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I am looking for a way of sharpening ridges and valleys in the following heightmap

Heightmap Terrain from heightmap

I think this problem is slightly different than a regular sharpen operation, because sharpening would affect first derivatives, but the sharpness of ridges and valleys must modify the second derivative.

  1. I have an idea to decompose into components that represent derivatives of different order up to including at least the second derivative:

$f(x, y) = f_0(x, y) + a_xf_x(x, y) + a_yf(x, y) + b_{xx}f_{xx}(x, y) + 2b_{xy}f_{xy}(x, y) + b_{yy}f_{yy}(x, y) + \text{H.O.T}$

and increasing the $b$-terms, which should affect the surface curvature. Would this work? If so, how to achieve such decomposition.

  1. It could work to use some polynomial decomposition. Notice for example that if

$p(x) = ax^2 + bx + c$

Then $p_xx(x) = 2a$. Thus, the second derivative is proportional to $a$. Some additional steps are then needed because $p(x)$ and $p_x(x)$ are also affected by $a$.

To give some information about mathematical properties of $f$ (continuity, differentiability, etc) i present the signal flow that produces the image below. Notice that "Make radial gradient" outputs the distance from the origin squared. Thus, exponential decay will in this case result in a Gaussian blur.

Image generator

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  • $\begingroup$ What's the tool you used with this nice flow of data? $\endgroup$
    – Royi
    Aug 29 '21 at 16:03
  • $\begingroup$ @Royi github.com/milasudril/texpainter $\endgroup$
    – user877329
    Aug 29 '21 at 19:17
  • $\begingroup$ Have you tried Unsharp Mask? $\endgroup$
    – Royi
    Aug 30 '21 at 5:03
  • $\begingroup$ @Royi From my post "I think this problem is slightly different than a regular sharpen operation, because sharpening would affect first derivatives". Unsharp mask would simply reintroduce the "bumps" removed by the gaussian blur. $\endgroup$
    – user877329
    Aug 30 '21 at 14:25

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