Topics in Particle and Dispersion Science

Point spread function and imaging in turbid medium

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The point spread function (PSF) of turbid medium, as defined by Mertens LE and Replogle 1977, is the contribution, due to light scattering by turbid medium, to the point spread function of an imaging system. Consider a linear and shift-invariant imaging system with the object space filled by a turbid medium characterized by PSF (ξ, r), where ξ is the unit vector of direction and r is the distance from the point source (Figure 1). As discussed in the caption of that figure, an unresolvable "point source" creates an irradiance distribution in the image plane

EI [PI (ξ)] = TL PSF(ξ - ξO) IO, max ΩLI

(1)

where P_I (ξ) is the point at the intersection of the image plane and direction ξ, T_L is the lens transmission, I_{O, max} is the maximum intensity at the object point P_O, and Ω_LI  is the solid angle subtended by the lens of the imaging system at the image. For simplicity, the paraxial imaging regime is assumed. Thus, variations of Ω_LI  with the projection angle can be neglected. Moreover, the PSF(ξ, r) can be taken to be the same for all points in the object. Hence, the distance argument, r, can be dropped. A contribution of the imaging system itself to its point spread function (PSF) is also neglected.

Note that Equation 1 implies that the irradiance distribution at the image plane is a convolution of the PSF and the irradiance distribution that would be created by a perfect imaging system working in a non-scattering medium.

By normalizing to unity the integral of E_I  over the entire image one can define an effective point spread function (PSF) of the imaging system in question:

CN ∫A EI [PI (ξ)] dAI	=  ∫APSFeff [PI (ξ)] dAI
	= 1	(2)

where C_N  is a normalization constant and A is the area of the image. Note that although such a point spread function is closely related to PSF, it is not identical with the latter due to a different (incomplete) normalization condition (see also Beam spread function: Definition). Here, the PSF is taken to be due solely to the presence of the turbid medium. However, in general, the PSF can account for imperfections of the imaging system and diffraction of light.

Consider now the image, formed by this imaging system, of an object with each elementary area, dA_O (Figure 1), being a unresolvable lambertian light source. In the absence of the turbid medium, light emitted by this object creates at the lens of the imaging system a directional radiance distribution, L_O (ξ) = ~E(P_O ) δ(ξ - ξ₀), where δ is the delta Dirac function (for example, http://mathworld.wolfram.com/DeltaFunction.html) of direction. The lens converts this radiance distribution into an irradiance distribution, E_IO, at the image plane as follows:

EIO [PI (ξ)] = TL LO (ξ, r) ΩLI

(3)

Introduction of a turbid medium into the object space of the imaging system blurs that "perfect" image of the object, because each elementary area, dA_O, of the object does not anymore contribute only to the corresponding image point but also to nearby image points (Figure 1). This blurring can be expressed as a convolution (for example, http://mathworld.wolfram.com/Convolution.html) of irradiance distribution E_IO with the PSF of the imaging system:

EIT [PI (ξ)]	=  ∫ EIO [PI (ξ)'] PSF (ξ - ξ', r) dAI [PI (ξ)']
	= EIO * PSF	(4)

where * denotes the operation of convolution. The effects of the imaging system imperfections and diffraction, PSF_S, can be combined with those of the the turbid medium, PSF, as follows:

EIT	= ( EIO * PSF ) * PSFS
	= ( EIO * PSFS ) * PSF
	= EIO * ( PSFS * PSF )	(5)

due to the associativity and commutativity of convolution (see also Point and beam spread functions of seawater).

CITATION: Jonasz M. 2009. Point spread function and imaging in turbid medium (www.tpdsci.com/Tpc/ImgTmPsfImgTbm.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 12-Jan-2009 Modified: 18-Jun-2009 Peer-reviewed: 12-Feb-2009

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