity) on list of x, y coordinates

I have a list of x-y coordinates corresponding to the points along the perimeter of a shape. MATLAB's Image Processing Toolbox seems to allow for shape factor analysis of closed shapes within binary images, but I don't have enough points to fully close the shape.

So far, I've just plotted the x-y coordinates, but I'm having difficulty analyzing further. Are there built in mathematical methods for calculating max/min feret diameter and circularity using just a set of points?

Alternatively, are there other methods in image processing that are generally used to analyze shapes described by individual points?

[Adam Danz]

Is the expected shape a circle? If so, you only need 3 points along the circumference to fit the circle (assuming center and radius is unknown).

If the shape is unknown and you do not have all of the points that define the shape, you can only compute the feret diamter of sections that are bounded at both ends of the diameter line.

If there's any expectation about the shape you might be able to do some fitting to estimate the missing coordinates.

[Suhaas Garre]

Unfortunately, these are all highly heterogeneous shapes (they are tumor boundaries) and so we are not trying to make any sort of assumptions about shape.

I am mainly trying to find ways to quantify the difference between more elongated shapes vs shapes with a more square-like aspect ratio.

[Adam Danz]

I wonder how you're getting an incomplete set of boundary coordinates. regionprops(), for example, returns perimeter coordinates, major & minor axis lengths, etc, of binary images.

I suppose there's noise in some of the images and if that's the case, I'd think the problem lies in reducing the noise in order to have a complete boundary.

[Suhaas Garre]

Sorry for the confusion, I am working with coordinates that are imported into MATLAB from a third-party application specific to the field.

I believe the incomplete coordinates are just from the somewhat imprecise way that the tumors are segmented. I have the original source images as well but the processing is already all done otherwise

[Adam Danz]

"I am mainly trying to find ways to quantify the difference between more elongated shapes vs shapes with a more square-like aspect ratio."

It sounds to me like you'll need to compute the length of the major and minor axes. If those two values are close to each other you can classify the object as a square-ish-blob. Otherwise you'd classify it as not-square-ish (note that it could be oblong, curved, u-shaped, etc).

If you're missing a big chunk of coordinates such as an entire side of the object, it may be obvious whether the object is square-ish or not. For example, we can assume the incomplete object below is not square-ish

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but we cannot make the same assumption with this object

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Perhaps sharing some examples may inspire more suggestions.

[Suhaas Garre]

Yes, I certainly see what you mean. I’ll post some scatter plots tomorrow with some examples of what I’m referring to.

Thanks for your help so far!

[Suhaas Garre]

I've updated the original question with an example of what I'm working with. I managed to close the shape yesterday, but as you can see, it is highly irregular. I've attached another example onto this comment as well.

Answer 1

[Adam Danz]

It looks like you're working with binary images in which case, you could use regionprops() along with the 'MajorAxisLength' and 'MinorAxisLength' properties to get the 'length' and 'width' of each blob. If those values are close to each other they can be categorized as square-ish. You'll have to choose a threshold such as

isSquareish = MinorAxisLength/MajorAxisLength > 0.95;

If you decide to go this route and get stuck, feel free to update us with your code so we can stay involved.

[Suhaas Garre]

I will give that a try, thank you for all your help!

I'll update this post as I make progress.

[Adam Danz]

Sure, good luck!

Answer 2

[Image Analyst]

There is a new function bwferet() that you might want to look into. You might want to look into edge linking algorithms to close the shapes.

A circle could have the same major and minor axis lengths. I've found that looking at circularities (perimeter squared divided by 4*pi*area) or feret diameters is not that great a way but it could be one factor in the decision if a blob is square. One of the more reliable ways to find a square is to find the centroid of the points and then get the distances of all perimeter points to the centroid, then plot that and look for 4 peaks. Attached is my demo for shape recognition where I do that for a variety of polygons.

[Adam Danz]

This is great.

BTW, I was assuming the OP wanted to classify elongated blobs from blobs without an obviously longer axis (hense the term square-ish). If the task is to identify shapes that are nearly square, then my simple minor/major axis idea isn't robust enough.

Answer 3

[Meg Noah]

For the final part of your question, 'Alternatively, are there other methods in image processing that are generally used to analyze shapes described by individual points?' one way is to raserize the data. This isn't a great way, but it gives ok results depending on the X, Y range and the accuracy you need:

% let inY and inX be your vectors <shapename>.Y, <shapename>.X
itheta = 0:360;
% fake data is a circle of radius R
R = 10.0;
inY = R.*sind(itheta); inX = R.*cosd(itheta);

h1 = figure(1);
gridDensity = 0.05; X1D = -2*R:gridDensity:2*R; Y1D = -2*R:gridDensity:2*R';
nX = numel(X1D); nY = numel(Y1D);
imagesc(X1D,Y1D,zeros(nY,nX),[0 255]);
set(h1,'position',[1 1 1000 800]);
set(gca,'units','pixels','position',[5 5 nX nY],'visible','off');
hold on;
colormap(gray);
axis equal
axis tight
plot(inY,inX,'color','w');
tim = getframe(gca); 
imorig = tim.cdata(:,:,1);
imorig(imorig>0) = 1;

% Note: If your data are in latitude/longitude values
%       then use vec2mtx to rasterize it
% rasterize the data values at grid size = 0.25
% gridDensity = 1/2; X1D = -2*R:0.25:2*R; Y1D = -2*R:0.25:2*R';
% [X2D,Y2D] = meshgrid(X1D,Y1D);
% [img, inRefVec] = vec2mtx(inY, inX, gridDensity);

% verify
h2 = figure(2);
imagesc(X1D,Y1D,imorig);

% apply standard blob detection and characterization
imblob = bwlabel(imorig,8);
stats = regionprops(imblob,'ALL');

diameterInPixels = norm(stats.BoundingBox([1 3]) - stats.Centroid);
diameterInDistance = diameterInPixels*gridDensity;

fprintf(1,'Diameter in Distance: %f\n',diameterInDistance);
 

The output is 10.015 whereas the input radius was 10.

If your data X, Y are latitude and longitude, then use the vec2mtx matlab command to rasterize.

[Suhaas Garre]

Hi Meg,

Thanks for your response! I actually got the points to fully close the shape so that doesn't appear to be an issue anymore, thankfully.

Answer 4

[Suhaas Garre]

As an update, I managed to close the shape, convert to greyscale, and binarize it.

I'm having some problems with the data itself (unrelated to MATLAB) so I will need to figure those out before I move along any further.

@Image Analyst: Thank you for the suggestions, I will give that a try once I get the rest working as well.

Thank you all for your help!