Minimizing profile curvature of 2D image
I am in search for (fast) algorithm to minimize profile curvature of this image on selected scales (in scale-space sense).
  
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  ... to minimize curvature at specified scale, [...], trying to apply this accross all scales (I am trying to utilize Gaussian / Laplacian image pyramids). It works by adjusting heights by adding height offset according to profile curvature at given point (height offset is negative in convex areas and positive in concave areas).


Modifications of curvature are basically modifications of a "signal's" dynamics.

In effect, what you are trying to say is "If the signal is rising (or falling) faster (or slower) than some threshold then...do something different". This "do something different" can be anything from "don't respond" to "respond slower" or "respond faster" and so on.

In one dimension, the Douglas-Peucker algorithm does more or less what you are after.

The algorithm takes only one parameter which you can relate to the scale of observation but essentially modifies the local slope so that it remains within certain limits.

Variants that do surface simplification, or mesh simplification exist too (or this, in the case of contours) which use some expression of "cost" or "stress" to decide which element to remove during the simplification.

If you are "tied" to a height map (i.e. implicit 3D data), then you are looking at removing data (i.e. decimation) and interpolating / filtering. For example, you can use your $P_c$ to "mark" the areas of high or low curving slope in one pass and compute the "fill-in" values of each patch on a second pass. For examples of these approaches, you might want to take a look at this or this or more generally this.

Hope this helps.