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Nonuniformity correction and correctability of infrared focal plane arrays
INFRAREDPHYSICS &TECHNOLOGY ELSEVIER
Infrared Phys. Technol. 36 (1995) 763-777
,
Nonuniformity correction and correctability of infrared focal plane arrays M. Schulz a, L. Caldwell b • Institute of Applied Physics, University of Erlangen-Narnberg, Staudstr. 7, D-91058 Erlangen, Germany bNight Vision Directorate, AMSEL RD NV TTP, 10235 Burbeck Rd Ste 110, Ft. Belvoir, VA 22060-5806, USA Received 12 September 1994
Abstract
Correction of photoresponse nonuniformity in infrared staring sensor arrays is investigated. A general nonuniformity correction procedure is proposed. The procedure is based on multiple irradiation sources and on least square fit approximations to the individual pixel response characteristics. Nonlinearities of the signal response are taken into account. A correctability figure of merit is defined which may be used to estimate the residual fixed-pattern noise after correction. The correction procedure and the correctability are applied to a real data set measured by a 64 x 64 element infrared focal plane array (FPA). It is shown that an offset correction is insufficient for this data set and that a linear correction reduces the fixed-pattern noise contribution to the magnitude of the temporal noise background. The residual uncorrected fixed pattern noise can be related to pixels showing large temporal noise.
1. Introduction A severe problem associated with the use of infrared focal plane arrays (FPAs) is the spatial photoresponse nonuniformity. The nonuniformity arises because each individual detector element in the FPA exhibits a response characteristics differing from those of its neighboring elements. The magnitude of the nonuniformity varies with the technology and with the detector material employed to fabricate the FPA. The responsivity variations are usually less than 1% for PtSi Schottky barrier FPAs; the variations may exceed 10% for compound semiconductor FPAs. The responsivity nonuniformities create a fixed pattern super-
imposed on the thermal image of the scene. In order to resolve temperature differences in the image of the scene to as low as AT ~ 0.01 K, which is required in many applications, the spatial nonuniformity must be less than 10 -4. In order to obtain this level of uniformity, a correction of the responsivity nonuniformity must be applied [1-7]. High-performance systems use multiple temperature irradiation sources to repetitively recalibrate the correction parameters [3]. Two irradiation sources are frequently used to correct for offset and gain nonuniformities. The frequency of these calibrations depends on the stability of the FPA. In scanning systems, the calibration can be performed within a single image frame. Depending
013.50-4495/95/$09..50 (~ 1995 Elsevier Science B.V. All rights reserved SSDI 1350-4495(94)00002- X
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M. Schulz and L. Caldwell/lnfrared Phys. Technol. 36 (I995) 763-777
on the technology and the design employed, staring FPAs typically must be recalibrated within a few minutes, although some remain stable for several hours. New technologies are evolving which perform a scene-based nonuniformity correction using algorithms based on a temporal high-pass filtering and an artificial neural network [4,5]. The importance of a nonuniformity figure of merit was pointed out in Ref. [6]. The future standard for modeling FPAs is the Night Vision and Electronic Sensors Directorate's " F L I R 92 Thermal Imaging System Performance Model" [8], which recognizes the importance of temporal and spatial noise contributions to the overall system performance. The modeling shows that a general scheme for optimized nonuniformity correction which covers the usable signal range, is still lacking. In this paper we propose a new scheme to evaluate the accuracy of nonuniformity correction methods. For this purpose, we introduce a general correction procedure based on multiple irradiation sources and on least-square curve fitting analysis for obtaining nonuniformity correction information in the usable signal range. Based on this analysis we are able to establish a figure of merit, which we call the "correctability". The general correction procedure and the correctability figure of merit have been tested on several different infrared FPAs. The procedure will be demonstrated in this paper on a data set measured by a long-wavelength infrared (LWIR) FPA. The correction procedure may be simplified to be applicable to practical system implementations.
2. Nonuniformity correction The nonuniformity correction should fulfill the following requirements: - All the detector elements should yield signal values after correction for which the scatter for uniform irradiation is less than a threshold value defined by the application. Usually, the threshold value is close to or less than the effective temporal noise. - T h e correction should be effective in the full signal range. The tolerable nonuniformity thres-
hold may be defined uniformly in the signal range or as a function of the signal value with the smallest threshold for small signal levels and with an increasing threshold for larger signals. - The number of parameters to be corrected should be kept as low as possible. Usually only one (offset correction) or two (linear correction) parameters are adjusted. A high number of correction parameters increases the memory and the computation requirements for the correction procedure, and thus the complexity of the system. - T h e correction procedure should be compatible with the nonlinear response characteristics of the FPA. These nonlinearities may lead to a correction which is not equally effective throughout the signal range specified and the computational level required may be too large to keep the residual spatial fluctuations below a threshold value. The general correction procedure proposed in this paper meets all the requirements stated above. The general method, however, requires a rather complex mathematical analysis. It will be shown below that the mathematical complexity can be reduced to a minimum so that the procedure can be applied in practice. In the following discussion the most general algorithms are presented in order to demonstrate the basic principles of the procedure. In contrast to the nonuniformity correction in Ref. [7] which models the FPA photoresponse characteristics by Mooney's linear infrared sensor model [3], our method is strictly based on data measured by the FPA under test. Typical data sets consist of a large number of signal values Yi.j obtained by the direct readout of the array for uniform infrared irradiation. The photoresponse output signal, the "amplitude value" yq may be a voltage or a current signal measured in arbitrary units. The subscript j = 1. . . . N refers to the N individual detector elements (pixels) in the array. The subscript i = 1. . . . . n indicates the temperature T~ employed in the uniform irradiation of the array by a blackbody source. The signal values are taken from one image frame under the usual imaging conditions. We call the set of the direct output amplitude values yq the raw signal data. The individual amplitude values yq exhibit a scatter
M. Schul= and L. CaMwell/lnfrared Phys~ TechnoL 36 (1995) 763-777
within the array in spite of uniform irradiation due to the temporal noise and due to the spatial nonuniformity (fixed pattern). The temporal noise in individual pixels may be determined by analyzing several image frames taken for the same irradiation. The photoresponse characteristics of the FPA is measured as a function of the irradiation by the multiple blackbody source temperatures used in the calibration. A high accuracy is obtained by taking the signal average over the whole array. This photoresponse characteristics measured in general is highly nonlinear. The nonlinearities originate in the optical detector system (Planck's law and limited spectral response), but are also caused in the electronic readout of the irradiation signal from the device. Especially, the readout cannot be adequately described by a theoretical modeling. The mean signal output value (y~) is defined by 1
= ~j.~y,j
(l)
Since the number N of pixels in the array usually is a large number N > 1000, a good approximation (standard error= tr/x/~) of the mean photoresponse characteristics is obtained by the averaging.
765
The overall FPA photoresponse characteristics R(T,.) is sampled at the n irradiation temperatures,
R(~) =.
(2)
A typical nonlinear photoresponse characteristics of an output signal sampled for a set of n = 6 blackbody irradiation temperatures as an example is depicted in Fig. l a. Instead of blackbody calibration sources, other irradiation sources of the FPA may be used. In the evaluation it is only assumed that the irradiation is uniform across the array and that the single variable (blackbody temperature, the photon flux, the irradiation power etc.) used to describe the infrared irradiation is representative for the spectral emittance of the thermal scene to be imaged, so that nonuniformities in the spectral response of the FPA are accounted for in the correction procedure. It is noted that the correction is only effective for this variable; nonuniformities may be incompletely corrected when the irradiation spectrum in the scene is different from that of the calibration sources. The bars in Fig. l a indicate the scatter range (maximum/minimum) of the data points. For large
7.5
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6.5
7
6.5
i'
>.,
Y
/
s
Pixel 9.=5.5 (34,49)
5.5
E
5
4.5 /
.>( unif.char.
a 3.5
........
290
' .........
300
' ........
310
i,
320 I T b
[K]
, ,,,
,~,
33O
,
....
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340
o ~
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,, 6
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b ,
) .... 6.5
7
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Fig. 1. (a) Array response characteristics of the mean raw signal value ( y > vs. blackbody irradiation temperature Tb. The open circles are data values, the continuous curve is a power law (4th order) fit. The bars indicate the full scatter range between y,~, and Ym~. (b) Signal output y as a function of the signal output (y> averaged over the whole FPA. The solid straight line of slope one represents the unified, photorespons¢ characteristics of the mean signal output. The data points depict the amplitude values of one spe~:ific pixd as indicated. The dashed line is a linear fit to the pixel data. The dotted line indicates the correction process of an arbitrary image signal value y to the unified characteristics. INFRA }6/4---C
766
M. Schul: and L. Caldwell/lnfrared Phys. Technol. 36 (1995) 763-777
arrays having a high number N of detector elements, the averaged response characteristics is improved by the factor ~ l/x/~. With a typical variance of a/~, 1% (see example demonstration below) and an array size of 64 x 64 elements an accuracy ofAy/ ~ 1.6 x l0 -~ sufficient for the correction is obtained. This accuracy is far better than that obtained by a theoretical modelling. If the FPA contains dead pixels showing no or only a poor photoresponse, the averaging may be improved by excluding these dead pixels from the analysis. It is also possible to analyze only a portion of the FPA. It is not important to know the explicit nonlinear mathematical function of the photoresponse characteristics. For the correction, we linearize the photoresponse characteristics by using the mean output signal as a new irradiation variable, R() = .
