Derivative of 2D dose (intensity) image wrt distance

I have a 2D image of size M×N representing a radiation dose map with a local dose , where is the spatial coordinate. I want to estimate both derivatives and with respect to r - how do I do that? If neccessary, the spatial resolution is 21 /pixel and isotropic.

Thanks in advance for any help or guidance!

Answer 1

[Aastha]

Hi @De Ar,

I understand that you have a 2D image representing a radiation dose map and you want to compute the first and second-order spatial derivatives. You have also mentioned that the spatial resolution is 21 μm/pixel.

To compute the derivatives, you may refer to the steps outlined below:

1. Use the “gradient” function to compute the first-order spatial derivatives using a finite difference approximation. Kindly refer to the MATLAB code snippet below to do so:

delta = 21e-6; % Spatial resolution in meters (21 micrometers)
[dose_dy, dose_dx] = gradient(dose, delta); % Compute gradient components

At each pixel, this returns two values corresponding to the partial derivatives ∂dose/∂y and ∂dose/∂x.

You may refer to MathWorks documentation of “gradient” function for any queries on it:

https://www.mathworks.com/help/releases/R2022a/matlab/ref/gradient.html

2. To compute the second-order spatial derivatives, apply the gradient function again to the first-order derivatives. This yields the entries of the Hessian matrix at each pixel:

[d2Iyy, d2Iyx] = gradient(dose_dy, delta);  % ∂²dose/∂y² and ∂²dose/∂y∂x
[d2Ixy, d2Ixx] = gradient(dose_dx, delta);  % ∂²dose/∂x∂y and ∂²dose/∂x²

These second-order derivatives describe how the gradient itself changes spatially and can be assembled into the Hessian matrix to analyze curvature and local structure.

I hope this helps!