Spreading Function

Spreading Function

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Surfaces Level 2 Spreading Function Effects

Figure 1: The cosine-2S spreading functions for
S = 2
and 20. Top panel: polar plot in φ ;
bottom panel: linear in φ .

Figure 2: A sea surface generated with the ECKV
omnidirectional spectrum and a cosine-2S spreading function with
S = 2 .
Compare with Fig. 3 . Generated by IDL routine cgPlot2Dsurf_3D.pro.
Table 1: Comparison of Cox-Munk mean square slopes and values for the
DFT-generated 2-D surface of Fig. 2 .
Figure 3: A sea surface generated with the ECKV
omnidirectional spectrum and a cosine-2S spreading function with
S = 2 0 .
Compare with Fig. 2 .

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Ocean Optics Web Book • All contents 2020 Creative Commons Attribution license.

Page updated:
March 22, 2021 Author: Curtis Mobley View PDF

The last ﬁgure on the previous page shows a contour plot of a two-dimensional, one-sided
variance spectrum Ψ 1 s ( k x , k y )
and a contour plot of a random surface generated from that variance spectrum. A
particular spreading function is implicitly contained in that two-dimensional variance
spectrum. The eﬀect on the generated sea surface of the spreading function contained
within Ψ 1 s ( k x , k y )
warrants discussion.

As we have seen (e.g. Eq. 4 of the Wave Variance Spectra: Examples page), a
2-D variance spectrum is usually partitioned as
Here 𝒮 ( k ) is the omnidirectional
spectrum, and Φ ( k , φ )
is the nondimensional spreading function, which shows how waves of diﬀerent
frequencies propagate (or “spread out”) relative to the downwind direction at
φ = 0 .

One commonly used family of spreading functions is given by the
“cosine-2S” functions of Longuet-Higgins et al. (1963) , which have the form
where S
is a spreading parameter that in general depends on
k , wind speed,
and wave age. C S
is a normalizing coeﬃcient that gives
Figure 1 shows the cosine-2S spreading functions for values of
S = 2
and 20. These spreading functions are strongly asymmetric in
φ , so
that more variance (wave energy) is associated with downwind directions
( | φ | < 9 0 d e g ) than upwind
( | φ | > 9 0 d e g ). The larger
the value of S ,
the more the waves propagate almost directly downwind
( φ = 0 ),
rather than at large angles relative to the downwind direction.
However, the cosine-2S spreading functions always have a least a tiny
bit of energy propagating in upwind directions, as can be seen for the
S = 2
curves. This is crucial for the generation of time-dependent surfaces, as will be
discussed on the next page.

Figure 2 shows a surface generated with the omnidirectional
variance spectrum of Elfouhaily et al. (1997) (ECKV) as used on the
previous page, combined with a cosine-2S spreading function ( 1 ) with
S = 2 for all
k values. The wind
speed is 1 0 m s − 1 . The
simulated region is 1 0 0 × 1 0 0 m
using 5 1 2 × 5 1 2
grid points. Note in this ﬁgure that the mean square slopes (mss) compare
well with the corresponding Cox-Munk values shown in Table . The mss
values for the generated surface are computed from ﬁnite diﬀerences, e.g.

averaged over all points of the 2-D surface grid. The
⟨ 𝜃 x ⟩ and
⟨ 𝜃 y ⟩ values
shown in the ﬁgure are the average angles of the surface from the horizontal in the
x and
y
directions). These are computed from
etc., averaged over all points of the surface.

The spreading function used in Fig 2 was chosen (with a bit of trial and error)
to give a distribution of along-wind and cross-wind slopes in close agreement with
the Cox-Munk values (except for a small amount of Monte-Carlo noise). Figure
3 shows a surface generated with a cosine-2S spreading function with
S = 2 0 ;
all other parameters were the same as for Fig. 2 . This
S value
gives wave propagation that is much more strongly in the downwind direction
φ = 0 , as
would be expected for long-wavelength gravity waves in a mature wave ﬁeld. The
surface waves thus have a visually more “linear” pattern, whereas the waves of Fig. 2
appear more “lumpy” because waves are propagating in a wider range of angles
φ from
the downwind.

As shown in on the Wave Variance Spectra: Theory page , the
total mean square slope depends only on the omnidirectional spectrum
𝒮 ( k ) . Thus
the total mss is the same (except for Monte Carlo noise) in both ﬁgures 2
and 3 , but most of the total slope is in the along-wind direction in Fig.
3 .

Real spreading functions are more complicated than the cosine-2S functions
used here. In particular, some observations Heron (2006) of long-wave gravity
waves tend to show a bimodal spreading about the downwind direction, which
transitions to a more isotropic, unimodal spreading at shorter wavelengths. Although
omnidirectional wave spectra are well grounded in observations, the choice of a
spreading function is still something of a black art. You are free to choose any
Φ ( k , φ ) so
long as it satisﬁes the normalization condition ( 2 ).

Ψ ( k x , k y ) = 1
k 𝒮 ( k ) Φ ( k , φ ) ≡ Ψ ( k , φ ) .