(3)
We thus obtain a "unified photoresponse characteristics" which is a straight line through zero and which has a slope of unity. This characteristics is easily and accurately interpolated between the sampled data sets defined by the various irradiation sources used in the calibration. It is also noted that the irradiation parameter directly measured for the calibration sources by the output signal averaged over the whole array exhibits an accuracy (Ay/~ l0 -4) superior to that usually obtained of the blackbody temperature T~(AT/T ,~ 10-3). The unified photoresponse characteristics of the same data as in Fig. la is shown in Fig. lb together with one characteristics of an individual detector element. The photoresponse characteristics of the individual pixel deviates from the unified characteristics due to the nonuniformity in the array. The data points also scatter around the response curve due to the temporal noise. The nonuniformity and the temporal noise are described by the signal output deviation, the amplitude deviation Ay~j, of individual data points
A),'i/= y~/-.
(4)
We perform a standard least square curve fitting with a polynom defined by the order of correction to the amplitude deviation of individual pixels.
Weighting factors wi may be applied if a closer approximation is desired to specific data sets. Specific data sets having a reduced temporal noise, e.g. those on the low end of the photoresponse characteristics irradiated with a reduced power, may also be favored by giving them an increased weighting in accordance with the magnitude of the temporal noise. The curve fitting of polynoms by the least square method including weighting is standard and may be found in the literature [9]. The amplitude deviation values Ayti~ calculated by the curve fitting are
Ay~i~ = aj+ b/ + cy(yi> 2 + ' ' ' .
(5)
For an offset correction only one parameter, a constant aj is determined for each pixel; for a linear correction two parameters, a constant offset aj and a gain bj, for a square correction three parameters aj, bj, and cj have to be determined and stored. The degree of correction may be arbitrarily chosen. It is noted that the amplitude deviation values are only approximated by the curve fitting procedure and that residual deviations remain after the curve fitting. The residuals of the approximation are due to the temporal noise in the individual pixels and due to the functional dependence in the individual photoresponse characteristics which is not accurately described by the curve fitting. The least square curve fitting procedure is the adequate method to approximate the individual pixel characteristics because it minimizes the error and takes into account the errors due to the temporal noise. For the correction of the nonuniformity, the data values determined by the curve fit to the amplitude deviation is subtracted.
Ayi~ = Ay~j- AyT~.
(6)
The procedure is explained in the schematic of Fig. 2 for one specific pixel. The amplitude deviations of this specific pixel are depicted by full squares. The data points of this example are all offset above zero. The deviations also show a trend to increase with increasing irradiation. The mean deviation value with offset correction is indicated by the horizontal line in Fig. 2. All the data points are lowered by the same amount (arrow marked "off") to yield the corrected amplitude deviations
M. Schul: and L. Caldwetl/lnfrared Phys. Teehnol. 36 (1995) 763-777
767
relations for the offset, the linear and the square correction,
0.1 • ............
"~ ~
7:" " "
•
...........
........-'""
0
....-I ......
•
off-~
•
lin
y jc ~- yj - a/,
offset correction,
c yj - aj YJ= l + b j '
linear correction,
l+bj /(l+bj): ~Y~=
o' 0
t-
4c~
c/
'
square correction
[]
"0.05
2cj
. . . .
4.5
I
5
. . . .
r
5.5
~
,
r
[B.u.]
I
6
,
,
,
,
I
,
8.5
Fig. 2. Signal amplitude deviation Ay as a function of the mean irradiation variable ( y ) for one specific pixel. The full squares are raw data points. The open squares are obtained after offset correction, the open circles are obtained after linear correction. The lines and arrows marked "off" and "lin" indicate the correction curve fitting for offset correction and linear correction, respectively.
represented by the open squares. With a linear correction a straight line, indicated by the dotted line in Fig. 2, is fit to the data points. This straight line is lowered and turned into the zero base line as indicated by the arrows marked "lin". The open circles depict the data points after the linear correction is applied. The linear correction yields a closer approximation to the zero base line than the offset correction. For a correction higher than linear (not shown in Fig. 2), a curve is fit to the data; for correction this curve is shifted, turned and straightened into the zero base line. The amplitude values y~/ after correction are obtained by adding the unified response function again. Y~j = ( Y i ) + Ay~j
(7)
For an arbitrary image for which the irradiation is nonuniform, the correction of the individual pixel amplitude is performed on the basis of the individual curve fit to the pixel with the correction parameters aj, bj, cj, etc. determined by the above procedure with multiple uniform irradiations. The corrected pixel amplitude is determined by eliminating the linearized irradiation parameter ( y ) from Eqs. (4-7) to obtain the
(8)
For corrections higher than of square order, the mathematical relations are more complex. The way of correction to determine the irradiation variable ( y ) from an arbitrary amplitude value y and to correct the amplitude to obtain yC is schematically indicated by the dotted line in Fig. lb. For the linear correction, the measured pixel amplitude is offset and then the slope is normalized to unity by the gain parameter bj. The gain parameter bj has to be small tb/I ,~ l for the correction to be linear. A correction line with b/= - I leading to an infinite correction factor 1/(1 + b j ) ~ oo is determined for a pixel having no response to the irradiation. Such a pixel is uncorrectable and has to be considered as a dead pixel. It is noted that for linear and higher order corrections, the temporal noise associated with the individual pixels leads to an error in the determination of the irradiation coordinate ( y ) in individual pixels for solving the above equations (4)-(7) to correct the arbitrary image. It can be shown that for the unified characteristics chosen the parameter averages (aj), (bj), (cj) etc. vanish and that thus the error due to the temporal noise in the correction is minimized; however, if a pixel exhibits a weak photoresponse and thus b j ~ - l, the temporal noise is amplified by the correction with the factor l/(l + b j)>> I. Such pixels may be uncorrectable or poorly correctable because the signal noise is amplified. It is further noted that the temporal noise which may vary from pixel to pixel, creates an error in the determination of the correction parameters aj, b/, etc. The determination of the fixed pattern and its correction are therefore limited by the temporal noise in the individual pixels. A pixel exhibiting excessive temporal noise, e.g. l/f-type noise, may
M. Schul: and L. Caldwell/Infrared Phys. Technol. 36 (1995) 763-777
768
therefore be poorly correctable or even uncorrectable. We will see later that for the test FPA this is the prevalent process which causes the uncorrected fixed pattern.
3. Correctability It was already noted in the previous section that the correction procedure applied to the amplitude values is never quite accurate and that residual spatial fluctuations remain. These residual fluctuations are due to errors in the measurement data e.g. due to the temporal noise, and to nonlinearities which are not accurately approximated by the regression analysis. The minimum signal resolved for a perfectly uniform array is defined by the temporal noise. The residual fixed pattern after correction in a nonuniform FPA further degrades this optimum resolution. An estimate of the goodness of the correction may be determined by relating the magnitude of the residual fluctuations in the array after correction to the temporal noise pattern. The goodness of the curve fitting for an individual pixel is described by the ;(2 value of the standard deviation normalized to the mean temporal noise )'~i averaged over the whole array for the various irradiation levels,
i-I
(9)
)'~i
The ;(a-distribution of a perfectly uniform FPA having only temporal (gaussian) noise is well known in statistics [9]. The probability distribution is, I
p(;('-) =
( ' l 2 -~(k - 2),2 e -~22
2,/2F(k/2) ,,.
,
(10)
where x = n - r is the number of the remaining degrees of freedom after curve fitting to n data sets with r parameters. For linear correction the degrees of freedom consumed in determining the offset and slope are r = 2; for offset correction only, it is r = 1. The maximum of the ;(2-probability distribution occurs at
2
;(~x = k - 2 .
(11)
Because of the high number of pixels in the array, the histogram of random ;(2-values approximates the ;(Z-probability function of Eq. (10) quite well. The position of the distribution maximum Eq. (1 !) can be used to obtain an estimate of the temporal noise in the FPA. Deviations of the ~(-'-data histogram from the ideal ;(Z-distribution are due to the residual spatial nonuniformities in the array. By subtracting the ideal ;(2-distribution due to temporal noise only from the real ;(-'-values of measured data for the array after correction, we obtain a number to estimate the goodness of the correction,
~.( c=
N 3 N - - " ) -(n-
r)-1
=Ii=1
(N--- l - ~ n : ~)
1.
(12)
The pixel number normalization is N - 1 because one degree of freedom is consumed for the averaging over the whole FPA to obtain the unified photoresponse characteristics. The goodness of the nonuniformity correction is perfect for c = 0, i.e. if there is only temporal noise. The goodness value is c = 1 when the fixed pattern noise after correction equals the temporal noise. For a fixed pattern noise far in excess of the temporal noise large c-values c >> 1 are determined. It is noted that a mean temporal noise of the detector elements is used in the normalization of the goodness and the correctability. The magnitude of the temporal noise strongly affects the correctability value. A small temporal noise level increases the c-value because the threshold for the fixed pattern noise is also small. Increasing the temporal noise makes the correction easier. The correctability therefore is only a good figure of merit when the mean temporal noise level is stated. It is further noted that a mean value of the temporal noise Y~i for the various irradiation conditions is used in the normalization of the ;(2 value to determine the goodness of the correction. A
M. Schul: and L. Caldwell/lnfrared Phys. Technoi. 36 (1995) 763-777
nonuniform distribution of the temporal noise, especially increased noise in specific pixels leads to large deviation values Ayjj which cannot or which can only be approximately corrected. Pixels having excessive noise are uncorrectable or are only partially correctable. The method proposed, however, minimizes the error due to the temporal noise by the regression analysis and by the use of the linearized irradiation variable using the average over the whole array. Pixels exhibiting temporal noise larger than the mean temporal noise degrade, i.e. increase, the correctability value c of the correctability. Very noisy pixels may be uncorrectable. A threshold Z2 value may be defined to select and to exclude specific pixels which are uncorrectable. In practice, the nonuniformity correction is frequently performed in a "two-point correction" by accurately measuring the signal amplitude at two irradiation levels by averaging the temporal noise for a long time. In this case the straight line fitting in the regression analysis can be assumed to be accurately known. The correctability is then obtained by a third or more measurement sets at a different irradiation level between the two correction points. The correctability is then determined as for the raw data without any degrees of freedom consumed to define correction parameters. The relation to be used is
c =,]j.t
"-
-- 1.
