Small moving infrared space target tracking algorithm based on probabilistic data association filter

Infrared Physics & Technology 63 (2014) 84–91

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Small moving infrared space target tracking algorithm based on probabilistic data association filter Zhengzhou Li a,b,⇑, Jing Chen a, Yuanshan Gu a, Lan Tang a, Zhen Dai a, Hongxia Fu a, Ruzhang Li b, Changju Liu b, Gang Jin c a

College of Communication Engineering, Chongqing University, Chongqing, China National Laboratory of Analogue Integrated Circuits, Sichuan Institute of Solid-State Circuits, No. 24 Research Institute of China Electronics Technology Group Corporation, Chongqing, China c China Aerodynamics Research & Development Center, Mianyang, China b

h i g h l i g h t s  The motion, amplitude and size of space target are afforded Gaussian distribution.  The motion, amplitude and size from false alarm are distributed uniformly.  Probabilities of the features are derived using probabilistic data association filter.

a r t i c l e

i n f o

Article history: Received 30 June 2013 Available online 13 December 2013 Keywords: Infrared space target LO moving target tracking Target state estimation Multi-data association PDA

a b s t r a c t Numerous false alarms for low signal-to-noise ratio (SNR) would seriously debase the performance for infrared low observable (LO) space target tracking. Due to the motion (i.e. azimuth, elevation and their derivative velocity), amplitude and size of infrared target are almost invariable and highly correlative, a multi-feature association approach based on probabilistic data association (PDA) is presented to track target in this paper. Firstly, the motion, amplitude and size of target are modeled as stationary random signal afforded Gaussian distribution. The probability of motion, amplitude and size of measurement originated as the target of interest is then estimated by Gaussian distribution, and that of false alarm is distributed uniformly. Subsequently, the combined probability of motion, amplitude and size is derived by PDA, and their weight coefficients are estimated adaptively according to their fluctuations. Finally, the relevant parameters including combination measurement are predicted and updated. Some experiments are included and the results show that the performance of target tracking by the proposed approach is significantly enhanced. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Ground-based electro-optical (EO) sensor is the primary technology of space surveillance network (SSN), which detects, tracks, identifies and even catalogs man-made objects flying in high earth orbit. The distance between the man-made object and ground-based EO sensor is usually more than 30,000 km, and the angle between the object and ground-based EO sensor is so small that the target on EO sensor is a small blob. Meanwhile, the energy of the object decays greatly for long distance propagation, and it is usually submerged in noise or/and clutter. During tracking such LO moving target, many measurements including multiple false alarms and/or the target of interest are ⇑ Corresponding author at: College of Communication Engineering, Chongqing University, Chongqing, China. Tel./fax: +86 23 65103544. E-mail address: [email protected] (Z. Li). 1350-4495/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.infrared.2013.12.003

detected in validation region for low probability of detection and high probability of false alarm. In this case, the strategy deciding how to use the received candidate measurements to associate the target’s trajectory formed in former scans is crucial to maintaining tracking and updating trajectory. At present, three kinds of strategy are applied to track LO moving target, namely, nearest-neighbor association, splitting tracking association and PDA. The nearest-neighbor filter only associates the nearest measurement with the existing trajectory, and it induces divergence very possibly. The splitting tracking filter splits the trajectory into as many sub-trajectories as candidate measurements, for example, 3-D velocity Match [1,2], multi-stage hypothesis testing (MHT) and dynamical programming algorithm (DPA) [3–5]. Evidently, it is likely to result in combination blast. The PDA utilizes all of the candidate measurements with different weights to form a combination measurement for maintaining tracking and update trajectory. The original PDA only uses the motion of target without

