An eigenvalue analysis of the noise-prone image leads to (a) an analysis of the eigenfunctions and eigenvalues of the sin<sup>2</sup>(<i>x</i>)/<i>x</i><sup>2</sup> kernel; and (b) an expression relating an effective number <i>N</i><sub>eff</sub> of degrees of freedom directly to the signal-to-noise ratio σ<sub>0</sub>/σ<sub><i>v</i></sub>. The latter are the variances of object and noise, respectively. For the particular case of incoherent, diffraction-limited imagery, <i>N</i><sub>eff</sub> is found to be reduced from its noise-free value, the Shannon number, by the factor (1-σ<sub><i>v</i></sub>/σ<sub>0</sub>). A maximum number <i>N</i><sub>max</sub> of degrees of freedom is also defined. Comparing one-dimensional objects illuminated alternatively by coherent and incoherent light, we find they have the same number <i>N</i><sub>max</sub> of degrees of freedom. However, for the corresponding two-dimensional case, the incoherent value for <i>N</i><sub>max</sub> is double that of the coherent value.
M. Bendinelli, A. Consortini, L. Ronchi, and B. Roy Frieden, "Degrees of freedom, and eigenfunctions, for the noisy image," J. Opt. Soc. Am. 64, 1498-1502 (1974)