Statistical detection of resolved targets in background clutter using optical/infrared imagery Larry B. Stotts1,* and Lawrence E. Hoff2 1
Science and Technology Associates, 4100 N. Fairfax Drive, Suite 910, Arlington, Virginia 22203, USA 2
Hoff Engineering, 3627 Tennyson Street, San Diego, California 92106, USA *Corresponding author: [email protected] Received 2 April 2014; revised 13 June 2014; accepted 30 June 2014; posted 1 July 2014 (Doc. ID 208754); published 30 July 2014
The use of optics to detect targets has been around for a long time. Early attempts at automatic target detection assumed target plus noise, which means that the targets were small compared to the pixel field of view and therefore unresolved. However, the advent of advanced focal plane technology has resulted in optical systems that can provide highly resolved target images. The intent of this paper is to develop a general solution for the detection of resolved targets in background clutter. We recognize that resolved targets obscure any background clutter that would have been visible if the targets were absent. An optimum detection algorithm is derived that compares a test statistic to a threshold and decides a target is present if the statistic is less than the threshold. We find that the detection performance depends upon (1) the apparent contrast rather than the signal to noise ratio and (2) is highly dependent on the background clutter to common system noise ratio. In fact, the target can still be detected even when the target contrast goes to zero provided the background clutter is greater than the common system noise. Computer simulations are shown to validate the theoretical detection and false alarm probabilities. The findings in this paper should be useful to engineers and scientists designing electro-optical and infrared sensors for finding resolved targets immersed in background-cluttered images. © 2014 Optical Society of America OCIS codes: (100.2960) Image analysis; (100.4997) Pattern recognition, nonlinear spatial filters; (280.4788) Optical sensing and sensors. http://dx.doi.org/10.1364/AO.53.005042
1. Introduction
The application of optics to detecting objects at a distance has been around since the earliest known working telescopes appeared in 1608, first credited to Hans Lippershey. Duntley and his colleagues, beginning at MIT and later at the Visibility Laboratory of the Scripps Institute of Oceanography, were instrumental in establishing the science of visibility that the optics community has relied upon since the 1940s [1,2]. What they found was that a resolved object of interest can be “seen” because it has a different radiance (or irradiance) than the other sources of
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APPLIED OPTICS / Vol. 53, No. 22 / 1 August 2014
radiance (irradiance) surrounding it. The accepted metric for an object’s visibility is the Weber contrast, defined as C
N0 − Nb ; Nb
(1a)
where N 0 is object’s radiance and N b is the background radiance. Under this definition, the contrast varies between −1 for N b ≫ N 0 and ∞ for N b ≪ N 0, looking at the extremes of these two conditions. The accepted minimum visibility of an object that can be seen is the metrological visibility, defined as 2%. However, in practice, some observers have been able to see objects with contrast down to 0.5% [3,4]. The associated signal-to-noise ratio (SNR) for object
detection is defined as the above contrast divided by the standard deviation of the noise [3]. Mathematically, this is written as SNR
C σ noise
;
(1b)
where C denotes the contrast and σ noise the standard deviation of the noise. (The electrical SNR is the square of the above since optical detectors create current proportional to the received optical radiance [5].) (It is interesting to note that C is dimensionless, but σ noise is in photoelectron counts, which means that SNR is in inverse photoelectron counts.) The exact value for the minimum detectable SNR depends on the size of the object; specifically, proportional to the square root of its area of the target [3]. The minimum value appears to be around one but may be higher for some observers [3,4]. To date, there has not been a formal mathematical connection between contrast and SNR in statistical hypothesis testing, only empirically like the above [6,7]. The standard approach for analyzing the optical detection of targets in clutter was first reported by Helstrom [6] in 1964, and it parallels the signal-plus-additive hypothesis testing developed for RF communications and sensing [7]. Received power, and background and system noise standard deviations, are their key performance variable, with no mention of contrast [7]. Papers employing these techniques still can be found in the literature today [8,9]. Unfortunately in these papers, the signalplus-noise assumption is inappropriate because a resolved target obscures any background clutter, violating the key premise of this hypothesis test. This can be clearly seen in Fig. 1, which shows two examples of such a situation [10]. Specifically, we see in these figures that the bar targets, asphalt, airplanes, and/or buildings mask any terrain signature underneath them, and that background terrain is found elsewhere. Schaum recently complained about the continued use of the above additive model in electro-optical image analysis [11]. Because of its “phenomenological inaccuracy,” he advocated the use of the more appropriate replacement target model. Specifically, he recognized that this model is seldom used because the classical analysis approach to this background replacement problem was intractable unless narrow restrictions are imposed, but it is the appropriate problem to solve. His proposed solution was to apply continuum fusion methodology [12], and he presented a detailed paper on its application, with examples to validate his approach to replacement model detection analysis [11]. As pointed out by Goudail in a private communication [13], researchers have been developing techniques for the replacement target model for some time under the topic of pattern recognition with nonoverlapping target and background clutter [14–20]. The overarching approach taken by these researchers was to make it an
Fig. 1. (a) Calibration targets from the photo resolution range, Edwards Air Force Base, and (b) a tri-bar array at Eglin Air Force Base (imagery from Google Earth) [10].
estimation and detection problem rather than trying to tackle the classical approach [13]. To first order, each starts with a multiple-hypotheses-test where each hypothesis had a target of unknown intensity plus system noise of the form a sx − xj nd xwx − xj
(2)
and background clutter/noise of the form nj x1 − wx − xj :
(3)
In the above, a is the unknown signal variation, s· is the received target signal, xj is the target location in the presence of spatially disjointed noise nj x, and w· is a unit-magnitude window function that defines the support of the target so that it is unity within the target and zero elsewhere [14,18]. Different hypotheses H j corresponds to a possible location xj and values of a. A test statistic equation is generated, differentiated, and then set to zero in order to solve for an estimate of the unknown parameter ˆ is inserted back into the test a. This estimate, a, statistic equation and used to maximize the conditional probability PrH j jrj using Bayes laws to decide which hypothesis is the most probable. A correlation plane results from the filter processing with peak(s) identifying the possible target location(s). Variations or modifications of the above are in the other references, e.g., gray-level target 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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detection [15], regularized filtering [16], and polarimetric contrast imaging [20]. In all cases, sufficient parameters to estimate target location(s) were described, but like the standard approach, no mention of Weber contrast was ever given or implied, i.e., Michelson contrast [21] and Weber contrast are different entities. Schaum’s work [11,12] used a similar initial formulation to develop their continuum fusion methods and might be considered a variation of this earlier work as well. The interested reader should peruse all these cited references to see the variety of the detection problems investigated [11,12,14–20]. This paper complements the cited works of Goudail, Javidi, and Réfrégier by tackling their proposed problem at its most basic level, determining target presence versus target absence. Specifically, we will solve the intractable problem referenced by Schaum in his recent paper; namely, the development of a general solution for the classical resolved target detection in background clutter problem and will do so without requiring the referenced assumptions and approximations cited by Schaum [11]. Like previous research, we initially will employ the Gaussian probability density function (PDF) assumption for each hypothesis. (Although most real images contain multimodal PDFs, many researchers have shown that high pass-like prefiltering of the background cluttered imagery can create locally Gaussian-distributed clutter under stationary conditions with minimal target loss, and the authors believe it is reasonable assumption here [22–30].) In this paper, we will specifically develop an optimum detector for the hypotheses of signal-only plus common system noise H 1 , and background clutter-only plus common system noise H 0 , the classical replacement model. Expressions for the resulting test statistic and its associated detection and false alarm probabilities will be established. Unlike conventional detectors, our optimum detector compares a test statistic to a threshold and decides a target is present if the statistic is LESS than the threshold, something different from classical approaches. In this new theory, the reader will find that (1) the apparent contrast, rather than target signal strength, is the primary determining characteristic of a target in image data, and (2) the detection probability is highly dependent on the background clutter to common system noise ratio, even in the case of zero contrast targets. Finally, we will find that the appropriate definition of SNR is proportional to the contrast and is inversely proportional to the clutter noise standard deviation divided by the background clutter mean, as opposed to the definition in Eq. (1b) where the contrast is divided by the background clutter noise standard deviation alone. Theoretical detection and false alarm probabilities are validated using computer simulation. This theory also provides the upper bound of detection performance by use of known signal parameters, in comparison to the unknown signal case. 5044
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Fig. 2. Example geometry.
