Efficient Beltrami Flow in Patch-Space Aaron Wetzler and Ron Kimmel Department of Computer Science, Technion, Israel Abstract. The Beltrami framework treats images as two dimensional manifolds embedded in a joint features-space domain. This way, a color image is considered to be a two dimensional surface embedded in a hybrid spacial-spectral five dimensional {x, y, R, G, B} space. Image selective smoothing, often referred to as a denoising filter, amounts to the process of area minimization of the image surface by mean curvature flow. One interesting variant of the Beltrami framework is treating local neighbor- ing pixels as the feature-space. A distance is defined by the amount of deformation a local patch undergoes while traversing its support in the spatial domain. The question we try to tackle in this note is how to per- form patch based denoising accurately, and efficiently. As a motivation we demonstrate the performance of the Beltrami filter in patch-space, and provide useful implementation considerations that allow for param- eter tuning and efficient implementation on hand-held devices like smart phones. Keywords: Beltrami flow, patch-space, denoising 1 Introduction Following the success of the Non Local Means denoising method as introduced by Buades et al. in [2] much attention has been devoted to developing various types of patch based denoising techniques. A patch, in terms of an image, is generally considered to be a square region of pixels of fixed size centered at the coordinates of an image pixel. Peyr` e in [6] studies patch based manifolds while a more specific analysis of a generalized patch based denoising framework is done by Tschumperl` e and Brun in [12]. They show that the NL means [2] and Bilat- eral [11] filters are isotropic versions of their patch based diffusion framework by choosing a specific patch size and metric. In much the same way Sochen et al. present the Beltrami framework and show in [8] how choices of different metrics can be used to produce filtering methods like the anisotropic diffusion process of Perona and Malik [5] as an example. Anisotropic diffusion was also shown by Barash in [1] to have a strong connection to the Bilateral filter through the adaptive smoothing filter and Elad in [4] demonstrated its connection to other classical filtering techniques. In [7] Maragos and Roussos, explore a generalization of the Beltrami flow using weighted patches. We will use a similar formulation while setting the weights of