- Email: [email protected]
An improved nonuniformity correction algorithm for infrared focal plane arrays which is easy to implement
Infrared Physics & Technology 39 Ž1998. 15–21
An improved nonuniformity correction algorithm for infrared focal plane arrays which is easy to implement Ruizhong Wang ) , Peiyi Chen, Peihsin Tsien Institute of Microelectronics, Tsinghua UniÕersity, Beijing 100084, China Received 7 April 1997
Abstract An improved nonuniformity correction algorithm is presented which is suitable for infrared focal plane arrays with a nonlinearity of the photoresponse characteristic. Compared to older algorithms, it is simpler, faster and easier to implement by software or hardware. At the same time, it is shown that the correction precision and the correction effect of the new algorithm are substantially the same as that of the old one through theoretical analysis and comparisons of simulation results. q 1998 Elsevier Science B.V. PACS: 42.82.Et Keywords: Infrared focal plane array; Nonuniformity correction
1. Introduction Infrared focal plane arrays ŽFPAs. are widely used in military systems of infrared detection and imaging. Generally, the responsivity of the detectors in the FPAs varies because of the nonuniformity of the process and bias, which causes fixed pattern noise in the infrared thermal image and degrades the temperature resolution of the system. Therefore, the image signal generated by the FPAs must be corrected by software or hardware in most current systems. If the response of the detectors has a linear relation to target temperature, the 2 point or multi-point linear correction algorithm w1x can be used. Since the correction functions are all linear, there are only two ) C orresponding tsinghua. edu.cn.
author.
E -m ail:
w rz@ sun5.im e.
correction parameters to store for each detector, so this algorithm is easy to implement. However, the response of the detectors usually has a nonlinear relation to target temperature. When the range of the target temperature is relatively large, the linear correction algorithm may cause relatively large errors because the above nonlinear relation becomes obvious. The piecewise linear correction algorithm w1x is good in this case, but it has two disadvantages: Ži. There are many correction parameters to store Žif the number of intervals is n, the number of parameters is 2 n for each detector.. Žii. The temperature interval which the response signal of a detector belongs to needs to be decided so that the right correction parameters can be selected. In 1994, a correction algorithm based on polynomial fitting was presented by Schulz and Caldwell w2x which is suitable for infrared focal plane arrays with a nonlinearity of the photoresponse characteristic and can avoid these disadvantages.
1350-4495r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 1 3 5 0 - 4 4 9 5 Ž 9 7 . 0 0 0 3 4 - 0
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
16
Schulz’s algorithm w2x is improved in this paper. The new algorithm is simpler, so it has a higher correction speed and is easier to implement. At the same time, it is shown that the correction precision and the correction effect of the new algorithm are substantially the same as that of the old one through theoretical analysis and comparisons of simulation results. 2. Improvement of the algorithm and analysis The basic principle of Schulz’s algorithm is as follows. The difference DYj between the response signal Yj of each detector in a FPA and the mean response signal ² Y : of all the N detectors to the same target is fitted by an nth order polynomial of ² Y :, i.e.
Ý
a i j ² Y :i
Ý a i j Yj i Ž j s 1, 2,
..., N.
is0
Ž 7. n
Ý a i j Yj i Ž j s 1, 2,
² Y : f Yj y
is 0
Ž j s 1, 2, . . . , N . Ž 1. The fitting coefficients a i j can be obtained from the measured data of the detectors at several points in the range of target temperature. We can solve Eq. Ž1. for ² Y : and get ² Y : f Fj Ž Yj . Ž j s 1, 2, . . . , N . Ž 2. Since the value of Yj after correction should be ² Y :, the correction formula of Yj is Yj c s Fj Ž Yj .
n
DYj s Yj y ² Y : f
thus
n
DYj s Yj y ² Y : f
This algorithm has an obvious disadvantage, i.e. it is difficult to implement, especially by hardware, because Ži. It has many operations of division and extracting of a root, especially when the correction order is high. Žii. When the correction order is higher than 3, we cannot get an analytic correction formula. Žiii. When the correction order is higher than 1, since Eq. Ž1. of ² Y : has multiple roots, the correct correction formula must be selected. In our algorithm, we fit DYj by the polynomial of Yj , not ² Y :, i.e.
