Generally, the unknown signal is mixed with noise, so the measurement of the SNR of an experimental image is not straightforward. Two ways of measuring the SNR of "raw" image data have been put forth: one is based on the dependence of the sample variance [equation (3.53)] of the average image on N (= the number of images averaged) and the other on the cross-correlation of two realizations of the image.
18.104.22.168. N-dependence of sample variance We assume that the noise is additive, uncorrelated, stationary (i.e., possessing shift-independent statistics), and Gaussian; and further, that it is uncorrelated with the signal (denoted by p). In that case, the variance of a "raw" image pt is, independently of i, var( p^ = var( p) + var(n) (3-57)
The variance of the average p^ of N images is var( p[N]) = var( p) + ^var(n) (3-58)
that is, with increasing N, the proportion of the noise variance in the variance of the average is reduced. This formula suggests the use of a plot of var(p^) versus 1/N as a means to obtain the unknown quantities var(p) and var(n) (Hanicke, 1981; Frank et al., 1981a; Hanicke et al., 1984). If the assumptions made at the beginning are correct, the measured values of var(p^) should lie on a straight line whose slope is the desired quantity var(n) and whose intersection with the var(p) axis (obtained by extrapolating it to 1/N = 0) gives the desired quantity var(p). Figure 3.20 shows such a variance plot obtained for a set of 81 images of the negatively stained 40S ribosomal subunit of HeLa cells (Frank et al., 1981a). It is seen that the linear dependency predicted by equation (3.58) is indeed a good approximation for this type of data, especially for large N.
22.214.171.124. Measurement by cross-correlation Another approach to the measurement of the SNR makes use of the definition of the cross-correlation coefficient (CCC). The CCC of two realizations of a noisy image, py and pky, is defined as [see equation (3.16)]
E [ py -<pi)][ pkj -<pk)] P12 = (-—--j/2 (3-59)
where <pi) and <pk) again are the sample means defined in the previous section.
When we substitute py = py + ny, pky = pk + nky, and observe that according to the assumptions both noise functions have the same variance var(n), we obtain the simple result (Frank and Al-Ali, 1975):
1 + a from which the SNR is obtained as
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