Chapter 13
Review of Sampling
13.1 Digitization
Digitization is the conversion of a continuous-tone and spatially continuous brightnessdistribution f [x, y] to an discrete array of integers fq[n,m] by two operations whichwill be discussed in turn:(A) SAMPLING — a function of continuous coordinates f [x, y] is evaluated on a
discrete matrix of samples indexed by [n,m].(B) QUANTIZATION — the continuously varying brightness f at each sample is
converted to a one of set of integers fq by some nonlinear thresholding process.The digital image is a matrix of picture elements, or pixels if your ancestors are
computers. Video descendents (and imaging science undergraduates) often speak ofpels (often misspelled pelz). Each matrix element is an integer which encodes thebrightness at that pixel. The integer value is called the gray value or digital count ofthe pixel.Computers store integers as BInary digiTS, or bits (0,1)
2 bits can represent: 004 = 0., 014 = 1, 104 = 2., 114 = 3.;a total of 22 = 4numbers.
(The symbol “4” denotes the binary analogue to the decimal point “.”, that is,the binary point divides the ordered bits with positive and negative powers of 2).
m BITS can represent 2m numbers =⇒ 8 BITS = 1 BYTE =⇒ 256 decimalnumbers, [0, 255]
Note that digitized image contains a finite amount of information: the numberof bits required to store the data. This will usually be less than the quantity ofinformation in the original image. In other words, digitization creates errors. We willdiscuss digitizing and reconstruction error after describing the image display process.
269
270 CHAPTER 13 REVIEW OF SAMPLING
13.2 Sampling
This operation derives a discrete set of data points at (usually) uniform spacing. In itssimplest form, sampling is expressed mathematically as multiplication of the originalimage by a function that measures the image brightness at discrete locations:
fs [n ·∆x] = f [x] · s [x;n ·∆x]
where:
f [x] = brightness distribution of input image
s [x;n ·∆x] = sampling function
fs [n ·∆x] = sampled input image defined at coordinates n ·∆x
The ideal sampling function for functions of continuous variables is generated fromthe so-called “Dirac delta function” δ [x], which is defined by many authors, includingGaskill. For the (somewhat less rigorous) purpose here, we may consider the samplingfunction to be the sum of uniformly spaced “discrete” Dirac delta functions, whichGaskill calls the COMB and Bracewell calls it the SHAH :
s [x;n ·∆x] ≡
⎧⎨⎩ 1 if x = n ·∆x(n = 0,±1,±2, . . .)
0 otherwise
The COMB function defined by Gaskill (called the SHAH function by Bracewell).
13.2.1 Ideal Sampling
Multiplication of the input f [x] by a COMB function merely evaluates f [x] on theuniform grid of points located at n ·∆x, where n is an integer. Because it measuresthe value of the input at an infinitesmal point, this is a mathematical idealizationthat cannot be implemented in practice. Even so, the discussion of ideal samplingusefully introduces some essential concepts.Consider ideal sampling of a sinusoidal input function with spatial period X0 that
13.2 SAMPLING 271
is ideally sampled at intervals separated by ∆x:
f [x] =1
2
∙1 + cos
∙2πx
X0+ φ
¸¸=⇒ fs [n ·∆x] =
1
2
∙1 + cos
∙2πx
X0+ φ
¸¸· COMB
h x
∆x
i
The amplitude of the function at the sample indexed by n is:
fs [n ·∆x] =1
2
µ1 + cos
∙2πx
X0+ φ)
¸¶· δ [x− n ·∆x]
= fs [n ·∆x] =1
2·µ1 + cos
∙2πn
µ∆x
X0
¶+ φ
¸¶The dimensionless parameter ∆x
X0is the ratio of the sampling interval to the spatial
period (wavelength) of the sinusoid and is a measurement of the fidelity of the sampledimage. Mathematical expressions for the sampled function fs obtained for severalvalues of ∆x
X0are:
Case I:∆x
X0=1
12, φ = 0 =⇒ fs [n] =
1
2·³1 + cos
hπn6
i´Case II:
∆x
X0=1
2, φ = 0 =⇒ fs [n] =
1
2· (1 + cos [πn]) = 1
2[1 + (−1)n]
Case III:∆x
X0=1
2, φ = −π
2=⇒ fs [n] =
1
2· (1 + sin [πn]) = 1
2
Case IV:∆x
X0=3
4, φ = 0 =⇒ fs [n] =
1
2·µ1 + cos
∙3πn
2
¸¶Case V:
∆x
X0=5
4, φ = 0 =⇒ fs [n] =
1
2·µ1 + cos
∙5πn
2
¸¶
272 CHAPTER 13 REVIEW OF SAMPLING
Illustration of sampling of a biased sinusoid, showing aliasing if the signal oscillateswith a period smaller than 2 ·∆x.
