Diffusion models are getting popular in generative image and video synthesis.
However, due to the diffusion process, they require a large number of steps to
converge. To tackle this issue, in this paper, we propose to perform the
diffusion process in the gradient domain, where the convergence becomes faster.
There are two reasons. First, thanks to the Poisson equation, the gradient
domain is mathematically equivalent to the original image domain. Therefore,
each diffusion step in the image domain has a unique corresponding gradient
domain representation. Second, the gradient domain is much sparser than the
image domain. As a result, gradient domain diffusion models converge faster.
Several numerical experiments confirm that the gradient domain diffusion models
are more efficient than the original diffusion models. The proposed method can
be applied in a wide range of applications such as image processing, computer
vision and machine learning tasks.

通过在梯度域内进行扩散过程，可以加快收敛速度，得到更高效的梯度域扩散模型，适用于图像处理、计算机视觉和机器学习任务。

图像合成的梯度域扩散模型

Gradient Domain Diffusion Models for Image Synthesis

Rain streak removal is an important issue in outdoor vision systems and has
recently been investigated extensively. In this paper, we propose a novel video
rain streak removal approach FastDeRain, which fully considers the
discriminative characteristics of rain streaks and the clean video in the
gradient domain. Specifically, on the one hand, rain streaks are sparse and
smooth along the direction of the raindrops, whereas on the other hand, clean
videos exhibit piecewise smoothness along the rain-perpendicular direction and
continuity along the temporal direction. Theses smoothness and continuity
results in the sparse distribution in the different directional gradient
domain, respectively. Thus, we minimize 1) the $\ell_1$ norm to enhance the
sparsity of the underlying rain streaks, 2) two $\ell_1$ norm of unidirectional
Total Variation (TV) regularizers to guarantee the anisotropic spatial
smoothness, and 3) an $\ell_1$ norm of the time-directional difference operator
to characterize the temporal continuity. A split augmented Lagrangian shrinkage
algorithm (SALSA) based algorithm is designed to solve the proposed
minimization model. Experiments conducted on synthetic and real data
demonstrate the effectiveness and efficiency of the proposed method. According
to comprehensive quantitative performance measures, our approach outperforms
other state-of-the-art methods especially on account of the running time. The
code of FastDeRain can be downloaded at
this https URL

本文提出了一种基于 split augmented Lagrangian shrinkage 算法的视频雨点去除方法 FastDeRain，在梯度域中充分考虑了雨点的鉴别特征和清晰视频，并通过三种不同的范数来最小化雨点的稀疏性、空间平滑性和时间连续性，在合成和真实数据上的实验表明该方法的有效性和高效性。