Maximum entropy models from phase harmonic covariances

The covariance of a stationary process X is diagonalized by Fourier transform. It does not take into account the complex phase and defines Gaussian maximum entropy models. We introduce general family harmonic moments, which rely on phases to capture non-Gaussian properties. They are defined as Hˆ(LX), where L linear operator Hˆ non-linear multiplies each coefficient integers. can also be calculated from rectifiers, relates Hˆ(LX) neural network coefficients. If transform then sparse matrix whose non-zero off-diagonal coefficients dependencies between frequencies. These have similarities with high order but smaller statistical variabilities because Lipschitz. wavelet reveal across scales, specify geometry local coherent structures. models conditioned these covariances. precision numerically evaluated synthesize images turbulent flows other processes.