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Z. Li et al. / Infrared Physics & Technology 63 (2014) 84–91

considering the other features [6]. Actually, motion, amplitude and size of target could be detected simultaneously by infrared imaging sensor, and it is likely to enhance the capability of target tracking if the amplitude and size of target are integrated into PDA. The maximum likelihood estimator in conjunction with probabilistic data association (ML-PDA) [7,8] combines motion with amplitude for target motion analysis (TMA), and it effectively enhances the performance of target tracking in the case of clutter. The received energy of man-made object on the EO sensor mainly depends on its light radiation, observation angle and the distance between object and ground-based EO sensor, Wang et al. [9]. As we known, the light radiation of space target can be classed as self-radiation and environmental radiation. Self-radiation mainly depends on object’s volume and surface character, and environmental radiation is basically decided by solar radiation, earth’s radiation and reflection, et al. These factors are stationary and always change slowly, and it brings about that the brightness and size of space target in EO detector are almost invariable and highly correlative. A modified ML-PDA [10] models the amplitude of target as stationary random signal, and significantly improves the performance of target tracking. Meanwhile, the features such as size, edge and even shape of target on EO sensor are also integrated into tracker and the target tacking performance is enhanced greatly when these features are continuous and relevant [11–13]. To make use of the characters that the motion, amplitude and size of object are almost invariable and highly correlative, a multi-feature association approach based on PDA is presented in this paper to further improve the tracking performance for LO target. The motion, amplitude and size of target of interest are modeled as stationary random signals afford Gaussian distribution, and the probability of measurement being target is estimated and derived by PDA, and their weight coefficients are estimated and updated adaptively according to their fluctuations. The remainder of the paper is organized as follows. Section 2 describes the image signal and the target signal characters, and Section 3 derives the target state estimation based on PDA by modeling the target’s motion, amplitude and size as Gaussian distribution. Section 4 presents the small target tracking based on target state estimation described above, and the receiver operation characteristics (ROC) of original ML-PDAF, modified ML-PDA associating motion and amplitude using Gaussian distribution mode, and the approach developed in this paper are compared using the criteria based on the Crame–Rao lower bound in Section 5. Some experiments are included and the results show that the performance of target tracking by the proposed approach is significantly enhanced in Section 6, and a summary of the paper is finally given in Section 7.

and vertical direction, respectively. dx(k) and dy(k) are the extent parameters at horizontal and vertical direction, respectively, and they are usually equal, which means that the target is often circle. a(k), dx(k), dy(k) and (xt, yt) are all prior unknown parameters. Fig. 1 is a frame of the image sequence captured outfield by an EO tracking system, and the white rectangle is the tracking window or validation region. The target is man-made satellite, and it is submerged in glint noise. The mean SNR is about 4 and the mean contrast is about 1.0 within the 20  20 pixels area (i.e. tracking window) around the target in the image sequence. Fig. 2 is the estimate amplitude of the target along time, and its mean is about 200 with variance of 2 within 8 bit quantization scale. From the figure, it is shown that the amplitude a(k) of target is correlative, and the amplitude could be estimated by mean and adding noise in a short period. Fig. 3 is the size or pixels that the target occupied, and the target occupies about 10 pixels. The pixels the target occupies could be modeled as mean with adding noise within a short period. To describe expediently, these parameters dx(k) and dy(k) of every candidate measurement are substituted with the size that the measurement occupies in this paper.

hi ðkÞ ¼ fzi ðkÞ; gi ðkÞ;ki ðkÞis the target  originated measurementg; i ¼ 1;...;mk

ð5Þ

2. Space target signal

h0 ðkÞ ¼ fnone of those measurements is from targetg

ð6Þ

3. Target state estimation The motion, amplitude and size of target are the main features, which are utilized to distinguish and track the target from the noise in infrared imaging tracking system. There maybe are several candidate measurements within the validation region for low SNR, and let us define the set of measurements at time k as m

zðkÞ ¼ fzi ðkÞ; gi ðkÞ; ki ðkÞgi¼1k

where mk, zi(k), gi(k) and ki ðkÞ are the total number of measurements, motion, amplitude and size of the ith measurement within the validation region at time k, respectively. There are many candidate measurements, and the issue now is how to use one or more than one of these received candidate measurement to further update the state of the target for target tracking. The motion, amplitude and size of space target are continuous, so the state of target at time k is relevant to that of the former scans. The cumulative set of measurements up to time k is k

Zk ¼ fzðjÞgj¼1

ð4Þ

The target of interest may or may not exist in these candidate measurements, and the corresponding association events are defined as

The image sequence with small target submerged in clutter can be modeled as

f ðx; y; kÞ ¼ sðx; y; kÞ þ nðx; y; kÞ

ð1Þ

where s(x, y, k) is the target intensity, n(x, y, k) represents the background clutter or noise, (x, y) is the spatial coordinates of image sequence, and k refers to the time or image frame. The small target concentrates itself relatively in a small other than pixel-sized object region with uniform amplitude, and could be described by point spread function (PSF) [14].