optical
surveillance
and
reconnaissance
2. Background
Figure 2 illustrates a basic optical surveillance or reconnaissance sensing scenario [31,32]. A continuous radiance field Fx; y; t, containing both a target and clutter (terrain) radiative sources, is imaged by a simple lens onto a two-dimensional optical focal plane array. The detector array creates a sampled output image Fm; n that measured the energy received at each sample picture element (pixel) during the array’s integration period. This output image is either fed into system hardware for immediate processing or into a storage medium that will be hooked into processing hardware at a later time. For purposes of analysis, it is often convenient to convert the above image matrix into a vector form by column (or row) scanning the matrix and then stringing the elements together in a long vector [31]. This is called vectorization. The reason for dealing with images in vector form is that it (1) provides a more compact notation and (2) allows us to apply results derived previously for one-dimensional signal processing applications [7]. For those reasons, we will be assuming images that already have been transformed into vector forms. 3. Nonzero Contrast Target Detection in Background Clutter
Let us assume we have an optical image containing a multipixel (resolved) target embedded in a spatially varying background. The target is only subject to system noise, and the background is subject to both spatially varying background clutter noise and system noise. Following Helstrom [6], we define N as the number of pixels completely encompassing the target. The target, background clutter mean, and pixel input vector currents then are written as
and
s fsn ; n 1; 2; …; Ng;
(4)
μb fμbn ; n 1; 2; …; Ng;
(5)
i fin ; n 1; 2; …; Ng:
(6)
The two hypotheses in this case are defined as: Hypothesis 0: background clutter plus system noise and Hypothesis 1: target present plus system noise. The PDF for Hypothesis H 0 is given by p0 i
1 2πσ 2T
N∕2
X N in − μbn − μt 2 exp − ; (7) 2σ 2T n1
where in ≡ nth pixel current value in image vector i, μt ≡ system noise mean, μbn ≡ nth pixel background mean, σ 2T σ 2b σ 2t ;
(8)
σ 2T ≡ sum of total background and system noise variances, σ 2b ≡ background noise variance, σ 2t ≡ system noise variance. The PDF for Hypothesis H 1 is given by
1 p1 i 2πσ 2t
N∕2
X N in − sn − μt 2 exp − : 2σ 2t n1
p0 m
1 2πσ 2T
N∕2
X N 2 mn exp − 2σ 2T n1
(9)
1 p1 m 2πσ 2t
N∕2
(10)
X N mn − μbn Cn 2 ; exp − 2σ 2t n1 (11)
respectively, where mn ≡ nth pixel value in the image vector m and Cn ≡ apparent pixel contrast
sn − μbn : μbn
1 1 − α 2σ 2t 2σ 2T
(12)
Using the above two PDFs, we can write the likelihood ratio as P1 m Λm P0 m N X N σT mn − μbn Cn 2 m2n exp − − 2 : σt 2σ 2t 2σ T n1
σ2 σ 2T − σ 2t 2b 2 > 0; 2 2 2σ T σ t 2σ T σ t
ln Λ
2 N X σ N μ C 2 ln T2 − α mn − bn 2n 2 σt 2ασ t n1 N X μbn Cn 2 σ 2
T
−
N X μbn Cn 2
2σ 2b σ 2t 2σ 2t n1 2 2 X N σb σT N μbn Cn 2 ln 2 − mn − 2 σt 2σ 2T σ 2t n1 2ασ 2t n1
N X μbn Cn 2
2σ 2b
:
(16)
Dividing Eq. (16) by −Nσ 2b ∕2σ 2T σ 2t and rearranging the terms, we are able to define the optimum detector to be q
N μbn Cn σ 2T 2 ≤ 1X mn − > q0 ; N n1 σ 2b
(17)
for deciding between hypotheses H 1 and H 0 . When both sides of the inequality equation are divided by a negative number, the sense of the inequality test reverses. The upper inequality/equal-to decides H 1 , and the lower inequality decides H 0. The decision level q0 in Eq. (17) is given by 2 2 2 N∕2 X N 2σ T σ t σt μbn Cn 2 ln Λ0 2 − : q0 − Nσ 2b σT 2σ 2b n1 (18)
(13)
(15)
since σ 2T > σ 2b > σ 2t , in general. Substituting α into Eq. (14), we have
n1
and
with
Redefining the current as mn in − μbn − μt , the above two PDFs can be rewritten as
Taking the logarithm of Eq. (13) gives P m ln Λm ln 1 P0 m 2 X N σ N ln T2 − mn − μbn Cn 2 ∕2σ 2t − m2n ∕2σ 2T 2 σt n1 2 X N σ N μ C 2 mn − bn 2n ln T2 − α 2 σt 2ασ t n1 N X μbn Cn 2 μbn Cn 2 − ; (14) 4ασ 4t 2σ 2t n1
To implement the Neyman–Pearson form of the optimum detector, we select q0 to achieve a preassigned false alarm rate in Eq. (17) rather than use Eq. (18). Let us look for the moment at a test statistic of the form shown in Eq. (17) but without the inverse-N term. By letting 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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4 X N μ2bn C2n σ S1 t2 ; σ b n1 σ 2b
xn ≡ zero mean random variable with variance σ 2 (19) and An ≡ constants;
(20)
under Hypothesis H 1 . Given the above, the probability of false alarm for the test statistic in Eq. (17) can be written as
then we define the test statistic to be of the form s
N X
Z Qf a
xn An 2 :
n1
Equation (22) is the (unnormalized) noncentral chi-squared density with N degrees of freedom, where the function I κ x is the modified Bessel function of the first kind and of order κ {Ref. [7], Eq. (4.87), p. 138}. The parameter S is given by the sum of the squares of Eyn Exn An , or in other words, S
N X
Exn An 2 :
(23)
n1
N 2σ 2T
Z
q0
0
(24)
If we let ν Nq∕σ 2T and define λ0 S0 ∕σ 2T , we obtain Qf a
1 2
Z
Nq0 ∕σ 2T
0
N−2∕4 p ν e−λ0 ν∕2 I N −1 λ0 ν dν: 2 λ0 (31)
The integrand of Eq. (31) is the standard form of the normalized noncentral chi-squared distribution with N degrees of freedom and noncentrality parameter λ0 {Ref. [7], Eq. (4.89), p. 139}. (It should be noted that this distribution also is known as the generalized Rayleigh, or Rayleigh–Rice distribution, or the Rice distribution.) If we define that distribution as f χ ν; N; λ0 , then the probability can be written concisely as Z Qf a
S0
σ 4T σ 2b
X N
μ2bn C2n σ 2b n1
:
5046
N σ 2T X C2n : 2 2 σ b n1 σ b ∕μ2bn
Z Qd
N 2σ 2t
Z
q0 0
q0
0
f 1 qdq
APPLIED OPTICS / Vol. 53, No. 22 / 1 August 2014
(33)
(34)
p S1 Nq Nq N−2∕4 −S1 Nq∕2σ2 t IN e dq −1 2 S1 σ 2t (35)
(27)
However, under Hypothesis H 1, the average of Eyn Exn An must include the nonzero average of xn, which is the term sn − μˆ bn . This implies that the S parameter in this case is given by
(32)
Similarly, the probability of detection can be written as
p NqS N Nq N−2∕4 −SNq∕2σ 2 dq: e IN −1 f q qdq 2 2 S σ2 2σ (26)
f χ ν; N; λ0 dν;
with
(25)
From Eq. (17) under Hypothesis H 0 , we get
Nq0 ∕σ 2T
0
λ0
is the Jacobian of the transformation from s to q. This means that Eq. (24) equals
(29)
(30)
where ∂s N J ∂q
f 0 qdq
N−2∕4 p Nq S0 Nq −S0 Nq∕2σ 2T dq: e I N −1 2 S0 σ 2T
To include the inverse-N term, we define a new random variable q s∕N with PDF f q q and make the following change of variables: f q q ps NqjJj;
q0 0
(21)
The test statistics in Eq. (21) have been analyzed by McDonough and Whalen [7]. By defining the variables yn xn An, they showed that the PDF in this case is given by p 1 s N−2∕4 −Ss∕2σ 2 sS e I N −1 ds: ps sds 2 2 σ2 2σ S (22)
(28)
Z
Nq0 ∕σ 2t 0
f χ ν; N; λ1 dν;
with the noncentrality parameter λ1 given by
(36)
λ1
N σ 2t X C2n : σ 2b n1 σ 2b ∕μ2bn
(37)