Φ ( k , φ ) = C S cos 2 S ( φ ∕ 2 ) ,

m s s x ( r , s ) = z ( r + 1 , s ) − z ( r , s )
x ( r + 1 ) − x ( r )

𝜃 x ( r , s ) = tan ( m s s x ( r , s ) ) ,

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An Experimental High Fidelity Perceptual Audio Coder
Implemented Model for Spreading Function
Click for http://ccrma.stanford.edu/~jos/bbt/Bark_Frequency_Scale.html
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An Experimental High Fidelity Perceptual Audio Coder
Implemented Model for Spreading Function

In [ 5 ][ 6 ] a special masking experiment is presented;
Two narrowband noise maskers, with a frequency difference ,
are masking a tone in between if the tone has low enough intensity.
The other way around is also tested, where two tones mask noise.
The experiment shows that the masking from the two maskers is approximately
constant as long as the is less than the critical bandwidth at
that frequency. Thus, a masker can mask almost uniformly within the
critical band.

Masking does not only occur within the critical band,
but also spreads to neighboring bands. A spreading function SF ( z , a )
can be defined, where z is the frequency and a the amplitude of a
masker. This function would give a masking threshold produced by a sinle
masker for neighboring frequencies. The simplest function would be a triangular
function with slopes of +25 and -10 dB / Bark, but a more sophisticated one
is highly nonlinear and depends on both frequency and amplitude of masker.

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^ Progress in Optics . Elsevier. 2008-01-25. p. 355. ISBN 978-0-08-055768-7 .

^ Jump up to: a b Ahi, Kiarash; Anwar, Mehdi (May 26, 2016). Anwar, Mehdi F; Crowe, Thomas W; Manzur, Tariq (eds.). "Developing terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolution" . Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N . Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense. 9856 : 98560N. Bibcode : 2016SPIE.9856E..0NA . doi : 10.1117/12.2228680 . S2CID 114994724 .

^ Ahi, Kiarash; Anwar, Mehdi (May 26, 2016). Anwar, Mehdi F; Crowe, Thomas W; Manzur, Tariq (eds.). "Modeling of terahertz images based on x-ray images: a novel approach for verification of terahertz images and identification of objects with fine details beyond terahertz resolution" . Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N . Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense. 9856 : 985610. Bibcode : 2016SPIE.9856E..10A . doi : 10.1117/12.2228685 . S2CID 124315172 .

^ Ahi, Kiarash; Shahbazmohamadi, Sina; Asadizanjani, Navid (July 2017). "Quality control and authentication of packaged integrated circuits using enhanced-spatial-resolution terahertz time-domain spectroscopy and imaging" . Optics and Lasers in Engineering . 104 : 274–284. Bibcode : 2018OptLE.104..274A . doi : 10.1016/j.optlaseng.2017.07.007 .

^ Ahi, K. (November 2017). "Mathematical Modeling of THz Point Spread Function and Simulation of THz Imaging Systems". IEEE Transactions on Terahertz Science and Technology . 7 (6): 747–754. Bibcode : 2017ITTST...7..747A . doi : 10.1109/tthz.2017.2750690 . ISSN 2156-342X . S2CID 11781848 .

^ Light transmitted through minute holes in a thin layer of silver vacuum or chemically deposited on a slide or cover-slip have also been used, as they are bright and do not photo-bleach.
S. Courty; C. Bouzigues; C. Luccardini; M-V Ehrensperger; S. Bonneau & M. Dahan (2006). "Tracking individual proteins in living cells using single quantum dot imaging" . In James Inglese (ed.). Methods in Enzymology: Measuring biological responses with automated microscopy, Volume 414 . Academic Press. pp. 223–224 . ISBN 9780121828196 .

^
P. J. Shaw & D. J. Rawlins (August 1991). "The point-spread function of a confocal microscope: its measurement and use in deconvolution of 3-D data". Journal of Microscopy . 163 (2): 151–165. doi : 10.1111/j.1365-2818.1991.tb03168.x . S2CID 95121909 .

^ "POINT SPREAD FUNCTION (PSF)" . www.telescope-optics.net . Retrieved 2017-12-30 .

^ Jump up to: a b The Natural Resolution

^ Principles and Practice of Light Microscopy

^ Corner Rounding and Line-end Shortening

^ Roorda, Austin; Romero-Borja, Fernando; Iii, William J. Donnelly; Queener, Hope; Hebert, Thomas J.; Campbell, Melanie C. W. (2002-05-06). "Adaptive optics scanning laser ophthalmoscopy" . Optics Express . 10 (9): 405–412. Bibcode : 2002OExpr..10..405R . doi : 10.1364/OE.10.000405 . ISSN 1094-4087 . PMID 19436374 . S2CID 21971504 .

The point spread function ( PSF ) describes the response of a focused optical imaging system to a point source or point object. A more general term for the PSF is the system's impulse response ; the PSF is the impulse response or impulse response function (IRF) of a focused optical imaging system. The PSF in many contexts can be thought of as the extended blob in an image that represents a single point object, that is considered as a spatial impulse. In functional terms, it is the spatial domain version (i.e., the inverse Fourier transform) of the optical transfer function (OTF) of an imaging system . It is a useful concept in Fourier optics , astronomical imaging , medical imaging , electron microscopy and other imaging techniques such as 3D microscopy (like in confocal laser scanning microscopy ) and fluorescence microscopy .

