image derivatives using gaussian

Question

this is the function I have for calculating the image derivatives. Please help me understand this code as I am new to this field. If anyone could give me some links to understand this concept, I'll be greatful. some doubts that i have -

Why are we using ndgrid here? What are the directions 'x', 'y', 'xx', ('xy', 'yx'), 'yy' here? And how and why does the formula for this gaussian change according to the directions? Why are we using imfilter at the end?

function D =calc_image_derivatives(I,sigma,direction)    
[x,y]=ndgrid(floor(-3*sigma):ceil(3*sigma),floor(-3*sigma):ceil(3*sigma));
switch(direction)
case 'x'
  DGauss=-(x./(2*pi*sigma^4)).*exp(-(x.^2+y.^2)/(2*sigma^2));
 case 'y'
  DGauss=-(y./(2*pi*sigma^4)).*exp(-(x.^2+y.^2)/(2*sigma^2));
 case 'xx'
  DGauss = 1/(2*pi*sigma^4) * (x.^2/sigma^2 - 1) .* exp(-(x.^2 + y.^2)/(2*sigma^2));
 case {'xy','yx'}
  DGauss = 1/(2*pi*sigma^6) * (x .* y)           .* exp(-(x.^2 + y.^2)/(2*sigma^2));
 case 'yy'
  DGauss = 1/(2*pi*sigma^4) * (y.^2/sigma^2 - 1) .* exp(-(x.^2 + y.^2)/(2*sigma^2));
 end
 D = imfilter(I,DGauss,'conv','replicate');

Answer 1

This code calculates various directional derivatives of the image - x/y are first directional derivative, xx/yy/xy are second derivatives. The digital filter used for derivation is a 2D Gaussian of standard variation sigma, derived by the appropriate partial derivative (for example, in the case 'xx', the Gaussian is derived twice by x). From your question, I'm not sure you're familiar with the notion of a partial derivative, you can Google it. ndgrid is used to create grid matrices - this is a very commonly used approach in Matlab. Perhaps you know the function meshgrid, it is the same, only ndgrid can also create grid matrices of higher dimensions.

imfilter is used to perform a convolution (correlation to be more precise) between the digital filters to the image. The result of the this is an estimation of the required derivative.