( sðx; y; kÞ ¼ aðkÞ  exp 

1 2

" 2  2 #) x  xt ðkÞ y  yt ðkÞ þ dx ðkÞ dy ðkÞ

ð2Þ

where a(k) is the target intensity amplitude, (xt, yt) denotes the target location at time k, and xt and yt represent the horizontal

ð3Þ

Fig. 1. Space target image.

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3.1. Target motion state estimation The state equation of target motion can be represented as the following formulas:

Xðkjk  1Þ ¼ Fðk; k  1ÞXðk  1Þ þ uX ðkÞ 0

_ _ yðkÞ; yðkÞ XðkÞ ¼ ½xðkÞ; xðkÞ;

ð11Þ ð12Þ

where X(k) is the target’s state vector including position and velocity information, F(k, k  1) is the state-transition matrix, and uX(k) is state noise with zero mean and covariance matrix QX(k). The measurement equation of motion is represented as

zðkÞ ¼ HðkÞXðkÞ þ wz ðkÞ

ð13Þ

0

zðkÞ ¼ ½zx ðkÞ; zy ðkÞ

ð14Þ

where z(k), H(k) and w(k) are the measurement vector, measurement matrix and measurement noise with zero mean and covariance matrix Rz(k), respectively. The theory of PDA assumes that the position of measurement originated from the target is afforded Gaussian distribution with its estimated value as mean, and the position of false measurement is distributed uniformly in the invalidation region. Therefore the probability density function of motion of measurement originated from target of interest and from false alarm can be estimated by the following two formulas (15) and (16), respectively.

Fig. 2. Target amplitude versus frame.

pT fzi ðkÞjhi ðkÞ; mk ; Zk1 g 1 exp ¼ 2pjsz ðkÞj1=2   1 0  ½zi ðkÞ  zðkjk  1Þ s1 z ðkÞ½zi ðkÞ  zðkjk  1Þ 2 pc fzi ðkÞjqi ðkÞ; mk ; Zk1 g ¼

So the conditional association probability is derived as

bi ¼ pfhi ðkÞjZk g ¼ pfhi ðkÞjzðkÞ; gðkÞ; kðkÞ; mk ; Zk1 g pfzðkÞ; gðkÞ; kðkÞjhi ðkÞ; mk ; Zk1 gpfhi ðkÞjmk ; Zk1 g pfzðkÞ; gðkÞ; kðkÞg

vzi ðkÞ ¼ zi ðkÞ  zðkjk  1Þ ð7Þ

Because the prior probability of each measurement originated from the target of interest is equal, the prior probability pfhi ðkÞjmk ; Zk1 g could be k1

pfhi jmk ; Z

g¼

8 m < 1PF k

i ¼ 1 . . . mk

:

i¼0

mk m

PF k

ð8Þ

ð9Þ

pfzi ðkÞ; gi ðkÞ; ki ðkÞjhi ðkÞ; mk ; Zk1 g ¼ pfzi ðkÞjhi ðkÞ; mk ; Zk1 gpfgi ðkÞjhi ðkÞ; mk ; Zk1 g pfki ðkÞjhi ðkÞ; mk ; Zk1 g

ð17Þ

From the above two assumptions about motion, and by combing the formulas (15)–(17), the probability about motion could be derived as

8   0 mk þ1 1 > exp  12 v zi ðkÞ s1 > z ðkÞv zi ðkÞ < VG 2pjsz ðkÞj1=2 pfzðkÞjhi ; mk ; Zk1 g ¼ i ¼ 1 . . . mk > > : mk VG i¼0

ð18Þ

where PF is the false alarm ratio. Assuming the motion, amplitude and size of measurement are independent, the following formulas could be get

pfzðkÞ; gðkÞ; kðkÞg ¼ pfzðkÞgpfgðkÞgpfkðkÞg

ð16Þ

where VG represents the column of validation region, and sz(k) is the expected error covariance innovation of the position corresponding to the estimated position of target. Let us define vzi(k) as position bias for the ith candidate measurement corresponding to the estimated position

Fig. 3. Target size (pixels) versus frame.

