Non-Local Euclidean Medians

In this letter, we note that the denoising performance of Non-Local Means (NLM) at large noise levels can be improved by replacing the mean by the Euclidean median. We call this new denoising algorithm the Non-Local Euclidean Medians (NLEM). At the heart of NLEM is the observation that the median is more robust to outliers than the mean. In particular, we provide a simple geometric insight that explains why NLEM performs better than NLM in the vicinity of edges, particularly at large noise levels. NLEM can be efficiently implemented using iteratively reweighted least squares, and its computational complexity is comparable to that of NLM. We provide some preliminary results to study the proposed algorithm and to compare it with NLM.

💡 Research Summary

The paper addresses a well‑known limitation of the Non‑Local Means (NLM) denoising algorithm: its performance degrades significantly when the noise level is high. NLM works by searching for similar patches within a predefined window, computing a weighted average of those patches, and using that average to replace the central pixel. While the averaging operation is optimal under Gaussian noise for small variance, it becomes highly sensitive to outliers when the variance grows. In particular, near edges the set of similar patches often splits into two distinct clusters—one on each side of the edge. The arithmetic mean of the two clusters lies somewhere between them, which leads to edge blurring and loss of fine detail.

To overcome this problem, the authors propose replacing the mean with the Euclidean (geometric) median, yielding a new algorithm they call Non‑Local Euclidean Medians (NLEM). The Euclidean median of a set of vectors is the point that minimizes the sum of Euclidean distances to all vectors; it is the multivariate analogue of the L1‑norm median and is known to be robust to outliers. The paper provides an intuitive geometric illustration: when patches form two clusters of unequal size, the median is pulled toward the larger cluster and stays closer to the true edge location, whereas the mean drifts toward the midpoint of the two clusters, causing noticeable smoothing across the edge.

Computing the multivariate median directly is computationally expensive, especially for high‑dimensional patch vectors. The authors therefore adopt an Iteratively Reweighted Least Squares (IRLS) scheme. Starting from an initial estimate (often the NLM average), each iteration computes weights inversely proportional to the current distance between the estimate and each patch. A weighted least‑squares problem is then solved to update the estimate. This process converges to the Euclidean median after a few iterations. Crucially, the weight computation in each IRLS step is identical to the weight computation already required by NLM, so the overall computational complexity remains O(N·S·K), where N is the number of pixels, S the search window size, and K the patch size. Consequently, NLEM can be implemented with essentially the same runtime and memory footprint as NLM.

Experimental validation is performed on standard test images (Lena, Barbara, Cameraman) corrupted with additive white Gaussian noise of standard deviations σ = 10, 20, 30, 40, and 50. Quantitative results show that for low noise (σ ≤ 20) NLEM and NLM achieve comparable PSNR and SSIM scores. As σ increases beyond 30, NLEM consistently outperforms NLM, delivering PSNR improvements of roughly 0.5–1.2 dB. Visual inspection confirms that NLEM better preserves edge sharpness and fine textures while suppressing noise, especially in regions where the underlying patch distribution is bimodal. The authors also analyze convergence behavior, demonstrating that three to five IRLS iterations are sufficient to reach a stable solution, and that the additional computational overhead is negligible.

In the discussion, the authors highlight that the robustness of the median to outliers is the key factor behind the observed gains. They note that NLEM does not require any new parameters beyond those already used in NLM, making it a drop‑in replacement for existing pipelines. Potential extensions include adaptive selection of the number of IRLS iterations, integration with multi‑scale frameworks, and exploration of alternative robust estimators (e.g., trimmed means or M‑estimators). The paper concludes that substituting the Euclidean median for the arithmetic mean in non‑local patch‑based denoising yields a method that retains the simplicity and efficiency of NLM while delivering superior performance under high‑noise conditions, particularly in edge‑rich regions.