Branchless vectorized median filtering

Median filtering is an important tool in signal or image processing. Based on the vector capabilities of modern hardware, which allows for vectorized min, max and mask operations, we provide a median algorithm of complexity O(NM) that is both branchless and vectorized. In contrast to conventional fast median filters, whose run-time is data-dependent and that can operate only on scalar data, its runtime is predictable and data-independent and it can simultaneously operate on several one-dimensional signals thereby making use of data-level parallelism. Our branchless vectorized median (BVM) filter keeps track of a sorted array from which values are deleted and to which new values are inserted. As a spin-off effect we could also use our work to provide a data sorting algorithm that is branchless and vectorized. Although it is of O(M²) computational complexity while other sort algorithms are of O(M lnM) computational complexity, at least for typical data, it may outperform other implementations for small M. This is mainly due to the fact that we have no unexpected branches which would stall the instruction pipeline and that we can simultaneously operate on vectors of data. Here, however, we focus on median filtering. BVM is compared to a median filter based on doubly linked lists (DLL) and to the median implementation of the Intel Performance Primitives (IPP) library. The comparison uses constant data and linear data, which are two unrealistic settings, and random data, which is more realistic. For constant data IPP is up to five times faster than BVM whose run-time does not depend on the data themselves. However, regarding the realistic case of median filtering noisy data BVM outperforms DLL by about a factor of 1.5 and IPP by about a factor of 2 in our CPU-based implementations, which is far more than we did expect initially.