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Talking about sharpening a contour in an image. What's the optimal direction for derivation? What's the maximum value of derivative?

I think that the optimal direction for derivation is the direction in which points the gradient. And the maximum value of derivative is the magnitude.

Is my answer to those questions right? Is there anything else you can suggest?

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Yes, you are correct.

The direction of the gradient vector (dx,dy) is (pretty much by definition) the "optimal direction of derivation".
The magnitude of the gradient is the "intensity" of the edge, the steepness of the derivative.

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