Influence of stochastic noise statistics on Kalman filter performance based on video target tracking

Vol.27 No.3 JOURNAL OF ELECTRONICS (CHINA) May 2010

INFLUENCE OF STOCHASTIC NOISE STATISTICS ON KALMAN FILTER PERFORMANCE BASED ON VIDEO TARGET TRACKING1

Chen Ken M. R. Napolitano* Zhang Yun Li Dong (College of Information Science and Engineering, Ningbo University, Zhejiang 315211, China)

*(College of Engineering, West Virginia University, WV 26506, USA)

Abstract The system stochastic noises involved in Kalman filtering are preconditioned on being ideally white and Gaussian distributed. In this research, efforts are exerted on exploring the influence of the noise statistics on Kalman filtering from the perspective of video target tracking quality. The correlation of tracking precision to both the process and measurement noise covariance is investigated; the signal-to-noise power density ratio is defined; the contribution of predicted states and measured outputs to Kalman filter behavior is discussed; the tracking precision relative sensitivity is derived and applied in this study case. The findings are expected to pave the way for future study on how the actual noise statistics deviating from the assumed ones impacts on the Kalman filter optimality and degra-dation in the application of video tracking.

Key words Stochastic noise; Kalman filter performance; Video tracking quality; Noise statistics

CLC index TP391

DOI 10.1007/s11767-010-0332-8

I. Introduction Since its debut in 1960’s, the Kalman filter has

witnessed a wide variety of applications in nu-merous fields such as radar target tracking, control systems, navigation, economics, etc., and later it was introduced to image processing in 1970’s[1] and commonly utilized in video target tracking algo-rithms[2]. For video target tracking, Kalman filter has been taken advantages mainly for its optimized noise-tolerance, Markov processes featuring mem-ory-saving and good estimation efficiency. As has been well noted, the very demanding preconditions for applying Kalman filter are that, for a given linear system, the process and measurement sto-chastic noise statistics must observe for being white and Gaussian distributed[3,4]. In an overwhelming number of literatures, the actually unknown sta-tistics are usually pre-assumed to be a given

1 Manuscript received date: December 23, 2009; revised

date: March 19, 2010. Supported by Science Foundation of Zhejiang Education Department (Y200804700), and Ningbo Natural Science Foundation of Zhejiang Province (201001A6001075). Communication author: Chen Ken, born in 1962, male, Ph.D., Associate Professor. The College of Information Science & Engineering, Ningbo University, Zhejiang 315211, China. Email: [email protected].

quantity prior to probing the subject of interest. As the author of this work has been aware, inadequate technical attention has up to date been directed to further studying the correlation between the Kalman filter performance and the varying al-though most likely unknown stochastic noise sta-tistics. A limited number of reports have come into the author’s attention concerning this pursued topic: A. Murakami, et al. made some endeavor on the role of system noise in Kalman filtering for the application of system parameter identification[5]. D. J. Jwo, et al. from theoretic perspective goes a great length on delving into the Kalman per-formance optimality and degradation associated with the discrepancy between the assumed sto-chastic noise statistics and actual ones[6].

In this work, the problem identified above is explored from video tracking quality perspective. The white and Gaussian noises with varying sta-tistics are mixed into the given system dynamics, based on which to view how these statistics impact on the tracking quality. The dynamic contribution of predicted states and accessible measurement to Kalman filter behavior is discussed; the concept of Signal-to-Noise Power-density Ratio (SNPR) and tracking Precision Relative Sensitivity (PRS) are introduced and applied; and conclusions as per the results from assigned number of tests are presented.

CHEN et al. Influence of Stochastic Noise Statistics on Kalman Filter Performance Based on Video Target Tracking 421

In the end, the future study issues are laid out.

II. Video Tracking and Dynamic Model Construction 1. Video-imaging set-up

To better probe how the stochastic noises arising from both process and readable measure-ment impact upon the video tracking quality, the videotaping environment is set up with great technical care in a bid to mitigate the possible inaccuracies resulting from the target extraction from image background. Therefore, a black target against white background is the prioritized con-sideration to serve this purpose. The target uses a rigid light-weight ball painted in black out of con-sideration of its orientation-invariant property and explicit centroid location relative to its circular profile, and the background takes a white board in an attempt to increase the gray contrast so as to best facilitate the target segmentation. The cen-troid determined in such a manner is believed suf-ficiently, if not completely, error-free, and later rendered as the nominal true target position. The video frame covers 800×600 pixels in size with 30 fps frame rate. The target ball is cast horizontally at a certain speed being assumed constant, and in vertical direction it moves downward followed by upward after hitting the rigid floor, upon which in the whole motioning course the gravity imposes its force upon the ball all the time with acceleration of 9.8 m/s2.

