Hierarchical Recursive Running Median

TL;DR

This paper introduces a new approximately constant-time algorithm for 2D median filtering that outperforms previous methods in speed and complexity, especially for high bit-depth images and hardware without SIMD support.

Contribution

A novel median filtering algorithm with lower theoretical complexity and natural scalability to higher precision data, improving speed for high bit-depth images and hardware constraints.

Findings

Outperforms previous constant-time median algorithms

Scales to higher precision data without modifications

Suitable for hardware without SIMD extensions

Abstract

To date, the histogram-based running median filter of Perreault and H\'ebert is considered the fastest for 8-bit images, being roughly O(1) in average case. We present here another approximately constant time algorithm which further improves the aforementioned one and exhibits lower associated constant, being at the time of writing the lowest theoretical complexity algorithm for calculation of 2D and higher dimensional median filters. The algorithm scales naturally to higher precision (e.g. 16-bit) integer data without any modifications. Its adaptive version offers additional speed-up for images showing compact modes in gray-value distribution. The experimental comparison to the previous constant-time algorithm defines the application domain of this new development, besides theoretical interest, as high bit depth data and/or hardware without SIMD extensions. The C/C++ implementation of the algorithm is available under GPL for research purposes.