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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.

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  • $\begingroup$ Do you want hints to which choice to use or what? $\endgroup$ – Royi Jun 28 at 18:04
  • $\begingroup$ @Royi i am asking for a terminology clarification ultimately. $\endgroup$ – IntegrateThis Jun 28 at 18:06
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Unless mentioned otherwise withing the context the classic interpretation of Second Derivative Gaussian Filter is indeed (a) in your question:

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

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