¼

1 VG

ð15Þ

ð10Þ

Therefore, the probability of every candidate measurement originated from target lies on the target state estimation including motion, amplitude and size using by some probabilistic distribution.

3.2. Target amplitude state estimation As described in Section 1, the received energy of man-made objects on the EO sensor mainly depends on objects light radiation, observation angle and the distance between object and groundbased electro-optical sensor. These factors change always slowly, and it brings about the amplitude of space target in EO detector is mostly invariable and highly relevant. The state of amplitude a(k) of target at time k could be modeled as that at previous time k  1 adding white Gaussian noise ua(k) with zero mean and unknown variance Qa(k), and it could be expressed as following formula:

aðkÞ ¼ aðk  1Þ þ ua ðkÞ

ð19Þ

Z. Li et al. / Infrared Physics & Technology 63 (2014) 84–91

And its measurement equation satisfies

gðkÞ ¼ aðkÞ þ wg ðkÞ

ð20Þ

where wg(k) is a white Gaussian noise with zero mean and unknown variance matrix Ra. Generally, target amplitude is afforded Gaussian distribution with its estimated value as mean, but false alarm is afforded uniform distribution. So the probability density functions of measurement from target of interest and from false alarm could be respectively denoted as ( ) 2 1 ½gi ðkÞ  aðkjk  1Þ pT ðgi ðkÞjhi ðkÞ; mk ; Zk1 Þ ¼ exp  ð21Þ 1=2 2sa ðkÞ 2psa ðkÞ

pc ðgi ðkÞjhi ðkÞ; mk ; Zk1 Þ ¼

1

ð22Þ

gmax ðkÞ  T g ðkÞ

where sa(k) is the expected error covariance innovation of the amplitude corresponding to the estimated amplitude of target, gmax(k) represents the maximum amplitude within the validation region, and Tg(k) denotes the threshold estimated by the criterion of constant false alarm ration (CFAR). In the same way, let us define vai(k) as the amplitude bias for the candidate measurement corresponding to estimated amplitude.

v ai ðkÞ ¼ gi ðkÞ  aðkjk  1Þ

8 m þ1 > ½g ðkÞ  T g ðkÞ k 2ps 1ðkÞ1=2 exp > < max a  1  k1 0 1 pfgðkÞjhi ;mk ;Z g ¼  v ðkÞ s ðkÞ v ðkÞ i ¼ 1 .. .mk ai ai a 2 > > : m ½gmax ðkÞ  T g ðkÞ k i¼0 ð24Þ 3.3. Target’s size state estimation As described in Section 1, the size of man-made objects on the EO sensor mainly depends on objects light radiation, observation angle and the distance between object and ground-based electrooptical sensor. These factors change always slowly, and it brings about the size of space target in EO detector is highly correlative. Simultaneously, the size of target on EO sensor is continuous and relevant, and d(k) could be modeled as previous state d(k  1) adding zero-mean white Gaussian noise ud(k).

dðkÞ ¼ dðk  1Þ þ ud ðkÞ

ð25Þ

where ud(k) is the zero-mean white Gaussian noise with covariance matrix Qd(k). Generally, target size is afforded Gaussian distribution with its estimated size of target of interest as mean, but false alarm is afforded uniform distribution. A size threshold T k ðkÞ is usually used to eliminate scattered false alarm and the probability density function of measurement from target of interest and from false alarm are formula (19) and (20), respectively. 1

(

½ki ðkÞ  dðkjk  1Þ pT ðki ðkÞjhi ðkÞ; mk ; Zk1 Þ ¼ exp  1=2 2sk ðkÞ 2psk ðkÞ

pc ðki ðkÞjhi ðkÞ; mk ; Z

k1

Let us define v ki ðkÞ as the size bias for the candidate measurement corresponding to estimated size.

v ki ðkÞ ¼ ki ðkÞ  dðkjk  1Þ

1 Þ¼ kmax ðkÞ  T k ðkÞ

2

)

ð26Þ

ð27Þ

where sk ðkÞ is the expected error covariance innovation of the size corresponding to the estimated size of target of interest, kmax ðkÞ represents the maximum size within the validation region.