It should be noted that there are some sensing scenarios where there is secondary “background clutter” noise source common to both target and its adjacent background clutter. An example is when an airborne camera looks through the ocean surface at an underwater target lying on the ocean floor, and the reflection of the solar and sky radiance by the capillary and gravity waves on the water surface adds additional statistical noise clutter to the image [32]. Stotts and Karp, among others, have shown the wave statistics can be successfully modeled as a Gaussian distribution [33]. Therefore, one can just add the resulting noise variance to both hypotheses’ PDFs just like we did with system noise variance. This makes any analysis of this situation straightforward. As a final observation, we note that there is an interesting relationship between the above two noncentrality parameters λ0 and λ1 . Let us define the difference and ratio of these two parameters as the contrast-to-noise ratio (CNR) and the background noise ratio (BNR), respectively, we have CNR λ0 − λ1
N X
C2n σ 2b ∕μ2bn n1
(38)
and BNR
λ0 σ 2T σ 2b 2 2 1: λ1 σt σt
(39)
Equation (38) is equivalent to the electrical SNR used to characterize detection performance of signal-plus-noise detectors. Its square root is proposed as the correct form of the SNR as opposed to the version in Eq. (1b). By replacing σ b by σ b ∕μb, SNR now is dimensionless as it should be, rather than being in inverse photoelectron counts. Equation (38) also shows that any minimum detectable signal using this equation will strongly depend on N, the number of pixels encompassing the target signature. That is, the larger the target, the easier it is to detect. This agrees with the SNR’s dependence on target area referenced in the introduction [3]. Rewriting Eq. (38) as λ1 λ0 − CNR and substituting it into Eq. (36), we find that Z Qd
BNRNq0 ∕σ 2T 0
f χ ν; N; λ0 − CNRdν:
(40)
The pair of equations (32) and (40) now defines the false alarm and detection probabilities in terms of CNR and BNR. It is clear from Eq. (40) that the higher CNR, the more the PDF under Hypothesis
H 1 shifts toward zero and more of the PDF area will be accumulated by the integration process, resulting in a higher detection rate. Alternately, as CNR goes down, the detection probability also goes down, but the exact detection rate will depend upon the BNR. In the next section, the case of zero contrast will be examined and we will show for the case of the background clutter being stronger than the system noise that a target presence still can be detected. 4. Zero Contrast Target Detection in Background Clutter
In the above development, we assumed a nonzero contrast. One might now ask if the above equations are valid for the situation where sn μbn for all values of n, the zero-contrast case. Again, the target only will be subject to system noise and the background to both spatially varying background clutter noise and the same system noise. Letting the pixel contrast equal zero in Eq. (15) yields q
2 2 2 N∕2 N ≤ 2σ T σ t 1X σ ln Λ0 2t q0 : (41) m2i > N i1 Nσ 2b σT
In this situation, the test statistic now is a measure of the data sample variance and is estimating the variability, or “roughness,” of the sample set. If the sample is sufficiently smooth, i.e., the test is less than the threshold, then the sample is likely a target. However, if the sample is rough, i.e., the test is greater than the threshold, the sample is most likely background clutter. Let us look at the false alarm probability in Eq. (31). Again assume that N is even, i.e., N 2M ≠ 0, where M is an integer. If we further assume the argument of the modified Bessel function is small, then we can write Eq. (31) as Qf a
N−2∕4 Nv λ0 0 pN−2 2 1 λ0 Nv × e−λ0 Nv∕2 dv 2 Z Nq ∕2σ 2 0 1 T −1 wN∕2 e−w dw → N Γ 2 0 N ≈ 2ΓN∕2
Z
q0
r N q0 N I ; −1 2 σ 2T 2
(42)
43
as λ → 0 [34], assuming the Hypothesis 0 noise variance. Here, Iu; p is the Pearson’s form of the incomplete gamma function [7] given by Iu; p
1 Γp 1
Z
up11∕2 0
yp e−y dy:
(44)
The integrand of Eq. (42) is the well-known chisquared distribution [35]. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
5047
Following the same procedure for the probability of detection, we find 1 Q d → N Γ 2 1 N Γ 2
Z
2 h σ
T σ2 t
Nq0 2σ 2 T
i
0
h
Z
BNR
Nq0 2σ 2 T
−1
wN∕2 e−w dw:
(45)
i −1
wN∕2 e−w dw
0
r N q0 N − 1 : ; I BNR 2 σ 2T 2
(46)
(47)
It should be noted that since the apparent contrast is zero and the target-detection CNR relationship in the previous section does not apply, Eq. (47) suggests that BNR takes on the role of the detection SNR, which is also dimensionless. This makes sense because the ratio of the two noise variances will be the deciding factors in the hypothesis test and detection probability. 5. Computer Simulation Results—Nonzero Contrast Case
Let us now see if our false alarm and detection probabilities agree with computer simulation results for the case of a nonzero contrast target. We will start with validating the former. Assume for the moment that N is an even integer. Since the argument of the modified Bessel function is large, then the probability of false alarm in Eq. (31) can be written as Qf a
N ≈ 2
Z
q0 0
N p 2 2π 1 p 8π
Z
Z
Nv λ0
q0 0
N−2∕4
e
NvN−3∕4 λN−1∕4 0
Nq0 ∕σ 2T 0
−λ0 Nv∕2
p e λ0 Nv p dv 2π λ0 Nv1∕4 (48)
p p 2 e− λ0 − Nv ∕2 dv
zN−3∕4
p p 2 z− λ0 ∕2
e− N−1∕4
λ0
dz;
Qf a
1 ≈ p 2π
Z p 2 Nq0 ∕σ 2T y N−1∕2 −y−p λ0 ∕2 p e dy: λ0 0 (51)
As we will soon see, the above also works for oddinteger values of N. Recall that the apparent pixel contrast Cn can be both positive and negative, depending on whether sn > μbn or sn < μbn, respectively. The result is that the test statistic can be of two forms, namely, N μbn jCn jσ 2T 2 ≤ 1X > q0 ; mn ∓ q N n1 σ 2b
(52)
where the upper minus sign applies when the contrast is positive, and the lower plus sign applies when the contrast is negative. Here, j…j denotes the absolute value of the enclosed argument. We begin by validating the above positive contrast probability of false alarm equations. Specifically, we processed four 8192 × 8192 Gaussian noise images using the formula N μbn jCn jσ 2T 2 1X mn − N n1 σ 2b
53
[Eq. (17)] to create an estimated PDF and then calculated its cumulative probability distribution against certain detection thresholds. For this simulation, we set sn 6 and μbn 2 for all values of n, which yields a contrast of 2. The total noise variance σ 2T was set to 1, with the background noise variance σ 2b equal to 0.9 and a system noise variance σ 2t equal to 0.1. Figure 3 is a comparison of the “minus-sign” test simulation results and the probability of false
(49)
(50)
using Eq. (9.7.1) in Abramowitz and Stegun [34]. If we let y
p v
and 1 dy p dν; 2 ν then Eq. (50) becomes 5048
APPLIED OPTICS / Vol. 53, No. 22 / 1 August 2014
Fig. 3. Comparison of “minus-sign” test simulation results with a noncentral chi-squared probability of false alarm as a function of threshold level for N 25, 49, 81, and 121. In the figure, the contrast is 2, the total noise variance 1.0, the background noise variance 0.9, and the system noise variance 0.1.
Fig. 4. Comparison of “plus-sign” test simulation results with a noncentral chi-squared probability of false alarm as a function of threshold level for N 25, 49, 81, and 121. In the figure, the contrast is 2, the total noise variance 1.0, the background noise variance 0.9, and the system noise variance 0.1.
Fig. 5. Comparison between the detection probability given in Eq. (36) and the cumulative signal-plus-system-noise distribution as a function of threshold for N 25, 49, 81, and 121. In the figure, the contrast is 2, the total noise variance 1.0, the background noise variance 0.9, and the system noise variance 0.1.
