Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.

两项关键思想已大大改善了图像增强和去噪技术：将图像数据提升到多方向分布以及应用非线性PDE（例如总变化流（TVF）和平均曲率流（MCF））。Chambolle和Pock（适用于TVF）和Citti等人最近结合了这两种想法。（对于MCF）用于二维图像。在这项工作中，我们扩展了它们的方法，以增强和去噪任意尺寸的图像，从而基于（子）黎曼几何创建了一个统一的几何和算法PDE框架。特别是，我们采用了不同的数值方法，通过应用Brezis-Komura梯度流理论证明了TVF的收敛性。我们的框架还允许通过使用局部自适应帧和相干增强技术来进行其他数据适应。我们通过定向得分将TVF和MCF应用于2D图像中细长结构的增强和去噪，并将结果与​​Perona–Malik扩散和BM3D进行比较。我们还演示了在DW-MRI中3D交叉纤维束的去噪和增强方面的技术。与数据驱动的扩散相比，我们看到更好地保留了束边界和交叉处纤维取向密度的角清晰度。我们还演示了在DW-MRI中3D交叉纤维束的去噪和增强方面的技术。与数据驱动的扩散相比，我们看到更好地保留了束边界和交叉处纤维取向密度的角清晰度。我们还演示了DW-MRI中3D交叉纤维束的降噪和增强技术。与数据驱动的扩散相比，我们看到更好地保留了束边界和交叉处纤维取向密度的角清晰度。