ð28Þ

In the same way, from the above assumptions about size, and by combing the formulas (26)–(28), the probability about size could be derived as

pfki ðkÞjhi ðkÞ; mk ; Zk1 g 8  ðki ðkÞdðkjk1ÞÞ2 < ½k ðkÞ  T ðkÞmk þ1 1 i ¼ 1. .. mk max k 1=2 exp  2s ðkÞ k p s ðkÞ 2 k ¼ mk : ½kmax ðkÞ  T k ðkÞ i¼0 ð29Þ

4. Target tracking based on PDAF Combing the formulas (18), (24), and (29) into (7), the normalized weight coefficient bi(k) are indicated as

bi ðkÞ ¼

ð23Þ

From the above assumptions about amplitude, and by combing the formulas (21)–(23), the probability about amplitude could be derived as

87

8 ei ðkÞ > i ¼ 1 . . . mk > mk X > > > > bðkÞþ e ðkÞ i > < i¼1

bðkÞ > i¼0 > mk > X > > > > e ðkÞ bðkÞþ i :

ð30Þ

i¼1

where m

bðkÞ ¼

ei ðkÞ ¼

PF k mk 1 1 m ð1  PF k ÞV G gmax ðkÞ  T g ðkÞ kmax ðkÞ  T k ðkÞ

 1 1 T 1 exp  v ðkÞ s ðkÞ v ðkÞ zi zi z 1=2 2 2pjsz ðkÞj1=2 2psa ðkÞ

 1 1 T  exp  v ai ðkÞ s1 a ðkÞv ai ðkÞ  1=2 2 2psk ðkÞ

 1 T  exp  v ki ðkÞ s1 k ðkÞv ki ðkÞ 2

ð31Þ

1

ð32Þ

Using the total probability theorem with respect to the above events, the associated conditional bias va(k) about target amplitude at time k can be calculated by combing these measurements’ bias with the different weights bi(k)

v ai ðkÞ ¼ gi ðkÞ  aðkjk  1Þ

ð33Þ

According to Kalman filter, the state of the amplitude could be updated by these measurements at the current sampling time

aðkjkÞ ¼ aðkjk  1Þ þ Ga ðkÞv a ðkÞ

ð34Þ

where Ga(k) are the associated gain matrix of the target amplitude, and it is defined as

Ga ðkÞ ¼ Pa ðkjk  1Þs1 a ðkÞ

ð35Þ

P a ðkjk  1Þ is the associated covariance matrix of target amplitude, which is updated by

Pa ðkjk  1Þ ¼ Pa ðk  1jk  1Þ þ Q a ðk  1Þ

ð36Þ

Pa ðkjkÞ ¼ b0 ðkÞPa ðkjk  1Þ þ ð1  b0 Þ½Pa ðkjk  1Þ  Ga ðkÞP a ðkjk  1Þ "m # k X 0 0 0 þ Ga ðkÞ bi v ai ðkÞv ai ðkÞ  v a ðkÞv a ðkÞ Ga ðkÞ

ð37Þ

i¼1

and sa(k) is the innovation covariance of target’s amplitude, which is calculated as

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Z. Li et al. / Infrared Physics & Technology 63 (2014) 84–91

sa ðkÞ ¼ Pa ðkjk  1Þ þ Ra ðkÞ

ð38Þ

Similarly, for the target size information and motion, the update state dðkjkÞ and X(k), the associated conditional bias v k ðkÞ and vz(k), associated covariance matrix Pk ðkjkÞ and Pz ðkjk  1Þ, and associated gain matrix Gk ðkÞ and Gz(k) are updated and estimated as same as the target amplitude. The parameters including state conditional mean, state covariance matrix and gain matrix of motion, amplitude and size at sample time k could be estimated by the theory of PDA [14], and then the LO moving space target could be tracked and its trajectory also could be updated at successive time.