The degree of spreading (blurring) in the image of a point object for an imaging system is a measure of the quality of the imaging system. In non-coherent imaging systems, such as fluorescent microscopes , telescopes or optical microscopes, the image formation process is linear in the image intensity and described by a linear system theory. This means that when two objects A and B are imaged simultaneously by a non-coherent imaging system, the resulting image is equal to the sum of the independently imaged objects. In other words: the imaging of A is unaffected by the imaging of B and vice versa , owing to the non-interacting property of photons. In space-invariant systems, i.e. those in which the PSF is the same everywhere in the imaging space, the image of a complex object is then the convolution of the that object and the PSF. The PSF can be derived from diffraction integrals. [1]

By virtue of the linearity property of optical non-coherent imaging systems, i.e.,

the image of an object in a microscope or telescope as a non-coherent imaging system can be computed by expressing the object-plane field as a weighted sum of 2D impulse functions, and then expressing the image plane field as a weighted sum of the images of these impulse functions. This is known as the superposition principle , valid for linear systems . The images of the individual object-plane impulse functions are called point spread functions (PSF), reflecting the fact that a mathematical point of light in the object plane is spread out to form a finite area in the image plane. (In some branches of mathematics and physics, these might be referred to as Green's functions or impulse response functions. PSFs are considered impulse response functions for imaging systems. )

When the object is divided into discrete point objects of varying intensity, the image is computed as a sum of the PSF of each point. As the PSF is typically determined entirely by the imaging system (that is, microscope or telescope), the entire image can be described by knowing the optical properties of the system. This imaging process is usually formulated by a convolution equation. In microscope image processing and astronomy , knowing the PSF of the measuring device is very important for restoring the (original) object with deconvolution . For the case of laser beams, the PSF can be mathematically modeled using the concepts of Gaussian beams . [3] For instance, deconvolution of the mathematically modeled PSF and the image, improves visibility of features and removes imaging noise. [2]

The point spread function may be independent of position in the object plane, in which case it is called shift invariant . In addition, if there is no distortion in the system, the image plane coordinates are linearly related to the object plane coordinates via the magnification M as:

If the imaging system produces an inverted image, we may simply regard the image plane coordinate axes as being reversed from the object plane axes. With these two assumptions, i.e., that the PSF is shift-invariant and that there is no distortion, calculating the image plane convolution integral is a straightforward process.

Mathematically, we may represent the object plane field as:

i.e., as a sum over weighted impulse functions, although this is also really just stating the sifting property of 2D delta functions (discussed further below). Rewriting the object transmittance function in the form above allows us to calculate the image plane field as the superposition of the images of each of the individual impulse functions, i.e., as a superposition over weighted point spread functions in the image plane using the same weighting function as in the object plane, i.e.,

O
(

x

o

,

y

o

)

{\displaystyle O(x_{o},y_{o})}

. Mathematically, the image is expressed as:

in which

PSF

(

x

i

/

M
−
u
,

y

i

/

M
−
v
)

{\textstyle {\mbox{PSF}}(x_{i}/M-u,y_{i}/M-v)}

is the image of the impulse function

δ
(

x

o

−
u
,

y

o

−
v
)

{\displaystyle \delta (x_{o}-u,y_{o}-v)}

.

The 2D impulse function may be regarded as the limit (as side dimension w tends to zero) of the "square post" function, shown in the figure below.

We imagine the object plane as being decomposed into square areas such as this, with each having its own associated square post function. If the height, h , of the post is maintained at 1/w 2 , then as the side dimension w tends to zero, the height, h , tends to infinity in such a way that the volume (integral) remains constant at 1. This gives the 2D impulse the sifting property (which is implied in the equation above), which says that when the 2D impulse function, δ( x − u , y − v ), is integrated against any other continuous function, f ( u , v ) , it "sifts out" the value of f at the location of the impulse, i . e ., at the point ( x , y ) .

The concept of a perfect point source object is central to the idea of PSF. However, there is no such thing in nature as a perfect mathematical point source radiator; the concept is completely non-physical and is rather a mathematical construct used to model and understand optical imaging systems. The utility of the point source concept comes from the fact that a point source in the 2D object plane can only radiate a perfect uniform-amplitude, spherical wave — a wave having perfectly spherical, outward travelling phase fronts with uniform intensity everywhere on the spheres (see Huygens–Fresnel principle ). Such a source of uniform spherical waves is shown in the figure below. We also note that a perfect point source radiator will not only radiate a uniform spectrum of propagating plane waves, but a uniform spectrum of exponentially decaying ( evanescent ) waves as well, and it is these which are responsible for resolution finer than one wavelength (see Fourier optics ). This follows from the following Fourier transform expression for a 2D impulse function,

The quadratic lens intercepts a portion of this spherical wave, and refocuses it onto a blurred point in the image plane. For a single lens , an on-axis point source in the object plane pr
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