(13)
( N - l)n
/~Z'(Yf-(Y))2~2ty~(N" - 1) 1 = ,2 -
scatter of the fixed pattern across the array. If both contributions to the noise are independent and random, the standard deviation a is given by a - ' = y ~ + a~
(15)
where a,z is the standard deviation of the fixed pattern only. Eq. (12) or the simplified form Eq. (14) can be used as a standard to measure and to specify correctabilities in FPAs. The measurement data is easily accessible by photoresponse measurements on FPAs.
4. Gradual spatial corrections The unified response function defined for the correction procedure assumes a uniform array photoresponse with random fluctuations superimposed. In practice, the array may show a gradual variation of the response or a nonuniform irradiation across the array due to limitations by cold shields. A more general response function may then be defined which takes into account gradual spatial variations of the response in the array, e.g. a linear or quadratic variation. In this case a tilted plane or a curved plane is fit to the array data for each irradiation temperature rather than a horizontal (constant) plane
R(y,) = (y,) +f~(j)
(Y,b)'/Yu
For a standard two-point correction by only wellaveraged correction points, a minimum of one additional point n = 1 is required to determine the correctability. For a single measurement data set (usually in the middle between the two correction points), the definition of the correctability simplifies to
c=~
769
l,
(14)
where tr is the standard deviation of the raw signal data set from the mean value obtained by averaging over the whole array. Note that tr 2 comprises the temporal noise of each individual pixel and the
(16)
The amplitude deviations to be corrected are then determined by the differences of the amplitude values to this higher order unified curved response plane f a ( J ) + (Y~). The starting amplitude deviations of the raw data are then reduced on the expense of additional spatial parameters rather than response parameters. The degrees of freedom /~ which are consumed in the fitting of the plane to the array have to be taken into account in the determination of the correctability. The correction procedure is similar to the one explained above. The gradual variation may be removed by correcting the spatial variation. The general correctability including this spatial curved plane fitting is
/j~(Y~-(Y~)--fct~(j))2/c c=
ij
- 1.
(17)
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M. Schul-- and L. Caldwell/Infrared Phys. Technol. 36 (I995) 763-777
T =295K
"d
b
Fig. 3. Raw signal data of the test array uniformly irradiated with a blackbody source at Tb = 295 K. (a) 3-Dimensional representati of the 64 x 64 signal data values. (b) Image representation of the 64 x 64 detector signals. The contrast ranges from Ym,. (black) Ym~ (white).
M. Schulz and L. Ca/dwell/Infrared Phys. Technol. 36 (1995) 763-777
771
IOIX~
The large mathematical effort for fitting curved planes to the array data is in most cases not justified. This more complex analysis may, however, be necessary if a uniform irradiation of a large FPA is impossible because of technical limitations e.g. given by the optics or cold shields.
T = 295K R a w Data b 1OOOO
t7 = 5.3x10-2
1000 e,
100
,
5. Correction and correctability of a test F P A
1
-
10
I
IJ
5.1. Raw signal data 1
A mercury cadmium telluride staring FPA comprising 64 x 64 detector elements is used to test the correction method and to determine the correctability in the long-wavelength atmospheric window. The data obtained is used to demonstrate the applicability of the method for nonuniformity correction and not to prove the state of the art of FPA fabrication. We therefore need not specify further details of the array. The correction method is independent of the type of array and of the operating conditions. For the test, the detector array is uniformly irradiated by a blackbody source. The blackbody temperature could be adjusted in the range from Tb=295K (RT) to Tb = 3 5 0 K in steps of AT = 9 K. Each individual detector element could be addressed by an electronic read-out. The output signal was recorded by a computer. For each read-out cycle, an image data set of 64 x 64 signal values was recorded in arbitrary units. The image
Table 1 Statistical evaluation of the 6 raw data sets taken for the test FPA. The columns from the left to fight list the data set number n, the blackbody irradiation temperature T b, the signal value
o'
Y~=
Ymi,
Y,
n
(K)
(a.u.)
(a~u.)
(a.u.)
(a.u.)
(10 -~ a.u.)
1 2 3 4 5 6
295 304 313 322 331 340
4.60 4.92 5.29 5.68 6.10 6.58
0.053 0.057 0.063 0.070 0.078 0.087
5.22 5.54 5.91 6.39 6.91 7.49
3.97 4.28 4.65 5.03 5.46 5.93
0.70 0.72 0.75 0.78 0.81 0.84
0.1 3.8
4
4.2
4.4
4.6
4.8
5
5.2
5.4
Amplitude y [a.u.] Fig. 4. Histogram of the raw signal output amplitude data y taken for the full test array irradiated by a blackbody at Tb -- 295 K. The axis marks indicate the counting intervals. The vertical line indicates the mean value ~y).
data set exhibits a scatter in the signal magnitude due to the temporal noise of each individual pixel and due to the fixed pattern of the spatial nonuniformity of the array. The signal amplitude distribution of the 64 x 64 raw data set taken for the test array irradiated with a room temperature blackbody is represented in the 3-dimensional plot of Fig. 3a. The image contrast of the raw data set is depicted in Fig. 3b. The data set exhibits a "noisy" pattern with a number of randomly distributed spikes. A regular peak pattern occurs on the left rim and at the top right corner indicating a read-out problem. A cluster of defects appears in the top left corner. There is no dead element in the array showing no photoresponse. The photoresponse characteristics of the array is that depicted in Fig. l a. The open circles represent the signal data values averaged over the whole array. The vertical bars indicate the scatter range of the data values in each set. The continuous line represents a power law fit to the data points. The data values are listed in Table l together with the standard deviation #; in each signal data set and with the rms temporal noise y, for each data set. The standard deviation values are on average only 1% of the amplitude value and are too small
772
,t4. Schulz and L. Caldwell/Infrared Phys. Technol. 36 (1995) 763-777
T
= b
295K -~-4.e,0~
4.6o5.
'k ,4, q
(a)
AT=0.02K b
Fig. 5--Caption on facing page
M. Schul= and L. Caldwell/lnfrared Phys. lrechnol. 36 (1995) 763-777
to be indicated in Fig. la. The standard deviations increase with increasing irradiation at the higher blockbody temperatures. It is noted that the mean values exhibit a nonlinear increase as a function of the blackbody temperature. The nonlinearity is only partially due to the nonlinear increase of the irradiation power with increasing blackbody temperature. An esti-. mate based on the calculated irradiation power shows that the nonlinearity is also partially due to the read-out electronics. It is therefore not possible to remove the nonlinearity with an accuracy better than 10-4 as is required for a high temperature resolution by replotting the data as a function of the irradiation power calculated by the blackbody temperature. The scatter of the raw signal data is also statistically evaluated by a histogram. The histogram of amplitude values taken at Tb = 295 K is depicted in Fig. 4. The semi-logarithmic representation indicates the approximately gaussian distribution around the maximum which is closely centered at the mean value. There are, however, assymmetric tails and small side lobes in the distribution. The histogram is useful to demonstrate the scatter of the signal values; however, the location of the extreme values and the clustering of deviations is only visible in the spatial representations of Figs. 3a and 3b. 5.2. Corrected signal data The spatial nonuniformities of the signal output in the test FPA have been corrected by an offset, a linear, and a square correction procedure. Weighting factors inverse to the temporal noise. values are applied to slightly favor the temperature sets that show a reduced noise. The distribution of the output signal amplitude values yC obtained after linear correction of the 64 x 64 data set of the test FPA is represented in a 3-dimensional plot of Fig. 5a. The image contrast of the corrected data
773
Table 2 Statistical evaluation of the 6 data sets taken for the test FPA after correction, The columns from left to fight list the data set number n, the blackbody irradiation temperature Tb, and the residual standard deviations after full correction by linear curve fitting ¢¢, after two-point correction pinned to the sets 1 and 3 cry.3, and after two-point correction pinned to the sets I and 6 ~.6
:rb
a°
~..~
~I.6
n
(K)
(I0-~ a.u.)
(I0-~ a.u.)
(10-3 a.u.)
1 2 3 4 5 6
295 304 313 322 331 340
0.50 0.58 0.80 0.73 0.91 0.69
0 0.87 0 1.49 2.48 3.25
0 0.85 1.11 1.03 1.33 0
set is depicted in Fig. 5b. Note that the amplitude scale is now expanded by almost a factor 100 compared to the scale in Fig. 3a. The correction procedure reduces the spatial fluctuations to 1% of those of the raw data. The spatial distribution yields a noisy pattern; however, there are no spikes superimposed as in the raw data. The image of the spatial noise in Fig. 5b also does not exhibit any clustering nonuniformities. In order to demonstrate the noise equivalent temperature resolution, a stripe with a temperature step of AT = 0.02 K is superimposed onto the image. This temperature step can be clearly resolved after correction. The uniformity improvement is quantified by the standard deviation values listed in Table 2. The magnitude of the standard deviation values a ¢ after full linear correction is reduced to approx. 1% to those of the raw data listed in Table 1. The magnitude of the standard deviation is now of the order of the effective temporal noise signal y~ listed in Table 1. The standard deviation values listed show an increasing trend for increasing blackbody irradiation. A two-point correction is simulated by giving two different data sets, e.g. the sets 1 and 3 (1/3) or the sets 1 and 6 (1/6), respectively, a high
Fig. 5. Signal data after linear correction of the FPA test array uniformly irradiated with a blackbody source at Tb = 295 K. (a) 3-Dimensional representation of the 64 x 64 signal data values. (b) Image representation of the 64 x 64 detector signals. The contrast ranges from 4.5965 (black) to 4.6008 (white). A signal step equivalent to a temperature difference AT = 0.02 K is superimposed in the shape of a stripe in the center.
