Degrees of freedom, and eigenfunctions, for the noisy image
M. Bendinelli, A. Consortini, Laura Ronchi, B. Roy Frieden · Journal of the Optical Society of America · 1974
An eigenvalue analysis of the noise-prone image leads to (a) an analysis of the eigenfunctions and eigenvalues of the sin2(x)/x2 kernel; and (b) an expression relating an effective number Neff of degrees of freedom directly to the signal-to-noise ratio σ0/σv. The latter are the variances of object and noise, respectively. For the particular case of incoherent, diffraction-limited imagery, Neff is found to be reduced from its noise-free value, the Shannon number, by the factor (1 − σv/σ0). A maximum number Nmax 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 Nmax of degrees of freedom. However, for the corresponding two-dimensional case, the incoherent value for Nmax is double that of the coherent value.