alarm using Eq. (51) as a function of the normalized threshold for N 25, 49, 81, and 121. Figure 4 is a similar comparison, but we are using the formula
create the estimated PDF for the test statistic. Figure 5 compares the detection probability given in Eq. (36) and the cumulative distribution derived from this PDF as a function of threshold. Again, we see very excellent agreement between theory and simulation results. This indicates that we have a consistent and valid theory for resolved target detection in background clutter. It is interesting to observe in Fig. 5 the effect the number of target pixels has on the probability of detection. The various curves all cross at a decision level q0 ≈ 0.3 in the region just above the 50% detection level. Above the crossover point, the detection rate increases with increasing values of the target mask size N; below that point, the opposite occurs. That is, the detection performance in the latter case deteriorates with increasing N. It can be shown that the average and variance of the normalized test statistic q are given by
N μbn jCn jσ 2T 2 1X mn : N n1 σ 2b
(54)
In both cases, the agreement between theory and simulation is excellent and clearly show that the larger the target, the better the detector performs. This latter fact is true for all statistical results contained in this paper. The next observation one can derive from these two figures is that our test statistic is valid whether sn > μbn (positive contrast) or sn < μbn (negative contrast). These figures also confirm that Eq. (51) is true for odd integer targets as well as even integer targets since all of the values of N employed were odd in showing the excellent agreement between theory and simulation. Let us now look at the validation of the probability of detection and what we can learn from that. As before, we set sn 6 and μbn 2 for all values of n; contrast equal to 2. The total noise variance σ 2T was set to 1, with the background noise variance σ 2b equal to 0.9 and a system noise variance σ 2t equal to 0.1. For the probability of detection simulation, we created four 8192 × 8192 Gaussian noise images that were zeromean variance σ 2T . We then placed p and had pnoise N -pixel-by- N -pixel targets with mean sn − μbn and Gaussian noise of variance σ 2t , at selected locations in those four images to guarantee no target interaction when filtering was done. In particular in each image, we placed a target at pixel location (50, 50) and subsequently placed other targets at locations 125 pixels apart in both direction x and y from that point, ending at location (8050, 8050). The images then were processed using Eq. (53) to
Eq
σ 2t λ1 N N
(55)
σ 4t 4λ1 2N ; N2
(56)
and Varq
respectively. For the noise and clutter parameters selected for this example, the noncentrality parameter λ1 equals 1.9753N. This means that the above equations reduce to Eq 0.29753
(57)
Varq 0.0990∕N;
(58)
and
1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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because it is below the output noise floor” [16]. We do not agree. The authors feel that their Fig. 1(c) clearly shows a target detection peak like in Fig. 7, although broadened by their estimation/detection method, and is essentially in the correct location of their initial tank target. 6. Computer Simulation Results—Zero Contrast Case
Fig. 6. Plots of the test statistic PDF as a function of the test statistic q for N 25, 49, 81, and 121.
respectively. Thus, we find for our PDFs that their test statistic average should be a constant for all values of N and that their variance falls off by inverse N. These aspects clearly can be seen in Fig. 6, which depict the processed data PDFs for N 25, 49, 81, and 121. These graphs illustrate the same mean value for all the plotted N values and that the width (variance) of the PDFs decreases as N values get bigger. It is clear that Eq is the same value of the crossover point in Fig. 5. This means that the Qd ’s for smaller N grow faster than the Qd ’s for larger N for small decision levels close to zero, but the larger N probabilities of detection catch up and exceed the lower N Qd ’s as one approaches and passes the crossover point; thus explaining the behavior noted above. Finally, Fig. 7 shows a one-sub-frame realization of a N 25 target detection used in the creation of Fig. 5. As implied by our test statistic having reversed comparison signs with the test statistic, the “matched filter peak” after processing is below the processed noise level, not above. Javidi et al. noted in one of their papers that a similar plot [Fig. 1(c)] implied that “the correlation peak cannot be seen
Fig. 7. One-sub-frame realization of a N 25 target detection used in the creation of Fig. 8. 5050
APPLIED OPTICS / Vol. 53, No. 22 / 1 August 2014
Let us now turn to the case of a zero contrast target. Specifically, let us compare computer simulation results with our zero-contrast false alarm and detection probabilities. Figure 8 is a comparison of the cumulative distribution from the test statistic given in Eq. (41) using four 8192 × 8192 processed images of Gaussian noise with variance equal to one (1) and the probability of false alarm from Eq. (43). On the other hand, Fig. 9 is a comparison of the cumulative distribution from the test statistic given in Eq. (41) using four 8192 × 8192 processed images of Gaussian noise with variance equal to 0.1 and the probability of detection from Eq. (47). Both figures show excellent agreement between simulation results and theory. Referring to Fig. 9, we again see a crossover point similar to the one in Fig. 5. However, its value is different. The average and variance of the normalized test statistic q in this case are given by Eq σ 2t
(59)
and Varq
2σ 4t ; N
(60)
respectively, since the noncentrality parameter is zero. Substituting in our simulation value for σ 2t , the above equations reduce to Eq 0.1
(61)
Fig. 8. Comparison of the cumulative distribution from the test statistic in Eq. (41) and the probability of false alarm from Eq. (43).
upon the BNR. Finally, we found that the above definition of CNR disagrees with the conventional definition of SNR and needs the additional multiplicative factor of background clutter mean to be correct. These are important findings for any scientist or engineer designing electro-optical and infrared sensors for finding resolved targets immersed in background-cluttered images. References and Note
Fig. 9. Comparison of the cumulative distribution from the test statistic in Eq. (39) and the probability of detection from Eq. (45).
and Varq 0.02∕N;
(62)
respectively. Clearly, Fig. 9 shows that the crossover point equals Eq. (61) and the curves exhibit the same behavior as in Fig. 5. This implies that the explanation for the probability of detection results in the nonzero contrast case equally apply here as well. 7. Summary
This paper developed a general solution for the resolved target detection in background clutter problem considered previously as an intractable problem. To do so, we had to recognize that a resolved target obscures any background clutter that would have been observable if the target were absent. Using Gaussian PDFs for background clutter and sensor noise, we developed an optimum detector deciding between the hypotheses of signal-only plus common system noise H 1 and background clutter-only plus common system noise H 0 . Counter to convention, this detector compares a test statistic to a threshold and decides a target is present if the statistic is LESS than the threshold, not greater than the threshold. Expressions for the resulting test statistic and its associated detection and false alarm probabilities were established and found to be in excellent agreement with computer simulation results, even for the case of zero contrast targets. Key findings for the nonzero contrast case are that the detection probability depends on (1) the contrast noise ratio (CNR) defined as the sum of the square of the apparent pixel contrast times the mean background clutter level, divided by the background clutter variance, and (2) the background noise ratio (BNR) defined as the ratio of the total noise variance divided by the system noise variance. Alternately, the key findings for the zero contrast target case were that (1) the target is detectable and (2) its detection probability depended solely
1. S. Q. Duntley, “Biography from University of California,” http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1229& context=sio/arch. 2. “History of the Scripps Visibility Lab,” http://www.physics .miami.edu/optics/ken/OtherPapers/A25_VA_OOXVI_2002.pdf. 3. S. W. Smith, “Special imaging techniques,” in The Scientist and Engineer’s Guide to Digital Signal Processing (California Technical, 1997), Chap. 25, pp. 433–434. 4. H. R. Blackwell, “Contrast thresholds of the human eye,” J. Opt. Soc. Am. 36, 624–643 (1946). 5. S. Karp and L. B. Stotts, “Visible/infrared imaging on the rise,” IEEE Signal Process. Mag. 30(6), 178–182 (2013). 6. C. W. Helstrom, “The detection and resolution of optical signals,” IEEE Trans. Inf. Theory 10, 275–287 (1964). 7. R. N. McDonough and A. D. Whalen, Detection of Signals in Noise, 2nd ed. (Academic 1995). 8. E. Hallenborg, D. L. Buck, and E. Daly, “Evaluation of automated algorithms for small target detection and non-natural terrain characterization using remote multi-band imagery,” Proc. SPIE 8137, 813711 (2011). 9. M. Kabate, M. R. Azimi-Sadjadi, and J. D. Tucker, “An underwater target detection system for electro-optical imagery data,” in OCEANS 2009, MTS/IEEE Biloxi—Marine Technology for Our Future: Global and Local Challenges, 26–29 October, 2009, pp. 1–8. 10. Fifteen Tri-bar Arrays, Photo Resolution Range, target at Eglin Air Force Base, and at Edwards Air Force Base, CA (from west to east), 2012, 15 images, 8 × 10 inches each, imagery from Google Earth. Scaled and cropped by the Center for Land Use Interpretation. Permanent Collection of UCR Sweeney Art Gallery at UCR ARTS block, University of California, Riverside. Web link with more information: http://sites.artsblock.ucr.edu/free‑enterprise/pclui/. 11. A. Schaum, “Enough with the additive target model,” Proc. SPIE 9088, 90880C (2014). 12. A. P. Schaum and B. J. Daniel, “Continuum fusion methods of spectral detection,” Opt. Eng. 51, 111718 (2012). 13. F. Goudail, Institut d’Optique Graduate School (private communication, 2014). 14. F. Goudail and P. Réfrégier, “Optimal and suboptimal detection of a target with random grey levels imbedded in nonoverlapping noise,” Opt. Commun. 125, 211–216 (1996). 15. F. Goudail and P. Réfrégier, “Influence of nonoverlapping noise on regularized linear filters for pattern recognition,” Opt. Lett. 20, 2237–2239 (1995). 16. F. Goudail and P. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint background noise,” Opt. Lett. 21, 495–497 (1996). 17. B. Javidi and J. Wang, “Design of filters to detect a noisy target in nonoverlapping background noise,” J. Opt. Soc. Am. A 11, 2604–2612 (1994). 18. B. Javidi, P. Réfrégier, and P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993). 19. B. Javidi and J. Wang, “Limitation of the classic definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992). 20. P. Réfrégier and F. Goudail, “Invariant polarimetric contrast parameters of coherent light,” J. Opt. Soc. Am. A 19, 1223–1233 (2002). 21. A. A. Michelson, Studies in Optics (University of Chicago, 1927). 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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22. B. R. Hunt and T. M. Cannon, “Nonstationary assumptions of Gaussian models of images,” IEEE Trans. Syst. Man Cybern. 6, 876–882 (1976). 23. H. J. Trussell and R. P. Kruger, “Comments on ‘Nonstationary assumptions for Gaussian models in images’,” IEEE Trans. Syst. Man Cybern. 8, 579–582 (1978). 24. M. R. Whiteley, B. M. Welsh, and M. C. Roggemann, “Limitations of Gaussian assumptions for the irradiance distribution in digital imagery: nonstationary image ensemble considerations,” J. Opt. Soc. Am. A 15, 802–810 (1998). 25. X. Yu, L. E. Hoff, I. S. Reed, A. M. Chen, and L. B. Stotts, “Automatic target detection and recognition in multiband imagery: a unified ML detection and estimation approach,” IEEE Trans. Image Process. 6, 143–156 (1997). 26. X. Yu, I. S. Reed, and A. D. Stocker, “Comparative performance analysis of adaptive multispectral detectors,” IEEE Trans. Signal Process. 41, 2639–2656 (1993). 27. E. Winter and M. J. Schlangen, “Detection of extended targets using multi-spectral infrared,” Proc. SPIE 1954, 173–180 (1993). 28. I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760–1770 (1990).
