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

固定过程X的协方差通过傅立叶变换对角化。它没有考虑复杂的傅立叶相位，而是定义了高斯最大熵模型。我们介绍了一般的相位谐波协方差矩家族，它们依赖于复杂的相位来捕获非高斯性质。它们被定义为的协方差$\hat{\mathcal{H}}（\mathrm{大号}X）$，其中L是一个复数线性算子，而$\hat{\mathcal{H}}$是一个非线性相位谐波算子，它将每个复数系数的相位乘以整数。运营商$\hat{\mathcal{H}}$ 也可以从整流器计算，这与 $\hat{\mathcal{H}}（\mathrm{大号}X）$神经网络系数。如果L是傅立叶变换，则协方差是一个稀疏矩阵，其非零非对角线系数捕获频率之间的相关性。这些系数与高阶矩具有相似性，但统计差异较小，因为$\hat{\mathcal{H}}$是Lipschitz。如果L是复数小波变换，则非对角线系数会揭示跨比例的依存关系，从而确定了局部相干结构的几何形状。我们介绍了以这些小波相位谐波协方差为条件的最大熵模型。对这些模型的精度进行了数值评估，以合成湍流和其他平稳过程的图像。