In this work we tackle the problem of estimating the density $f_X$ of a random variable $X$ by successive smoothing, such that the smoothed random variable $Y$ fulfills $(\partial_t - \Delta_1)f_Y(\,\cdot\,, t) = 0$, $f_Y(\,\cdot\,, 0) = f_X$. With a focus on image processing, we propose a product/fields of experts model with Gaussian mixture experts that admits an analytic expression for $f_Y (\,\cdot\,, t)$ under an orthogonality constraint on the filters. This construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show preliminary results on image denoising where our model leads to competitive results while being tractable, interpretable, and having only a small number of learnable parameters. As a byproduct, our model can be used for reliable noise estimation, allowing blind denoising of images corrupted by heteroscedastic noise.

翻译：本文通过连续平滑方法解决随机变量$X$的密度$f_X$估计问题，其中平滑后随机变量$Y$满足$(\partial_t - \Delta_1)f_Y(\,\cdot\,, t) = 0$，$f_Y(\,\cdot\,, 0) = f_X$。聚焦于图像处理任务，我们提出一种采用高斯混合专家的产品/专家场模型，该模型在滤波器正交约束条件下可对$f_Y(\,\cdot\,, t)$给出解析表达式。这一构造天然允许模型通过经验贝叶斯方法在整个扩散时域上同步训练。我们展示了图像去噪的初步实验结果：在保持可解释性、可计算性且仅含少量可学习参数的前提下，该模型达到了具有竞争力的性能。作为副产品，本模型还可用于可靠噪声估计，从而实现异方差噪声污染图像的盲去噪。