What Is an Oriented Gaussian Second Derivative Filter In the paper: Detecting and Localizing Edges Composed of Steps, Peaks and Roofs, the authors refer to an image filter as an oriented second-derivative Gaussian filter. I'm trying to figure out what this means.
From my understanding a Gaussian filter for a given standard deviation $\sigma$ of size $n \times n$ for some odd positive integer $n$ is given by the formula $g(x,y)  = \frac {1}{2 \pi \sigma^2} e^{\frac{-(x^2+y^2)}{2 \sigma ^2}}$ applied to image co-ordinates of a rectangular region of a 2-D image (ignoring padding for now).
Then there are Guassian first-derivative filters consisting of $g_x(x,y)$ and $g_y(x,y)$, the partial derivatives, which for a given angle $\theta$ can be combined into an overall oriented filter, say:
$t(x,y, \theta) = cos (\theta) g_x(x,y) + sin (\theta) g_y(x,y)$
as per the answer to my question here.
Now it seems to me there are some choices for what could be considered by the term oriented second-derivative Gaussian filter (which after some Google searching I could not find a definition of):
a) An orietned Laplacian of Gaussian (since it involves second derivatives), a.k.a $L(x,y, \theta) = cos (\theta) g_{xx} (x,y) + sin (\theta) g_{yy}(x,y)$ (where $g_{xx},g_{yy}$ are the partial derivatives twice with respect to $x,y$ respectively). This seems the most likely choice.
b) Some kind of mixed partials derivative filter like $t(x,y, \theta) = cos (\theta) g_{xy} + sin(\theta) g_{yx}$, although the mixed partials should be equal by Clairaut's theorem.
Any insights appreciated.
Unless mentioned otherwise withing the context the classic interpretation of Second Derivative Gaussian Filter is indeed (a) in your question:
$$ L \left( x, y, \theta \right) = \cos \left( \theta \right) {g}_{xx} \left( x, y \right) + \sin \left( \theta \right) {g}_{yy} \left( x, y \right) $$