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29. L. B. Stotts, S. R. Yool, J. R. Evans, D. N. Williams, I. S. Reed, and Y. Barniv, “Two- and three-dimensional matched filtering techniques for locating dim air vehicles in highly cluttered staring IR sensor data,” Proceedings of the Twelfth Annual DARPA Strategic Systems Symposium, Naval Postgraduate School, Monterey, California, October 28–30, 1986. 30. L. B. Stotts, S. R. Yool, J. R. Evans, and D. N. Williams, “Techniques for detecting dim air vehicles in highly-cluttered staring IR sensor data,” Proceedings of the National IRIS, Orlando, Florida, October, 1987, pp. 20–23. 31. W. K. Pratt, Digital Image Processing, 4th ed. (Wiley, 2007). 32. S. Karp and L. B. Stotts, Fundamentals of Electro-Optic Systems Design: Communications, Lidar, and Imaging (Cambridge 2013). 33. L. B. Stotts and S. Karp, “Wave statistics derived from optical radiance measurements below the sea surface,” Appl. Opt. 21, 978–981 (1982). 34. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, 1970). 35. C. W. Helstrom, “Multiple random variables,” in Probability and Stochastic Process for Engineers, 2nd ed. (Macmillan, 1991), Chap. 4, p. 220.
Science and Technology Associates, 4100 N. Fairfax Drive, Suite 910, Arlington, Virginia 22203, USA 2
Hoff Engineering, 3627 Tennyson Street, San Diego, California 92106, USA *Corresponding author: [email protected] Received 2 April 2014; revised 13 June 2014; accepted 30 June 2014; posted 1 July 2014 (Doc. ID 208754); published 30 July 2014
The use of optics to detect targets has been around for a long time. Early attempts at automatic target detection assumed target plus noise, which means that the targets were small compared to the pixel field of view and therefore unresolved. However, the advent of advanced focal plane technology has resulted in optical systems that can provide highly resolved target images. The intent of this paper is to develop a general solution for the detection of resolved targets in background clutter. We recognize that resolved targets obscure any background clutter that would have been visible if the targets were absent. An optimum detection algorithm is derived that compares a test statistic to a threshold and decides a target is present if the statistic is less than the threshold. We find that the detection performance depends upon (1) the apparent contrast rather than the signal to noise ratio and (2) is highly dependent on the background clutter to common system noise ratio. In fact, the target can still be detected even when the target contrast goes to zero provided the background clutter is greater than the common system noise. Computer simulations are shown to validate the theoretical detection and false alarm probabilities. The findings in this paper should be useful to engineers and scientists designing electro-optical and infrared sensors for finding resolved targets immersed in background-cluttered images. © 2014 Optical Society of America OCIS codes: (100.2960) Image analysis; (100.4997) Pattern recognition, nonlinear spatial filters; (280.4788) Optical sensing and sensors. http://dx.doi.org/10.1364/AO.53.005042
1. Introduction
The application of optics to detecting objects at a distance has been around since the earliest known working telescopes appeared in 1608, first credited to Hans Lippershey. Duntley and his colleagues, beginning at MIT and later at the Visibility Laboratory of the Scripps Institute of Oceanography, were instrumental in establishing the science of visibility that the optics community has relied upon since the 1940s [1,2]. What they found was that a resolved object of interest can be “seen” because it has a different radiance (or irradiance) than the other sources of
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radiance (irradiance) surrounding it. The accepted metric for an object’s visibility is the Weber contrast, defined as C
N0 − Nb ; Nb
(1a)
where N 0 is object’s radiance and N b is the background radiance. Under this definition, the contrast varies between −1 for N b ≫ N 0 and ∞ for N b ≪ N 0, looking at the extremes of these two conditions. The accepted minimum visibility of an object that can be seen is the metrological visibility, defined as 2%. However, in practice, some observers have been able to see objects with contrast down to 0.5% [3,4]. The associated signal-to-noise ratio (SNR) for object
detection is defined as the above contrast divided by the standard deviation of the noise [3]. Mathematically, this is written as SNR
C σ noise
;
(1b)
where C denotes the contrast and σ noise the standard deviation of the noise. (The electrical SNR is the square of the above since optical detectors create current proportional to the received optical radiance [5].) (It is interesting to note that C is dimensionless, but σ noise is in photoelectron counts, which means that SNR is in inverse photoelectron counts.) The exact value for the minimum detectable SNR depends on the size of the object; specifically, proportional to the square root of its area of the target [3]. The minimum value appears to be around one but may be higher for some observers [3,4]. To date, there has not been a formal mathematical connection between contrast and SNR in statistical hypothesis testing, only empirically like the above [6,7]. The standard approach for analyzing the optical detection of targets in clutter was first reported by Helstrom [6] in 1964, and it parallels the signal-plus-additive hypothesis testing developed for RF communications and sensing [7]. Received power, and background and system noise standard deviations, are their key performance variable, with no mention of contrast [7]. Papers employing these techniques still can be found in the literature today [8,9]. Unfortunately in these papers, the signalplus-noise assumption is inappropriate because a resolved target obscures any background clutter, violating the key premise of this hypothesis test. This can be clearly seen in Fig. 1, which shows two examples of such a situation [10]. Specifically, we see in these figures that the bar targets, asphalt, airplanes, and/or buildings mask any terrain signature underneath them, and that background terrain is found elsewhere. Schaum recently complained about the continued use of the above additive model in electro-optical image analysis [11]. Because of its “phenomenological inaccuracy,” he advocated the use of the more appropriate replacement target model. Specifically, he recognized that this model is seldom used because the classical analysis approach to this background replacement problem was intractable unless narrow restrictions are imposed, but it is the appropriate problem to solve. His proposed solution was to apply continuum fusion methodology [12], and he presented a detailed paper on its application, with examples to validate his approach to replacement model detection analysis [11]. As pointed out by Goudail in a private communication [13], researchers have been developing techniques for the replacement target model for some time under the topic of pattern recognition with nonoverlapping target and background clutter [14–20]. The overarching approach taken by these researchers was to make it an
Fig. 1. (a) Calibration targets from the photo resolution range, Edwards Air Force Base, and (b) a tri-bar array at Eglin Air Force Base (imagery from Google Earth) [10].
