A convergence proof for local mode filtering

In this paper, we present a convergence proof for an iterative procedure of local mode filtering. We formulate the filtering as quadratic optimization problem based on Legendre transform convex function, from which two closed-form expressions at each iteration step are derived variables to be optimized. Those analytical solutions ensure that value objective function increases monotonically with progress procedure. also show experimental results using grayscale image, support our theoretical practically.