Eight video sequences in total are videotaped separately indoor and tagged from Sequences 1 to 8. Figs. 1(a) and 1(b) exhibits the moving trajectory of the target ball for Sequences 1 and 2, respectively. Given the required statistic nature of the noises under investigation, in order to make the noises better Gaussian-wise distributed, the number of video frames is expected to be sufficiently large so as to enable the mean of the noises to tend to its unbiased estimate[7]. Therefore, the eight video sequences are concatenated in such a way that the sequences with odd number are taken as they are while the even-number ones are flapped horizaon-tally or “mirrowed” prior to being inserted between the odd-numbered sequences, resulting with altogeter 572 frames that can be used in this

research. Fig. 2(a) shows the trajectory from the concatenation of Sequences 1 and 2. Note that for Sequence 2, the target ball actually horizontally traverses in the frame from right to left due to the mirrowing effect. As is shown in Fig. 2(b), from 81st frame and on, with all of which corresponding to Sequence 2, the target ball vertically jumps higher-and-higher in stead of lower-and-lower, opposing to the manner it does in Sequence 1 in which all the frames counte for less than 81 st.

Fig. 1 Moving trajectory of Sequences 1 and 2

422 JOURNAL OF ELECTRONICS (CHINA), Vol.27 No.3, May 2010

Fig. 2 Concatenated trajectory of Sequences 1 and 2

2. Dynamic modeling and Kalman filter prelimi-naries

As has stated previously, the target ball’s dy-namic motion is decomposed into that in horizontal and vertical directions. Numerous tracking tests indicate that the horizontal tracking errors are significantly trivial, so the effort in this work en-tirely focus on analyzing the tracking quality in vertical direction, which involves much more com-plex kinetics especially when taking the floor-hit instances into account. Eqs. (1) and (2) are the linear stochastic difference equations governing discrete-time controlled system with state .n∈X ℜ

1 1 1+ +k k k k− − −=X AX Bu W (1)

= +k k kZ HX V (2)

where the vectors are assigned as follows:

( ) 01 0 0

0( ) 0 1 0= =

00 0 1 0( )

0 0 0 1( )

x

y

kx

y

c k t

c k t

s k

ts k

⎡ ⎤ ⎡ ⎤Δ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦

X A B

1

( 1)

( 1)

( 1)

( 1)

x

y

k-sx

sy

w k

w k

w k

w k

⎡ ⎤−⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

W

( ) 1 0 0 0 ( )= = =

0 1 0 0( ) ( )

x x

k ky y

z k v k

z k v k

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

Z H V

In the above assignations, xc and yc respec-tively denotes the horizontal and vertical centroid coordinate of the ball in the frame; tΔ stands for time interval between frames, and 1/30tΔ = second for frame rate of 30 fps; u represents the gravity acceleration acting upon the ball and

9.8;u = xz and yz are the accessible measurements with ,n∈Z ℜ designated as the measured hori-zontal and vertical position in the current frame, respectively. The vector 1k−W and kV is the process noise vector n∈W ℜ and measurement noise vec-tor ,n∈V ℜ respectively, which are zero-mean Gaussian white sequences having zero cross-cor-relation with each other, and satisfy

T =k i ikE δ⎡ ⎤⎣ ⎦WW Q (3)

Tk i ikE δ⎡ ⎤ =⎣ ⎦VV R (4)

T =0, for all andk iE k i⎡ ⎤⎣ ⎦WV (5)

where E denotes the expectation, and Q and R corresponds to the process noise and measurement noise covariance matrix. ikδ symbolizes the Kronecker delta function and

1,=

0,ik

i k

i kδ

⎧ =⎪⎪⎨⎪ ≠⎪⎩ (6)

The discrete Kalman filter loop algorithm it-eratively applies two stages of computations[3]: Stage 1 Prediction phase

11 kk k u− −

−−= +X AX B (7)

T1= +k k

− −−P AP A Q (8)

Stage 2 Correction phase

( ) 1T T= +k k k

−− −K P H HP H R (9)

( )= k kk k k

− −+ −X X K Z HX (10)

( )k k k−= −P I K H P (11)

In the above equations, “^” implies the esti-mated, the superscript “–” indicates a priori, K is the Kalman gain, and P is estimate error covari-ance with

( )( )T

k k k kk

⎡ ⎤= − −⎢ ⎥

⎢ ⎥⎣ ⎦P E X X X X (12)


It is of imperative necessity to mention that, through Eq. (1) to Eq. (12), all the length units concerning movement of the target ball, such as distance, velocity, acceleration, are gauged in pixels within the 800×600 video frame instead of metric units. In all due respects, the computational results with this regulation will be in agreement with that performed in metric units since the constant pixel-to-millimeter conversion factor can be cali-brated if required[8].