774
M. Schulz and L. Caldwell/Infrared Phys. Technol. 36 (1995) 763-777
weighting factor. The standard deviation values for these two-point corrections are also listed in Table 2. The standard deviation values for two-point correction vanish in accordance with the pinning for the sets 1/3 and 1/6, respectively. It is noted that the standard deviations for twopoint correction are always higher than those for the full linear regression to all the data points. Between the pinning points, the increase is approximately 50%, however, outside the pinning range, the magnitude of the standard deviation a~/3 assumes values up to 5 times larger than the full correction (rc. The histograms of the data after linear correction depicted in Fig. 6 show that the wide tails of the raw signal are fully corrected. The distributions are still gaussian near the maximum. The maximum of the distributions is always located at the unified response function value. However, the twopoint corrections show a wider spread than the full regression correction. It is noted that the two-point corrections slightly degrade the correction but they are still comparable to the full correction taking into account the. 6 data sets available which yields the optimum correction for the data available. The improvement mainly reflects the improved averaging of the temporal noise by the increased number of data points. The full correction by the least-square fitting method covers the full usable signal range and yields an optimum correction.
100000 T = 331K corrected b
10000
1
1000
a
1/3
=24.8x10 4
100 10
0.1
6.09
6.1
6.11
6.12
Amplitude y¢ [a.u.]
Fig. 6. Histogram of the signal output data y" after linear correction. The test array is irradiated by a blackbody at Tb=331 K. The three distributions are for the full linear correction (narrow), two-point linear correction pinned in the sets l and 6 (medium). and for the two-point correction pinned in the sets 1 and 3 (wide). The standard deviations are indicated in the insert.
erly corrected to a magnitude below the temporal noise value. The x~-distribution of the square correction scheme is depicted in Fig. 8 and of the offset correction in Fig. 9, respectively. The histogram
•'•
1000
5.3. Correctability of test array
'
-,
8
linear correction c =0.53
,~
100
The correctability of the 64 x 64 test FPA was tested by plotting the x:-distribution after linear correction in Fig. 7 with the normalization to the mean temporal noise of the data sets given in Table 1. Figure 7 also depicts the ideal X: distribution calculated by Eq. ( 1 0 ) with k = 6 - 2 = 4 for linear correction of the 6 data sets. It is noted that the histogram observed is well represented by the calculated curve. The maximum is in the correct position Z~=x= k - 2 = 2. However, there appears to be a tail in the measured data due to an uncorrected fixed pattern in the array. The correctability determined is c = 0.53 thus indicating that the fixed pattern noise has been prop-
a=9.1x104
i
\"L~
10
°
\\if2 0.1
0
10
2O
3O
4O
50
6O
70
X=
Fig. 7. Histogram of the Z-'-values of the test array determined by linear correction. The effective temporal noise value used in the normalization is Yt = 7 x l0 -=, The continuous curve is the calculated ;('=-distribution for temporal noise.
M. Schulz and L. Caldwell/Infrared Phys. Technoi. 36 (1995) 763-777
775
10000
1000
offset correction c=35
looo 100
100 10
, 0.1
, . . . . . . . . . . . . . . .
0
5
. . . . . . . .
,,
....
0.1
:
IIH ........
~
"
10 15 20 25 30 35 40 45 50 X2
X2
Fig. 8. Histogram of the X'-values of the test array determined after square correction. The effective temporal noise value is y t = 7 x l0 -4. The continuous curve is the calculated X2distribution for the temporal noise.
of the square correction fits the calculated ;(2. distribution. The distribution of the offset correction is very broad so that the calculated ;(-'-distribution degenerates into the vertical axis of the figure and cannot be shown on the same diagram. All the correctability values determined are listed in Table 3. The high c-value of the raw data indicates that the non-uniformity predominates the noise in the image. The offset correction is not sufficient to correct the nonuniformity into the magnitude of the temporal noise level. After linear and even more so by square correction, the fixed-pattern noise is corrected to a level better than the temporal noise. The
Fig. 9. Histogram of the X2-value of the test array determined after offset correction. The effective temporal noise value is y t = 7 X 10 -4.
square correction, however, does not improve the uniformity to a large extent compared to the linear correction. The contributions to the correction listed in the columns 3-5 of Table 3 demonstrate that the offset and the slope correction contribute about the same fraction to the correction. The square correction yields only about 10%. 5.3. Uncorrectable pixels The histograms of the ;(2-distribution in Figs. 7-9 permit to pinpoint specific pixels which exhibit a large ;(2-value and which are thus poorly corrected. The threshold of a large ;(2-value may be
Table 3 Correctability c value determined for the raw data and after correction to first (offset correction), second (linear correction), and third (square correction) order. The columns 3 to 5 list the maximum standard deviations of the contributions to the correction. The parameters a,, ab, a# are the standard deviations of the correction parameters a, b, c, respectively, and (Y)mx is the maximum of the mean FPA output signal
raw data offset correction linear correction square correction
c
tra (10 -3 a.u.)
~rb
~¢
88 35 0.53 0.48
46.2 52.8 52.8
45.7 45.2
3.8
M. Schulz and L. Caldwell/lnfrared Phys. Technol. 36 (1995) 763-777
776
defined by the calculated distribution depicted by the continuous curves. The calculated ;(:-distribution shows that based on the mean temporal noise, the probability of pixels having Z: >20 is negligible. Pixels having 7.:>20 are poorly corrected. Figure 10 shows a 2-dimensional scatter plot of the poorly corrected pixels having 7.2> 20 as a function of the correction parameters a, b determined by linear correction. The selected poorly corrected pixels cannot be correlated with large correction parameters indicating large deviations; in contrast to this assumption, they even accumulate in the region around a, b ~ 0 and thus exhibit a photoresponse around the average. The photoresponse characteristics of the specific pixel at coordinates (34,49) shown as an example in Fig. lb is the pixel having the largest ;(:-value Z2= 58 after linear correction and Z 2 =49 after square correction in the whole data set. The photoresponse characteristics of this specific pixel does not deviate to a large extent from the unified characteristics. The reason of the poor correction is visible in Fig. I 1 where the residuals of the linear curve fit are depicted as a function of the irradiation variable ( y ) . The residuals exhibit a random scatter. The curve fit therefore cannot be improved I~y using a higher order polynom. The large residuals indicate an excessive temporal noise
,-2
,., O
0
"o
~-2 03 -3
pixel (34,49)
-4 -5 4.5
5
5.5
6
6.5
[a.u.]
Fig. 1 I. Residuals of the uncorreetable pixel with coordinates (34,49) as a function of the mean output signal ( y ) .
for this pixel, five times as high as the mean value. Poorly corrected pixels are therefore especially those pixels that exhibit excessive noise, probably of l/f-type noise. The curve fitting for these noisy pixels is inaccurate and a large scatter remains as a fixed pattern. It is noted that the residuals are not visible in Fig. lb. The accuracy required for complete correction into the magnitude of the ,verage temporal noise is 10 -4. Such a high accuracy is not resolved in the graphical representation of the figure; however, the nonuniformity correction is limited by this small effect.
0.25 X2>20 0.2
6. Conclusions 0.15 .o
0.1
.C
¢~ 0.05
tie
,#'.
0
. "*
-0.05 -0.1
,, "r
-0.2
-0.1
,,
i,
0
I
0.1
0.2
,'
0.3
0.4
0.5
offset a [a.u.]
Fig. I0. Scatter plot of uncorrectable pixels having a large ;C2-value X~> 20. The variables are the correction parameters for offset a and the gain b, respectively.
An evaluation procedure is proposed for the nonuniformity correction of FPAs. The procedure is based on multiple irradiation sources. The nonlinearity of the photoresponse curve which so far limited the correction to only two calibration sources, the "two-point correction", is removed by defining a unified photoresponse function which is a straight line of slope one through the origin. The deviations from this unified response function are corrected for all the individual pixels by an offset correction, a linear correction, and a square-law or higher order correction. The correction procedure is applied to a real data set measured for an FPA.
M. Schul- and L. Caldwell/lnfrared Phys. Technol. 36 (1995) 763-777
A correctability is defined which may be used to estimate the residual fixed-pattern noise after correction. The correctability is also tested by the measured data set. The linear correction reduces the spatial fluctuations of the raw data (as = 88)'0 down to the temporal noise y, level (~s=0.5yt). The residual uncorrected pattern can be related to pixels having excessive temporal noise. The correction of the fixed pattern is mainly limited by the nonuniformity of the temporal noise.
References [1] R.W. Helfrich, Programmable compensation technique for staring arrays, in: Smart Sensors, D. F. Barbe Ed., Proc. SPIE 178 (1979) 110--121. [2] K. Hartnett, C. Marshall, N. Butler, J. Stobie and S. lwasa, Compensation electronics for staring focal plane arrays, in: Technical .Issues in Focal Plane Development, E. Krikorian and W.S. Chan Eds., Proc. SPIE 282 (1981) 73-78.
