R. Walraven, “Polarization imagery,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. SPIE112, 164–167 (1977).
The four Stokes parameters can be determined (up to an ambiguity in sign of one of the parameters) with only three intensity measurements for monochromatic radiation.22 Because circular polarization and unpolarized light are indistinguishable with a system that detects only linear polarization states, their effects on the final representation should be similar, and partially circularly polarized monochromatic radiation is analyzed here for mathematical clarity.
This representation of the incident field is exact up to the choice of a constant. Although the linearly polarized portion of the radiation is constrained to have unit amplitude here, in general it can have arbitrary amplitude. In the general case, A is the ratio between the amplitudes of the circularly polarized and linearly polarized portions of the incident radiation.
Nonbirefringent, nonoptically active photoreceptors are considered for simplicity, since most linear polarization analyzers satisfy this requirement. In a biological PVS, the photoreceptors may in fact be birefringent or optically active, but this can be accounted for by adding appropriate phase delay terms to the P and Q matrices (birefringence) or adding off-diagonal terms to these matrices (optical activity). As mentioned in Section 2, to sense the complete state of polarization, at least one birefringent receptor is needed either to yield the intensity of a circular polarization state or to give a relative phase value.
The matrix given by Eq. (19) [as well as Eq. (10)] is real symmetric and therefore an example of a self-adjoint matrix. Self-adjoint M×M matrices always produce M orthogonal eigenvectors that span the vector space operated on by the matrix.20 Correlation matrices like the ones treated here are always self-adjoint, even if the random variables are complex.16 The fact that any M×M correlation matrix necessarily spawns M orthogonal eigenvectors is the basis of principal-components analysis.18
In their study Bernard and Wehner13 assumed that the output of the individual receptors was proportional to the logarithm of the input. In this investigation the receptors are assumed to respond linearly within a particular range of intensities.
J. S. Tyo, “Polarization-difference imaging: a means for seeing through scattering media,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, 1997).
Photoreceptors actually capture radiation in discrete quanta; however, if the intensity of the radiation is strong enough, the discretization can be ignored, and the above formulation can be applied.
S. M. Kay, Modern Spectral Estimation: Theory And Application (Prentice-Hall, Englewood Cliffs, N.J., 1988).
In this section the ensemble is assumed to be uniformly distributed in angle of polarization with a fixed ellipticity described by the parameter ∊. These results can be easily generalized to include other ensembles by simply altering the pdf’s used to calculate the entries in the correlation matrix.
T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1984), Chap. 11.
A. Bermann, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic, New York, 1979), Chap. 4.
C. E. Shannon, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).
C. H. Papas, Theory of Electromagnetic Radiation (Dover, New York, 1988), pp. 118–134.
Although choosing N>3 does not seem to make sense in the context explored herein, some simple, fast, nonlinear systems may be proposed that operate optimally with N>3.
G. F. J. Garlick, C. A. Steigman, W. E. Lamb, “Differential optical polarization detectors,” U.S. patent3,992,571 (November16, 1976).
G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-I–A-III-10 (1966).
L. B. Wolff, T. A. Mancini, “Liquid crystal polarization camera,” in Proceedings of the IEEE Workshop on Applications of Computer Vision (IEEE, New York, 1992), pp. 120–127.