Image smoothing viaL₀gradient minimization

Abstract

We present a new image editing method, particularly effective for sharpening major edges by increasing the steepness of transition while eliminating a manageable degree of low-amplitude structures. The seemingly contradictive effect is achieved in an optimization framework making use of L 0 gradient minimization, which can globally control how many non-zero gradients are resulted in to approximate prominent structure in a sparsity-control manner. Unlike other edge-preserving smoothing approaches, our method does not depend on local features, but instead globally locates important edges. It, as a fundamental tool, finds many applications and is particularly beneficial to edge extraction, clip-art JPEG artifact removal, and non-photorealistic effect generation.

Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional

“…In contrast, results of the L 0 -Smoothing always have smooth variations, which can only be avoided at considerable increase of run time. Also, [21] approximates (1) worse and worse when increasing the edge set penalization . This leads to artifacts such failing to get rid of small scale structures (grass) despite an overall smooth solution.…”

“…The L 0 -Smoothing Approach of Xu et al For the piecewise constant case, Xu et al [21] recently proposed a fast approximating method. Assuming the image domain has been discretized into a finite rectangular grid, again denoted by ⌦, the piecewise constant Mumford-Shah limit corresponds to L 0 penalization of the gradient:…”

“…Similar to [21] we work with an already discretized image domain ⌦. We propose to minimize the energy…”

“…Applying this regularizer in (6) yields piecewise constant approximations of the input image. This model is the same as (3) which is considered in [21]. Note that in contrast to convex relaxation methods [16,18] we do not need to discretize the range of u : ⌦ !…”

“…The method [21] is devised for the piecewise constant case ↵ = 1, but the solutions depend on a parameter  > 1. To reduce smooth variations, it must be chosen near 1, which increases the run time significantly.…”