induced by false alarms. It is distinct that greater I1(PD, VG, mk) or J1 is, more accurate the state estimate would be. For the modified ML-PDA associating motion and amplitude using Gaussian distribution model, the likelihood probability function pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g is given by mk

pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g ¼ ð1  PD ÞV G m

þ PD V G k (



mk

1

gmax ðkÞ  sðkÞ



mk 1

1

gmax ðkÞ  sðkÞ

 exp 

 2  2 ) 1 zðkÞ  HðkÞXðkÞ 1 gðkÞ  at ðkjk  1Þ  2 sz ðkÞ 2 sa ðkÞ

ð45Þ 5. Receiver operating characteristic (ROC) with Crame–Rao lower bound

And also substituting Eq. (45) into Eq. (40) gives FIM J2 of the modified ML-PDA

The target motion state estimated by PDA is unbiased estimate, and there should be Cramer–Rao lower bound given by [15]

J2 ¼

I2 ðPD ; V G ; mk Þ jsz ðkÞj

where

0

Ef½XðkÞ  XðkÞ½XðkÞ  XðkÞ g P J 1

ð39Þ

where J is Fisher information matrix (FIM), and it is defined as

I2 ðPD ;V G ; mk Þ ¼

J ¼ Ef½rX ln pfzðkÞjXðkÞg½r #2 Z gmax Z " rX pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g ¼ pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g s z2V

denotes

ð40Þ

unbiased

estimate,

and

rX ln pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g is the differential coefficient to the state vector X. To describe expediently, the FIMs of the original ML-PDA proposed in [6], the ML-PDA associating motion and amplitude using Gaussian distribution model presented in [10], and the ML-PDA associating motion, amplitude and size in this paper, are denoted as J1, J2 and J3, respectively. For original ML-PDA, the likelihood probability function pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g is given by m

PD V G k m pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk g ¼ ð1  P D ÞV G k þ 2pjsz ðkÞ (

2 ) 1 zðkÞ  HðkÞXðkÞ PF  exp  2 sz ðkÞ PD  2  1 g ðkÞd  exp 1þd 2ð1 þ dÞ

where H(k) is the measurement matrix. Substituting Eq. (41) into Eq. (40) gives FIM J1 of original MLPDA

ð42Þ

I1 ðPD ; V G ; mk Þ ¼

s



2

zðkÞHðkÞxðkÞ sz ðkÞ

z2V pfzðkÞjxðkÞ; gðkÞ; mk g (  2   ) zðkÞ  HðkÞxðkÞ gðkÞ  at ðkjk  1Þ 2  exp  dzdg  sz ðkÞ sa ðkÞ

1mk

pfzðkÞjhðkÞ; Zk1 ; gðkÞ; dðkÞ; mk g

 mk

 mk 1 1 m ¼ ð1  PD ÞV G k kmax ðkÞ  T k ðkÞ gmax ðkÞ  sðkÞ

mk 1

mk 1 1 1 m þ PD V G k kmax ðkÞ  T k ðkÞ gmax ðkÞ  sðkÞ (  2  2 1 zðkÞ  HðkÞxðkÞ 1 gðkÞ  at ðkjk  1Þ  exp   2 sz ðkÞ 2 sa ðkÞ  2 ) 1 kðkÞ  dðkjk  1Þ ð48Þ  2 sk ðkÞ And also substituting Eq. (48) into Eq. (40) gives FIM J3

