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According to Parseval's theorem, the variance of a band-limited function can be expressed as an integral (or its discrete equivalent) over its squared Fourier transform:

JBa where Bq denotes a modified version of the resolution domain B, that is, the bounded domain in Fourier space representing the signal information. The modification symbolized by the □ subscript symbol is that the integration exempts the term |P(0)|2 at the origin. (Parseval's theorem expresses the fact that the norm in Hilbert space is conserved on switching from one set to another set of orthonormalized basis functions.)

When we apply this theorem to both signal and additive noise portions of the image, p(r) = o(r) + n(r), we obtain var(o) /bd |O(k)|2|H(k)|2dk var(ra) = Rb |N(k)|2dk

Often the domain, B', within which the signal portion of the image possesses appreciable values, is considerably smaller than B. In that case, it is obvious that low-pass filtration of the image to band limit B' leads to an increased SNR without signal being sacrificed. For uniform spectral density of the noise power up to the boundary of B, the gain in SNR on low-pass filtration to the true band limit B' is, according to Parseval's theorem, equation (3.55), equal to the ratio area{B}/area{B'}. Similarly, it often happens that the noise power spectrum |N(k)|2 is uniform, whereas the transform of the signal transferred by the instrument, | O(k)|21 H(k)|2, falls off radially. In that case, the elimination, through low-pass filtration, of a high-frequency band may boost the SNR considerably without affecting the interpretable resolution.

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