The staircasing effect in neighborhood filters and its solution

• Published in: IEEE transactions on image processing 2006-06, Vol.15 (6), p.1499-1505
• Main Authors: Buades, A., Coll, B., Morel, J.-M.
• Format: Article
• Language: English
• Subjects: Algorithms; Applied sciences; Artificial Intelligence; Color; Colored noise; Computer Science; Detection, estimation, filtering, equalization, prediction; Difference equations; Digital filters; Exact sciences and technology; Filtering algorithms; Filtration - methods; Image denoising; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image Processing; Image restoration; Imaging, Three-Dimensional - methods; Information Storage and Retrieval - methods; Information, signal and communications theory; Low pass filters; Mathematical models; Nonlinear equations; Nonlinear filtering and enhancement; Partial differential equations; Reproducibility of Results; restoration; Sensitivity and Specificity; Signal and communications theory; Signal processing; Signal Processing, Computer-Assisted; Signal to noise ratio; Signal, noise; Staircases; Telecommunications and information theory
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2006.871137

Many classical image denoising methods are based on a local averaging of the color, which increases the signal/noise ratio. One of the most used algorithms is the neighborhood filter by Yaroslavsky or sigma filter by Lee, also called in a variant "SUSAN" by Smith and Brady or "Bilateral filter" by Tomasi and Manduchi. These filters replace the actual value of the color at a point by an average of all values of points which are simultaneously close in space and in color. Unfortunately, these filters show a "staircase effect," that is, the creation in the image of flat regions separated by artifact boundaries. In this paper, we first explain the staircase effect by finding the subjacent partial differential equation (PDE) of the filter. We show that this ill-posed PDE is a variant of another famous image processing model, the Perona-Malik equation, which suffers the same artifacts. As we prove, a simple variant of the neighborhood filter solves the problem. We find the subjacent stable PDE of this variant. Finally, we apply the same correction to the recently introduced NL-means algorithm which had the same staircase effect, for the same reason.