I3 ðPD ; V G ; mk Þ jsz ðkÞj

J3 ¼

ð49Þ

where I3 ðPD ;V G ;mk Þ Z

kmax Tk

Z

gmax

s

Z

8 > > > <

9 > > > = pfzðkÞjxðkÞ;gðkÞ;dðkÞ;mk g dzdgdk    2  2  2 > z2V > > > > > : exp  zðkÞHðkÞxðkÞ  gðkÞat ðkjk1Þ  kðkÞdðkjk1Þ ; s2



zðkÞHðkÞxðkÞ sz ðkÞ

2

sz ðkÞ

sa ðkÞ

sk ðkÞ

ð50Þ

where gmax

s2

and s ¼ g  sðkÞ . For the ML-PDA associating motion, amplitude and size using Gaussian distribution model, the likelihood probability function pfzðkÞjhðkÞ; Zk1 ; gðkÞ; dðkÞ; mk g is given by

¼

I1 ðPD ; V G ; mk Þ jsz ðkÞ

Z

Z

m PD V G k ½ max ðkÞ

ð41Þ

J1 ¼

gmax

ð47Þ

 pfzðkÞjhðkÞ; Zk1 ; gðkÞ; mk gdzdg X(k) = Xtrue(k)

Z s

0 X ln pfzðkÞjXðkÞg jXðkÞ¼X ðkÞ true

where

ð46Þ

Z

n2 q2 z2V



2

zðkÞHðkÞXðkÞ sz ðkÞ

TðkÞ

(

  ) g ðkÞd zðkÞ  HðkÞXðkÞ 2 dzdg exp  1þd sz ðkÞ 2

ð43Þ (

 2 )  2  1 zðkÞ  HðkÞXðkÞ g ðkÞd TðkÞ ¼ m þ nq exp  exp 2 sz ðkÞ 2ð1 þ dÞ mk

and m ¼ ð1  PD ÞV G

P V

mk

; n ¼ 2pDjszGðkÞj ; q ¼ PPDF

ð44Þ

1 . 1þd

The variable I1(PD, VG, mk) is called as information attenuation factor, which describes the incertitude of the target state estimate

m PD V G k ½ max ðkÞ

1mk

1mk

And s ¼ g  sðkÞ ½kmax ðkÞ  T k ðkÞ . To evaluate the estimate precision of the original ML-PDA, modified ML-PDA and the ML-PDA associating motion, amplitude and size, the information attenuation factors I1(PD, VG, mk), I2(PD, VG, mk) and I3(PD, VG, mk) versus measurement number mk are shown in Figs. 4–6 under the following conditions, respectively. The amplitude average of target a(k) and noise is equal to 0.8 and 0.3, respectively, the SNR d is equal to 1 db, the detection probability PD is equal to 0.9, the maximum amplitude gmax, the maximum size kmax , the target amplitude variance sa(k), the target size variance sk ðkÞ and the target location variance |sz(k)| are equal to one, and the gate probability PG is equal to 0.5.

Z. Li et al. / Infrared Physics & Technology 63 (2014) 84–91

89

Fig. 4. I1(PD, VG, mk) versus mk.

Fig. 5. I2(PD, VG, mk) versus mk.

Fig. 7. Infrared images (a) original image and (b) validation region.

I3(PD, VG, mk) is bigger than that of I2(PD, VG, mk) and I1(PD, VG, mk) when mk has same value. So the Cramer–Rao lower bound of the PDA-AI associating motion, amplitude and size is less than that of original PDA-AI and modified PDA-AI associating motion and amplitude, and the motion state estimate by the former is more accurate and more reliable than the latter two. 6. Experiments and analysis

Fig. 6. I3(PD, VG, mk) versus mk.

In Figs. 4–6, the black lines are I1(PD, VG, mk), I2(PD, VG, mk) and I3(PD, VG, mk) versus mk, respectively. The range of I1(PD, VG, mk) is very narrow, and it would be decreased to zero quickly when mk increases. Whereas, the range of I2(PD, VG, mk) and I3(PD, VG, mk) are wider than that of I1(PD, VG, mk), and they decrease slowly with mk increases comparatively. Simultaneously, the value of

