Efficient Beltrami Flow in Patch-Space

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 special-spectral five dimensional $\left\{ x,y,R,G,B\right\}$ 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 neighboring 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 perform 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 parameter tuning and efficient implementation on hand-held devices like smart phones.