How to graph/plot 2D Laplacian of Gaussian (LoG) function in MATLAB or python?

Question

Hello.

I am trying to make a 3-D graph similar to the one below, that illustrates the 2-D Laplacian of Gaussian (LoG) function. How can I accomplish this through MATLAB or python? Code snippets would be greatly appreciated.

I have found that we can plot the gaussian using this method, but I am looking for how to plot the laplacian of gaussian.

Answer 1

You can use the discrete laplacian function del2:

N = 3;
x=linspace(-N, N,30);
[X,Y]=meshgrid(x,x);
z=del2((1000/sqrt(2*pi).*exp(-(X.^2/2)-(Y.^2/2))));
surf(X,Y,z);

Results:

Answer 2

Using del2 applied to a Gaussian one obtains an approximation to the true Laplacian function (it uses a discrete approximation to the derivative). This is not necessary, we can easily compute the expression for the second derivative of the Gaussian, and use that.

First we define a 1D Gaussian:

x = linspace(-4,4,41);
G = exp(-x.^2/2)/sqrt(2*pi);

Next, we compute the 2nd derivative of the 1D Gaussian:

Gxx = G .* (x.^2-1);

The Gaussian has a nice property that you can multiply two 1D functions together to get the 2D function. Thus,

data = G .* Gxx.';

is the 2nd derivative along the y-axis of a 2D Gaussian. The transposed of data is the 2nd derivative along the x-axis.

The Laplace is defined as the sum of partial derivatives along each axis:

data = data + data.';

Plotting this leads to (I tried replicating the point of view of the original graph also):

Here's the full code:

x = linspace(-4,4,41);
G = exp(-x.^2/2)/sqrt(2*pi);
Gxx = G .* (x.^2-1);
data = G .* Gxx.';
data = data + data.';
surf(x,x,data,'facecolor','white')
view(45,13)
set(gca,'dataaspectratio',[1,1,0.08])
grid off
xlabel('X')
ylabel('Y')