. Where does an edge appear in the second derivative of image intensity, as opposed to the first derivative?
The first derivative peaks at an edge; the second derivative crosses zero exactly at the edge location.
. Why does the Marr-Hildreth detector smooth the image with a Gaussian before applying the Laplacian?
Marr and Hildreth's fix is to smooth first, then differentiate -- equivalent to convolving with a single combined Laplacian-of-Gaussian kernel.
. What does a Difference of Gaussians (DoG) approximate?
Subtracting two Gaussian blurs at different scales approximates a scaled LoG using only two blurs and a subtraction, instead of an explicit second-derivative kernel.
. What does a Laplacian pyramid store at each level that a Gaussian pyramid does not, allowing exact reconstruction?
Each Laplacian level captures exactly what blurring/downsampling would discard, so summing back up the pyramid reconstructs the original exactly.