Reversing the Order of Operators for Edge Detection?
In the classic framework both the Smoothing and the Difference Filter are applied using Convolution.
Since it is done using convolution it implies the operation is Linear Spatially Invariant (LSI).
LSI operators can be applied in any order and the result will be the same. This is also a result of the commutativity property of the convolution operator.
Let's say $ f $ is the smoothing operator and $ g $ is the difference operator then:
$$ y = f \ast g \ast x = g \ast f \ast x $$
In practice, a more computational efficient way is define $ h = f \ast g = g \ast f $.
See Difference of Gaussians.