Non-local means

⚠️ Curated Limitations & Negative Results

• The discrete Z(i) display in Section 3 repeats v(N_i)−v(N_i), whereas its numerator compares N_i with N_j. Taken literally, that denominator would not normalize the displayed kernels for nonidentical patches. The method here uses the sum of the numerator kernels, consistent with the explicit unit-sum requirement and correctly indexed continuous normalizer on PDF p. 1. This is a documented printed-source discrepancy, not a claimed published author erratum or an audited implementation correction.
• The stated expected noisy patch-distance offset of 2sigma^2 requires normalized spatial weights, distinct corresponding image pixels and independent zero-mean noise of equal variance. Ordinary overlap between different patches correlates some summands but does not itself invalidate that expectation. The self-comparison i=j has distance zero; padding or boundary rules that repeat indices need separate treatment. Expected distance ordering does not imply the same ordering in a particular noisy image, unbiased exponential weights or unbiased restored pixels.
• Section 4 considers center-excluded neighboring values as conditional-expectation covariates and assumes a strictly stationary mixing process, with complete hypotheses and proof referred to an unread external source. Those statements do not establish unconditional consistency of a fixed-bandwidth finite-image implementation whose practical patches include the center. Equality of noisy and clean conditional means needs zero conditional mean of center noise given the surrounding observations; the stated signal-independent, spatially independent, zero-mean Gaussian noise model is sufficient. Squared-error optimality of the exact conditional mean additionally requires finite second moments and does not make every finite kernel estimate optimal.
• Bandwidth, patch size, search set, self-weight convention and image boundaries affect smoothing, bias, noise sensitivity and computation. The exponential formula requires h>0; h=0 is not a defined direct substitution. Finite exact arithmetic gives a positive normalizer with an included self-patch, but numerical underflow and other implementation choices need care. The source’s fixed experimental settings do not specify an automatic noise estimator, a universally optimal parameter choice or a guarantee for all noise distributions.
• This identity uses the original 2005 pixelwise Gaussian-spatially weighted formulation. The later IPOL presentation includes RGB distances, subtraction and clipping of a noise-distance offset, self-weight replacement prose and patchwise overlapping-estimate aggregation; these are not silently incorporated into the original definition. Shared patch similarity does not make the method identical to BM3D transform-domain collaborative filtering, a denoising autoencoder or a learned non-local neural block.
• The finite image demonstrations and reported errors do not establish universal reconstruction quality, benchmark superiority or a general runtime advantage. Evidence uses the exact six-page university-mirrored CVPR 2005 paper and its local page locators. Historical inspection included the later IPOL manuscript as context, but did not include author software, the external consistency proof, the full journal expansion or benchmark reproduction. Bibliographic publication year, PDF build time and actual source access remain distinct.