The output evaluated for ∆xX0= 1
2depends on the phase of the sinusoid; if sampled
at the extrema, then the sampled signal has the same dynamic range as f [x] (i.e.,it is fully modulated), show no modulation, or any intermediate value. The interval∆x = X0
2defines the Nyquist sampling limit. If ∆x
X0> 1
2sample per period, then the
same set of samples could have been obt5ained from a sinusoid with a longer periodand a different sampling interval ∆x. For example, if ∆x
X0= 3
4, then the reconstructed
function appears as though obtained from a sinudoid with periodX 00 = 3X0 if sampled
with ∆xX00= 1
4. In other words, the data set of samples is ambiguous; the same samples
13.3 ALIASING —WHITTAKER-SHANNONSAMPLINGTHEOREM273
could be obtained from more than one input, and thus we cannot distinguish amongthe possible inputs based only on knowledge of the samples.
13.3 Aliasing —Whittaker-Shannon Sampling The-orem
As just demonstrated, the sample values obtained from a sinusoid which has beensampled fewer than two times per period will be identical to those from a sinusoidwith a longer period. This ambiguity is called aliasingin sampling, but similar effects show up whenever periodic functions are multiplied
or added. In other disciplines, these go by different names such as beats, Moiré fringes,and heterodyning. To illustrate, consider the product of two sinusoidal functions withthe different periods X1and X2(and thus spatial frequencies ξ1 = 1
X1, ξ2 =
1X2).
cos [2πξ1x] · cos [2πξ2x] =1
2cos [2π(ξ1 + ξ2)x] +
1
2cos [2π(ξ1 − ξ2)x]
The second term oscillates slowly and is the analog of the aliased signal.Though the proof is beyond our mathematical scope at this time, we state that
a sinusoidal signal that has been sampled without aliasing can be perfectly recon-structed from its ideal samples. This will be demonstrated in the section on imagedisplays. Also without proof, we make the following claim:
Any function can be expressed as a unique sum of sinusoidal componentswith (generally) different amplitudes, frequencies, and phases.
If the sinusoidal representation of f [x] has a component with a maximum spatialfrequency ξmax, and if we sample f [x] so that this component is sampled without alias-ing, then all sinusoidal components of f [x] will be adequately sampled and f [x]canbe perfectly reconstructed from its samples. Such a function is band-limited andξmax is the cutoff frequency of f [x]. The corresponding minimum spatial period isXmin =
1ξmax
. Thus the sampling interval ∆x can be found from:
∆x
Xmin<1
2=⇒ ∆x <
Xmin
2=⇒ ∆x <
1
2ξmax
This is the Whittaker-Shannon sampling theorem. The limiting value of the sam-pling interval ∆x = 1
2ξmaxdefines the Nyquist sampling limit. Sampling more or less
frequently than the Nyquist limit is oversampling or undersampling, respectively.
∆x >1
2ξmax=⇒ undersampling
∆x <1
2ξmax=⇒ oversampling
274 CHAPTER 13 REVIEW OF SAMPLING
The Whittaker-Shannon Sampling Theorem is valid for all types of sampled sig-nals. An increasingly familiar example is digital recording of audio signals (e.g., forcompact discs or digital audio tape). The sampling interval is determined by themaximum audible frequency of the human ear, which is generally accepted to beapproximately 20kHz. The sampling frequency of digital audio recorders is 44,000samplessecond which translates to a sampling interval of
144,000 s
= 22.7µs. At this samplingrate, sounds with periods greater than 2 · 22.7µs = 45.4µs (or frequencies less than(45.4µs)−1 = 22 kHz) can theoretically be reconstructed perfectly, assuming thatf [t] is sampled perfectly (i.e., at a point). Note that if the input signal frequency isgreater than the Nyquist frequency of 22 kHz, the signal will be aliased and will ap-pear as a lower-frequency signal in the audible range. Thus the reconstructed signalwill be wrong. This is prevented by ensuring that no signals with frequencies abovethe Nyquist limit is allowed to reach the sampler; higher frequencies are filtered outbefore sampling.
13.4 Realistic Sampling — Averaging by the Detec-tor
Signals cannot really be sampled at infinitesimal points by multiplication by a COMB;this would mean that the signal would be measued by a detector that has infinitesimalarea; such a measurement would have infinitesimal magnitude. In realistic sampling,the continuous input is measured at uniformly spaced samples by using a detectorwith finite spatial (or temporal) size. The measured signal is an average of the inputthe detector area, and the image structure is blurred by the averaging process:
Realistic sampling averages the signal over a finite area and blurs informationabout fine structure that existed in the original continuous image.
The discrete samples are obtained by averaging the input at the sample coordi-nates. This is mathematically equivalent to averaging the continuous input f [x] withthe detector weighting function h1 [x] and sampling the result by multiplication witha COMB function. Spatial averaging may be expressed as the integral of the productof the input function and the averaging (weighting) function h1 [x], and is called aconvolution. The averaging process is sometimes called prefiltering, or antialiasing:Z +∞
−∞f [x− x0] · h1 [x] dx ≡ (f [x] ∗ h1 [x]) |x=x0
The sampled signal is obtained by multiplying the averaged signal by the COMB:
13.4 REALISTIC SAMPLING — AVERAGING BY THE DETECTOR275
fs [n ·∆x] = (f [x] ∗ h1 [x]) · COMBh x
∆x
iwhere:
f [x] = brightness distribution of input image
h1 [x] = antialiasing prefilter
fs [n ·∆x] = sampled input image defined at coordinates n ·∆x
realistic sampling is composed of two cascaded operations:
(1) averaging (convolution, prefiltering) over a detector function, and
(2) multiplication by an ideal sampling function COMB£
x∆x
¤
The nature of the antialiasing prefilter determines the effect of realistic sampling onthe output. This is typically characterized by measuring the effect on the modulationof a sinusoidal wave f [x] = 1
2(1 + cos [2πξ0x]). The modulation of a sinusoid is
defined as:
m =fmax − fmin
fmax + fminfor 0 ≤ m ≤ 1
Note that modulation is defined for nonnegative (i.e., biased) sinusoids ONLY. Theanalogous quantity for a nonnegative square wave is called contrast. For example,consider a sinusoid with unit modulation that is sampled by an array with elementsof width d spaced at intervals of width ∆x as shown:
Schematic of sampling of a biased nonnegative sinusoid with detectors of width dspaced at intervals of ∆x.
The signal is averaged over the detector area, e.g., the sampled value at n = 0 is:
276 CHAPTER 13 REVIEW OF SAMPLING
fs [n = 0] =1
d
Z d2
−d2
f [x] dx
=
Z +∞
−∞f [x] ·
µ1
dRECT
hxd
i¶dx
where: RECThxd
i≡
⎧⎪⎪⎪⎨⎪⎪⎪⎩1 if |x| < d
2
12if |x| = d
2
0 if |x| > d2
For f [x] as defined above, the set of samples is derived by integrating f [x] overthe area of width d centered at coordinates that are integer multiples of ∆x:
1
d
Z n·∆x+d2
n·∆x− d2
1
2
µ1 + cos
∙2πx
X0+ φ
¸¶dx =
1
2d
ÃZ n·∆x+ d2
n·∆x− d2
dx+
Z n·∆x+d2
n·∆x− d2
cos
∙2πx
X0+ φ
¸dx
!
=1
2d
∙µn ·∆x+
d
2
¶−µn ·∆x− d
2
¶¸+1
2d
sinh2πxX0+ φ
i2πX0
¯̄̄̄¯̄x=n·∆x+d
2
x=n·∆x−d2
=1
2+1
2d
sinh2πn · ∆x
X0+ πd
X0+ φ
i− sin
h2πn · ∆x
X0− πd
X0+ φ
i³2πX0
´By defining α = 2πn · ∆x
X0+ φ and β = πd
X0, and by using the trigonometric identity:
sin [α+ β]− sin [α− β] = 2 cosα sinβ,
we find an expression for the integral over the detector area:
fs [n] =1
2+1
2d
⎛⎝2 cos ∙2πn · ∆x
X0+ φ
¸ sin h πdX0
i2πX0
⎞⎠≡ 12+1
2cos
∙2πn · ∆x
X0+ φ
¸SINC
∙d
X0
¸where SINC [α] ≡ sin[πα]
πα:
13.4 REALISTIC SAMPLING — AVERAGING BY THE DETECTOR277
Graph of SINC [x] ≡ sin[πx]πx
Note that for constant functions X0 =∞ and SINC³
dX0
´→ 1; uniform weighted
averaging has no effect on constant inputs. The samples of cosine of period X0
obtained with sampling interval ∆x in the two cases are:
Realistic:fs [n] =1
2·µ1 +
1
2SINC
∙d
X0
¸· cos
∙2πn · ∆x
X0+ φ
¸¶Ideal : fs [n] =
1
2·µ1 + cos
∙2n
µ∆x
X0
¶+ φ
¸¶where d is the width of the detector. The amplitude of the realistic case is mul-
tiplied by a factor of SINCh
dX0
i, which is less than unity everywhere except at the
origin, , i.e., where d = 0 or X0 = ∞. As the detector size increases relative to thespatial period of the cosine ( i.e., as d
X0increases) , then SINC
hdX0
i→ 0 and the
modulation of the sinusoid decreases.
