How can I find the sum of the area in a specified region in the image according to the pixel intensity value?

Suppose two lines with theta angle are drawn from the point (1,1) in an image. Let's consider two arcs with radii of r1 and r2, which have the origin point (1,1) and intersect these lines. I want to find the sum of the area between the four of them in terms of pixels. However, arcs or lines are dividing some squares. I want to add the remaining value in the field to the total. For example, if 1/3 of a pixel is in the area and the pixel value is 120, I want to add 30 to the total. Which method should I use?

First of all, thank you for your comments. The indices of the matrix defined for an image are actually the midpoint of a pixel that we can think of as that square. The subject is a bit complicated, I don't want to go into too much detail. I can show the attached picture as an example. Let's assume that each pixel writes the intensity values inside. We will use this as a field. If we consider each pixel as a square, I need the sum of the area between two lines and two arcs in the picture. The area for the split pixels needs to be recalculated at the same rate. So I need to sum the areas of pixels multiplied by their intensities, not just the area of the pixels in between. Thank you for your interest.

[DGM]

You would need to create an accurately weighted antialiased mask. I don't know of any existing tools, and I have yet to finish the geometric masking tools for MIMT.

Steve describes a method in this series on creating accurate binarized masks. See also the prior posts on poly2mask() and this discussion.

Answer 1

[Image Analyst]

I would get the list of (x,y) coordinates for each arc, then call poly2mask. Then call nnz or bwarea or regionprops or polyarea

[DGM]

OP is trying to find the sum of the image pixels selected by an antialiased mask, not the area of a mask. Binarized methods might be approximate, but provide no edge weighting. The problem is generating the mask. The crude aa mask tools I've thrown around aren't accurately based on pixel overlap.

[Image Analyst]

True, I didn't understand it or read it carefully enough the first time. I'd need to know more context about the use case. Seems like he's mixing up analytical equations with digital coordinates. And it really depends on what he needs to know. And that depends on how the pixels are interpreted, like as points with no extent, or as little boxes with some spatial extent. I need to know why he thinks he needs such a high precision, super resolution answer. It could be that a digital answer is good enough for his needs. Otherwise it could get quite complicated with messy geometric equations or the symbolic toolbox to find out how much area of a little pixel box is covered by the analytical curve passing through it.

[DGM]

To be fair, I misread it the first time and had to completely change my response.

The last link I posted is from Steve's blog, and discusses various rationales for considering differently-shaped pixels.

Building off of poly2mask() code might be a way to get part of the work done, but IIRC, a lot of the simplifications that make it elegant only work because the result is binary. Maybe I'd have to look at it again.

Still, if perfection is required, it's worth noting that even with poly2mask(), the subpixel area is not exact, but quantized to 25 levels.

[Selim ARAS]

I added some explanation. The dots at the top right of the image can be thought of as the actual index points of the pixels. Thank you in advance for your help.

[Image Analyst]

It seems like you're considering the pixels as little squares. So this is more a geometry problem than a digital imaging and MATLAB problem. The geometrical and algebraic equations will be complicated and I can't afford to donate that much time to you. We'll leave it up to you to get the equations. After that we can help if you still need it. Sorry, it would have been so much easier if it had been a digital image.

And, by the way, you never explained the use case, meaning WHY do you need to do this?

[Selim ARAS]

I solved my problem myself and I will write how I solved it later. However, I will explain why I use it first after I publish my article. To solve this, you need an advanced knowledge of mathematics and geometry, you are right. It took quite a while to figure this out. I was also asking to gain this time. Thanks.

[DGM]

Yeah, those headaches are kind of why I still haven't added it to MIMT yet :)