Unless mentioned otherwise withing the context the classic interpretation of Second Derivative Gaussian Filter is indeed (a) in your question: $$ L \left( x, y, \theta \right) = \cos \left( \theta \right) {g}_{xx} \left( x, y \right) + \sin \left( \theta \right) {g}_{yy} \left( x, y \right) $$


When using a randomized pattern in BRIEF, this means that you computed random positions inside the patch once in an offline procedure, then used these random locations every time you computed the descriptors. This makes sense, as it means that when comparing descriptors you will actually compare the same locations, it's simply that the sampling pattern was ...

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