III. Gaussian Noises Addition and Sig-nal-to-Noise Power-density Ratio (SNPR)

In this research, the stochastic noises arising from both process and measurement are assumed to observe Gaussian distribution with zero mean and covariance of Q and R, respectively, that is,

( ) ~ (0, )p NW Q (13)

( ) ~ (0, )p NV R (14)

From Eqs. (1) and (2) with the matrix assig-nations, the equation below governs the vertical positioning

( ) ( ) ( 1) ( )y y y yz k c k w k v k= + − + (15)

In Eq. (15), yc acts to be the noise-free vertical reference position, also termed the nominal true target position, it is computed through extracting the black-colored target ball from the light-colored background. Also from Eq. (15), the independent stochastic process noise yw and measurement noise

yv are added to the nominal true target position (reference signal) ,yc forming the corrupted signal

.yz To describe the signal corruption level, an

SNPR is derived as follows: To the discrete reference signal ,yc apply the

discrete Fourier transform

[ ]1

0

1( ) ( )exp 2 / ,

for 0, 1, 2, , 1

N

cy yk

F u c k j uk NN

u N

π−

=

= −

= −

∑ (16)

the power density is formulated as 2 2 2( ) Re ( ) Im ( ),

for 0, 1, 2, , 1

cyF u u u

u N

= +

= − (17)

where “Re”and “Im” denotes the real and imagi-nary part, respectively[9].

Similarly, the power density for the noise signal yw and ,yv denoted respectively as

2| ( ) |wyF u and 2| ( ) | ,vyF u can also be sought. Thus the logarithmized SNPR is defined as

12

01 1

2 2

0 0

( )SNPR log

( ) ( )

N

cyu

N N

wy vyu u

F u

F u F u

−

=− −

= =

=+

∑

∑ ∑ (18)

IV. Influence of Varying Q and R on Kalman Filter Performance

To begin with, define some relevant vertical tracking quality descriptors. The vertical tracking precision is explicitly quantized by the standard deviation, noted for STDtrack, resulting from the residual between a posteriori state estimate ( )yc k in Eq. (10) and the nominal true vertical position

( ),yc k computed with 122

Track1

1STD ( ) ( )

N

yyk

c k c kN

∧

=

⎡ ⎤⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦∑ (19)

In Eq. (10), there are two variables, that is, a priori estimate k

−X and measurement or observa-

tion ,kZ which simultaneously contribute to the a posteriori estimate .kX In other words, Kalman filter assigns the varying trust weight concurrently on both k

−X and kZ with the varying noise co-

variance Q and R. To investigate this dynamic weight of trust from Kalman filter upon the two, let Eq. (10) be regrouped into two terms of interest: term one, denoted as _pri,k

−X including the a priori

estimate ;k−

X and term two, expressed as _obs,k−

X containing the observation .kZ Rearrange the right side of Eq. (10) for

( )k kk k k

−= − +X I K H X K Z (20)

Let _pri ( ) ,k kk

− −= −X I K H X _obs .k k k

−=X K Z

Based on Eq. (20) and taking the state element concerning merely vertical motion into the study interest, the Root of Mean Square Errors (RMSE) is calculated from the residuals between kX and

_pri,k−

X denoted as post_priRMSE , which is utilized to quantize the “closeness” between the two as the noise statistics varies. Likewise, the RMSE can also be computed based on the residuals between kX


and _obs,k−

X , denoted as post_obsRMSE , to check how _obsk

−X is closely associated with kX along

with noise statistical variation. Considering that the average vertical trajec-

tory height measured in pixels for the target ball counts for approximate 300 pixels, in a loose sense representing the signal strength, eight stochastic noise co-variances for both Q and R are thus se-lected in an increasing order for 1, 100, 400, 900, 2500, 10000, 90000, 160000 (unit: pixel2), corre-sponding to the standard deviation of 1, 10, 20, 30, 50, 100, 300, 400 (unit: pixel) for the assumed

Gaussian distribution with zero mean, among which the Q-and-R value of 100, 400 and 900 are considered “reasonable” and 10000, 90000, 160000 “impractically extreme” when compared with the average signal strength represented by 300 pixels.

