基于分段线性模型的卡尔曼滤波非均匀校正算法

第34卷增刊JOURNAL OF XIDIAN UNIVERSITY Vol.34 Sup.基于分段线性模型的卡尔曼滤波非均匀校正算法秦翰林1，周慧鑫1，刘上乾1，简耀波23,(1. 西安电子科技大学 技术物理学院，陕西 西安 710071；2. 华中科技大学 电子工程系，湖北 武汉 430074；3. 深圳大学 工程技术学院，广东 深圳 518060）摘要：为了降低探测器响应的漂移以及非线性对非均匀性校正的影响，提出了一种基于分段线性模型的卡尔曼滤波红外焦平面阵列非均匀性校正算法。

该算法引入探测器响应曲线的分段线性模型，对基于卡尔曼滤波器的红外焦平面非均匀性校正算法进行了扩展和改进，不仅能减小探测器的偏置和增益随时间漂移对校正的影响，而且还能消除探测器响应非线性对非均匀性校正性能的影响。

一组利用真实红外图像序列进行的实验验证了该算法可获得较好的校正性能。

关键词：非均匀校正；卡尔曼滤波；非线性模型；红外焦平面阵列中图分类号：TN21 文献标识码：A 文章编号：1001-2400(2007)S1-0168-05Nonuniformity correction algorithm based on the Kalman-filter witha piecewise model for infrared image sequencesQIN Han-lin1, ZHOU Hui-Xin1, LIU Shang-qian1, JIAN Yao-bo2,3（1 School of Technical Physics, Xidian University, Xi'an 710071， China;2 Institute of Optoelectronics Science and Engineering, HUST, Wu'han 430074, China;3 School of Engineer and Technology, ShenZhen University, Shenzhen 518060, China)Abstract:To reduce the effect that the detector nonlinear response imposes on the nonuniformity correction, anonuniformity correction algorithm based on the Kalman-filter with a piecewise model is presented. This paperintroduces a piecewise linear model of a detector response curve. And the Kalman-filter based nonuniformitycorrection algorithm is improved, which adapts to infrared focal plane arrays with nonlinearity of the responsecharacteristic. Compared with the original algorithm, it can solve not only the problems of detector offset andgain drift with time, but also the influence of the detector nonlinear response to the nonuniformity correctionperformance to some extent. Therefore it achieves a better effect of the nonuniformity correction than the originalalgorithm. The nonuniformity correction capability of the improved algorithm is validated with real infraredimage sequences.Key word: nonuniformity correction; Kalman filter; nonlinear model; infrared focal plane arrays At present, the infrared imaging system is more and more utilized in military and civil applications. Over those available infrared imaging system, the infrared focal plane arrays (IRFPA) imaging system, which has many advantages, such as compact-structure and high-sensitivity, is paid more and more attention to. It is well known that response nonuniformity in IRFPA can markedly degrade the quality of infrared images. It results in a fixed pattern noise superimposed on the true image, and this deteriorates the temperature and spatial resolutions of the IRFPA imaging systems seriously. So a corresponding nonuniformity correction (NUC) technology must be employed in nearly all IRFPA imaging systems applied for engineering[1,2] .Since the 1980’s, numerous nonuniformity correction techniques have been developed, which can fall into two main types. In the first type of NUC algorithm, such as two-point or multi-point correction techniques [1-3], the known blackbody calibration-source is necessary. Although these techniques have a high correcting accuracy, they can not effectively eliminate the drift of IRFPA’s response[1-3]. The other group of NUC techniques is scene-based algorithms which rely on signal processing to remove the nonuniformity noise, such as constant-statistical algorithm[4], neural-network algorithm[5], temporal high-pass filter technique[6], Kalman filter algorithm[7,8], etc. Scene-based NUC techniques are more desirable than the traditional calibration-based NUC techniques in many IRFPA imaging applications since they do not require a blackbody calibration-source and do not interrupt the normal operation of the IRFPA camera. In recent years, special attention has been paid to developing scene-based techniques ——————————————投稿日期：2007-06-09基金项目：国家自然科学基金资助项目(60377034)作者简介：秦翰林（1982-），男，西安电子科技大学博士研究生in NUC.Special relevance to the NUC algorithm developed in this paper is the method developed by Torres and Hayat, which employs a Gauss-Markov model for the NUC parameters as a means to capture the drift in the nonuniformity. Their technique utilizes such a dynamical model to estimate the gain and bias of each detector in the IRFPA from an imaging sequence using a Kalman-filter. The NUC method based on the Kalman-filter is deducted under the linear region of the detector response [7,8]. However, in practice when the dynamic range of a detector is wide, the non-linearity is more notable, and then the nonuniformity correction performance will be degraded markedly.In this paper, the above-mentioned NUC algorithm based on the Kalman-filter is extended and improved by utilizing a piecewise linear model of detector response. It can overcome effectively the influence of the nonlinear response of the IRFPA detector on the NUC accuracy. Experimental results have shown that the improved algorithm achieves a better effect of NUC in resolving detector’s offset and gain drift with time, and nonlinear effect on it. 1 Nonuniformity correction algorithm based on the Kalman filterTorres and Hayat have proposed the NUC algorithm of IRFPA based on Kalman-Filtering. The algorithm assumes that the response characteristic of each detector in the IRFPA is linear. For the (i , j )th detector in the IRFPA at time n, the read-out signal is given by the approximate linear relation)(n Y ij )()()()()(n V n O n T n G n Y ij ij ij ij ij ++=, (1)where and are the gain and the offset of the (i , j )th detector at time n , respectively, and is the average irradiance collected by the detector during the integration time of the IRFPA camera associated with the n -th frame. The represents the additive electronic noise associated with the (i , j )th detector at time n . The subscript ij will be omitted for sententiousness, because all operations are performed pixel-by-pixel.)