# Nonuniformity correction algorithm with efficient pixel offset estimation for infrared focal plane arrays

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## Abstract

This paper presents an infrared focal plane array (IRFPA) response nonuniformity correction (NUC) algorithm which is easy to implement by hardware. The proposed NUC algorithm is based on the linear correction scheme with the useful method of pixel offset correction coefficients update. The new approach to IRFPA response nonuniformity correction consists in the use of pixel response change determined at the actual operating conditions in relation to the reference ones by means of shutter to compensate a pixel offset temporal drift. Moreover, it permits to remove any optics shading effect in the output image as well. To show efficiency of the proposed NUC algorithm some test results for microbolometer IRFPA are presented.

## Keywords

Infrared focal plane array Nonuniformity correction Fixed-pattern noise## Background

Infrared focal plane arrays are widely used in various military and civil systems for thermal imaging. However they suffer from pixel-to-pixel responsivity (gain) and offset variations which induce a spatial noise called a fixed-pattern noise (FPN) in the image obtained from the detector array (Mooney et al. 1989). For instance, cooled HgCdTe IRFPAs offer the high electro-optical performance at the operating temperature of 77 K but for the long wavelength infrared (LWIR) region they exhibit a higher response nonuniformity than type-II InAs/GaInSb superlattice structures or quantum well infrared photoconductors (QWIPs) (Rogalski 2011). Modern uncooled microbolometer IRFPAs attain high performance and they become a good choice for cost-effective thermal imaging systems operating in LWIR range (Trouilleau et al. 2009). However they need some additional compensation due to inherent temporal drift of detector characteristics and the impact of housing temperature change on the detector array response. In order to obtain high thermal resolution of the infrared imagery, the IRFPA response nonuniformity must be reduced an order of magnitude below pixel temporal noise (Mooney and Shepherd 1996). For instance, to get a thermal resolution of 20 mK in the system operating in LWIR region where the scene contrast is about 2 %/K, the detector array response nonuniformity must be <0.04 % (*σ*/*m*) (Rogalski 2011).

Typical IRFPA response nonuniformity correction (NUC) relies on the signal processing of detector array output in order to remove FPN from the obtained image. In general, NUC methods are divided on reference-based and scene-based ones. The former use extended surface IR references as the uniform temperature sources to determine the suitable correction coefficients (Orżanowski and Madura 2010). The latter are reference-free and the coefficients for detector signal correction are obtained by the statistical analysis of pixel response in real-scene image sequences acquired by the thermal camera (Hayat et al. 1999). The integration of reference-based and scene-based technique into the radiometrically accurate NUC algorithm for IRFPA sensors is presented in the paper by Ratliff et al. (2003). The scene-based NUC methods are more sophisticated and they need the special operations to reduce “ghosting” artifacts appearing in the image after correction when the observed scene gives strong edges or slow global motion (Rossi et al. 2010).

The commonly used reference-based NUC method is the linear two-point calibration (TPC) (Perry and Dereniak 1993). The TPC algorithm is well known and it allows to compensate both gain and offset variations of particular pixels in the array. Moreover, it is easy to implement by hardware and quite sufficient in many applications. Even though this basic NUC algorithm is elaborated in detail, the efficient method of correction coefficients update, especially for pixel offsets, is dissembled or the one-point calibration by means of IR reference is suggested only.

In this paper a modified TPC algorithm enabling pixel offset correction coefficients update and removing optics shading effect by the proper usage of temporally averaged IRFPA response determined at closed inner shutter is presented. Since the IRFPA response on infrared radiance coming from the inner shutter does not include the impact of camera housing and optics infrared radiance then the direct using of that detector response as the offset correction coefficients can lead to the insufficient NUC results appearing as shading effect on the output image. In the presented correction scheme, the pixel response change determined at the actual operating conditions in relation to the reference ones at closed shutter is used to compensate a pixel offset temporal drift. It will be shown further that the proposed NUC algorithm offers some advantages in signal processing path and hardware implementation.

## Proposed NUC algorithm

*y*

_{ i }(

*ϕ*) and

*x*

_{ i }(

*ϕ*) are corrected and uncorrected response of

*i*th pixel in the array, respectively,

*x*

_{ i }(

*ϕ*

_{1}) is the individual pixel response and

*x*(

*ϕ*

_{1}) is the average pixel response in the array at first calibration point

*ϕ*

_{1},

*g*

_{ i }is the pixel gain correction coefficient. Since subtraction

*x*

_{ i }(

*ϕ*) −

*x*

_{ i }(

*ϕ*

_{1}) results in pixel offset equalization in the array then

*x*

_{ i }(

*ϕ*

_{1}) term in Eq. 1 is considered as the required pixel offset correction coefficient. Accordingly, the

*x*(

*ϕ*

_{1}) term in Eq. 1 determines a global pixel offset value in the array. The pixel gain correction coefficient,

*g*

_{ i }, is defined as a ratio of the average pixel response change to the individual pixel response change within irradiance flux range from

