Spira Alon, Kimmel Ron, Sochen Nir
Computer Science Department, The Technion-Israel Institute of Technology, Haifa 32000, Israel.
IEEE Trans Image Process. 2007 Jun;16(6):1628-36. doi: 10.1109/tip.2007.894253.
We introduce a short-time kernel for the Beltrami image enhancing flow. The flow is implemented by "convolving" the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. The kernel is appropriate for smoothing regular (flat) 2-D images, for smoothing images painted on manifolds, and for simultaneously smoothing images and the manifolds they are painted on. The kernel combines the geometry of the image and that of the manifold into one metric tensor, thus enabling a natural unified approach for the manipulation of both. Additionally, the derivation of the kernel gives a better geometrical understanding of the Beltrami flow and shows that the bilateral filter is a Euclidean approximation of it. On a practical level, the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the Beltrami flow. In addition, the kernel works with equal ease on regular 2-D images and on images painted on parametric or triangulated manifolds. We demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds.
我们为Beltrami图像增强流引入了一种短时核。该流是通过以类似于用高斯核对热方程进行卷积求解的方式，用与空间相关的核对图像进行“卷积”来实现的。该核适用于对规则（平坦）二维图像进行平滑处理，对绘制在流形上的图像进行平滑处理，以及同时对图像及其绘制所在的流形进行平滑处理。该核将图像的几何结构和流形的几何结构合并到一个度量张量中，从而为两者的处理提供了一种自然统一的方法。此外，核的推导让我们对Beltrami流有了更好的几何理解，并表明双边滤波器是它的欧几里得近似。在实际层面上，与现有的Beltrami流显式数值方案不同，使用该核允许采用任意大的时间步长。此外，该核在规则二维图像以及绘制在参数化或三角化流形上的图像上都能同样轻松地工作。我们通过将其应用于各种类型的图像和流形来展示该核的去噪特性。
