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The point spread function (PSF) of an imaging system is closely related to key incoherent image-transfer properties of the system, i.e. the optical transfer function (OTF) and the modulation transfer function (MTF). This relationship emerges as follows. In the theory of linear shift-invariant optical systems (for example, Goodman JW 1985), the PSF is the normalized system’s response to a point light source:
| PSF(xI -xI ' , yI -yI ') = CN Π [ L δ(xO -xO' , yO -yO') ] | (1) |
where CN is a normalization constant, Π is a projection operator representing the imaging system, L is the radiance of light emitted by the point source, δ is the delta Dirac function (for example, http://mathworld.wolfram.com/DeltaFunction.html), the subscripts I and O denote the image and object planes, respectively, and the object plane point (xO' , yO'), where the point source is located, is mapped by the imaging system to the image point (xI ' , yI '). The normalization constant, CN [W-1], is obtained by setting ∫APSF dA to unity, where A is the area of the image and dA is the elementary area of that image (see also Point spread function and imaging in turbid medium).
Various types of the point light source can be considered (Gordon HR 1994). Here it is assumed here that the point source is lambertian. Hence, the radiance of the point source is independent of direction.
Consider now a lambertian object, represented by a radiance distribution, LO (xO , yO ), in the object plane. With a perfect imaging system (working in a scatterless medium), represented by a projection operator, Π0, it is imaged into a "perfect" irradiance distribution, EIp
| EIp (xI , yI ) = Π 0[ LO (xO , yO ) ] | (2) |
As discussed in Point spread function and imaging in turbid medium, the image of an object, represented by irradiance distribution, EI , in the image plane can be expressed as a convolution:
| EI = EIp * PSF | (3) |
Consider the spatial frequency represention of image EI , i.e., a two-dimensional Fourier transform, EI, F (u, v), of the image, where u and v, are the spatial frequencies in the x, and y directions, respectively. According to the convolution theorem (for example, http://mathworld.wolfram.com/ConvolutionTheorem.html), one can express EI, F as follows:
| EI, F (u, v ) = EIp, F ( u, v ) PSFF ( u, v ) | (4) |
where the subscript F denotes the Fourier transform. Thus, the Fourier transform of PSF, i.e. PSFF, performs the role of a system transfer function in the spatial frequency domain. When PSFF is normalized as follows:
| OTF ( u, v ) = PSFF ( u, v ) / PSFF ( 0, 0 ) | (5) |
it is called the optical transfer function (OTF; for example, Williams CS and Becklund 2002, p. 39, Goodman JW 1996, pp. 131-141). Hence, the OTF and PSF are effectively a Fourier transform pair.
The OTF is, in general, a complex quantity that can be represented as follows:
| OTF ( u, v ) = MTF ( u, v ) exp[ i PTF ( u, v ) ] | (6) |
where MTF is the modulation transfer function and PTF is the phase transfer function. Given that MTF ≡ | OTF |, it modulates the power spectrum of spatial frequencies of the image. Indeed, the power spectrum, P(g), of function g is defined (for example, Bracewell RN 2000):
| P ( g ) = | gF |2 | (7) |
where gF is the Fourier transform of g. Hence, using equations 4, 5, and 6, one has:
| P ( EI, F ) | = | EIp, F ( u, v ) PSFF ( u, v ) |2 | |
| = P ( EIp, F ) | OTF ( u, v ) |2 | PSFF ( 0, 0 ) |2 | ||
| = P ( f ) MTF 2 ( u, v ) | PSFFT ( 0, 0 ) |2 | (8) |
When a turbid medium produces an axially symmetrical PSF (which is typically the case for media such as natural waters and the atmosphere), the two-dimensional Fourier transform of the PSF reduces to the Hankel transform (a one-dimensional transform, for example, Goodman JW 1996, pp. 10-11). Given that symmetry, the phase transfer function (Equation 6) vanishes (for example, Kabanov MV 1968) and the OTF becomes real and identical to MTF. Note that the MTF is sometimes referred to as the OTF in the light-scattering literature (for example, Zege EP and Kokhanovsky 1994).
| CITATION: Jonasz M. 2009. Point spread function vs. optical and modulation transfer functions (www.tpdsci.com/Tpc/ImgTmPsfOtf.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 12-Jan-2009 Modified: 02-Jan-2012 Peer-reviewed: 12-Feb-2009 |
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