There is an image sequences included in the experiment to test the models mentioned in this paper. And also it is used to evaluate the target tracking performance of original ML-PDA, modified MLPDA and the derived approach in this paper. It is captured outfield by infrared imaging system. Fig. 7(a) is one of the captured image sequence, and the target of interest is submerged in noise, which is at the center of the validation region in Fig. 7(b). The other bright points are false alarms. The mean SNR and the mean contrast are about 2.0 and about 0.5, respectively. The capacity of the validation region is 50  50 pixels. The noise is white noise afforded Gaussian distribution. The procedure to detect and track small moving target is as followings. To improve the ratio of target to noise, the successive images are accumulated and averaged. And the averaged image is segmented by the constant false alarm ratio criteria (CFAR), and there may be many candidate measurements. Then these candidate measurements should then be verified with successive images by the

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Fig. 8. Amplitude of target (a) amplitude versus frame. (b) Amplitude distribution. Fig. 9. Size of target (a) size versus frame and (b) size distribution.

M/N criteria, namely the candidate measurement would be a target if it exits for more than M times in N successive frames. Also the target trajectory would be initialized by extracting these information including motion, amplitude and size. Once the target trajectory has been initialized, the subsequent image would be accumulated and averaged along the predicted target trajectory, and then also is segmented by CFAR, there are more than one candidate measurements in the validation region. The target would be tracked using ML-PDAF and the revised PDAF to estimate target state by associating these candidate measurements with different weights. Fig. 8(a) is the curve of the amplitude of target in the image sequence, and it is shown that the amplitude of target is relevant. Fig. 8(b) shows the frequency of every amplitude level, and it is approximately afforded Gaussian distribution with mean of 100 and variance of 5, so it is reasonable to model the amplitude of target as stationary random signal afforded Gaussian distribution in this paper. Fig. 9(a) is the curve of the size of target in the image sequence. The sizes of target of interest fluctuate in the image sequence, but it is afforded Gaussian distribution with mean of 7 pixels shown in Fig. 9(b). Therefore it is also reasonable to model the size of target as stationary random signal afforded Gaussian distribution. The fluctuation of size of target may be induced by prior procedures, such as false segment. From Figs. 8 and 9, it is shown that modeling the amplitude and size of target of interest as stationary random signal is also

Fig. 10. Target in collision with false alarms.

reasonable, and it is feasible that the amplitude and size of target are utilized to enhance the performance of target tracking. During tracking the small space moving target, there are many targets of no interest or false alarms, which crosses the validation region of the target of interest shown in Fig. 10. The bright point at

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these bring about that the amplitude and size of target in EO detector are almost invariable and highly correlative. It is reasonable that they are modeled as stationary random signal, and it is feasible that the amplitude and the size of target are utilized to enhance the performance of target tracking based on multiple feature association using PDA. Acknowledgements This research was supported by the National Natural Science Foundation of China under Grant No. 61071191, Natural Science Foundation of Chongqing under Grant No. CSTC 2011BB2048 and Fundamental Research Funds for the Central Universities under Grant No. CDJZR10160004. And we are also grateful to the reviewers for their suggestion. References Fig. 11. Target tracking error.

center of the white rectangle is the dim target of interest, and a brighter and greater false target slowly runs across the validation region. Fig. 11 shows the mean square error (MSE) of the original MLPDA tracker, ML-PDA tracker modeling only amplitude as Gaussian distribution and ML-PDA tracker modeling both of amplitude and size as Gaussian distribution, respectively. Tracking error has been normalized between 0 and 2. From Fig. 11, it is shown that tracker of original ML-PDA tracker has been pulled away by some brighter, bigger and moving false alarm, the mean error of trajectory is increasing and it is divergent. The mean error of ML-PDA tracker with amplitude information can track the target of interest, but the error of target tracking is greater than that of the ML-PDA tracker with amplitude and size. One of the crucial reasons, which results in the failure of the original ML-PDA tracker, is that it assumes that the probability of the brighter candidate measurement being the target of interest would be greater, and then it will be pulled over by the brighter false alarm after only a few frames. The ML-PDA tracker with amplitude and the ML-PDA tracker with amplitude and size can correctively track the target, and integrating the size of target into ML-PDA can further improve the performance of target tracking. 7. Conclusions During dim small moving targets detection and tracking, many false alarms for low SNR seriously debase the performance of target tracking. These factors including space object light radiation, observation angle and the distance between object and groundbased EO sensor are stationary and always change slowly, and

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