Abstract

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.

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