estimation and detection problem rather than trying to tackle the classical approach [13]. To first order, each starts with a multiple-hypotheses-test where each hypothesis had a target of unknown intensity plus system noise of the form a sx − xj nd xwx − xj
(2)
and background clutter/noise of the form nj x1 − wx − xj :
(3)
In the above, a is the unknown signal variation, s· is the received target signal, xj is the target location in the presence of spatially disjointed noise nj x, and w· is a unit-magnitude window function that defines the support of the target so that it is unity within the target and zero elsewhere [14,18]. Different hypotheses H j corresponds to a possible location xj and values of a. A test statistic equation is generated, differentiated, and then set to zero in order to solve for an estimate of the unknown parameter ˆ is inserted back into the test a. This estimate, a, statistic equation and used to maximize the conditional probability PrH j jrj using Bayes laws to decide which hypothesis is the most probable. A correlation plane results from the filter processing with peak(s) identifying the possible target location(s). Variations or modifications of the above are in the other references, e.g., gray-level target 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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detection [15], regularized filtering [16], and polarimetric contrast imaging [20]. In all cases, sufficient parameters to estimate target location(s) were described, but like the standard approach, no mention of Weber contrast was ever given or implied, i.e., Michelson contrast [21] and Weber contrast are different entities. Schaum’s work [11,12] used a similar initial formulation to develop their continuum fusion methods and might be considered a variation of this earlier work as well. The interested reader should peruse all these cited references to see the variety of the detection problems investigated [11,12,14–20]. This paper complements the cited works of Goudail, Javidi, and Réfrégier by tackling their proposed problem at its most basic level, determining target presence versus target absence. Specifically, we will solve the intractable problem referenced by Schaum in his recent paper; namely, the development of a general solution for the classical resolved target detection in background clutter problem and will do so without requiring the referenced assumptions and approximations cited by Schaum [11]. Like previous research, we initially will employ the Gaussian probability density function (PDF) assumption for each hypothesis. (Although most real images contain multimodal PDFs, many researchers have shown that high pass-like prefiltering of the background cluttered imagery can create locally Gaussian-distributed clutter under stationary conditions with minimal target loss, and the authors believe it is reasonable assumption here [22–30].) In this paper, we will specifically develop an optimum detector for the hypotheses of signal-only plus common system noise H 1 , and background clutter-only plus common system noise H 0 , the classical replacement model. Expressions for the resulting test statistic and its associated detection and false alarm probabilities will be established. Unlike conventional detectors, our optimum detector compares a test statistic to a threshold and decides a target is present if the statistic is LESS than the threshold, something different from classical approaches. In this new theory, the reader will find that (1) the apparent contrast, rather than target signal strength, is the primary determining characteristic of a target in image data, and (2) the detection probability is highly dependent on the background clutter to common system noise ratio, even in the case of zero contrast targets. Finally, we will find that the appropriate definition of SNR is proportional to the contrast and is inversely proportional to the clutter noise standard deviation divided by the background clutter mean, as opposed to the definition in Eq. (1b) where the contrast is divided by the background clutter noise standard deviation alone. Theoretical detection and false alarm probabilities are validated using computer simulation. This theory also provides the upper bound of detection performance by use of known signal parameters, in comparison to the unknown signal case. 5044
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Fig. 2. Example geometry.
optical
surveillance
and
reconnaissance
2. Background
Figure 2 illustrates a basic optical surveillance or reconnaissance sensing scenario [31,32]. A continuous radiance field Fx; y; t, containing both a target and clutter (terrain) radiative sources, is imaged by a simple lens onto a two-dimensional optical focal plane array. The detector array creates a sampled output image Fm; n that measured the energy received at each sample picture element (pixel) during the array’s integration period. This output image is either fed into system hardware for immediate processing or into a storage medium that will be hooked into processing hardware at a later time. For purposes of analysis, it is often convenient to convert the above image matrix into a vector form by column (or row) scanning the matrix and then stringing the elements together in a long vector [31]. This is called vectorization. The reason for dealing with images in vector form is that it (1) provides a more compact notation and (2) allows us to apply results derived previously for one-dimensional signal processing applications [7]. For those reasons, we will be assuming images that already have been transformed into vector forms. 3. Nonzero Contrast Target Detection in Background Clutter
Let us assume we have an optical image containing a multipixel (resolved) target embedded in a spatially varying background. The target is only subject to system noise, and the background is subject to both spatially varying background clutter noise and system noise. Following Helstrom [6], we define N as the number of pixels completely encompassing the target. The target, background clutter mean, and pixel input vector currents then are written as
and
s fsn ; n 1; 2; …; Ng;
(4)
μb fμbn ; n 1; 2; …; Ng;
(5)
i fin ; n 1; 2; …; Ng:
(6)
The two hypotheses in this case are defined as: Hypothesis 0: background clutter plus system noise and Hypothesis 1: target present plus system noise. The PDF for Hypothesis H 0 is given by p0 i
1 2πσ 2T
N∕2
X N in − μbn − μt 2 exp − ; (7) 2σ 2T n1
where in ≡ nth pixel current value in image vector i, μt ≡ system noise mean, μbn ≡ nth pixel background mean, σ 2T σ 2b σ 2t ;
(8)
σ 2T ≡ sum of total background and system noise variances, σ 2b ≡ background noise variance, σ 2t ≡ system noise variance. The PDF for Hypothesis H 1 is given by
1 p1 i 2πσ 2t
N∕2
X N in − sn − μt 2 exp − : 2σ 2t n1
p0 m
1 2πσ 2T
N∕2
X N 2 mn exp − 2σ 2T n1
(9)
1 p1 m 2πσ 2t
N∕2
(10)
X N mn − μbn Cn 2 ; exp − 2σ 2t n1 (11)
respectively, where mn ≡ nth pixel value in the image vector m and Cn ≡ apparent pixel contrast
sn − μbn : μbn
1 1 − α 2σ 2t 2σ 2T
(12)
Using the above two PDFs, we can write the likelihood ratio as P1 m Λm P0 m N X N σT mn − μbn Cn 2 m2n exp − − 2 : σt 2σ 2t 2σ T n1
σ2 σ 2T − σ 2t 2b 2 > 0; 2 2 2σ T σ t 2σ T σ t
ln Λ
2 N X σ N μ C 2 ln T2 − α mn − bn 2n 2 σt 2ασ t n1 N X μbn Cn 2 σ 2
T
−
N X μbn Cn 2
2σ 2b σ 2t 2σ 2t n1 2 2 X N σb σT N μbn Cn 2 ln 2 − mn − 2 σt 2σ 2T σ 2t n1 2ασ 2t n1
N X μbn Cn 2
2σ 2b
:
(16)
Dividing Eq. (16) by −Nσ 2b ∕2σ 2T σ 2t and rearranging the terms, we are able to define the optimum detector to be q
N μbn Cn σ 2T 2 ≤ 1X mn − > q0 ; N n1 σ 2b
(17)
for deciding between hypotheses H 1 and H 0 . When both sides of the inequality equation are divided by a negative number, the sense of the inequality test reverses. The upper inequality/equal-to decides H 1 , and the lower inequality decides H 0. The decision level q0 in Eq. (17) is given by 2 2 2 N∕2 X N 2σ T σ t σt μbn Cn 2 ln Λ0 2 − : q0 − Nσ 2b σT 2σ 2b n1 (18)
(13)
(15)
since σ 2T > σ 2b > σ 2t , in general. Substituting α into Eq. (14), we have
n1
and
with
Redefining the current as mn in − μbn − μt , the above two PDFs can be rewritten as
Taking the logarithm of Eq. (13) gives P m ln Λm ln 1 P0 m 2 X N σ N ln T2 − mn − μbn Cn 2 ∕2σ 2t − m2n ∕2σ 2T 2 σt n1 2 X N σ N μ C 2 mn − bn 2n ln T2 − α 2 σt 2ασ t n1 N X μbn Cn 2 μbn Cn 2 − ; (14) 4ασ 4t 2σ 2t n1
To implement the Neyman–Pearson form of the optimum detector, we select q0 to achieve a preassigned false alarm rate in Eq. (17) rather than use Eq. (18). Let us look for the moment at a test statistic of the form shown in Eq. (17) but without the inverse-N term. By letting 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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4 X N μ2bn C2n σ S1 t2 ; σ b n1 σ 2b
xn ≡ zero mean random variable with variance σ 2 (19) and An ≡ constants;
(20)
under Hypothesis H 1 . Given the above, the probability of false alarm for the test statistic in Eq. (17) can be written as
then we define the test statistic to be of the form s
N X
Z Qf a
xn An 2 :
n1