Ž j s 1, 2, . . . , N .
..., N.
Ž 8.
is0
Therefore, the correction formula of Yj is n
Yj c s Yj y
Ý a i j Yj i Ž j s 1, 2,
..., N.
Ž 9.
is0
Since Eq. Ž9. is a polynomial of Yj , the improved algorithm avoids the above problems and it is faster and easier to implement.
Ž 3.
c
where Yj is the value of Yj after correction. For example, when the correction order is 0, the correction formula is Yj c s Yj y a0 j Ž j s 1, 2, . . . , N . Ž 4. which is an offset correction. When the correction order is 1, the correction formula is Yj y a 0 j Yj c s Ž j s 1, 2, . . . , N . Ž 5. 1 q a1 j which is a linear correction. When the correction order is 2, the correction formula is c
Yj s y
1 q a1 j 2 a2 j
"
)
Ž 1 q a1 j .
2
4 a 22 j
Ž j s 1, 2, . . . , N . which is a quadratic correction.
q
Yj y a0 j a2 j
Ž 6.
Fig. 1. The model of ² Y : used in the simulation and two models of Yj Žcurve 1: the model of ² Y :; curves 2 and 3: two arbitrary models of Yj ..
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
17
Fig. 2. Ža. The target temperature image used in the simulation. ŽThe temperature of the eight strips from left to right is 293, 304.4, 315.9, 327.3, 338.7, 350.1, 361.6, 373 K respectively.. Žb. The infrared thermal image before nonuniformity correction. Žc. The infrared thermal image after the first order 5 point correction by the improved algorithm. Žd. The infrared thermal image after the first order 5 point correction by the old algorithm. Že. The infrared thermal image after the second order 5 point correction by the improved algorithm. Žf. The infrared thermal image after the second order 5 point correction by the old algorithm.
18
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
Fig. 2 Žcontinued..
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
Fig. 2 Žcontinued..
19
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
20
Now we analyze the correction precision of the improved algorithm and the old one concisely. Because Yj Ž j s 1, 2, . . . , N . and ² Y : are both monotonic functions of target temperature T, DYj is a function of T and also a function of Yj or ² Y :. The essential difference between the improved algorithm and the old one is that DYj is fitted by the polynomial of Yj in the former, while it is fitted by the polynomial of ² Y : in the latter. Yj ŽT . can be expressed as Yj Ž T . s ² Y : Ž 1 q ´ j Ž T . .
Ž j s 1, 2, . . . , N .
Ž 10 .
where ´ j ŽT . is a random function of T and its absolute value is usually much less than 1. As the base functions in the fitting, Yj ŽT . and ² Y ŽT .: are substantially identical. Therefore, the precision of fitting DYj ŽT . by the polynomial of either of them is substantially the same and the correction precision of the two algorithms should be substantially the same.
Fig. 3. Comparison between the nonuniformity before and after correction. Žcurve 1: before correction; curve 2: after the first order 5 point correction by the improved algorithm; curve 3: after the first order 5 point correction by the old algorithm; curve 4: after the second order 5 point correction by the improved algorithm; curve 5: after the second order 5 point correction by the old algorithm..
