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In the Wikipedia article on ridge detection, there is this symbol $\partial_p$.

I know that $\frac{\partial f}{\partial p}$ is sometimes denoted as $\partial_p f$, but I've never seen $\partial_p$ written alone before.

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    $\begingroup$ Most likely they mean the differentation operator $\frac{\partial}{\partial p}$. $\endgroup$ – Piotr Benedysiuk Jun 21 '16 at 12:08
  • $\begingroup$ probably be they choose $p$ to take either the symbolic value $x$ or $y$ $\endgroup$ – Max Jun 21 '16 at 12:10
  • $\begingroup$ en.wikipedia.org/wiki/Partial_derivative#Notation might help $\endgroup$ – Merkh Jun 21 '16 at 12:10
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The notation $$ \partial_p = \sin \beta \partial_x - \cos \beta \partial_y \\ \partial_q = \cos \beta \partial_x + \sin \beta \partial_y $$ means this is true applied to any suitable function $f$, e.g. $$ \partial_p f = \sin \beta \partial_x f - \cos \beta \partial_y f\\ \partial_q f = \cos \beta \partial_x f + \sin \beta \partial_y f $$ or alternatively, if you like, $$ f_p = \sin \beta f_x - \cos \beta f_y \\ f_q = \cos \beta f_x + \sin \beta f_y $$

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$\partial_p$ is just that, the differential operator. It is written alone sometimes to emphasize it's own properties as a map, without needing a specific function to act on. In the article they discuss the properties of differentiating in a rotated frame. This changes the differential operator itself in those coordinates. So it makes sense to write it all out.

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