  We examine the affine Wigner distribution from a quantization perspective
with an emphasis on the underlying group structure. One of our main results
expresses the scalogram as (affine) convolution of affine Wigner distributions.
We strive to unite the literature on affine Wigner distributions and we provide
the connection to the Mellin transform in a rigorous manner. Moreover, we
present an affine ambiguity function and show how this can be used to
illuminate properties of the affine Wigner distribution. In contrast with the
usual Wigner distribution, we demonstrate that the affine Wigner distribution
is never an analytic function. Our approach naturally leads to several
applications, one of which is an approximation problem for the affine Wigner
distribution. We show that the deviation for a symbol to be an affine Wigner
distribution can be expressed purely in terms of intrinsic operator-related
properties of the symbol. Finally, we present an affine positivity conjecture
regarding the non-negativity of the affine Wigner distribution.