(n G ij )(n O ij )(n T ij )(n V ij The algorithm represents the gain and offset of each detector in a given frame with two state variables. In addition, due to the slow random variation of the gain and offset from block to block, the state variables are modeled by a Gauss-Markov process. It is assumed that the gain and the offset are constant in a given block of frames. In particular, they considered disjoint blocks of frames and assumed that drift in the gain and bias occurs only between blocks. Mathematically, the Gauss-Markov state e uation is gi en by q v k k k k k W G X X G G G +Φ=+1, (2)where 1+k X K = is the state vector at the k -th video-time. ⎥⎦⎤⎢⎣⎡++11k k O G k Φ =, is the transition diagonal matrix between the states at k -1 and k , a ⎥⎦⎤⎢⎣⎡k k b a 00k and b k represent the level of drift in the gain and bias between consecutive videos, respectively. The variable () is chosen according to the magnitude of drift between the gain (offset) at time kand the gain (offset) at time k +1. = is an identity matrix that randomly relates the driving noise vector W k a k b k G ⎥⎦⎤⎢⎣⎡1001k to the state vector X k . =, and are the driving noise for the gain process and the offsetprocess, respectively.k W ⎥⎦⎤⎢⎣⎡)2()1(k k W W )1(k W )2(k W The observation model for the detector response is given byk k k k V X H Y G G G +=, (3) where is the observation vector at the k -th block. The length of the observation vector of the k -th block is . = is the observation matrix at the k -th block. k Y G k A k H ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡111k l k kφφ##k V G = is the vector of independent, additive temporal noise elements in the k -th block. The main assumption in the observation model (3) is that the input irradiance values ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡k l k k V V #1k φ in the k -th block of frames is an independent sequence of uniformly-distributed random variables in the range [фmin k , фmax k ]. In particular, the range includes all possible radiation levels before the detectorbecomes saturated and is common to all the detectors in each block of frames. From the initial system state has a known mean:0X G []000ˆX X E X ==G . (4) And a known covariance matrix: T 0000()()0E X X X X ⎡Λ=−−=⎣P ⎤⎦G G . (5)Initially, the algorithm set the predictor of the state −1ˆX 1X G to10ˆX Φ−=0X . (6) This completes the required initial conditions for the Kalman-filter. We now introduce the subsequent estimates.For k ≥1, the state estimates based on observations k X ˆk Y Y G "G ,,11ˆ are obtained by the recursive process describedby the following equations: 1ˆk k k X X Φ−−−= , (7)T 11111k k k k k k k P P G Q G ΦΦ−T 1−−−−−−=+ , (8)T T (k k k k k k k K P H H P H R 1)−−=−+ , (9))ˆ(ˆˆ−−−+=kk k k k k X H Y K X X G , (10) −−=k k k k P H K I P )( , (11) where k H is the mean of the observation matrix at time k .In this model, the offset and the gain of each detector in the IRFPA can be estimated by an optimized recursion algorithm of a Kalman-filter. When a new frame block is coming, the filter makes use of the current scene information to refresh the old estimation value of the detector gain and offset. Once the gain and offset of the detector are estimated, the NUC of the image can be accomplished by subtracting the corresponding offset from the output of every detector unit, and divided by the corresponding gain.2 Improved algorithm based piecewise modelIn the above Torres’s algorithm, it is assumed that the response of the infrared detector is linear. In fact, the response is nonlinear. In practical applications where the wide dynamic range occurs, the infrared detector element will operate in a wide dynamic response range and so that the nonlinearity of detectors will become notable. If a linear model is still utilized in the algorithm, the nonlinearity will induce a large error in NUC. So a piecewise linear model is put forward in this paper.In the improved algorithm, the response curve of the detector element is divided into several segments reasonably. In each segment, the response curve is approximated by a linear model. Then we will apply the above-mentioned NUC algorithm based on the linear Kalman-Filter in each segment. The program is as follows:123(S 1+m response curve ofthe detector piecewise linear approximation linear approximationFig. 1 The piecewise linear model of the detector response in IRFPA(1) In the dynamic range of detector response, we divide the detector response curve into m segments by usingm +1 partition-point 1φ、2φ…1+m φ（1φ<2φ<…<1+m φ）, as shown in Fig. 1. In each segment, the detector responsecan be approximated by the linear model.(2) Measure and store the response output )(,p j i S φ of each element in IRFPA at different partition-points. (3) Calculate the mean matrix of the observation matrix k H .According to the assumption of the NUC algorithm based on the Kalman-filter in a linear model, to the random n -th observation, the received radiation )(n φ is a random variable with uniform distribution in the range[,] that includes all possible radiation levels before the detector becomes saturated. The min k φmax k ϕk H ismin max min max (,2)()/21 ,()/21k k k k k k l H φφφφ⎡⎤+⎢=⎢⎢⎥+⎣⎦##⎥⎥ （12）However, in the piecewise model, for the illumination φ in the section [p φ，1+p φ], the k H is11(,2)()/21 ()/21k p p k p p l H φφφφ++⎡⎤+⎢=⎢⎢⎥+⎣⎦##。