Find the volume of a nx3 Dataset

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Jose Andrés 2015-6-12

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https://ww2.mathworks.cn/matlabcentral/answers/223608-find-the-volume-of-a-nx3-dataset

评论： Jose Andrés 2015-6-21

Hello everybody, I have an easy question:

I have seen this great explanation about how to integrate the volume underneath a set of nonuniformly spaced data: http://blogs.mathworks.com/videos/2009/09/08/integrating-to-find-the-volume-underneath-a-set-of-nonuniformly-spaced-data/

but the interpolation here it´s done between 0-1 because of his dataset. My question is: if my dataset has a large number of different values (like a ball), how should I do this interpolation? I have thought about to change the

interpZ(0.5,0.5) %test interpolation
vol = quad2d(interpZ,0,1,0,1) %volume should be close to 1

like this:

interpZ(¿?,¿?) %test interpolation
vol = quad2d(interpZ,min(min(z)),max(max(z)),min(min(z)),max(max(z)))

Thank you.

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Answer 1

Image Analyst 2015-6-13

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https://ww2.mathworks.cn/matlabcentral/answers/223608-find-the-volume-of-a-nx3-dataset#answer_182566

You have to decide what constitutes the "volume". Let's say your N by 3 data are the (x,y,z) coordinates in a scatter cloud/cluster that looks roughly like a peanut. Now, is your 3D volume the bounding box of the peanut? Or do you want it to be the volume of only the peanut itself? Finding the bounding box of the whole peanut is trivial = just use max() and min() on each of the three dimensions, x, y, and z. If you want a peanut shaped volume, then you have to decide if some arbitrary (x,y,z) point is to be included inside the peanut or outside of it, so that if you have a regular grid (like a CT or MRI volumetric image) then you can find the volume of the irregular peanut shape.

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Image Analyst 2015-6-16

You said "integrate the volume underneath a set of nonuniformly spaced data". Do you want the volume under only that data - those locations - or do you want to "fill out" the data to other missing (x,y) locations, and get the volume under those locations also? If you only want data under the existing data, then no, you don't need the other points. Imagine your data was like skyscrapers of a city on some flat plain, and you want the volume of the skyscrapers. Assume unit width of one block for each skyscraper. Do you need to know where the sky scraper is located in the city to know its volume? No, you don't.

But what if you want to drape a huge blanket over the tops of the skyscrapers and assume that the blanket height represents the height of unknown buildings in between the known buildings? Now you will need to interpolate to get/estimate those missing heights, and that will require knowing which (x,y) each building is at. But once you have the building height at each block, you can simply add up the heights to get the volume. At least that's one way. The other way is to assume triangular buildings instead of rectangular blocks, and then you'd have to do something more complicated like using trapz or something.

Image Analyst 2015-6-17

spatial_calibration_demo.m

If you want the volume in real world units you have to know the width of a voxel. What is your field of view? What is the number of voxels across it? Divide those to get the real world units of a voxel - essentially the cross sectional area. See my attached spatial calibration demo.

Jose Andrés 2015-6-21