Thread Subject: Derivatives of 2D Gaussians

Hi everyone,

I'm trying to create a first and a second derivatives of a 2D Gaussian
function, however the results seem not to be as expected. Could anyone
kindly have a look at it? I checked if I did the derivation correctly,
but it seems to be OK.

Cheers,

Tin

function g = gauss2d(sigmax, sigmay, order)

    sigmax2 = sigmax * sigmax;
    sigmay2 = sigmay * sigmay;
    dx = floor(3.0*sigmax);
    dy = floor(3.0*sigmay);
    [X,Y] = meshgrid(-dx:dx,-dy:dy);
    f = 1/sqrt(2*pi*sigmax*sigmay);

    switch order
        case 0
            g = f * exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
        case 1
            g = f * (- (X.^2*sigmay2 + Y.^2*sigmax2)/
(sigmax2*sigmay2) ) * exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
        case 2
            g = f * ((X.^2 + Y.^2 - 2 * sigmax*sigmay)/
(sigmax2*sigmay2)) * exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
    end

    g = g ./ sum(sum(abs(g)));
end

On Oct 14, 12:30 pm, Martin Alegre <tin.ale...@gmail.com> wrote:
> Hi everyone,
>
> I'm trying to create a first and a second derivatives of a 2D Gaussian
> function, however the results seem not to be as expected. Could anyone
> kindly have a look at it? I checked if I did the derivation correctly,
> but it seems to be OK.
>
> Cheers,
>
> Tin
>
> function g = gauss2d(sigmax, sigmay, order)
>
>     sigmax2 = sigmax * sigmax;
>     sigmay2 = sigmay * sigmay;
>     dx = floor(3.0*sigmax);
>     dy = floor(3.0*sigmay);
>     [X,Y] = meshgrid(-dx:dx,-dy:dy);
>     f = 1/sqrt(2*pi*sigmax*sigmay);
>
>     switch order
>         case 0
>             g =  f * exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
>         case 1
>             g = f *  (- (X.^2*sigmay2  + Y.^2*sigmax2)/
> (sigmax2*sigmay2) )  *  exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
>         case 2
>             g = f *  ((X.^2 + Y.^2 - 2 * sigmax*sigmay)/
> (sigmax2*sigmay2))  *  exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
>     end
>
>     g = g ./ sum(sum(abs(g)));
> end

Martin,

When you write "the results seem not to be as expected", what was the
expected result?

Besides that, because you are dealing with a function of two
variables, the function has two first-order partial derivatives. One
with respect to each variable. Likewise, the function has three second
order partial derivatives. One with respect to x twice, one with
respect to y twice, and one with respect to x and then to y or vice
versa. Check you assignment make sure you understand what is being
asked.

Also, the last statement before the end of the function can be written
as

g = g/sum(abs(g(:));

1) g(:) is treated as a one-dimensional array thus obviating the need
for nested 'sum' functions.
2) No need for the ./ operator because the denominator is a scalar

HTH


Jomar

On Oct 14, 8:50 pm, Jomar Bueyes <jomarbue...@hotmail.com> wrote:
> On Oct 14, 12:30 pm, Martin Alegre <tin.ale...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > Hi everyone,
>
> > I'm trying to create a first and a second derivatives of a 2D Gaussian
> > function, however the results seem not to be as expected. Could anyone
> > kindly have a look at it? I checked if I did the derivation correctly,
> > but it seems to be OK.
>
> > Cheers,
>
> > Tin
>
> > function g = gauss2d(sigmax, sigmay, order)
>
> >     sigmax2 = sigmax * sigmax;
> >     sigmay2 = sigmay * sigmay;
> >     dx = floor(3.0*sigmax);
> >     dy = floor(3.0*sigmay);
> >     [X,Y] = meshgrid(-dx:dx,-dy:dy);
> >     f = 1/sqrt(2*pi*sigmax*sigmay);
>
> >     switch order
> >         case 0
> >             g =  f * exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
> >         case 1
> >             g = f *  (- (X.^2*sigmay2  + Y.^2*sigmax2)/
> > (sigmax2*sigmay2) )  *  exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
> >         case 2
> >             g = f *  ((X.^2 + Y.^2 - 2 * sigmax*sigmay)/
> > (sigmax2*sigmay2))  *  exp(-0.5 * ((X.^2/sigmax2)+(Y.^2/sigmay2)));
> >     end
>
> >     g = g ./ sum(sum(abs(g)));
> > end
>
> Martin,
>
> When you write "the results seem not to be as expected", what was the
> expected result?
>
> Besides that, because you are dealing with a function of two
> variables, the function has two first-order partial derivatives. One
> with respect to each variable. Likewise, the function has three second
> order partial derivatives. One with respect to x twice, one with
> respect to y twice, and one with respect to x and then to y or vice
> versa.  Check you assignment make sure you understand what is being
> asked.
>
> Also, the last statement before the end of the function can be written
> as
>
> g = g/sum(abs(g(:));
>
> 1) g(:) is treated as a one-dimensional array thus obviating the need
> for nested 'sum' functions.
> 2) No need for the ./ operator because the denominator is a scalar
>
> HTH
>
> Jomar

Hi Jomar,

Thanks for your reply

> When you write "the results seem not to be as expected", what was the
> expected result?
I was refering to the visual result. I'm studying Gaussian derivative
filters by my own. So, it's not for an assignment.
I wanted to reproduce some plots that I found in the following 2
webpages:
http://fourier.eng.hmc.edu/e161/lectures/gradient/node10.html (there's
a 2D plot for the Laplacian of Gaussian)

http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/OWENS/LECT6/node2.html
There's a subsection "Second order derivative operators" and again
there are 2D plots for Gaussian derivatives.

Best,

Tin