An optimization formulation for designing signal-dependent kernels that are based on radially Gaussian functions is presented. The method is based on optimality criteria and is not ad hoc. The procedure is automatic. The optimization criteria are formulated so that the resulting time-frequency distribution (TFD) is insensitive to the time scale and orientation of the signal in time-frequency. Examples demonstrate that the optimal-kernel TFD offers excellent performance for a larger class of signals than any current fixed-kernel representation. The technique performs well in the presence of substantial additive noise, which suggests that it may prove useful for automatic detection of unknown signals in noise. The cost of this technique is only a few times greater than that of the fixed-kernel methods and the 1/0 optimal kernel method.