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.

• Do you want hints to which choice to use or what?
– Royi
Jun 28 '20 at 18:04
• @Royi i am asking for a terminology clarification ultimately. Jun 28 '20 at 18:06

$$L \left( x, y, \theta \right) = \cos \left( \theta \right) {g}_{xx} \left( x, y \right) + \sin \left( \theta \right) {g}_{yy} \left( x, y \right)$$