Equation (22) is the (unnormalized) noncentral chi-squared density with N degrees of freedom, where the function I κ x is the modified Bessel function of the first kind and of order κ {Ref. [7], Eq. (4.87), p. 138}. The parameter S is given by the sum of the squares of Eyn Exn An , or in other words, S
N X
Exn An 2 :
(23)
n1
N 2σ 2T
Z
q0
0
(24)
If we let ν Nq∕σ 2T and define λ0 S0 ∕σ 2T , we obtain Qf a
1 2
Z
Nq0 ∕σ 2T
0
N−2∕4 p ν e−λ0 ν∕2 I N −1 λ0 ν dν: 2 λ0 (31)
The integrand of Eq. (31) is the standard form of the normalized noncentral chi-squared distribution with N degrees of freedom and noncentrality parameter λ0 {Ref. [7], Eq. (4.89), p. 139}. (It should be noted that this distribution also is known as the generalized Rayleigh, or Rayleigh–Rice distribution, or the Rice distribution.) If we define that distribution as f χ ν; N; λ0 , then the probability can be written concisely as Z Qf a
S0
σ 4T σ 2b
X N
μ2bn C2n σ 2b n1
:
5046
N σ 2T X C2n : 2 2 σ b n1 σ b ∕μ2bn
Z Qd
N 2σ 2t
Z
q0 0
q0
0
f 1 qdq
APPLIED OPTICS / Vol. 53, No. 22 / 1 August 2014
(33)
(34)
p S1 Nq Nq N−2∕4 −S1 Nq∕2σ2 t IN e dq −1 2 S1 σ 2t (35)
(27)
However, under Hypothesis H 1, the average of Eyn Exn An must include the nonzero average of xn, which is the term sn − μˆ bn . This implies that the S parameter in this case is given by
(32)
Similarly, the probability of detection can be written as
p NqS N Nq N−2∕4 −SNq∕2σ 2 dq: e IN −1 f q qdq 2 2 S σ2 2σ (26)
f χ ν; N; λ0 dν;
with
(25)
From Eq. (17) under Hypothesis H 0 , we get
Nq0 ∕σ 2T
0
λ0
is the Jacobian of the transformation from s to q. This means that Eq. (24) equals
(29)
(30)
where ∂s N J ∂q
f 0 qdq
N−2∕4 p Nq S0 Nq −S0 Nq∕2σ 2T dq: e I N −1 2 S0 σ 2T
To include the inverse-N term, we define a new random variable q s∕N with PDF f q q and make the following change of variables: f q q ps NqjJj;
q0 0
(21)
The test statistics in Eq. (21) have been analyzed by McDonough and Whalen [7]. By defining the variables yn xn An, they showed that the PDF in this case is given by p 1 s N−2∕4 −Ss∕2σ 2 sS e I N −1 ds: ps sds 2 2 σ2 2σ S (22)
(28)
Z
Nq0 ∕σ 2t 0
f χ ν; N; λ1 dν;
with the noncentrality parameter λ1 given by
(36)
λ1
N σ 2t X C2n : σ 2b n1 σ 2b ∕μ2bn
(37)
It should be noted that there are some sensing scenarios where there is secondary “background clutter” noise source common to both target and its adjacent background clutter. An example is when an airborne camera looks through the ocean surface at an underwater target lying on the ocean floor, and the reflection of the solar and sky radiance by the capillary and gravity waves on the water surface adds additional statistical noise clutter to the image [32]. Stotts and Karp, among others, have shown the wave statistics can be successfully modeled as a Gaussian distribution [33]. Therefore, one can just add the resulting noise variance to both hypotheses’ PDFs just like we did with system noise variance. This makes any analysis of this situation straightforward. As a final observation, we note that there is an interesting relationship between the above two noncentrality parameters λ0 and λ1 . Let us define the difference and ratio of these two parameters as the contrast-to-noise ratio (CNR) and the background noise ratio (BNR), respectively, we have CNR λ0 − λ1
N X
C2n σ 2b ∕μ2bn n1
(38)
and BNR
λ0 σ 2T σ 2b 2 2 1: λ1 σt σt
(39)
Equation (38) is equivalent to the electrical SNR used to characterize detection performance of signal-plus-noise detectors. Its square root is proposed as the correct form of the SNR as opposed to the version in Eq. (1b). By replacing σ b by σ b ∕μb, SNR now is dimensionless as it should be, rather than being in inverse photoelectron counts. Equation (38) also shows that any minimum detectable signal using this equation will strongly depend on N, the number of pixels encompassing the target signature. That is, the larger the target, the easier it is to detect. This agrees with the SNR’s dependence on target area referenced in the introduction [3]. Rewriting Eq. (38) as λ1 λ0 − CNR and substituting it into Eq. (36), we find that Z Qd
BNRNq0 ∕σ 2T 0
f χ ν; N; λ0 − CNRdν:
(40)
The pair of equations (32) and (40) now defines the false alarm and detection probabilities in terms of CNR and BNR. It is clear from Eq. (40) that the higher CNR, the more the PDF under Hypothesis
H 1 shifts toward zero and more of the PDF area will be accumulated by the integration process, resulting in a higher detection rate. Alternately, as CNR goes down, the detection probability also goes down, but the exact detection rate will depend upon the BNR. In the next section, the case of zero contrast will be examined and we will show for the case of the background clutter being stronger than the system noise that a target presence still can be detected. 4. Zero Contrast Target Detection in Background Clutter
In the above development, we assumed a nonzero contrast. One might now ask if the above equations are valid for the situation where sn μbn for all values of n, the zero-contrast case. Again, the target only will be subject to system noise and the background to both spatially varying background clutter noise and the same system noise. Letting the pixel contrast equal zero in Eq. (15) yields q
2 2 2 N∕2 N ≤ 2σ T σ t 1X σ ln Λ0 2t q0 : (41) m2i > N i1 Nσ 2b σT
In this situation, the test statistic now is a measure of the data sample variance and is estimating the variability, or “roughness,” of the sample set. If the sample is sufficiently smooth, i.e., the test is less than the threshold, then the sample is likely a target. However, if the sample is rough, i.e., the test is greater than the threshold, the sample is most likely background clutter. Let us look at the false alarm probability in Eq. (31). Again assume that N is even, i.e., N 2M ≠ 0, where M is an integer. If we further assume the argument of the modified Bessel function is small, then we can write Eq. (31) as Qf a
N−2∕4 Nv λ0 0 pN−2 2 1 λ0 Nv × e−λ0 Nv∕2 dv 2 Z Nq ∕2σ 2 0 1 T −1 wN∕2 e−w dw → N Γ 2 0 N ≈ 2ΓN∕2
Z
q0
r N q0 N I ; −1 2 σ 2T 2
(42)
43
as λ → 0 [34], assuming the Hypothesis 0 noise variance. Here, Iu; p is the Pearson’s form of the incomplete gamma function [7] given by Iu; p
1 Γp 1
Z
up11∕2 0
yp e−y dy:
(44)
The integrand of Eq. (42) is the well-known chisquared distribution [35]. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Following the same procedure for the probability of detection, we find 1 Q d → N Γ 2 1 N Γ 2
Z
2 h σ
T σ2 t
Nq0 2σ 2 T
i
0
h
Z
BNR
Nq0 2σ 2 T
−1
wN∕2 e−w dw:
(45)
i −1
wN∕2 e−w dw
0
r N q0 N − 1 : ; I BNR 2 σ 2T 2
(46)
(47)
It should be noted that since the apparent contrast is zero and the target-detection CNR relationship in the previous section does not apply, Eq. (47) suggests that BNR takes on the role of the detection SNR, which is also dimensionless. This makes sense because the ratio of the two noise variances will be the deciding factors in the hypothesis test and detection probability. 5. Computer Simulation Results—Nonzero Contrast Case
Let us now see if our false alarm and detection probabilities agree with computer simulation results for the case of a nonzero contrast target. We will start with validating the former. Assume for the moment that N is an even integer. Since the argument of the modified Bessel function is large, then the probability of false alarm in Eq. (31) can be written as Qf a
N ≈ 2
Z
q0 0
N p 2 2π 1 p 8π
Z
Z
Nv λ0
q0 0
N−2∕4
e
NvN−3∕4 λN−1∕4 0
Nq0 ∕σ 2T 0
−λ0 Nv∕2
p e λ0 Nv p dv 2π λ0 Nv1∕4 (48)
p p 2 e− λ0 − Nv ∕2 dv
zN−3∕4
p p 2 z− λ0 ∕2
e− N−1∕4
λ0
dz;
Qf a
1 ≈ p 2π
Z p 2 Nq0 ∕σ 2T y N−1∕2 −y−p λ0 ∕2 p e dy: λ0 0 (51)
As we will soon see, the above also works for oddinteger values of N. Recall that the apparent pixel contrast Cn can be both positive and negative, depending on whether sn > μbn or sn < μbn, respectively. The result is that the test statistic can be of two forms, namely, N μbn jCn jσ 2T 2 ≤ 1X > q0 ; mn ∓ q N n1 σ 2b
(52)
where the upper minus sign applies when the contrast is positive, and the lower plus sign applies when the contrast is negative. Here, j…j denotes the absolute value of the enclosed argument. We begin by validating the above positive contrast probability of false alarm equations. Specifically, we processed four 8192 × 8192 Gaussian noise images using the formula N μbn jCn jσ 2T 2 1X mn − N n1 σ 2b
53
[Eq. (17)] to create an estimated PDF and then calculated its cumulative probability distribution against certain detection thresholds. For this simulation, we set sn 6 and μbn 2 for all values of n, which yields a contrast of 2. The total noise variance σ 2T was set to 1, with the background noise variance σ 2b equal to 0.9 and a system noise variance σ 2t equal to 0.1. Figure 3 is a comparison of the “minus-sign” test simulation results and the probability of false
(49)
(50)
using Eq. (9.7.1) in Abramowitz and Stegun [34]. If we let y
p v
and 1 dy p dν; 2 ν then Eq. (50) becomes 5048
APPLIED OPTICS / Vol. 53, No. 22 / 1 August 2014
Fig. 3. Comparison of “minus-sign” test simulation results with a noncentral chi-squared probability of false alarm as a function of threshold level for N 25, 49, 81, and 121. In the figure, the contrast is 2, the total noise variance 1.0, the background noise variance 0.9, and the system noise variance 0.1.