In the study procedure, a total of eight tests are conducted, and each test is completed with Q being one of the eight quantities while R being all eight quantities as specified afore. For the exem-plary sake, Tab. 1 herein presents the running re-sults in which Q =400, R =[1, 100, 400, 900, 2500, 10000, 90000, 160000].

Tab. 1 Tracking quality descriptors by varying R with Q=400 (pixel2)

R (pixel2) SNPR STDTrack (pixel) RMSEpost_pri (pixel) RMSEpost_obs (pixel) K

1 4.8064 19.6182 362.2100 0.8987 0.9975

100 3.1123 20.91353 302.0244 62.0852 0.8340

400 2.5124 32.4859 228.6036 139.4486 0.6303

900 2.1560 40.7875 179.9154 190.5389 0.4971

2500 1.7461 61.6884 127.1974 251.6319 0.3491

10000 1.1133 89.5786 76.4263 320.8340 0.2059

90000 0.1331 178.1988 40.6507 452.1809 0.0899

160000 –0.0554 223.4626 38.6061 509.4052 0.0735

In the above specific case, which may be

viewed for generality in a sense, Tab.1 exhibits that, along with increasing strength of measurement noise quantized with incremental R, also agreeably indicated by SNPR, the tracking precision is dete-riorating, as demonstrated by STDtrack. From the figures in both 4th and 5th column, it is noteworthy that, with smaller R, the Kalman filter “trusts” the measurement more; but gradually weighs its trust more favorably on the a priori estimate as R in-creases. The Kalman gain, K in 6th column, implies that, as the measurement noise power goes more dominant, it scales down the contribution of the measurement, in other words, it shifts the Kalman filter’s favor to the a priori estimate.

In the same fashion, altogether eight tests are run with each Q to be 1, 100, 400, 900, 2500, 10000, 90000, and 160000 (unit: pixel2), respectively. To probe how variant Q impacts on the vertical tracking quality, three standard deviations, corre-sponding to the three reasonable quantities taken by R equal respectively to 100, 400, and 900, are averaged to represent the integral STDtrack. Fig. 3(a)

exhibits that, along with increasing Q, the signal becomes more corrupted as manifested with de-creasing non-logarithmic SNPR, and the vertical tracking precision starts with improvement wit-nessed by standard deviation curve going down until 10log ( ) 3Q ≈ and then deteriorates indicated by the same curve shooting up, as marked with circle “ ”, forming an open-upward hyperbolic pattern. Similar trend can be observed in Fig. 3(b) by the influence of varying R. However, a note ought to be taken in this case that, looking into the standard deviation curve prior to 10log ( ) 3,R ≈ it traverses in a monotonous ascent fashion instead of going downward as the curve with Q does, and the similar phenomenon persists as noted in each in-dividual test running.

Fig. 4(a) implies how the dependence (or “trust”) weight from the Kalman filter changes concurrently on both the a priori state estimate and the actual measurement along with the in-creasing Q. Having a look at the curve marked with square “ ”, it suggests that the association between Kalman filter performance and actual measure-


ment, post_obsRMSE , becomes stronger, inferring that with the increasing Q the Kalman filter more and more trusts the actual measurement, mean-while less and less depends on the a priori state estimate, as implied by the curve tagged with tri-angle “ ”. Fig. 4(b) gives the opposite results for the case where the Kalman filter performance is recorded in association, as R increments, with both the a priori state estimate and the actual meas-urement. With R increasing, the Kalman filter more and more relies on the a priori state estimate,

,kX−

in the meantime gradually discarding the contribution from the measured readings, .kZ

Fig. 3 Averaged tracking precision

To quantify the degree of liaison of varying Q and R with the tracking precision for comparison purposes, a tracking Precision Relative Sensitivity (PRS) is defined, aiming to gauge how sensitive the tracking quality, expressed previously as trackSTD , is to the varying stochastic noise statistics. To do so, the data points of the RMSE curve pertaining both Q and R in Fig. 3(a) and Fig. 3(b) are curve-fitted

respectively with a 3-order polynomial, resulting in 3 2( ) 12.31 49.87 65.86 57.40Qf x x x x= − + + related

to varying Q, and 3 2( ) 8.02 22.56Rf x x x= − 62.53 203.21,x− + associated with varying R in

this particular case, where x acts as the common variable for being 10log ( )Q and 10log ( )R as is shown in Fig. 5.

