  In this paper, we design mother wavelets for the 1D continuous wavelet
transform with some optimality properties. An optimal mother wavelet here is
one that has an ambiguity function with minimal spread in the continuous
coefficient space (also called phase space). Since the ambiguity function is
the reproducing kernel of the coefficient space, optimal windows lead to phase
space representations which are "optimally sharp." Namely, the wavelet
coefficients have minimal correlations with each other. Such a construction
also promotes sparsity in phase space. The spread of the ambiguity function is
modeled as the sum of variances along the axes in phase space. In order to
optimize the mother wavelet directly as a 1D signal, we pull-back the
variances, defined on the 2D phase space, to the so called window-signal space.
This is done using the recently developed wavelet-Plancharel theory. The
approach allows formulating the optimization problem of the 2D ambiguity
function as a minimization problem of the 1D mother wavelet. The resulting 1D
formulation is more efficient and does not involve complicated constraints on
the 2D ambiguity function. We optimize the mother wavelet using gradient
descent, which yields a locally optimal mother wavelet.