777
[3] A.F. Milton, F.R. Barone and M.R. Kruer, Influence of nonuniformity on infrared focal plane array performance, Opt. Eng. 2,) (1985) 855-862. [4] M.D. Nelson, J.F. Johnson and T.S. Lomheim, General noise processes in hybrid infrared focal plane arrays, Opt. Eng. 30 (1991) 1682-1700. [5] D.A. Scribner, K.A. Sarkady, LT. Caulfield, M.R. Kruer, G. Katz and C.J. Gridley, Nonuniformity correction for staring IR focal plane arrays using scene-based techniques, Proc. SPIE 1308 (1990) 224--233. [6] J.M. Mooney, F.D. Shephard, W.S. Ewing, J.E. Marguia and J. Silverman, "'Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. 28(11) (1989) I151-1161. [7] D.L. Perry, E.L. Dereniak. Linear theory of nonuniformity correction in infrared staring sensors, Opt. Eng. 32(8)(1993) 1854-1859. [8] US Army Night Vision and Electronic Sensors Directorate, System Noise, Chap. 4 in FL1R92 Thermal Imaging Systems Performance Model, Document RG5008993 (1993) pp. ARG-8-ARG-I 1. [9] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences. (McGraw Hill Book Co., New York, 1969).
Infrared Phys. Technol. 36 (1995) 763-777
,
Nonuniformity correction and correctability of infrared focal plane arrays M. Schulz a, L. Caldwell b • Institute of Applied Physics, University of Erlangen-Narnberg, Staudstr. 7, D-91058 Erlangen, Germany bNight Vision Directorate, AMSEL RD NV TTP, 10235 Burbeck Rd Ste 110, Ft. Belvoir, VA 22060-5806, USA Received 12 September 1994
Abstract
Correction of photoresponse nonuniformity in infrared staring sensor arrays is investigated. A general nonuniformity correction procedure is proposed. The procedure is based on multiple irradiation sources and on least square fit approximations to the individual pixel response characteristics. Nonlinearities of the signal response are taken into account. A correctability figure of merit is defined which may be used to estimate the residual fixed-pattern noise after correction. The correction procedure and the correctability are applied to a real data set measured by a 64 x 64 element infrared focal plane array (FPA). It is shown that an offset correction is insufficient for this data set and that a linear correction reduces the fixed-pattern noise contribution to the magnitude of the temporal noise background. The residual uncorrected fixed pattern noise can be related to pixels showing large temporal noise.
1. Introduction A severe problem associated with the use of infrared focal plane arrays (FPAs) is the spatial photoresponse nonuniformity. The nonuniformity arises because each individual detector element in the FPA exhibits a response characteristics differing from those of its neighboring elements. The magnitude of the nonuniformity varies with the technology and with the detector material employed to fabricate the FPA. The responsivity variations are usually less than 1% for PtSi Schottky barrier FPAs; the variations may exceed 10% for compound semiconductor FPAs. The responsivity nonuniformities create a fixed pattern super-
imposed on the thermal image of the scene. In order to resolve temperature differences in the image of the scene to as low as AT ~ 0.01 K, which is required in many applications, the spatial nonuniformity must be less than 10 -4. In order to obtain this level of uniformity, a correction of the responsivity nonuniformity must be applied [1-7]. High-performance systems use multiple temperature irradiation sources to repetitively recalibrate the correction parameters [3]. Two irradiation sources are frequently used to correct for offset and gain nonuniformities. The frequency of these calibrations depends on the stability of the FPA. In scanning systems, the calibration can be performed within a single image frame. Depending
013.50-4495/95/$09..50 (~ 1995 Elsevier Science B.V. All rights reserved SSDI 1350-4495(94)00002- X
764
M. Schulz and L. Caldwell/lnfrared Phys. Technol. 36 (I995) 763-777
on the technology and the design employed, staring FPAs typically must be recalibrated within a few minutes, although some remain stable for several hours. New technologies are evolving which perform a scene-based nonuniformity correction using algorithms based on a temporal high-pass filtering and an artificial neural network [4,5]. The importance of a nonuniformity figure of merit was pointed out in Ref. [6]. The future standard for modeling FPAs is the Night Vision and Electronic Sensors Directorate's " F L I R 92 Thermal Imaging System Performance Model" [8], which recognizes the importance of temporal and spatial noise contributions to the overall system performance. The modeling shows that a general scheme for optimized nonuniformity correction which covers the usable signal range, is still lacking. In this paper we propose a new scheme to evaluate the accuracy of nonuniformity correction methods. For this purpose, we introduce a general correction procedure based on multiple irradiation sources and on least-square curve fitting analysis for obtaining nonuniformity correction information in the usable signal range. Based on this analysis we are able to establish a figure of merit, which we call the "correctability". The general correction procedure and the correctability figure of merit have been tested on several different infrared FPAs. The procedure will be demonstrated in this paper on a data set measured by a long-wavelength infrared (LWIR) FPA. The correction procedure may be simplified to be applicable to practical system implementations.
2. Nonuniformity correction The nonuniformity correction should fulfill the following requirements: - All the detector elements should yield signal values after correction for which the scatter for uniform irradiation is less than a threshold value defined by the application. Usually, the threshold value is close to or less than the effective temporal noise. - T h e correction should be effective in the full signal range. The tolerable nonuniformity thres-
hold may be defined uniformly in the signal range or as a function of the signal value with the smallest threshold for small signal levels and with an increasing threshold for larger signals. - The number of parameters to be corrected should be kept as low as possible. Usually only one (offset correction) or two (linear correction) parameters are adjusted. A high number of correction parameters increases the memory and the computation requirements for the correction procedure, and thus the complexity of the system. - T h e correction procedure should be compatible with the nonlinear response characteristics of the FPA. These nonlinearities may lead to a correction which is not equally effective throughout the signal range specified and the computational level required may be too large to keep the residual spatial fluctuations below a threshold value. The general correction procedure proposed in this paper meets all the requirements stated above. The general method, however, requires a rather complex mathematical analysis. It will be shown below that the mathematical complexity can be reduced to a minimum so that the procedure can be applied in practice. In the following discussion the most general algorithms are presented in order to demonstrate the basic principles of the procedure. In contrast to the nonuniformity correction in Ref. [7] which models the FPA photoresponse characteristics by Mooney's linear infrared sensor model [3], our method is strictly based on data measured by the FPA under test. Typical data sets consist of a large number of signal values Yi.j obtained by the direct readout of the array for uniform infrared irradiation. The photoresponse output signal, the "amplitude value" yq may be a voltage or a current signal measured in arbitrary units. The subscript j = 1. . . . N refers to the N individual detector elements (pixels) in the array. The subscript i = 1. . . . . n indicates the temperature T~ employed in the uniform irradiation of the array by a blackbody source. The signal values are taken from one image frame under the usual imaging conditions. We call the set of the direct output amplitude values yq the raw signal data. The individual amplitude values yq exhibit a scatter
M. Schul= and L. CaMwell/lnfrared Phys~ TechnoL 36 (1995) 763-777
within the array in spite of uniform irradiation due to the temporal noise and due to the spatial nonuniformity (fixed pattern). The temporal noise in individual pixels may be determined by analyzing several image frames taken for the same irradiation. The photoresponse characteristics of the FPA is measured as a function of the irradiation by the multiple blackbody source temperatures used in the calibration. A high accuracy is obtained by taking the signal average over the whole array. This photoresponse characteristics measured in general is highly nonlinear. The nonlinearities originate in the optical detector system (Planck's law and limited spectral response), but are also caused in the electronic readout of the irradiation signal from the device. Especially, the readout cannot be adequately described by a theoretical modeling. The mean signal output value (y~) is defined by 1
(l)
Since the number N of pixels in the array usually is a large number N > 1000, a good approximation (standard error= tr/x/~) of the mean photoresponse characteristics is obtained by the averaging.
765
The overall FPA photoresponse characteristics R(T,.) is sampled at the n irradiation temperatures,
R(~) =
(2)
A typical nonlinear photoresponse characteristics of an output signal sampled for a set of n = 6 blackbody irradiation temperatures as an example is depicted in Fig. l a. Instead of blackbody calibration sources, other irradiation sources of the FPA may be used. In the evaluation it is only assumed that the irradiation is uniform across the array and that the single variable (blackbody temperature, the photon flux, the irradiation power etc.) used to describe the infrared irradiation is representative for the spectral emittance of the thermal scene to be imaged, so that nonuniformities in the spectral response of the FPA are accounted for in the correction procedure. It is noted that the correction is only effective for this variable; nonuniformities may be incompletely corrected when the irradiation spectrum in the scene is different from that of the calibration sources. The bars in Fig. l a indicate the scatter range (maximum/minimum) of the data points. For large
7.5
/#
6.5
7
6.5
i'
>.,
Y
/
s
Pixel 9.=5.5 (34,49)
5.5
E
5
4.5 /
.>( unif.char.
a 3.5
........
290
' .........
300
' ........
310
i,
320 I T b
[K]
, ,,,
,~,
33O
,
....
4.5
i ......
340
o ~
35O
4.5
5
,, 6
5.5 <~y>
b ,
) .... 6.5
7
[~.Uo]
Fig. 1. (a) Array response characteristics of the mean raw signal value ( y > vs. blackbody irradiation temperature Tb. The open circles are data values, the continuous curve is a power law (4th order) fit. The bars indicate the full scatter range between y,~, and Ym~. (b) Signal output y as a function of the signal output (y> averaged over the whole FPA. The solid straight line of slope one represents the unified, photorespons¢ characteristics of the mean signal output. The data points depict the amplitude values of one spe~:ific pixd as indicated. The dashed line is a linear fit to the pixel data. The dotted line indicates the correction process of an arbitrary image signal value y to the unified characteristics. INFRA }6/4---C
766
M. Schul: and L. Caldwell/lnfrared Phys. Technol. 36 (1995) 763-777
arrays having a high number N of detector elements, the averaged response characteristics is improved by the factor ~ l/x/~. With a typical variance of a/
(3)
We thus obtain a "unified photoresponse characteristics" which is a straight line through zero and which has a slope of unity. This characteristics is easily and accurately interpolated between the sampled data sets
A),'i/= y~/-
(4)
We perform a standard least square curve fitting with a polynom defined by the order of correction to the amplitude deviation of individual pixels.