3. Comparison of the simulation results The correction effect of the improved algorithm and the old one was compared through software simulation. The simulation comprises the following steps: Ži. A nonlinear model of the mean response characteristic of a FPA in a range of target temperature, i.e. the model of ² Y :, is constructed first. Žii. The model of response characteristic of each detector, i.e. the model of Yj , is randomly constructed in a certain nonuniformity based on the model of ² Y :. Žiii. The correction parameters of each detector are decided according to the correction algorithm used. Živ. A target temperature image is constructed for the simulation. Žv. The infrared thermal images before and after correction are displayed in the form of a gray-scale. The model of ² Y : used in the simulation and two models randomly constructed according to it are shown in Fig. 1. To illustrate the correction effect in the whole range of the target temperature conveniently, eight
gray strips which represent eight target temperatures in the range 293–373 K are selected as the target temperature image, as shown in Fig. 2Ža.. The scale of the FPA used in the simulation is 64 = 40. The infrared thermal image before correction is shown in Fig. 2Žb.. The images after the first order 5 point correction by the improved algorithm and by the old one are shown in Fig. 2Žc. and Žd., respectively, and the images after the second order 5 point correction by the improved algorithm and by the old one are shown in Fig. 3Že. and Žf., respectively. From Fig. 2 we can see that Ži. The image is obviously improved after the correction. Žii. The effect of the second order correction is better than that of the first order correction, which shows that the usual linear correction algorithm cannot give a good result because of the nonlinearity of the response characteristic of the FPA. Žiii. The effect of the improved algorithm and the old one are substantially the same both in the case of the first order correction and in the case of the second order correction.
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
The nonuniformity of the FPA obtained in the simulation before and after correction is shown in Fig. 3, which quantitatively illustrates the above conclusion. The nonuniformity d is defined as w1x N
ds
Ý < Yj y ² Y :
Ž 11 .
21
proved algorithm is simpler, faster and easier to implement than the old one. At the same time, it is shown that the correction precision and the correction effect of the improved algorithm are substantially the same as that of the old one through theoretical analysis and comparisons of the simulation results.
js1
4. Conclusion The old algorithm used in nonuniformity correction of infrared FPAs with a nonlinearity of the photoresponse characteristic is improved. The im-
References w1x Y. Gao, M. Wu, Q. Zhou, Chin. J. Infrared Millimeter Wave 12 Ž1993. 169 Žin Chinese.. w2x M. Schulz, L. Caldwell, Infrared Phys. Technol. 36 Ž1995. 763.
An improved nonuniformity correction algorithm for infrared focal plane arrays which is easy to implement Ruizhong Wang ) , Peiyi Chen, Peihsin Tsien Institute of Microelectronics, Tsinghua UniÕersity, Beijing 100084, China Received 7 April 1997
Abstract An improved nonuniformity correction algorithm is presented which is suitable for infrared focal plane arrays with a nonlinearity of the photoresponse characteristic. Compared to older algorithms, it is simpler, faster and easier to implement by software or hardware. At the same time, it is shown that the correction precision and the correction effect of the new algorithm are substantially the same as that of the old one through theoretical analysis and comparisons of simulation results. q 1998 Elsevier Science B.V. PACS: 42.82.Et Keywords: Infrared focal plane array; Nonuniformity correction
1. Introduction Infrared focal plane arrays ŽFPAs. are widely used in military systems of infrared detection and imaging. Generally, the responsivity of the detectors in the FPAs varies because of the nonuniformity of the process and bias, which causes fixed pattern noise in the infrared thermal image and degrades the temperature resolution of the system. Therefore, the image signal generated by the FPAs must be corrected by software or hardware in most current systems. If the response of the detectors has a linear relation to target temperature, the 2 point or multi-point linear correction algorithm w1x can be used. Since the correction functions are all linear, there are only two ) C orresponding tsinghua. edu.cn.
author.
E -m ail:
w rz@ sun5.im e.
correction parameters to store for each detector, so this algorithm is easy to implement. However, the response of the detectors usually has a nonlinear relation to target temperature. When the range of the target temperature is relatively large, the linear correction algorithm may cause relatively large errors because the above nonlinear relation becomes obvious. The piecewise linear correction algorithm w1x is good in this case, but it has two disadvantages: Ži. There are many correction parameters to store Žif the number of intervals is n, the number of parameters is 2 n for each detector.. Žii. The temperature interval which the response signal of a detector belongs to needs to be decided so that the right correction parameters can be selected. In 1994, a correction algorithm based on polynomial fitting was presented by Schulz and Caldwell w2x which is suitable for infrared focal plane arrays with a nonlinearity of the photoresponse characteristic and can avoid these disadvantages.
