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Fig. 1. Point spread function (PSF) and imaging in turbid medium. For simplicity, (1) the refractive index of the object space equals that of the image space, and (2) the paraxial imaging regime is assumed, i.e. cos(projection angle) ≈ 1. Distance f is the focal length of the lens. The subscript O stands for the object, I stands for the image.
Consider a lambertian object, whose elementary area, dAO at PO, generates radiance LO, represented by a red semi-circle centered at PO. Each such area is taken to be the “unresolvable point source”. Its size can be operationally defined by reversely projecting onto the object plane the elementary area (nominal resolution), dAI, of an imaging device, located at the image plane (see also The size of an incoherent point source). In the absence of a turbid medium, dAO is imaged into dAI . If all other points in the object plane are dark, dAI is the only elementary area of the image plane that is illuminated.
When a turbid medium is present in the object space and only dAO emits light, the entire image plane is illuminated to a varying degree, with the maximum at PO(ξO), where ξO is the unit vector that indicates a direction from PO to the lens (projection) center. The turbid medium converts, via multiple scattering (see Single and multiple scattering), the angular radiance distribution at PO into an angular distribution of radiance, LT (ξ, r), at the lens, where ξ is the direction unit vector and r is the distance of the lens from the object plane. This radiance distribution, when normalized by the maximum intensity, I0(PO) = IO, max , emitted by dAO, is the point spread function (PSF) of turbid medium.
Radiance distribution, LT , is converted by the lens into an irradiance distribution, EI [PI(ξ)] = TL LT (ξ) ΩLI = TL PSF(ξ - ξO) IO, max ΩLI , where TL is the lens transmission, and ΩLI is the solid angle that the lens subtends at the image plane. Consider the Cartesian coordinates in the image plane, relative to PI (ξO), i.e. x’I = xIO - xI and y’I = yIO - yI . It follows that EI , when expressed in these coordinates and normalized by a factor of TL IO, max ΩLI , equals the point spread function of the turbid medium (PSF). The inherent point spread function of the imaging system, observed even in the absence of the turbid medium, is neglected for simplicity.
| CITATION: Jonasz M. 2009. Point spread function and imaging in turbid medium: Geometry (www.tpdsci.com/tpc/ImgTmPsfImgTbmFig.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 19-Jan-2009 Modified: 27-Jun-2009 Peer-reviewed: 12-Feb-2009 |
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