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I've been told that they are influenced by noise considerably more than that of a first order edge detector such as a Gaussian. But why?

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Generally 2nd derivative is more sensitive to noise than 1st derivative. The 2nd derivative is usually accompanied by zero crossing detection, so it works better when grey level transitions are smooth. When the noise level is high, the gray level fluctuates and may result in many zero crossings.

Besides, there is less chance of multiple edge responses with 2nd derivative edge detectors.

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