*ϕ*

_{1}to

*ϕ*

_{2}and it is given by

*x*

_{ i }(

*ϕ*

_{1}) and

*x*

_{ i }(

*ϕ*

_{2}) are temporally averaged responses of

*i*th pixel in the array at first and second calibration point, respectively,

*x*(

*ϕ*

_{1}) and

*x*(

*ϕ*

_{2}) are spatially averaged array responses corresponding to the calibration points. Temporal averaging of the reference image sequences helps to reduce the impact of random noise (Ratliff et al. 2003). In order to update offset correction coefficients,

*x*

_{ i }(

*ϕ*

_{1}), the calibration by means of shutter as the IR reference is performed. The temporally averaged IRFPA response obtained at closed shutter is used as the new table of pixel offset correction coefficients in TPC algorithm according to Eq. 1. It is important to take the shutter image sequence between the calibration points. This approach to update of offset correction coefficients provides efficient NUC performance when the external shutter rather than internal one is applied. The internal shutter does not include entire optics path and the computed offset correction coefficients are not accurate. As a result the optics shading effect in the image after correction is observed (see Fig. 2a). Hence, it is needed to modify the conventional TPC algorithm in order to decrease this image shading effect and to provide efficient method of offset correction coefficients update.

*s*

_{ i }(

*ϕ*) and

*s*

_{ i }(

*ϕ*

_{ r }) are temporally averaged responses of

*i*th pixel in the array on incident irradiance flux,

*ϕ*, obtained with closed shutter at actual and reference operating conditions, respectively. Since pixel response change, Δ

*s*

_{ i }(

*ϕ*), is related to the pixel offset drift value then it can be introduced directly into the basic TPC algorithm formula as follows

*s*(

*ϕ*) and

*s*(

*ϕ*

_{ r }) are spatially averaged array responses determined with closed shutter at actual and reference operating conditions, respectively. Substituting Eq. 3 into Eq. 4 and arranging the resulting equation we obtain the final formula of the proposed NUC algorithm in the form

*g*

_{ i }and

*o*

_{ i }are fixed gain and offset correction coefficients, respectively. The pixel gain correction coefficient,

*g*

_{ i }, is determined according to Eq. 2 while the fixed pixel offset correction coefficient,

*o*

_{ i }, is given by

*Update*depicted in Fig. 1 is closed then the temporal average IRFPA response at closed shutter is calculated from the

*N*consecutive image frames. In many cases the averaging of 50–100 image frames is sufficient to reduce the influence of pixel temporal noise. Additionally, if

*N*is equal to the power of two we can obtain the average value by simple operations of data accumulating and bit shifting. However some external memory is needed to store acquired image data and intermediate results of the averaging process. Using a modern field-programmable gate array (FPGA) device that includes hardware multipliers and adders, it is easy to obtain a real-time implementation of the proposed NUC algorithm.

## Test results

*σ*

_{ t }, and spatial noise,

*σ*

_{ s }. It is estimated by the standard deviation of pixel values in the flat-field image. The temporal noise is calculated as the mean standard deviation of pixel response in the array on uniform blackbody radiance registered in time, typically at 50 image frames. Then the spatial noise value is evaluated according to Eq. 7. The estimated values of noise components occurring in the single IRFPA response on uniform blackbody radiance after NUC correction are listed in Tables 1 and 2, respectively. The statistics of IRFPA response are stated in ADU (ADC units). The experimental results show that the proposed modification of TPC algorithm provides the improvement of FPN compensation for tested IRFPA sensor by factor of 2.9 at 28.5 °C flat-field image.

Performance of the conventional TPC algorithm

Blackbody temperature (°C) | | | | | |
---|---|---|---|---|---|

17.5 | 3976.1 | 13.6 | 0.34 | 1.7 | 13.5 |

23.5 | 4101.7 | 13.6 | 0.33 | 1.8 | 13.5 |

28.5 | 4186.2 | 14.2 | 0.34 | 1.8 | 14.1 |

39.5 | 4450.8 | 13.9 | 0.31 | 1.8 | 13.8 |

Performance of the modified TPC algorithm

Blackbody temperature (°C) | | | | | |
---|---|---|---|---|---|

17.5 | 3976.0 | 4.0 | 0.10 | 1.7 | 3.6 |

23.5 | 4101.6 | 4.2 | 0.10 | 1.8 | 3.8 |

28.5 | 4186.1 | 5.2 | 0.12 | 1.8 | 4.9 |

39.5 | 4450.7 | 4.7 | 0.11 | 1.8 | 4.3 |

*σ*/

*m*). In both cases the shutter was activated about 20 min after performing of two-point calibration procedure.

## Conclusions

The modified two-point NUC algorithm enabling pixel offset correction coefficients update by the proper usage of temporally averaged IRFPA response determined with closed shutter is proposed. The use of pixel response change determined by the shutter at the actual IRFPA operating conditions in relation to the reference ones provides good detector offset temporal drift compensation and optics shading effect removing as well. The performed tests with microbolometer IRFPA confirm a high efficiency of the proposed NUC algorithm that is easy to implement by hardware too. In case of the thermal imager operating within wide ambient temperature range, the several fixed gain and offset correction coefficients tables are required.

## Notes

### Acknowledgements

This work was supported by Ministry of Science and Higher Education of Poland (Grant No. ON515006333) and National Centre for Research and Development, Poland.

### Competing interests

The author declares that he has no competing interests.

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