^{}Laboratoire d’Optique P.M. Duffieux, Unité de Recherche Associée 214 CNRS Faculté des Sciences et des Techniques, Université de Franche Comté, Route de Gray 25030, Besançon Cedex, France

J. Duvernoy, "Volterra–Wiener partially coherent imaging systems: three-dimensional objects and generalized G functionals," J. Opt. Soc. Am. A 7, 809-819 (1990)

The partially coherent imaging of a three-dimensional object is described by
introducing Volterra-series representations of the mutual intensity and the
image intensity. According to the Wiener theory of nonlinear systems, an optimal
analysis is then achieved by defining generalized Wiener G
functional (GWGF’s) that characterize the partially coherent transfer of
the mutual spectral density and the image spectral density. These functionals
are thus expanded into series of orthogonal functions in order to facilitate the
computation, and permit the identification, of the GWGF kernels. The bilinear
properties of partially coherent imaging are shown to be solved within the
framework of linear and tensor algebra by involving the image spectral density
vector, the mutual spectral density matrix, and characteristic matrices and
tensors of the imaging system.

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Generalized G-Functional Representation of the Image
Mutual Spectral Density $\stackrel{\sim}{J}({u}_{1},{u}_{2})$ for a Partially Coherent Imaging System
with Volterra Kernels ${\stackrel{\sim}{h}}_{0},{\stackrel{\sim}{h}}_{1}$, and ${\stackrel{\sim}{h}}_{2}$

Generalized G-Functional Representation of the Image
Spectral Density Ĩ(u) for a Partially Coherent
Imaging System with Volterra Kernels ${\stackrel{\sim}{h}}_{0}$, ${\stackrel{\sim}{h}}_{1}$, and ${\stackrel{\sim}{h}}_{2}$

Matrices A^{(0i)},
i = 0,…, 2, third-order tensors
B^{(1j)},
j = 1, 2, and fourth-order tensor
C^{22} are obtained from the orthogonal
representations of the kernels of the GWFG’s. Notice that
D^{(2)} shows a generalized bilinear form
of the object spectral density a.

Table 4

Representation of the Image Spectral Density by Means of the Spectral
Density Vector c^{a}

Vectors a^{(0i)},
i = 0,…,2, matrices
K^{(1j)},
j = 1, 2, and tensor
T^{(22)} are obtained from the orthogonal
representations of the kernels of the GWGF’s. Notice that
c^{(2)} is a generalized bilinear form of
the object spectral density vector a.

Tables (4)

Table 1

Generalized G-Functional Representation of the Image
Mutual Spectral Density $\stackrel{\sim}{J}({u}_{1},{u}_{2})$ for a Partially Coherent Imaging System
with Volterra Kernels ${\stackrel{\sim}{h}}_{0},{\stackrel{\sim}{h}}_{1}$, and ${\stackrel{\sim}{h}}_{2}$

Generalized G-Functional Representation of the Image
Spectral Density Ĩ(u) for a Partially Coherent
Imaging System with Volterra Kernels ${\stackrel{\sim}{h}}_{0}$, ${\stackrel{\sim}{h}}_{1}$, and ${\stackrel{\sim}{h}}_{2}$

Matrices A^{(0i)},
i = 0,…, 2, third-order tensors
B^{(1j)},
j = 1, 2, and fourth-order tensor
C^{22} are obtained from the orthogonal
representations of the kernels of the GWFG’s. Notice that
D^{(2)} shows a generalized bilinear form
of the object spectral density a.

Table 4

Representation of the Image Spectral Density by Means of the Spectral
Density Vector c^{a}

Vectors a^{(0i)},
i = 0,…,2, matrices
K^{(1j)},
j = 1, 2, and tensor
T^{(22)} are obtained from the orthogonal
representations of the kernels of the GWGF’s. Notice that
c^{(2)} is a generalized bilinear form of
the object spectral density vector a.