Derivation of Noise Correlations across Scales

In this section we provide a detailed derivation of the expected noise correlations across scales, E[< nsc, n >], referred to in Section 3 of the paper. For simplicity, let us first derive the expression for the 1D case. Let us denote a 1D noise patch, containingM = 2m+1 pixels and centered around a specific coordinate l in the noisy image N by: n = (Nl−m, ..., Nl+m). Let us denote by lsc the same relative coordinate at coarser scale sc and the respective patch centered around lsc coordinate in Nsc by: nsc = (Nsc,lsc−m, ..., Nsc,lsc+m). The noise image Nsc results from blurring and subsampling the original noise image N . Fig. 1.a (see next page) displays the creation of a coarse scale (e.g., half the original scale) from the fine scale via blur and subsample (the 3-tap blur kernel shown in the figure is for simplicity of illustration only). To illustrate the effect of blurring, each coarser pixel is colored according to the relative contribution (weight) of the original fine-scale pixels that were involved in the creation of this coarser pixel (e.g., the middle coarse pixel = 0.25·red pixel +0.5·green pixel+0.25·blue pixel). The different weights are illustrated by the different sizes of colored areas within each coarse pixel. Fig. 1.b (see next page) displays the pixel-wise correlation between a patch n in the original scale (the red dashed rectangle) and the coarse patch nsc (the blue dashed rectangle). For simplicity, l = 0. Assuming that the noise pixels in the original scale are independent of each other (they are sampled from i.i.d. Gaussian noise), the correlation between a pixel n(i) from the original patch and the coarse pixel nsc(i) is non-zero only if n(i) took part in the creation of nsc(i). In this example, the fine-scale cyan pixel n(−2) has zero correlation to the coarse-scale pixel nsc(−2), since it was not involved in its creation (no cyan color in nsc(−2)). The same holds for the fine-scale gray pixel n(2) versus the coarse-scale pixel nsc(2). On the other hand, the fine-scale green pixel n(0) has high correlation to nsc(0), since it had a high weight in the creation of nsc(0) (and accordingly, the largest portion of nsc(0) is colored green). The fine-scale red pixel n(−1) has a little correlation to nsc(−1) (and the same holds for the fine-scale blue pixel n(1) versus nsc(1)). More formally: due to the blurring (prior to subsampling), any single pixel in a coarser scale is a linear combination of pixels from the original scale. Therefore, Nlsc+i = Σkα lsc+i k Nk, where {Nk} are pixels in the original scale (k is a general spatial coordinate in the original scale), and {αsc k } are their respective weights in the blur process. Using the above relation and the linearity of the expectation operator: