sampling, aliasing, and target appearance

ELSEVIER Infrared Physics & Technology 37 (1996) 627-634

INFRARED PHYSICS & TECHNOLOGY

Sampling, aliasing, and target appearance

Gerald C. Hoist EO Techru~logies. 2932 Cove Trail, Winter Park, FL 32789-1159. USA

Received 15 October 1995

Abstract

Most infrared test targets consist of periodic bars with 3-bar and 4-bar targets being the most popular. These targets are characterized by the bar-pattern fundamental frequency. While the bar-pattern fundamental frequency may be below the detector array Nyquist frequency, the higher harmonics may not. Aliasing creates distortions that elevate minimum resolvable temperature measurement test results. Historically, sampled-data systems contained a post-reconstruction filter that limited the reproduced image spectrum to the Nyquist frequency. Many infrared imaging system designs tend to omit this filter and this allows beat frequencies to appear in reproduced bar patterns.

Keywords: Sampling; Aliasing; Sampled-data system; Bar target; Minimum resolvable temperature; MRT

1. Introduct ion

Sampling (digitization) is an inherent feature of all infrared imaging systems [1]. The detector pro- vides a spatially sampled representation of the scene. Sampling creates ambiguity in target edge location, and diagonal lines appear to have jagged edges or 'jaggies'. Moir6 patterns are produced when viewing periodic structures. Periodic structures are rare in nature and aliasing is seldom reported when viewing natural scenery although aliasing is always present. It may become apparent when viewing periodic targets such as picket fences, plowed fields, and railroad tracks. The appearance of the reproduced scene depends upon: (a) the relationship between the optical and detector cutoff frequencies, (b) the relationship between the target and the Nyquist frequencies, and (c) subsequent electronic filtering.

The highest frequency that can be reconstructed faithfully is one-half the sampling rate. Any input signal above the Nyquist frequency, fN, (which is

defined as one-half the sampling frequency, fs ) will be aliased down to a lower frequency [1,2]. That is, an undersampled signal will appear as a lower frequency after reconstruction (Fig. 1). After aliasing, the original signal can never be recovered.

Signals can be undersampled or oversampled. Un- dersampling is a term used to denote that the input frequency is greater than the Nyquist frequency. It does not imply that the sampling rate is inadequate for any specific application. Similarly, oversampling does not imply that there is excessive sampling. It simply means that there are more samples available than that required by the Nyquist criterion. The Nyquist criterion requires two samples per input frequency.

The square wave (bar pattern) is the most popular infrared test target. It is usually described by its fundamental frequency only. The expansion of a square wave into a Fourier series clearly shows that it consists of an infinite number of sinusoidal frequencies. Although the square wave fundamental

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628 G.C. Hoist / Infrared Physics & Technology 37 (1996) 627-634

input I l l ls ld signal

0 2T 3"1" 4T ST

Fig. 1. An undersampled sinusoid will appear as a lower frequency after reconstruction. The sampling frequency is f s = 1 / T. When T is the measured in mrad, mm, or time, the sampling frequency is in cy /mrad, c y / m m , or Hz respectively.

frequency may be oversampled, some higher harmonics will not. During digitization, the content of the higher order frequencies will be aliased down to lower frequencies and the square wave will change its appearance. There will be intensity variations from bar-to-bar and the bar width will not remain constant. These variations elevate minimum resolvable temperature measurement test results.

Sampling theory is traditionally presented from a modulation transfer function (MTF) viewpoint. Lin- ear system theory is used for system analysis because of the wealth of mathematical tools available. Approximations are used to account for sampling effects. While these approximations are adequate to describe the MTFs, they do not indicate how imagery may be distorted. Aliasing is eliminated when the input signal is band-limited. Since the aliasing occurs at the detector, the signal must be band-limited by the optical system.

2. Sampled-data systems

In a sampled-data system, the sampling frequency interacts with the signal to create sum and difference frequencies. Any input frequency, f , will appear as nfs+ f after sampling ( n = - o c to +oc). For a band-limited system (highest frequency is fH), the baseband ( - f H to fH) is replicated at nfs. To avoid distortion, the lowest possible sampling frequency is that value where the baseband adjoins the first side band (n = 1). This leads to the sampling theorem that a band-limited system must be sampled at twice the highest frequency (fs > 2fH) to avoid distortion in the reconstructed image.

As the sampling frequency decreases, the first side band starts to overlap the baseband and the power spectra add. The overlapping region creates distortion in the reconstructed image. It is impossible to tell whether the reconstructed frequency resulted from an input frequency of f or nfs +_f (see Fig. 1). This is aliasing. Once aliasing has occurred, it cannot be removed.

After sampling, the data is simply an array of integers residing in a memory. Individual data points are represented by memory locations or indices. It is the relationship between the data array and the cam- era field-of-view or clock timing that creates dis- tances and units between sample points. Since moni- tors are analog devices, the data is transformed into an analog signal using a sample-and-hold circuit. This creates a 'blocky' image and the individual blocks are labeled as pixeis on a screen. The 'blockiness' is removed by a post-reconstruction filter. If the original signal was oversampled ( f N >fH) and if the post-reconstruction filter limits frequencies to fN, then the reconstructed image can be identical to the original image.

The appropriate post-reconstruction filter is essen- tial to remove the 'blockiness' that appears in the reproduced scene. On the other hand, square waves contain many frequencies. If the post-reconstruction filter removes higher order frequencies, the square wave will appear as a sinusoid [3]. As a result, many infrared imaging systems do not contain a post-reconstruction filter. The sampled data after the sample-and-hold circuitry is fed directly to a display. This preserves edge sharpness and square waves continue to look like square waves. However, the square wave fundamental frequency in the baseband will interact with its replicated frequency in the first side band to create a beat pattern.

3. The detector as a sampler

In one-dimension, a rectangular detector MTF is

MTFaetector = sinc( a f ) (1) where sinc(x) is equal to sin(~rx)/(Irx). Using the small angle approximation, the detector angular sub- tense (DAS), is

d detector size

ot = DAS = fl effective focal length (2)

G.C. Hoist/Infrared Physics & Technology 37 (1996) 627-634 629

0.8

0 . 6

~_ 0.4

• o.2 0 -

-0.2

-0.4

NORMALIZED SPATIAL FREQUENCY

Fig. 2. Detector MTF as a function of normalized spatial fre-

quency f/.Coc" Detector cutoff is ./PC = l / a . The negative MTF values represent contrast reversal: periodic dark bars will appear as light bars. The MTF is usually plotted only up to the first zero.

zero. Unfortunately, this representation may lead the analyst to believe that there is no response above fDC. The absolute highest spatial frequency that can be imaged is limited by the optical system cutoff, foc. It is equal to D/A where D is the aperture diameter and A is the average wavelength, foc may be either higher (optically-limited system) or lower (detector-limited system) than foc. Most imaging systems are detector-limited (foc <foe) .

4. The detector array as a sampler

Fig. 2 illustrates the MTF in one dimension. The MTF is equal to zero when f = k/a. The first zero (k = l) is considered the detector cutoff, foc , because any higher frequency will not be faithfully reproduced. As a sampler, fs = 2foc and fN = fDC" It is customary to plot the MTF only up to the first

I

o.I

u. o. I I - • 0.4

0.2

0 0

f-F--Yq

S, ,

Nyquist Sampling

0.2 0.4 0.$ 0.8 1

NORMAliZED SPATIAL FREQUENCY (a)

The discrete location of detectors in an array act as a sampler whose angular sampling rate is Scc = dcc/fl where dec is the detector center-to-center spacing (detector pitch). Detector arrays can faithfully reproduce signals up to fN = 1/2Scc. Al- though the individual detectors can reproduce higher spatial frequencies, the spectrum is sampled at fs = l /See (Fig. 3). Staring arrays are inherently undersampled when compared to the detector spatial frequency cutoff. As d/dcc decreases, the detector array Nyquist frequency also decreases. This poten- tially leads to more aliasing and image distortion.

Staring systems, because of detector location symmetry, tend to have equal sampling rates in both the horizontal and vertical directions. With scanning systems, the detector output in the scan direction can be electronically digitized at any rate whereas in the cross scan direction, the detector locations define the sampling rate. Therefore, in a scanning system, the sampling rate may be significantly different in the

D E 2 0 I--s,~-I

Nycluist Sampling

0.8

" 0.6 I,- • 0.4

0.2

0 0 0.2 0.4 0.6 0.8 1

NORMALIZED SPATIAL FREQUENCY (b)

Fig. 3. Two arrays with different center-to-center spacings. The

DASs are the same for both. (a) Scc = ct or d = dec. (b) Scc = 2 a o r d c c = 2 d .

1

0.8 I -

0 0.6 LU

0.4

~ 0 . 2 e~

0 I I A A 2 3 4 5 $ 7 8 9

NORMALIZED FREQUENCY

Fig. 4. Fourier transform of a square wave. It consists of an infinite number of odd harmonics. Since the pulse amplitude was positive only, a DC component is present. The fundamental frequency is 0.923. The harmonics occur at 2.77, 4.62, 6.46, g.31 . . . . .

630 G.C. Hoist/Infrared Physics & Technology 37 (1996) 627-634

1

0.8 ...m

0 0.6 UJ

0.4

0.2

0

I

~AA~AA~ 0 1 2 3 5

RELATIVE FREQUENCY

Fig. 5. Fourier transform of a square wave after sampling with a very small detector (d<< dec), )'s = 2 and fN = 1. The DC component is replicated at )'s, 2fs , 3fs . . . . .

1

~ 0.8

0 0.6 tU

~ 0 . 2 e~

. . . . . A _ 0

1 2 3 5


Fig. 7. Fourier transform of a square wave after sampling when d = dec. The detector MTF reduces the amplitude of the higher order harmonics.

horizontal and vertical directions. Scanning systems can satisfy the Nyquist criterion in the scan direction.

5. MTF analysis

To understand sampling effects, a one-dimen- sional square wave of infinite extent is evaluated analytically. Fig. 4 illustrates the Fourier transform of the computer generated square wave. The fundamental frequency is fo = 0.923. Following theoreti- cal predictions, the bar target contains all the odd harmonics. An additional peak occurs at zero frequency (DC component) because the signal amplitude ranged from zero to one.

The imagery is modified by the optical MTF. For this study, the optical MTF was assumed to be unity over the spatial frequencies of interest. Since most infrared imaging systems are detector-limited, this is a reasonable assumption. Thus, the full square wave falls on the detector array without modification. The signal was then sampled (fs = 2) with a very small detector (d<<dcc) . This is equivalent to using a

flash converter in electronic circuitry and is the representation seen in most text books. Fig. 5 illustrates the aliasing of 3fo, 5fo, 7fo, 9fo, and 1 lfo into the baseband. The baseband spectrum is replicated at fs and is symmetrical about fN" This is the spectrum as it resides in a computer memory.

To see an image, the data points are clocked out in a fashion consistent with monitor timing require- ments. The sample-and-hold circuitry is represented by a low pass analog filter whose MTF is

Fig. 6 illustrates the resultant spectrum of the analog signal.

Real detectors have a finite width and modify the input frequency spectra according the detector MTF (Eq. (1)). The detector output represents a spatial integration of the input signal. Fig. 7 portrays the spectrum after sampling by a detector whose width is equal to the center-to-center spacing (d =dcc ) . The detector MTF reduced the higher order harmonics. Fig. 8 shows the analog spectrum after the sample-

1 I -

0.8 l - --t O 0.6

0.4

0.2

0 1 2 3 4 5 6


1 I -

0.8 I -

0 0.6 tu ~ 0.4

2 0 . 2

0 1 2 3 4 5 6

RELATIVE FREQUENCY

Fig. 6. Fourier transform after a sample-and-hold circuit. This is Fig. 8. Fourier transform after a sample-and-hold circuit, d = dec. the spectrum in Fig. 5 multiplied by MTFs_ H . This is the spectrum in Fig. 7 multiplied by MTFs_ H .

G.C. Hoist / Infrared Physics & Technology 37 (1996) 627-634 631

o ~ ~ , , 0.5 0.75 1 1.25 1.5


Fig. 9. Number of cycles that must be present to see the complete beat frequency.

Fig. 11. Ideal staring array output when fo ~iN = 0.522, N = 0.54, d = d c c . The light line is the input and the heavy line is the detector output. The output nearly replicates the input when

J:o/]N < 0.6.

and-hold circuit. If an ideal reconstruction filter was added, the spectrum would only contain frequencies

up to fN"

6. Target appearance

The MTF approach only shows how the frequency spectrum is modified by the sampling process. It does not indicate target appearance. When two frequencies are close together, they create a beat frequency that is equal to the difference. If the system does not have a post-reconstruction filter, the fundamental and its replication in the first side band create a beat frequency of fb~,t = ( f s - - f o ) - - f o = 2(fN--fo). The beat frequency period lasts for N input frequency cycles:

L N (4)

2(fN --fo)

AS the input frequency approaches the Nyquist frequency, more bars (one cycle equals a bar plus a space) are required [4] to see the beat frequency (Fig. 9). Since standard targets consist of three or four

'--out of phase-

Fig. 10. Beat frequency produced by an ideal staring array when ]o/fN = 0.952, N = 9.9, d = dEc. The light line is the input and the heavy line is the detector output. The beat frequency envelope is also shown. A 4-bar pattern may either be replicated or there may be a negligible output depending upon the phase.

bars, the beat frequency is not obvious when per- forming most laboratory measurements.

Figs. 10-12 illustrate a staring array output after the sample-and-hold circuitry and without an ideal post-reconstruction filter. The optical MTF is unity for all spatial frequencies. That is, all higher harmonics that would have been attenuated by the optical system are available for aliasing by the detector. The detector width is equal to the center-to-center spacing. If fo/fN = 0.952, the beat frequency is equal to 9.9 cycles of the input frequency (Fig. 10). Here, the target must contain at least 10 cycles to see the entire beat pattern. When fo/fN is less than about 0.6 (Fig. 11), the beat frequency is not obvious. Now the output nearly replicates the input but there is some slight variation in pulse width and amplitude. In the region where fo/fN is approximately between 0.6 and 0.9, adjacent bar amplitudes are always less than the input amplitude (Fig. 12). The variation in pulse width and amplitude is due to aliasing.

Standard characterization targets, however, consist of several bars. Therefore the beat pattern may not be seen when viewing a 3-bar or 4-bar target. The bar pattern would have to be moved + 1 /2 DAS to change the output from a maximum value (in-phase) to a minimum value (out-of-phase). This

Fig. 12. Ideal staring army output when fo/ fN = 0.811, N = 2.14, d = dcc. The light line is the input and the heavy line is the detector output. The output never looL~ quite right when -to/JN is between 0.6 and 0.9.

632 G.C. Hoist/Infrared Physics & Technology 37 (1996) 627-634

can be proven by selecting four adjacent bars in Fig. I0. When fo/fN is less than about 0.6, a 4-bar pattern will always be seen (select any four adjoining bars in Fig. 11). When fo/fN < 0.6, phasing effects are minimal and a phase adjustment of + 1 /2 DAS will not affect an observer's ability to resolve a 4-bar target. In the region where fo/fN is approximately between 0.6 and 0.9, 4-bar targets will never look correct (Fig. 12). One or two bars may be either much wider than the others or one or two bars may be of lower intensity than the others. Since a 4-bar target in this region never looks right, the measured minimum resolvable temperature (MRT) is elevated in this region [5].

Input frequencies of fo =fN/k, where k is an integer, are faithfully reproduced (i.e., no beat frequencies). When k = 1, as the target moves from in-phase to out-of-phase, the output will vary from a maximum to zero. Selection of fo/k targets avoids the beat frequency problem but significantly limits the number of spatial frequencies selected.

Signals whose frequencies are above Nyquist frequency will be aliased down to lower frequencies. This would be evident if an infinitely long periodic target was viewed. However, when fo is less than about l . l fN, it is possible to select a phase such that four adjoining bars appear to be faithfully reproduced (Fig. 13). These targets can be resolved although the underlying fundamental frequency has been aliased to a lower frequency.

Figs. 10-12 represent the output when d = dcc. As d/dcc decreases, the detector MTF increases and the spectrum slowly changes from Fig. 8 to Fig. 6. The output wave forms are illustrated in Fig. 14. The finite width detector emphasizes the beat frequencies. Fig. 14(e) typifies a flash analog-to-digital

(*)

(b)

(c)

(d)

(e)

Fig. 14. Ideal staring array output when fo/ fN =0.952 when d/dcc is (a) 0.8, (b) 0.6, (c) 0.4, (d) 0.3, and (e) 0.03. The light line is the input and the heavy line is the detector output.

converter. This also shows that the ambiguity in edge location exists for all sampling systems.

Fig. 13. Ideal staring array output when fo/ fN = 1.094, N = 5.8, d = dcc. The light line is the input and the heavy line is the detector output. By selecting the appropriate phase, the output appears to replicate an input 4-bar pattern.

7. Sample-scene phase

Although distortions were demonstrated with bar patterns, they also exist in reproduced sinusoids. Sampled data systems are nonlinear and do not have a unique MTF [6-11 ]. The 'MTF' depends upon the phase relationships of the scene with the sampling lattice. Superposition does not hold and any MTF

G.C. Holst / Infrared Physics & Technology 37 (1996) 627-634 633

0 . . . . : . . . . 0 0.5 1.5 2

NORMAUZED SPATIAL FREQUENCY Fig. 15. Average and median scene-sample phase MTFs normalized to f/Jh. The MTF is defined only up to the Nyquist frequency.

derived for the sampler cannot, in principle, be used to predict results for general scenery. To account for the nonlinear sampling process, a sample-scene MTF is used as an approximation. As the sampling rate increases, the MTF becomes better defined. As the sampling rate approaches infinity, the system becomes an analog system and the MTF is well defined.

A wide range of MTF values is possible [12-14] for any given spatial frequency input depending upon the phase. In general, the 'MTF' is [15]

MTFphase = c o s ( ~ 0) (5)

where 0 is the phase angle between the target and the sampling lattice. For example, when f=fN, the MTF is a maximum when 0 = 0 (in-phase) and a zero when 0 = 7r/2 (out-of-phase). To approximate a median value for phasing, 0 is set to ~-/4. Here, approximately one-half of the time the MTF will be higher and one-half of the time the MTF will be lower. The median sampling MTF is

M T F m e d i a n ~ c o s 7 " (6)

An MTF averaged over all phases is sometimes represented by

MTFa .... ge = s i n c ( ~ ) (7)

Fig. 15 illustrates the difference between the two equations. Since these are approximations, they may be considered roughly equal over the range of interest (zero to the Nyquist frequency). For laboratory measurements where the target phase is adjusted to obtain the maximum output, M T F p h a s e = 1.

8. Summary

With electrical circuitry, a filter is placed just before the analog-to-digital converter to band-limit the signal so that the Nyquist criterion is met. This filter is called an anti-aliasing filter. The ideal filter will have unity MTF up to f s and zero there after.

With infrared imaging system, the sampling frequency depends upon the detector-to-detector spacing. Since aliasing occurs at the detector, it can only be avoided if the system is optically band-limited. Optical band-limiting can be achieved by using small diameter optics, by blurring the image through defo- cusing, or an anti-alias optical filter [16]. Unfortu- nately, these approaches also degrade the MTF in the baseband and typically are considered undesirable.

The appearance of a reproduced bar target depends upon the optical MTF, detector MTF, the detector pitch, and post-reconstruction filter. The imagery shown in Figs. 10-12 was for a system where the optical MTF was unity for all spatial frequencies. As the optical cutoff decreases, the optical MTF will attenuate the higher order frequencies. If fo is greater then f o e / 3 , then the third harmonic will be attenuated by the optical system and only the fundamental frequency will reach the detector array. The detector output will be a digitized sinusoidal signal. It is the post-reconstruction filter that removes the replicated spectra. Without this filter, beat frequencies are present in the reproduced image. Aliasing and beat frequencies become obvious when viewing periodic structures and are seldom reported when viewing general scenery.

If a system is Nyquist frequency limited, then the Nyquist frequency is used as a measure of resolution.

1 ~ q u l S t 0.8

u . 0 .6 I - - 5 0 .4

0.2 0

0.5 1 1.5 2 NORMALIZED FREQUENCY

Fig . 16. M T F represen ta t ion o f an u n d e r s a m p l e d s y s t e m as a

function of f/JN. Since no frequency can exist above the Nyquist frequency, some researchers represent the MTF as zero above the Nyquist frequency.

634 G.C. Hoist~Infrared Physics & Technology 37 (1996) 627-634

Since no frequency can exist above the Nyquist frequency, many researchers represent the MTF as zero above the Nyquist frequency (Fig. 16). This representation may be too restrictive for modeling purposes. Systems can detect signals whose spatial frequencies are above cutoff but cannot faithfully reproduce them. For example, patterns above the Nyquist frequency are aliased to a frequency below Nyquist and a 4-bar pattern may appear as a distorted 3-bar pattern.

Finally, the phasing effects shown here become obvious when f/fN is not an integer. Many computer simulations and hand drawn diagrams use integers so that the detector locations are either in-phase or out-of-phase with respect to the target. Therefore beat frequencies and aliasing effects are often not seen with quickly drawn diagrams.

References

[1] J.M. Lloyd, Thermal Imaging Systems (Plenum Press, New York, 1975) pp. 369-387.

[2] L.M. Biberman and B.H. Tsou, in: The Infrared and Electro- Optical Systems Handbook, Vol. 3, ed. W.D. Rogatto, E l e c -

tro-Optical Components (Environmental Research Institute of Michigan, Ann Arbor, MI, 1993) pp. 450-462.

[3] G.C. Hoist, Testing and Evaluation of Infrared Imaging Systems (JCD Publishing, Winter Park, FL, 1993) pp. 238- 245.

[4] T.S. Lomheim, L.W. Schumann, R.M. Shima, J.S. Thomp- son and W.F. Woodward, Opt. Eng. 29(8) (1990) 911-927.

[5] G.C. Hoist, Testing and Evaluation of Infrared Imaging Systems (JCD Publishing, Winter Park, FL, 1993) pp. 325- 327.

[6] S.K. Park and R.A. Schowengerdt, Appl. Opt. 21(17) (1982) 3142-3151.

[7] S.E. Reichenbach, S.K. Park and R. Narayanswamy, Opt. Eng. 30(2) (1991) 170-177.

[8] W. Wittenstein, J.C. Fontanella, A.R. Newberry and J. Baars, Opt. Acta 29(1) (1982) 41-50.

[9] J.C. Felz, Opt. Eng. 29(8) (1990) 893-904. [10] S.K. Park, R.A. Schowengerdt and M. Kaczynski, Appl. Opt.

23(15) (1984) 2572-2582. [11] L. deLuca and G. Cardone, Appl. Opt. 30(13) (1991) 1659-

1664. [12] J.C. Feltz and M.A. Karim, Appl. Opt. 29(5) (1990) 717-722. [13] J.C. Feltz, Opt. Eng. 29(8) (1990) 893-904. [14] L. de Luca and G. Cardone, Appl. Opt. 30(13) (1991)

1 6 5 9 - 1664 .

[15] F.A. Rosell, in: The Fundamentals of Thermal Imaging Systems, ed. F. Rosell and G. Harvey, NRL Report 8311 (Naval Research Laboratory, Washington, DC, 1979) p. 217.

[16] J.E. Greivenkamp, Appl. Opt. 29(5) (t990) 676-684.

sampling, aliasing, and target appearance

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