Separable Anisotropic Diffusion

Anisotropic diffusion has many applications in image processing, but the high computational cost usually requires accuracy trade-offs in order to grant its applicability in practical problems. This is specially true when dealing with 3D images, where anisotropic diffusion should be able to provide interesting results for many applications, but the usual implementation methods greatly scale in complexity with the additional dimension. Here we propose a separable implementation of the most general anisotropic diffusion formulation, based on Gaussian convolutions, whose favorable computational complexity scales linearly with the number of dimensions, without any assumptions about specific parameterizations. We also present variants that bend the Gaussian kernels for improved results when dealing with highly anisotropic curved or sharp structures. We test the accuracy, speed, stability, and scale-space properties of the proposed methods, and present some results (both synthetic and real) which show their advantages, including up to 60 times faster computation in 3D with respect to the explicit method, improved accuracy and stability, and min–max preservation.