1350-4495r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 1 3 5 0 - 4 4 9 5 Ž 9 7 . 0 0 0 3 4 - 0
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
16
Schulz’s algorithm w2x is improved in this paper. The new algorithm is simpler, so it has a higher correction speed and is easier to implement. At the same time, it is shown that the correction precision and the correction effect of the new algorithm are substantially the same as that of the old one through theoretical analysis and comparisons of simulation results. 2. Improvement of the algorithm and analysis The basic principle of Schulz’s algorithm is as follows. The difference DYj between the response signal Yj of each detector in a FPA and the mean response signal ² Y : of all the N detectors to the same target is fitted by an nth order polynomial of ² Y :, i.e.
Ý
a i j ² Y :i
Ý a i j Yj i Ž j s 1, 2,
..., N.
is0
Ž 7. n
Ý a i j Yj i Ž j s 1, 2,
² Y : f Yj y
is 0
Ž j s 1, 2, . . . , N . Ž 1. The fitting coefficients a i j can be obtained from the measured data of the detectors at several points in the range of target temperature. We can solve Eq. Ž1. for ² Y : and get ² Y : f Fj Ž Yj . Ž j s 1, 2, . . . , N . Ž 2. Since the value of Yj after correction should be ² Y :, the correction formula of Yj is Yj c s Fj Ž Yj .
n
DYj s Yj y ² Y : f
thus
n
DYj s Yj y ² Y : f
This algorithm has an obvious disadvantage, i.e. it is difficult to implement, especially by hardware, because Ži. It has many operations of division and extracting of a root, especially when the correction order is high. Žii. When the correction order is higher than 3, we cannot get an analytic correction formula. Žiii. When the correction order is higher than 1, since Eq. Ž1. of ² Y : has multiple roots, the correct correction formula must be selected. In our algorithm, we fit DYj by the polynomial of Yj , not ² Y :, i.e.
Ž j s 1, 2, . . . , N .
..., N.
Ž 8.
is0
Therefore, the correction formula of Yj is n
Yj c s Yj y
Ý a i j Yj i Ž j s 1, 2,
..., N.
Ž 9.
is0
Since Eq. Ž9. is a polynomial of Yj , the improved algorithm avoids the above problems and it is faster and easier to implement.
Ž 3.
c
where Yj is the value of Yj after correction. For example, when the correction order is 0, the correction formula is Yj c s Yj y a0 j Ž j s 1, 2, . . . , N . Ž 4. which is an offset correction. When the correction order is 1, the correction formula is Yj y a 0 j Yj c s Ž j s 1, 2, . . . , N . Ž 5. 1 q a1 j which is a linear correction. When the correction order is 2, the correction formula is c
Yj s y
1 q a1 j 2 a2 j
"
)
Ž 1 q a1 j .
2
4 a 22 j
Ž j s 1, 2, . . . , N . which is a quadratic correction.
q
Yj y a0 j a2 j
Ž 6.
Fig. 1. The model of ² Y : used in the simulation and two models of Yj Žcurve 1: the model of ² Y :; curves 2 and 3: two arbitrary models of Yj ..
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
17
Fig. 2. Ža. The target temperature image used in the simulation. ŽThe temperature of the eight strips from left to right is 293, 304.4, 315.9, 327.3, 338.7, 350.1, 361.6, 373 K respectively.. Žb. The infrared thermal image before nonuniformity correction. Žc. The infrared thermal image after the first order 5 point correction by the improved algorithm. Žd. The infrared thermal image after the first order 5 point correction by the old algorithm. Že. The infrared thermal image after the second order 5 point correction by the improved algorithm. Žf. The infrared thermal image after the second order 5 point correction by the old algorithm.