Fig. 4 Trust weight of the Kalman filter on predicted states and measurement

For the fitting curve with variable Q, as pre-sented with solid line in Fig. 5, the first-order de-rivative, ( )/ ,Qdf x dx is taken with respect to x. A total of 11 points are selected with its value starting with 0, incrementally stepped by 0.5 and ending with 5, that is, 1 0.5i ix x −= + with 1 0x = and

1, 2, , 11.i = At each of these 11 designated points, the ( )/

iQ x xdf x dx =⏐ is calculated point-wise. The PRS is then computed first by summing up the absolute value of each resultant derivative followed by taking its average, expressed as PRSQ, and


Eq. (21) gives its calculative formula.

1

( )1PRS

i

MQ

Qi x x

df x

M dx= =

= ∑ (21)

The PRSR can also be obtained by the similar token, that is,

1

( )1PRS

i

MR

Ri x x

df x

M dx= =

= ∑ (22)

where M=11 for obtaining overall PRS, denoted as

totalPRS , which is computed as a whole for de-scriptive purposes.

Fig. 5 Curve-fit of averaged tracking precision with varying Q and R

By setting a “watershed” point on the abscissa in Fig. 5 where 10log ( ) 3Q ≈ (or 10log ( ) 3),R ≈ the break-down PRS can be sought separately before and after this dividing point, denoted for

priPRS and postPRS , respectively. Tab. 2 presents these three resulting sensitivities.

Tab. 2 Tracking precision relative sensitivity

STDtrack vs PRSpri PRSpost PRStotal

Q 63.81 153.93 104.78

R 26.03 276.21 139.75

Tab. 2 suggests that the tracking precision turns

out to be more sensitive to varying R than to varying Q in an overall sense. However, prior to the “bottom of the hyperbola” formed by the curve of tracking error against Q, which corresponds to the more applicable scope, the tracking precision is more sensitive to the varying Q with RPS being

63.81 than to R with RPS being 26.03. Based on a number of tests, this occurrence is all the time conspicuous. This phenomenon may to a certain degree agree with the remarks made in Ref. [6] concerning this characteristics, stating that de-creasing value of Q is more sensitive than the in-creasing one. This agreement can be comprehended in a way to see that within the reasonable selection range of Q, i.e., 900Q ≤ or 10log ( ) 3,Q < the curve slope decreases from some magnitude to zero as Q increases.

Fig. 6 Tracking precision with varying Q and R

Figs. 6(a) and 6(b) reveal the vertical tracking precision responding to the varying Q and R for the whole eight tests conducted in this research. Fig. 6(a) indicates that, when Q is reasonably small compared with the averaged signal strength, that is to say, less than 900 pixel2, the tracking precision is deteriorated relatively faster as R increases, but this “fastness” is relaxed as Q increases. Fig. 6(b) fur-


ther substantiates the remarks just made, i.e., with Q becoming further stronger, which should be deemed impractically extreme in real-world oc-currences, though, the variation of tracking preci-sion responds in a more dampened manner as R turns more dominant against signal strength. At the “outrageously extreme” level as Q exceeds 90000 pixel2, the tracking precision stays at the worst level but is in a loose sense invariant to varying R.

V. Conclusions and Future Research In this work, the influence of stochastic noises

existing in both the process and measurement on the video target tracking quality is presented. It is obvious that, the Kalman filter performance de-pends more on measured outputs as the process noise gets stronger, and weighs more on predicted states as the system is more corrupted by the measurement noises. The accuracy of target tracking is more sensitive to process noise statistics when both noises are sufficiently insignificant, while more sensitive to measurement noise statis-tics as both noises become excessively stronger. In this particular research, the measurement noise statistics contributes more to the tracking quality as a whole.

This research is conducted on the assumption that the noise statistics used in Kalman filter al-gorithm perfectly reflects its true counterpart, which may not even moderately be satisfied in real-world cases. Therefore, the future relevant study needs extending to deal with the situations in which there are disparities between the noise sta-tistics assumed for Kalman filter computation and

that existing in real world.

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Influence of stochastic noise statistics on Kalman filter performance based on video target tracking