Fig. 4. Comparison of “plus-sign” test simulation results with a noncentral chi-squared probability of false alarm as a function of threshold level for N 25, 49, 81, and 121. In the figure, the contrast is 2, the total noise variance 1.0, the background noise variance 0.9, and the system noise variance 0.1.
Fig. 5. Comparison between the detection probability given in Eq. (36) and the cumulative signal-plus-system-noise distribution as a function of threshold for N 25, 49, 81, and 121. In the figure, the contrast is 2, the total noise variance 1.0, the background noise variance 0.9, and the system noise variance 0.1.
alarm using Eq. (51) as a function of the normalized threshold for N 25, 49, 81, and 121. Figure 4 is a similar comparison, but we are using the formula
create the estimated PDF for the test statistic. Figure 5 compares the detection probability given in Eq. (36) and the cumulative distribution derived from this PDF as a function of threshold. Again, we see very excellent agreement between theory and simulation results. This indicates that we have a consistent and valid theory for resolved target detection in background clutter. It is interesting to observe in Fig. 5 the effect the number of target pixels has on the probability of detection. The various curves all cross at a decision level q0 ≈ 0.3 in the region just above the 50% detection level. Above the crossover point, the detection rate increases with increasing values of the target mask size N; below that point, the opposite occurs. That is, the detection performance in the latter case deteriorates with increasing N. It can be shown that the average and variance of the normalized test statistic q are given by
N μbn jCn jσ 2T 2 1X mn : N n1 σ 2b
(54)
In both cases, the agreement between theory and simulation is excellent and clearly show that the larger the target, the better the detector performs. This latter fact is true for all statistical results contained in this paper. The next observation one can derive from these two figures is that our test statistic is valid whether sn > μbn (positive contrast) or sn < μbn (negative contrast). These figures also confirm that Eq. (51) is true for odd integer targets as well as even integer targets since all of the values of N employed were odd in showing the excellent agreement between theory and simulation. Let us now look at the validation of the probability of detection and what we can learn from that. As before, we set sn 6 and μbn 2 for all values of n; contrast equal to 2. The total noise variance σ 2T was set to 1, with the background noise variance σ 2b equal to 0.9 and a system noise variance σ 2t equal to 0.1. For the probability of detection simulation, we created four 8192 × 8192 Gaussian noise images that were zeromean variance σ 2T . We then placed p and had pnoise N -pixel-by- N -pixel targets with mean sn − μbn and Gaussian noise of variance σ 2t , at selected locations in those four images to guarantee no target interaction when filtering was done. In particular in each image, we placed a target at pixel location (50, 50) and subsequently placed other targets at locations 125 pixels apart in both direction x and y from that point, ending at location (8050, 8050). The images then were processed using Eq. (53) to
Eq
σ 2t λ1 N N
(55)
σ 4t 4λ1 2N ; N2
(56)
and Varq
respectively. For the noise and clutter parameters selected for this example, the noncentrality parameter λ1 equals 1.9753N. This means that the above equations reduce to Eq 0.29753
(57)
Varq 0.0990∕N;
(58)
and
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because it is below the output noise floor” [16]. We do not agree. The authors feel that their Fig. 1(c) clearly shows a target detection peak like in Fig. 7, although broadened by their estimation/detection method, and is essentially in the correct location of their initial tank target. 6. Computer Simulation Results—Zero Contrast Case
Fig. 6. Plots of the test statistic PDF as a function of the test statistic q for N 25, 49, 81, and 121.
respectively. Thus, we find for our PDFs that their test statistic average should be a constant for all values of N and that their variance falls off by inverse N. These aspects clearly can be seen in Fig. 6, which depict the processed data PDFs for N 25, 49, 81, and 121. These graphs illustrate the same mean value for all the plotted N values and that the width (variance) of the PDFs decreases as N values get bigger. It is clear that Eq is the same value of the crossover point in Fig. 5. This means that the Qd ’s for smaller N grow faster than the Qd ’s for larger N for small decision levels close to zero, but the larger N probabilities of detection catch up and exceed the lower N Qd ’s as one approaches and passes the crossover point; thus explaining the behavior noted above. Finally, Fig. 7 shows a one-sub-frame realization of a N 25 target detection used in the creation of Fig. 5. As implied by our test statistic having reversed comparison signs with the test statistic, the “matched filter peak” after processing is below the processed noise level, not above. Javidi et al. noted in one of their papers that a similar plot [Fig. 1(c)] implied that “the correlation peak cannot be seen
Fig. 7. One-sub-frame realization of a N 25 target detection used in the creation of Fig. 8. 5050
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Let us now turn to the case of a zero contrast target. Specifically, let us compare computer simulation results with our zero-contrast false alarm and detection probabilities. Figure 8 is a comparison of the cumulative distribution from the test statistic given in Eq. (41) using four 8192 × 8192 processed images of Gaussian noise with variance equal to one (1) and the probability of false alarm from Eq. (43). On the other hand, Fig. 9 is a comparison of the cumulative distribution from the test statistic given in Eq. (41) using four 8192 × 8192 processed images of Gaussian noise with variance equal to 0.1 and the probability of detection from Eq. (47). Both figures show excellent agreement between simulation results and theory. Referring to Fig. 9, we again see a crossover point similar to the one in Fig. 5. However, its value is different. The average and variance of the normalized test statistic q in this case are given by Eq σ 2t
(59)
and Varq
2σ 4t ; N
(60)
respectively, since the noncentrality parameter is zero. Substituting in our simulation value for σ 2t , the above equations reduce to Eq 0.1
(61)
Fig. 8. Comparison of the cumulative distribution from the test statistic in Eq. (41) and the probability of false alarm from Eq. (43).
upon the BNR. Finally, we found that the above definition of CNR disagrees with the conventional definition of SNR and needs the additional multiplicative factor of background clutter mean to be correct. These are important findings for any scientist or engineer designing electro-optical and infrared sensors for finding resolved targets immersed in background-cluttered images. References and Note
Fig. 9. Comparison of the cumulative distribution from the test statistic in Eq. (39) and the probability of detection from Eq. (45).
and Varq 0.02∕N;
(62)
respectively. Clearly, Fig. 9 shows that the crossover point equals Eq. (61) and the curves exhibit the same behavior as in Fig. 5. This implies that the explanation for the probability of detection results in the nonzero contrast case equally apply here as well. 7. Summary
This paper developed a general solution for the resolved target detection in background clutter problem considered previously as an intractable problem. To do so, we had to recognize that a resolved target obscures any background clutter that would have been observable if the target were absent. Using Gaussian PDFs for background clutter and sensor noise, we developed an optimum detector deciding between the hypotheses of signal-only plus common system noise H 1 and background clutter-only plus common system noise H 0 . Counter to convention, this detector compares a test statistic to a threshold and decides a target is present if the statistic is LESS than the threshold, not greater than the threshold. Expressions for the resulting test statistic and its associated detection and false alarm probabilities were established and found to be in excellent agreement with computer simulation results, even for the case of zero contrast targets. Key findings for the nonzero contrast case are that the detection probability depends on (1) the contrast noise ratio (CNR) defined as the sum of the square of the apparent pixel contrast times the mean background clutter level, divided by the background clutter variance, and (2) the background noise ratio (BNR) defined as the ratio of the total noise variance divided by the system noise variance. Alternately, the key findings for the zero contrast target case were that (1) the target is detectable and (2) its detection probability depended solely
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