Weighting factors wi may be applied if a closer approximation is desired to specific data sets. Specific data sets having a reduced temporal noise, e.g. those on the low end of the photoresponse characteristics irradiated with a reduced power, may also be favored by giving them an increased weighting in accordance with the magnitude of the temporal noise. The curve fitting of polynoms by the least square method including weighting is standard and may be found in the literature [9]. The amplitude deviation values Ayti~ calculated by the curve fitting are
Ay~i~ = aj+ b/
(5)
For an offset correction only one parameter, a constant aj is determined for each pixel; for a linear correction two parameters, a constant offset aj and a gain bj, for a square correction three parameters aj, bj, and cj have to be determined and stored. The degree of correction may be arbitrarily chosen. It is noted that the amplitude deviation values are only approximated by the curve fitting procedure and that residual deviations remain after the curve fitting. The residuals of the approximation are due to the temporal noise in the individual pixels and due to the functional dependence in the individual photoresponse characteristics which is not accurately described by the curve fitting. The least square curve fitting procedure is the adequate method to approximate the individual pixel characteristics because it minimizes the error and takes into account the errors due to the temporal noise. For the correction of the nonuniformity, the data values determined by the curve fit to the amplitude deviation is subtracted.
Ayi~ = Ay~j- AyT~.
(6)
The procedure is explained in the schematic of Fig. 2 for one specific pixel. The amplitude deviations of this specific pixel are depicted by full squares. The data points of this example are all offset above zero. The deviations also show a trend to increase with increasing irradiation. The mean deviation value with offset correction is indicated by the horizontal line in Fig. 2. All the data points are lowered by the same amount (arrow marked "off") to yield the corrected amplitude deviations
M. Schul: and L. Caldwetl/lnfrared Phys. Teehnol. 36 (1995) 763-777
767
relations for the offset, the linear and the square correction,
0.1 • ............
"~ ~
7:" " "
•
...........
........-'""
0
....-I ......
•
off-~
•
lin
y jc ~- yj - a/,
offset correction,
c yj - aj YJ= l + b j '
linear correction,
l+bj /(l+bj): ~Y~=
o' 0
t-
4c~
c/
'
square correction
[]
"0.05
2cj
. . . .
4.5
I
5
. . . .
r
5.5
~
,
r
I
6
,
,
,
,
I
,
8.5
Fig. 2. Signal amplitude deviation Ay as a function of the mean irradiation variable ( y ) for one specific pixel. The full squares are raw data points. The open squares are obtained after offset correction, the open circles are obtained after linear correction. The lines and arrows marked "off" and "lin" indicate the correction curve fitting for offset correction and linear correction, respectively.
represented by the open squares. With a linear correction a straight line, indicated by the dotted line in Fig. 2, is fit to the data points. This straight line is lowered and turned into the zero base line as indicated by the arrows marked "lin". The open circles depict the data points after the linear correction is applied. The linear correction yields a closer approximation to the zero base line than the offset correction. For a correction higher than linear (not shown in Fig. 2), a curve is fit to the data; for correction this curve is shifted, turned and straightened into the zero base line. The amplitude values y~/ after correction are obtained by adding the unified response function again. Y~j = ( Y i ) + Ay~j
(7)
For an arbitrary image for which the irradiation is nonuniform, the correction of the individual pixel amplitude is performed on the basis of the individual curve fit to the pixel with the correction parameters aj, bj, cj, etc. determined by the above procedure with multiple uniform irradiations. The corrected pixel amplitude is determined by eliminating the linearized irradiation parameter ( y ) from Eqs. (4-7) to obtain the
(8)
For corrections higher than of square order, the mathematical relations are more complex. The way of correction to determine the irradiation variable ( y ) from an arbitrary amplitude value y and to correct the amplitude to obtain yC is schematically indicated by the dotted line in Fig. lb. For the linear correction, the measured pixel amplitude is offset and then the slope is normalized to unity by the gain parameter bj. The gain parameter bj has to be small tb/I ,~ l for the correction to be linear. A correction line with b/= - I leading to an infinite correction factor 1/(1 + b j ) ~ oo is determined for a pixel having no response to the irradiation. Such a pixel is uncorrectable and has to be considered as a dead pixel. It is noted that for linear and higher order corrections, the temporal noise associated with the individual pixels leads to an error in the determination of the irradiation coordinate ( y ) in individual pixels for solving the above equations (4)-(7) to correct the arbitrary image. It can be shown that for the unified characteristics chosen the parameter averages (aj), (bj), (cj) etc. vanish and that thus the error due to the temporal noise in the correction is minimized; however, if a pixel exhibits a weak photoresponse and thus b j ~ - l, the temporal noise is amplified by the correction with the factor l/(l + b j)>> I. Such pixels may be uncorrectable or poorly correctable because the signal noise is amplified. It is further noted that the temporal noise which may vary from pixel to pixel, creates an error in the determination of the correction parameters aj, b/, etc. The determination of the fixed pattern and its correction are therefore limited by the temporal noise in the individual pixels. A pixel exhibiting excessive temporal noise, e.g. l/f-type noise, may
M. Schul: and L. Caldwell/Infrared Phys. Technol. 36 (1995) 763-777
768
therefore be poorly correctable or even uncorrectable. We will see later that for the test FPA this is the prevalent process which causes the uncorrected fixed pattern.
3. Correctability It was already noted in the previous section that the correction procedure applied to the amplitude values is never quite accurate and that residual spatial fluctuations remain. These residual fluctuations are due to errors in the measurement data e.g. due to the temporal noise, and to nonlinearities which are not accurately approximated by the regression analysis. The minimum signal resolved for a perfectly uniform array is defined by the temporal noise. The residual fixed pattern after correction in a nonuniform FPA further degrades this optimum resolution. An estimate of the goodness of the correction may be determined by relating the magnitude of the residual fluctuations in the array after correction to the temporal noise pattern. The goodness of the curve fitting for an individual pixel is described by the ;(2 value of the standard deviation normalized to the mean temporal noise )'~i averaged over the whole array for the various irradiation levels,
i-I
(9)
)'~i
The ;(a-distribution of a perfectly uniform FPA having only temporal (gaussian) noise is well known in statistics [9]. The probability distribution is, I
p(;('-) =
( ' l 2 -~(k - 2),2 e -~22
2,/2F(k/2) ,,.
,
(10)
where x = n - r is the number of the remaining degrees of freedom after curve fitting to n data sets with r parameters. For linear correction the degrees of freedom consumed in determining the offset and slope are r = 2; for offset correction only, it is r = 1. The maximum of the ;(2-probability distribution occurs at
2
;(~x = k - 2 .
(11)
Because of the high number of pixels in the array, the histogram of random ;(2-values approximates the ;(Z-probability function of Eq. (10) quite well. The position of the distribution maximum Eq. (1 !) can be used to obtain an estimate of the temporal noise in the FPA. Deviations of the ~(-'-data histogram from the ideal ;(Z-distribution are due to the residual spatial nonuniformities in the array. By subtracting the ideal ;(2-distribution due to temporal noise only from the real ;(-'-values of measured data for the array after correction, we obtain a number to estimate the goodness of the correction,
~.( c=
N 3 N - - " ) -(n-
r)-1
=Ii=1
(N--- l - ~ n : ~)
1.
(12)
The pixel number normalization is N - 1 because one degree of freedom is consumed for the averaging over the whole FPA to obtain the unified photoresponse characteristics. The goodness of the nonuniformity correction is perfect for c = 0, i.e. if there is only temporal noise. The goodness value is c = 1 when the fixed pattern noise after correction equals the temporal noise. For a fixed pattern noise far in excess of the temporal noise large c-values c >> 1 are determined. It is noted that a mean temporal noise of the detector elements is used in the normalization of the goodness and the correctability. The magnitude of the temporal noise strongly affects the correctability value. A small temporal noise level increases the c-value because the threshold for the fixed pattern noise is also small. Increasing the temporal noise makes the correction easier. The correctability therefore is only a good figure of merit when the mean temporal noise level is stated. It is further noted that a mean value of the temporal noise Y~i for the various irradiation conditions is used in the normalization of the ;(2 value to determine the goodness of the correction. A
M. Schul: and L. Caldwell/lnfrared Phys. Technoi. 36 (1995) 763-777
nonuniform distribution of the temporal noise, especially increased noise in specific pixels leads to large deviation values Ayjj which cannot or which can only be approximately corrected. Pixels having excessive noise are uncorrectable or are only partially correctable. The method proposed, however, minimizes the error due to the temporal noise by the regression analysis and by the use of the linearized irradiation variable using the average over the whole array. Pixels exhibiting temporal noise larger than the mean temporal noise degrade, i.e. increase, the correctability value c of the correctability. Very noisy pixels may be uncorrectable. A threshold Z2 value may be defined to select and to exclude specific pixels which are uncorrectable. In practice, the nonuniformity correction is frequently performed in a "two-point correction" by accurately measuring the signal amplitude at two irradiation levels by averaging the temporal noise for a long time. In this case the straight line fitting in the regression analysis can be assumed to be accurately known. The correctability is then obtained by a third or more measurement sets at a different irradiation level between the two correction points. The correctability is then determined as for the raw data without any degrees of freedom consumed to define correction parameters. The relation to be used is
c =,]j.t
"-
-- 1.
(13)
( N - l)n
/~Z'(Yf-(Y))2~2ty~(N" - 1) 1 = ,2 -
scatter of the fixed pattern across the array. If both contributions to the noise are independent and random, the standard deviation a is given by a - ' = y ~ + a~
(15)
where a,z is the standard deviation of the fixed pattern only. Eq. (12) or the simplified form Eq. (14) can be used as a standard to measure and to specify correctabilities in FPAs. The measurement data is easily accessible by photoresponse measurements on FPAs.