The modulation of the image of a sine-wave of period X0, or spatial frequencyξ = 1
X0, is reduced by a factor SINC
hdX0
i= SINC [dξ0].
Example of Reduced Modulation due to Prefiltering
The input function f [x] has a period of 128 units with two periods plotted. It is thesum of six sinusoidal components plus a constant:
f [x] =1
2+1
2
6Xn=1
(−1)n−1
nsin
∙2π(2n− 1)x256
¸.
278 CHAPTER 13 REVIEW OF SAMPLING
The periods of the component sinusoids are:
X1 =128
1units =⇒ ξ1 =
1
128
cyclesunit
' 0.0078cyclesunit
X2 =128
3units ' 42.7 units =⇒ ξ2 =
3
128
cyclesunit
' 0.023cyclesunit
X3 =128
5units = 25.6 units =⇒ ξ3 =
5
128
cyclesunit
' 0.039cyclesunit
X4 =128
7units ' 18.3 units =⇒ ξ4 =
7
128
cyclesunit
' 0.055cyclesunit
X5 =128
9units ' 14.2 units =⇒ ξ4 =
9
128
cyclesunit
' 0.070cyclesunit
X6 =128
11units ' 11.7 units =⇒ ξ4 =
11
128
cyclesunit
' 0.086cyclesunit
The constant bias of 0.5 ensures that the function is positive. The first sinusoidalcomponent (X01 = 128 units) is the fundamental and carries most of the modulationof the image; the other components (the higher harmonics) have less amplitude. Thespatial frequency of each component is much less than the Nyquist limit of 0.5.
SINC
∙d
X01
¸= SINC [dξ1] = SINC
∙8 · 1128
¸' 0.994
SINC
∙d
X02
¸= SINC [dξ2] = SINC
∙8 · 3128
¸' 0.943
SINC
∙d
X03
¸= SINC [dξ3] = SINC
∙8 · 5128
¸' 0.847
SINC
∙d
X04
¸= SINC [dξ4] = SINC
∙8 · 7128
¸' 0.714
SINC
∙d
X05
¸= SINC [dξ5] = SINC
∙8 · 9128
¸' 0.555
SINC
∙d
X06
¸= SINC [dξ6] = SINC
∙8 · 11128
¸' 0.385
Note that the modulation of sinusoidal components with shorter periods (higherfrequencies) are diminished more severely by the averaging. A set of prefiltered imagesfor several different averaging widths is shown on a following page. If the detectorwidth is 32 units, the resulting modulations are:
13.4 REALISTIC SAMPLING — AVERAGING BY THE DETECTOR279
SINC [dξ1] = SINC
∙32 · 1
128
¸' 0.900
SINC [dξ2] = SINC
∙32 · 3
128
¸' 0.300
SINC [dξ3] ' −0.180SINC [dξ4] ' −0.129SINC [dξ5] ' −0.100SINC [dξ6] ' +0.082
Note that the components with periods X04 and X05 have negative modulation,, i.e., fmax < fmin. The contrast of those components is reversed. As shown, thesampled image looks like a sawtooth with a period of 128 units.
If the detector size is 128, each component is averaged over an integral number ofperiods and the result is just the constant bias; the modulation of the output is zero:
SINC [dξ1] = SINC
∙128
128
¸= SINC [1] = 0
SINC [dξ2] = SINC
∙128· 7
42
¸= SINC [3] = 0
For a detector width of 170 units, the modulations are:
SINC [dξ1] = SINC
∙170 · 1
128
¸' −0.206
SINC [dξ2] = SINC
∙170 · 3
128
¸' −0.004
SINC [dξ3] = SINC
∙170 · 5
128
¸' +0.043
SINC [dξ4] = SINC
∙170 · 7
128
¸' −0.028
SINC [dξ5] ' −0.004SINC [dξ6] ' +0.021
Because the first (largest amplitude) sinusoidal component has negative modula-tion, so does the resulting image. The overall image contrast is reversed; darker areasof the input become brighter in the image.
280 CHAPTER 13 REVIEW OF SAMPLING
Illustration of the reduction in modulation due to “prefiltering”: (a) input functionf [n]; (b) result of prefiltering with uniform averagers of width d = 0, d = X0
16, and
d = X0
8; (c) magnified view of (b), showing the change in the signal; (d) result
offiltering with uniform averagers of width d = X0
2, d = X0, and d = X0
0.75, showing
the “contrast reversal” in the last case.