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Image Sampling:Sampling aperture, Sampling for area measurements

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...Image Processing Fundamentals
Figure 15: Optical Illusions
The left illusion induces the illusion of gray values in the eye that the brain
"knows" does not exist. Further, there is a sense of dynamic change in the image
due, in part, to the saccadic movements of the eye. The right illusion, Kanizsa's
triangle, shows enhanced contrast and false contours [14] neither of which can be
explained by the system-oriented aspects of visual perception described above.
5.
Image Sampling
Converting from a continuous image a(x,y) to its digital representation b[m,n]
requires the process of sampling. In the ideal sampling system a(x,y) is multiplied
by an ideal 2D impulse train:
+∞
+∞
∑ ∑δ (x - m Xo , y - nYo )
= a( x, y ) ·
bideal [ m.n]
m =-∞ n =-∞
(52)
+∞
+∞
∑ ∑  a( mXo , nYo )δ ( x - mX  o, y - nYo )
=
m= - ∞ n=-∞
where Xo and Yo are the sampling distances or intervals and δ(·,·) is the ideal
impulse function. (At some point, of course, the impulse function δ(x,y) is
converted to the discrete impulse function δ[m,n].) Square sampling implies that Xo
=Yo. Sampling with an impulse function corresponds to sampling with an
infinitesimally small point. This, however, does not correspond to the usual
situation as illustrated in Figure 1. To take the effects of a finite sampling aperture
p(x,y) into account, we can modify the sampling model as follows:
+∞
+∞
∑ ∑  δ ( x - m Xo , y - nYo )
b[m.n ] = ( a(x, y) p( x, y)) ·
(53)
m=-∞ n=-∞
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...Image Processing Fundamentals
The combined effect of the aperture and sampling are best understood by
examining the Fourier domain representation.
+∞
+∞
1
∑ ∑ A(Ω- m s, Ψ - nΨs ) · P (Ω - m s , Ψ - nΨs )
B(, Ψ ) =
(54)
4š 2
m=-∞ n =-∞
where s = 2š/Xo is the sampling frequency in the x direction and Ψs = 2š/Yo is
the sampling frequency in the y direction. The aperture p(x,y) is frequently square,
circular, or Gaussian with the associated P (,Ψ). (See Table 4.) The periodic
nature of the spectrum, described in eq. (21) is clear from eq. (54).
5.1 SAMPLING DENSITY FOR IMAGE P  ROCESSING
To prevent the possible aliasing (overlapping) of spectral terms that is inherent in
eq. (54) two conditions must hold:
· Bandlimited A(u,v) ­
A (u, v) 0
u > uc
v > vc
for
and
(55)
· Nyquist sampling frequency ­
 s > 2 · uc
and Ψs > 2 · vc
(56)
where uc and vc are the cutoff frequencies in the x and y direction, respectively.
Images that are acquired through lenses that are circularly-symmetric, aberration-
free, and diffraction-limited will, in general, be bandlimited. The lens acts as a
lowpass filter with a cutoff frequency in the frequency domain (eq. (11)) given by:
2 NA
uc = v  c =
(57)
λ
where NA is the numerical aperture of the lens and λ is the shortest wavelength of
light used with the lens [16]. If the lens does not meet one or more of these
assumptions then it will still be bandlimited but at lower cutoff frequencies than
those given in eq. (57). When working with the F-number (F ) of the optics instead
of the NA and in air (with index of refraction = 1.0), eq. (57) becomes:
2
1
uc = v  c =  
(58)
λ  4F2 + 1
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...Image Processing Fundamentals
5.1.1 Sampling aperture
The aperture p (x,y) described above will have only a marginal effect on the final
signal if the two conditions eqs. (56) and (57) are satisfied. Given, for example, the
distance between samples Xo equals Yo and a sampling aperture that is not wider
than Xo, the effect on the overall spectrum--due to the A(u,v)P (u,v) behavior
implied by eq.(53)--is illustrated in Figure 16 for square and Gaussian apertures.
The spectra are evaluated along one axis of the 2D Fourier transform. The Gaussian
aperture in Figure 16 has a width such that the sampling interval Xo contains ±3σ
(99.7%) of the Gaussian. The rectangular apertures have a width such that one
occupies 95% of the sampling interval and the other occupies 50% of the sampling
interval. The 95% width translates to a fill factor of 90% and the 50% width to a fill
factor of 25%. The fill factor is discussed in Section 7.5.2.
1.0
-- Square aperture,
0.9
fill = 25%
-- Gaussian
aperture
0.8
0.7
-- Square aperture,
fill = 90%
0.6
0.0
0.1
0.2
0.3
0.4
0.5
Fraction of Nyquist frequency
Figure 16: Aperture spectra P (u,v=0) for frequencies up to half the Nyquist
frequency. For explanation of "fill" see text.
5.2 SAMPLING DENSITY FOR IMAGE ANALYSIS
The "rules" for choosing the sampling density when the goal is image analysis--as
opposed to image processing--are different. The fundamental difference is that the
digitization of objects in an image into a collection of pixels introduces a form of
spatial quantization noise that is not bandlimited. This leads to the following results
for the choice of sampling density when one is interested in the measurement of
area and (perimeter) length.
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...Image Processing Fundamentals
5.2.1 Sampling for area measurements
Assuming square sampling, Xo = Yo and the unbiased algorithm for estimating area
which involves simple pixel counting, the CV (see eq. (38)) of the area
measurement is related to the sampling density by [17]:
lim CV ( S) = k2S  - 3 2
lim CV (S ) = k3S  - 2
2D :
3D :
(59)
S →∞
S →∞
and in D dimensions:
lim CV (S ) = k  DS- ( D+1 ) 2
(60)
S →∞
where S is the number of samples per object diameter. In 2D the measurement is
area, in 3D volume, and in D-dimensions hypervolume.
5.2.2 Sampling for length measurements
Again assuming square sampling and algorithms for estimating length based upon
the Freeman chain-code representation (see Section 3.6.1), the CV of the length
measurement is related to the sampling density per unit length as shown in Figure
17 (see [18, 19].)
100.0%
Pixel Count
10.0%
Freeman
Kulpa
1.0%
Corner Count
0.1%
1
10
100
1000
Sampling Density / Unit Length
Figure 17: CV of length measurement for various algorithms.
The curves in Figure 17 were developed in the context of straight lines but similar
results have been found for curves and closed contours. The specific formulas for
length estimation use a chain code representation of a line and are based upon a
linear combination of three numbers:
L = α · Ne + β · No + γ · Nc
(61)
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