4. Gradual spatial corrections The unified response function defined for the correction procedure assumes a uniform array photoresponse with random fluctuations superimposed. In practice, the array may show a gradual variation of the response or a nonuniform irradiation across the array due to limitations by cold shields. A more general response function may then be defined which takes into account gradual spatial variations of the response in the array, e.g. a linear or quadratic variation. In this case a tilted plane or a curved plane is fit to the array data for each irradiation temperature rather than a horizontal (constant) plane
R(y,) = (y,) +f~(j)
(Y,b)'/Yu
For a standard two-point correction by only wellaveraged correction points, a minimum of one additional point n = 1 is required to determine the correctability. For a single measurement data set (usually in the middle between the two correction points), the definition of the correctability simplifies to
c=~
769
l,
(14)
where tr is the standard deviation of the raw signal data set from the mean value obtained by averaging over the whole array. Note that tr 2 comprises the temporal noise of each individual pixel and the
(16)
The amplitude deviations to be corrected are then determined by the differences of the amplitude values to this higher order unified curved response plane f a ( J ) + (Y~). The starting amplitude deviations of the raw data are then reduced on the expense of additional spatial parameters rather than response parameters. The degrees of freedom /~ which are consumed in the fitting of the plane to the array have to be taken into account in the determination of the correctability. The correction procedure is similar to the one explained above. The gradual variation may be removed by correcting the spatial variation. The general correctability including this spatial curved plane fitting is
/j~(Y~-(Y~)--fct~(j))2/c c=
ij
- 1.
(17)
770
M. Schul-- and L. Caldwell/Infrared Phys. Technol. 36 (I995) 763-777
T =295K
"d
b
Fig. 3. Raw signal data of the test array uniformly irradiated with a blackbody source at Tb = 295 K. (a) 3-Dimensional representati of the 64 x 64 signal data values. (b) Image representation of the 64 x 64 detector signals. The contrast ranges from Ym,. (black) Ym~ (white).
M. Schulz and L. Ca/dwell/Infrared Phys. Technol. 36 (1995) 763-777
771
IOIX~
The large mathematical effort for fitting curved planes to the array data is in most cases not justified. This more complex analysis may, however, be necessary if a uniform irradiation of a large FPA is impossible because of technical limitations e.g. given by the optics or cold shields.
T = 295K R a w Data b 1OOOO
t7 = 5.3x10-2
1000 e,
100
,
5. Correction and correctability of a test F P A
1
-
10
I
IJ
5.1. Raw signal data 1
A mercury cadmium telluride staring FPA comprising 64 x 64 detector elements is used to test the correction method and to determine the correctability in the long-wavelength atmospheric window. The data obtained is used to demonstrate the applicability of the method for nonuniformity correction and not to prove the state of the art of FPA fabrication. We therefore need not specify further details of the array. The correction method is independent of the type of array and of the operating conditions. For the test, the detector array is uniformly irradiated by a blackbody source. The blackbody temperature could be adjusted in the range from Tb=295K (RT) to Tb = 3 5 0 K in steps of AT = 9 K. Each individual detector element could be addressed by an electronic read-out. The output signal was recorded by a computer. For each read-out cycle, an image data set of 64 x 64 signal values was recorded in arbitrary units. The image
Table 1 Statistical evaluation of the 6 raw data sets taken for the test FPA. The columns from the left to fight list the data set number n, the blackbody irradiation temperature T b, the signal value
o'
Y~=
Ymi,
Y,
n
(K)
(a.u.)
(a~u.)
(a.u.)
(a.u.)
(10 -~ a.u.)
1 2 3 4 5 6
295 304 313 322 331 340
4.60 4.92 5.29 5.68 6.10 6.58
0.053 0.057 0.063 0.070 0.078 0.087
5.22 5.54 5.91 6.39 6.91 7.49
3.97 4.28 4.65 5.03 5.46 5.93
0.70 0.72 0.75 0.78 0.81 0.84
0.1 3.8
4
4.2
4.4
4.6
4.8
5
5.2
5.4
Amplitude y [a.u.] Fig. 4. Histogram of the raw signal output amplitude data y taken for the full test array irradiated by a blackbody at Tb -- 295 K. The axis marks indicate the counting intervals. The vertical line indicates the mean value ~y).
data set exhibits a scatter in the signal magnitude due to the temporal noise of each individual pixel and due to the fixed pattern of the spatial nonuniformity of the array. The signal amplitude distribution of the 64 x 64 raw data set taken for the test array irradiated with a room temperature blackbody is represented in the 3-dimensional plot of Fig. 3a. The image contrast of the raw data set is depicted in Fig. 3b. The data set exhibits a "noisy" pattern with a number of randomly distributed spikes. A regular peak pattern occurs on the left rim and at the top right corner indicating a read-out problem. A cluster of defects appears in the top left corner. There is no dead element in the array showing no photoresponse. The photoresponse characteristics of the array is that depicted in Fig. l a. The open circles represent the signal data values averaged over the whole array. The vertical bars indicate the scatter range of the data values in each set. The continuous line represents a power law fit to the data points. The data values are listed in Table l together with the standard deviation #; in each signal data set and with the rms temporal noise y, for each data set. The standard deviation values are on average only 1% of the amplitude value and are too small
772
,t4. Schulz and L. Caldwell/Infrared Phys. Technol. 36 (1995) 763-777
T
= b
295K -~-4.e,0~
4.6o5.
'k ,4, q
(a)
AT=0.02K b
Fig. 5--Caption on facing page
M. Schul= and L. Caldwell/lnfrared Phys. lrechnol. 36 (1995) 763-777
to be indicated in Fig. la. The standard deviations increase with increasing irradiation at the higher blockbody temperatures. It is noted that the mean values exhibit a nonlinear increase as a function of the blackbody temperature. The nonlinearity is only partially due to the nonlinear increase of the irradiation power with increasing blackbody temperature. An esti-. mate based on the calculated irradiation power shows that the nonlinearity is also partially due to the read-out electronics. It is therefore not possible to remove the nonlinearity with an accuracy better than 10-4 as is required for a high temperature resolution by replotting the data as a function of the irradiation power calculated by the blackbody temperature. The scatter of the raw signal data is also statistically evaluated by a histogram. The histogram of amplitude values taken at Tb = 295 K is depicted in Fig. 4. The semi-logarithmic representation indicates the approximately gaussian distribution around the maximum which is closely centered at the mean value. There are, however, assymmetric tails and small side lobes in the distribution. The histogram is useful to demonstrate the scatter of the signal values; however, the location of the extreme values and the clustering of deviations is only visible in the spatial representations of Figs. 3a and 3b. 5.2. Corrected signal data The spatial nonuniformities of the signal output in the test FPA have been corrected by an offset, a linear, and a square correction procedure. Weighting factors inverse to the temporal noise. values are applied to slightly favor the temperature sets that show a reduced noise. The distribution of the output signal amplitude values yC obtained after linear correction of the 64 x 64 data set of the test FPA is represented in a 3-dimensional plot of Fig. 5a. The image contrast of the corrected data
773
Table 2 Statistical evaluation of the 6 data sets taken for the test FPA after correction, The columns from left to fight list the data set number n, the blackbody irradiation temperature Tb, and the residual standard deviations after full correction by linear curve fitting ¢¢, after two-point correction pinned to the sets 1 and 3 cry.3, and after two-point correction pinned to the sets I and 6 ~.6
:rb
a°
~..~
~I.6
n
(K)
(I0-~ a.u.)
(I0-~ a.u.)
(10-3 a.u.)
1 2 3 4 5 6
295 304 313 322 331 340
0.50 0.58 0.80 0.73 0.91 0.69
0 0.87 0 1.49 2.48 3.25
0 0.85 1.11 1.03 1.33 0
set is depicted in Fig. 5b. Note that the amplitude scale is now expanded by almost a factor 100 compared to the scale in Fig. 3a. The correction procedure reduces the spatial fluctuations to 1% of those of the raw data. The spatial distribution yields a noisy pattern; however, there are no spikes superimposed as in the raw data. The image of the spatial noise in Fig. 5b also does not exhibit any clustering nonuniformities. In order to demonstrate the noise equivalent temperature resolution, a stripe with a temperature step of AT = 0.02 K is superimposed onto the image. This temperature step can be clearly resolved after correction. The uniformity improvement is quantified by the standard deviation values listed in Table 2. The magnitude of the standard deviation values a ¢ after full linear correction is reduced to approx. 1% to those of the raw data listed in Table 1. The magnitude of the standard deviation is now of the order of the effective temporal noise signal y~ listed in Table 1. The standard deviation values listed show an increasing trend for increasing blackbody irradiation. A two-point correction is simulated by giving two different data sets, e.g. the sets 1 and 3 (1/3) or the sets 1 and 6 (1/6), respectively, a high
Fig. 5. Signal data after linear correction of the FPA test array uniformly irradiated with a blackbody source at Tb = 295 K. (a) 3-Dimensional representation of the 64 x 64 signal data values. (b) Image representation of the 64 x 64 detector signals. The contrast ranges from 4.5965 (black) to 4.6008 (white). A signal step equivalent to a temperature difference AT = 0.02 K is superimposed in the shape of a stripe in the center.