18
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
Fig. 2 Žcontinued..
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
Fig. 2 Žcontinued..
19
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
20
Now we analyze the correction precision of the improved algorithm and the old one concisely. Because Yj Ž j s 1, 2, . . . , N . and ² Y : are both monotonic functions of target temperature T, DYj is a function of T and also a function of Yj or ² Y :. The essential difference between the improved algorithm and the old one is that DYj is fitted by the polynomial of Yj in the former, while it is fitted by the polynomial of ² Y : in the latter. Yj ŽT . can be expressed as Yj Ž T . s ² Y : Ž 1 q ´ j Ž T . .
Ž j s 1, 2, . . . , N .
Ž 10 .
where ´ j ŽT . is a random function of T and its absolute value is usually much less than 1. As the base functions in the fitting, Yj ŽT . and ² Y ŽT .: are substantially identical. Therefore, the precision of fitting DYj ŽT . by the polynomial of either of them is substantially the same and the correction precision of the two algorithms should be substantially the same.
Fig. 3. Comparison between the nonuniformity before and after correction. Žcurve 1: before correction; curve 2: after the first order 5 point correction by the improved algorithm; curve 3: after the first order 5 point correction by the old algorithm; curve 4: after the second order 5 point correction by the improved algorithm; curve 5: after the second order 5 point correction by the old algorithm..
3. Comparison of the simulation results The correction effect of the improved algorithm and the old one was compared through software simulation. The simulation comprises the following steps: Ži. A nonlinear model of the mean response characteristic of a FPA in a range of target temperature, i.e. the model of ² Y :, is constructed first. Žii. The model of response characteristic of each detector, i.e. the model of Yj , is randomly constructed in a certain nonuniformity based on the model of ² Y :. Žiii. The correction parameters of each detector are decided according to the correction algorithm used. Živ. A target temperature image is constructed for the simulation. Žv. The infrared thermal images before and after correction are displayed in the form of a gray-scale. The model of ² Y : used in the simulation and two models randomly constructed according to it are shown in Fig. 1. To illustrate the correction effect in the whole range of the target temperature conveniently, eight
gray strips which represent eight target temperatures in the range 293–373 K are selected as the target temperature image, as shown in Fig. 2Ža.. The scale of the FPA used in the simulation is 64 = 40. The infrared thermal image before correction is shown in Fig. 2Žb.. The images after the first order 5 point correction by the improved algorithm and by the old one are shown in Fig. 2Žc. and Žd., respectively, and the images after the second order 5 point correction by the improved algorithm and by the old one are shown in Fig. 3Že. and Žf., respectively. From Fig. 2 we can see that Ži. The image is obviously improved after the correction. Žii. The effect of the second order correction is better than that of the first order correction, which shows that the usual linear correction algorithm cannot give a good result because of the nonlinearity of the response characteristic of the FPA. Žiii. The effect of the improved algorithm and the old one are substantially the same both in the case of the first order correction and in the case of the second order correction.
R. Wang et al.r Infrared Physics & Technology 39 (1998) 15–21
The nonuniformity of the FPA obtained in the simulation before and after correction is shown in Fig. 3, which quantitatively illustrates the above conclusion. The nonuniformity d is defined as w1x N
ds
Ý < Yj y ² Y :
Ž 11 .
21
proved algorithm is simpler, faster and easier to implement than the old one. At the same time, it is shown that the correction precision and the correction effect of the improved algorithm are substantially the same as that of the old one through theoretical analysis and comparisons of the simulation results.
js1
4. Conclusion The old algorithm used in nonuniformity correction of infrared FPAs with a nonlinearity of the photoresponse characteristic is improved. The im-
References w1x Y. Gao, M. Wu, Q. Zhou, Chin. J. Infrared Millimeter Wave 12 Ž1993. 169 Žin Chinese.. w2x M. Schulz, L. Caldwell, Infrared Phys. Technol. 36 Ž1995. 763.