774
M. Schulz and L. Caldwell/Infrared Phys. Technol. 36 (1995) 763-777
weighting factor. The standard deviation values for these two-point corrections are also listed in Table 2. The standard deviation values for two-point correction vanish in accordance with the pinning for the sets 1/3 and 1/6, respectively. It is noted that the standard deviations for twopoint correction are always higher than those for the full linear regression to all the data points. Between the pinning points, the increase is approximately 50%, however, outside the pinning range, the magnitude of the standard deviation a~/3 assumes values up to 5 times larger than the full correction (rc. The histograms of the data after linear correction depicted in Fig. 6 show that the wide tails of the raw signal are fully corrected. The distributions are still gaussian near the maximum. The maximum of the distributions is always located at the unified response function value. However, the twopoint corrections show a wider spread than the full regression correction. It is noted that the two-point corrections slightly degrade the correction but they are still comparable to the full correction taking into account the. 6 data sets available which yields the optimum correction for the data available. The improvement mainly reflects the improved averaging of the temporal noise by the increased number of data points. The full correction by the least-square fitting method covers the full usable signal range and yields an optimum correction.
100000 T = 331K corrected b
10000
1
1000
a
1/3
=24.8x10 4
100 10
0.1
6.09
6.1
6.11
6.12
Amplitude y¢ [a.u.]
Fig. 6. Histogram of the signal output data y" after linear correction. The test array is irradiated by a blackbody at Tb=331 K. The three distributions are for the full linear correction (narrow), two-point linear correction pinned in the sets l and 6 (medium). and for the two-point correction pinned in the sets 1 and 3 (wide). The standard deviations are indicated in the insert.
erly corrected to a magnitude below the temporal noise value. The x~-distribution of the square correction scheme is depicted in Fig. 8 and of the offset correction in Fig. 9, respectively. The histogram
•'•
1000
5.3. Correctability of test array
'
-,
8
linear correction c =0.53
,~
100
The correctability of the 64 x 64 test FPA was tested by plotting the x:-distribution after linear correction in Fig. 7 with the normalization to the mean temporal noise of the data sets given in Table 1. Figure 7 also depicts the ideal X: distribution calculated by Eq. ( 1 0 ) with k = 6 - 2 = 4 for linear correction of the 6 data sets. It is noted that the histogram observed is well represented by the calculated curve. The maximum is in the correct position Z~=x= k - 2 = 2. However, there appears to be a tail in the measured data due to an uncorrected fixed pattern in the array. The correctability determined is c = 0.53 thus indicating that the fixed pattern noise has been prop-
a=9.1x104
i
\"L~
10
°
\\if2 0.1
0
10
2O
3O
4O
50
6O
70
X=
Fig. 7. Histogram of the Z-'-values of the test array determined by linear correction. The effective temporal noise value used in the normalization is Yt = 7 x l0 -=, The continuous curve is the calculated ;('=-distribution for temporal noise.
M. Schulz and L. Caldwell/Infrared Phys. Technoi. 36 (1995) 763-777
775
10000
1000
offset correction c=35
looo 100
100 10
, 0.1
, . . . . . . . . . . . . . . .
0
5
. . . . . . . .
,,
....
0.1
:
IIH ........
~
"
10 15 20 25 30 35 40 45 50 X2
X2
Fig. 8. Histogram of the X'-values of the test array determined after square correction. The effective temporal noise value is y t = 7 x l0 -4. The continuous curve is the calculated X2distribution for the temporal noise.
of the square correction fits the calculated ;(2. distribution. The distribution of the offset correction is very broad so that the calculated ;(-'-distribution degenerates into the vertical axis of the figure and cannot be shown on the same diagram. All the correctability values determined are listed in Table 3. The high c-value of the raw data indicates that the non-uniformity predominates the noise in the image. The offset correction is not sufficient to correct the nonuniformity into the magnitude of the temporal noise level. After linear and even more so by square correction, the fixed-pattern noise is corrected to a level better than the temporal noise. The
Fig. 9. Histogram of the X2-value of the test array determined after offset correction. The effective temporal noise value is y t = 7 X 10 -4.
square correction, however, does not improve the uniformity to a large extent compared to the linear correction. The contributions to the correction listed in the columns 3-5 of Table 3 demonstrate that the offset and the slope correction contribute about the same fraction to the correction. The square correction yields only about 10%. 5.3. Uncorrectable pixels The histograms of the ;(2-distribution in Figs. 7-9 permit to pinpoint specific pixels which exhibit a large ;(2-value and which are thus poorly corrected. The threshold of a large ;(2-value may be
Table 3 Correctability c value determined for the raw data and after correction to first (offset correction), second (linear correction), and third (square correction) order. The columns 3 to 5 list the maximum standard deviations of the contributions to the correction. The parameters a,, ab, a# are the standard deviations of the correction parameters a, b, c, respectively, and (Y)mx is the maximum of the mean FPA output signal
raw data offset correction linear correction square correction
c
tra (10 -3 a.u.)
~rb
~¢
88 35 0.53 0.48
46.2 52.8 52.8
45.7 45.2
3.8
M. Schulz and L. Caldwell/lnfrared Phys. Technol. 36 (1995) 763-777
776
defined by the calculated distribution depicted by the continuous curves. The calculated ;(:-distribution shows that based on the mean temporal noise, the probability of pixels having Z: >20 is negligible. Pixels having 7.:>20 are poorly corrected. Figure 10 shows a 2-dimensional scatter plot of the poorly corrected pixels having 7.2> 20 as a function of the correction parameters a, b determined by linear correction. The selected poorly corrected pixels cannot be correlated with large correction parameters indicating large deviations; in contrast to this assumption, they even accumulate in the region around a, b ~ 0 and thus exhibit a photoresponse around the average. The photoresponse characteristics of the specific pixel at coordinates (34,49) shown as an example in Fig. lb is the pixel having the largest ;(:-value Z2= 58 after linear correction and Z 2 =49 after square correction in the whole data set. The photoresponse characteristics of this specific pixel does not deviate to a large extent from the unified characteristics. The reason of the poor correction is visible in Fig. I 1 where the residuals of the linear curve fit are depicted as a function of the irradiation variable ( y ) . The residuals exhibit a random scatter. The curve fit therefore cannot be improved I~y using a higher order polynom. The large residuals indicate an excessive temporal noise
,-2
,., O
0
"o
~-2 03 -3
pixel (34,49)
-4 -5 4.5
5
5.5
6
6.5
Fig. 1 I. Residuals of the uncorreetable pixel with coordinates (34,49) as a function of the mean output signal ( y ) .
for this pixel, five times as high as the mean value. Poorly corrected pixels are therefore especially those pixels that exhibit excessive noise, probably of l/f-type noise. The curve fitting for these noisy pixels is inaccurate and a large scatter remains as a fixed pattern. It is noted that the residuals are not visible in Fig. lb. The accuracy required for complete correction into the magnitude of the ,verage temporal noise is 10 -4. Such a high accuracy is not resolved in the graphical representation of the figure; however, the nonuniformity correction is limited by this small effect.
0.25 X2>20 0.2
6. Conclusions 0.15 .o
0.1
.C
¢~ 0.05
tie
,#'.
0
. "*
-0.05 -0.1
,, "r
-0.2
-0.1
,,
i,
0
I
0.1
0.2
,'
0.3
0.4
0.5
offset a [a.u.]
Fig. I0. Scatter plot of uncorrectable pixels having a large ;C2-value X~> 20. The variables are the correction parameters for offset a and the gain b, respectively.
An evaluation procedure is proposed for the nonuniformity correction of FPAs. The procedure is based on multiple irradiation sources. The nonlinearity of the photoresponse curve which so far limited the correction to only two calibration sources, the "two-point correction", is removed by defining a unified photoresponse function which is a straight line of slope one through the origin. The deviations from this unified response function are corrected for all the individual pixels by an offset correction, a linear correction, and a square-law or higher order correction. The correction procedure is applied to a real data set measured for an FPA.
M. Schul- and L. Caldwell/lnfrared Phys. Technol. 36 (1995) 763-777
A correctability is defined which may be used to estimate the residual fixed-pattern noise after correction. The correctability is also tested by the measured data set. The linear correction reduces the spatial fluctuations of the raw data (as = 88)'0 down to the temporal noise y, level (~s=0.5yt). The residual uncorrected pattern can be related to pixels having excessive temporal noise. The correction of the fixed pattern is mainly limited by the nonuniformity of the temporal noise.
References [1] R.W. Helfrich, Programmable compensation technique for staring arrays, in: Smart Sensors, D. F. Barbe Ed., Proc. SPIE 178 (1979) 110--121. [2] K. Hartnett, C. Marshall, N. Butler, J. Stobie and S. lwasa, Compensation electronics for staring focal plane arrays, in: Technical .Issues in Focal Plane Development, E. Krikorian and W.S. Chan Eds., Proc. SPIE 282 (1981) 73-78.
777
[3] A.F. Milton, F.R. Barone and M.R. Kruer, Influence of nonuniformity on infrared focal plane array performance, Opt. Eng. 2,) (1985) 855-862. [4] M.D. Nelson, J.F. Johnson and T.S. Lomheim, General noise processes in hybrid infrared focal plane arrays, Opt. Eng. 30 (1991) 1682-1700. [5] D.A. Scribner, K.A. Sarkady, LT. Caulfield, M.R. Kruer, G. Katz and C.J. Gridley, Nonuniformity correction for staring IR focal plane arrays using scene-based techniques, Proc. SPIE 1308 (1990) 224--233. [6] J.M. Mooney, F.D. Shephard, W.S. Ewing, J.E. Marguia and J. Silverman, "'Responsivity nonuniformity limited performance of infrared staring cameras," Opt. Eng. 28(11) (1989) I151-1161. [7] D.L. Perry, E.L. Dereniak. Linear theory of nonuniformity correction in infrared staring sensors, Opt. Eng. 32(8)(1993) 1854-1859. [8] US Army Night Vision and Electronic Sensors Directorate, System Noise, Chap. 4 in FL1R92 Thermal Imaging Systems Performance Model, Document RG5008993 (1993) pp. ARG-8-ARG-I 1. [9] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences. (McGraw Hill Book Co., New York, 1969).