Application of a Strong Tracking Finite-Difference Extended Kalman Filter to Eye Tracking

Non-intrusive methods for eye tracking are important for many applications of vision-based human computer interaction, such as driver fatigue detection, eye gaze replacing the hand operating mouse, eye typing instead of manually depressing keys as a virtual keyboard, eye gaze correction for video conferencing, interactive assistant application for disabled users, etc. However, due to the eye motion be the high nonlinearity, the obstacles of robustness of external interference and accuracy of eye tracking, these tend to significantly limit their scope of application. In this paper, we present a strong tracking finite-difference extended Kalman filter algorithm, and overcome the modeling of nonlinear eye tracking. In filtering calculation, strong tracking factor is introduced to modify prior covariance matrix to improve the accuracy of the filter. The filter uses finite-difference method to calculate partial derivatives of nonlinear functions to eye tracking. The last experimental results show validity of our method for eye tracking under realistic conditions.


Introduction
Since eye tracking was first introduced by Mowrer in 1936, it has been gaining in popularity over the past decades as a window into observers' visual and cognitive process.For instance, researchers have utilized eye tracking to study behavior in such domains as driver fatigue detection (Qiang et al., 2004;Horng et al., 2004;Dong et al., 2004 ), eye typing for helping users with movement disabilities interact with computers (Majaranta & Raiha, 2002), eye tracking analysis of user behavior in WWW search (Laura et al., 2004), using eye tracking techniques to study collaboration on physical tasks for medical research, VR system for measuring inspection methods, and image scanning (Noton & Stark, 1971).Above all applications, two types of human-computer interfaces utilize eye tracking, passive and active interfaces.Passive interfaces monitor the user's eye movements and automatically adapt themselves to the user.For example in driver fatigue detection, the researchers track the driver eyes to fatigue detection, because the human eyes express the most direct reaction when dozing, inattention and yawning.On the other hand, Active interfaces allow users to explicitly control the interface though the use of eye movements.For example, eye typing has users look at keys on a virtual keyboard to type instead of manually depressing keys as on a traditional keyboard (Majaranta & Raiha, 2002;Takehiko et al., 2003).Such active interfaces have been quite effective at helping users with movement disabilities interact with computers.Not surprisingly, eye tracking has attracted the interest of many researchers, and eye trackers have been commercially available for many years (Qiang et al., 2004;Horng et al., 2004;Takehiko et al., 2003;John et al., 2005).In the past decades, many researchers have paid attention to the eye tracking in human computer interaction.There have been many methods that support non-invasive eye tracking.In (Li et al., 2005), all of these eye tracking algorithms can be classified into two approaches: feature-based and model-based approaches.Feature-based approaches detect and localize image features related to the position of the eye.A commonality among featurebased approaches is that a criteria (e.g., a threshold) is needed to decide when a feature is present or absent.The determination of an appropriate threshold is typically left as a free parameter that is adjusted by the user.The tracked features vary widely across algorithms but most often rely on intensity levels or intensity gradients.For example, in infrared (IR) images created with the dark-pupil technique, an appropriately set intensity threshold can dynamic system, parameters estimation for nonlinear system identification and dual estimation where both states and parameters are estimated simultaneously.However, EKF simply linearizes all nonlinear functions to the first order by using the Talyor series expansions.At the same time, EKF may cause more errors for the nonlinear system while estimating system state and its variance.Moreover, the linearization may lead to divergence of filtering process.In a nonlinear mismatched model and limited applications scope, EKF filter will lead the divergence problem of state estimation.For these reasons, two improved EKF algorithms are introduced to tackle some of those problems.

Suboptimal fading extended Kalman filter
In this section, an adaptive extended Kalman filter -a suboptimal fading extended Kalman filter (SFEKF) is presented.The derivation of the filter is presented (Zhou et al., 1991;Zhou et al., 1990) in detail.SFEKF has the following good properties: 1) lower sensitivity to the statistics of the initial states and the statistics of the system and/or measurement noise, 2) stronger tracking ability to the suddenly changing states and bias no matter whether the filter operates in dynamic or stationary fashion, 3) acceptable computational complexity.Considering a class of nonlinear discrete-time dynamical system, 1 (,,) where, k x is the state vector, k y is the measurement vector, k u is control input vector, k v is process noise and k w is measurement noise.k v and k w are statistically independent.
The equations of mean and covariance are as follow.

[] ,
kk Ev q = cov[ , ] ( ) The extended Kalman filter is based on the assumption that sensor noises and, propagation errors are driven by zero-mean, Gaussian-distributed, white, random process.Retaining only the first-order terms in the Taylor series expansion, one obtains where () x Fk and () v F k are the partial derivatives of () () w Fk are the partial derivatives of () f i to x and w .
So the suboptimal fading extended Kalman filter (SFEKF) is deduced as follows: The predicted state estimation equations are 1 (, ,) The predicted covariance estimation equations are Where (1 ) 1 k λ +≥ is the suboptimal fading factor, which is used to fade the bypast datum and adjust predictable state estimation covariance matrix.
With this model (Zhou et al., 1993), (1 ) k λ + can be directly determined as follows: with 01 ρ ≤≤ is the preselected forgetting, it may be selected according to the real processes.For fast changing processes, a smaller ρ should be selected, and vice versa.As that pointed out in the paper (Zhou et al., 1993), (1 ) k λ + is insensitive to the value of ρ .

Strong tracking finite-difference extended Kalman filter
Deriving the ideas in papers (Fan et al., 2006;Zhou et al., 1997), the authors proposed a finite-difference method to replace partial derivatives of nonlinear functions.From further improving the self-covariance and between-covariance, we obtain the algorithm based on strong tracking filter-difference enhanced kalman filter.
The predicted covariance matrix, gain matrix and covariance estimate of suboptimal fading extended Kalman filter (SFEKF) are mended as follows From before-mentioned deduction, we can infer to that all calculations of above include the process noise impact and the error problem of model linearization.The step number which nonlinear function is linearized also changes with last time covariance matrix, process noise and observation noise.The filter becomes very simple because of replacing partial derivatives calculation using finite-difference value.The new strong finite-difference Kalman filter (STFDEKF) has more accuracy and covariance estimation, and improves the robustness of target tracking.The last experiment results show that the STFDEKF can be used for the high nonlinear stochastic systems such as eye tracking.

STFDEKF based eye tracking and results
In this section, we develop the following eye tracking using STFDEKF.Because the eye motion is the high nonlinearity of the likelihood model, it's very difficult to model human eye movement dynamics.In our tracking system, the following nonlinear equations are used to model the eye movement dynamics.
where, the initial value 0 x and 0 v are zero.The acceleration a follows the sine distribution, and a will be considered process noise

Conclusion
This paper proposes a new eye tracking method using strong finite-difference Kalman filter.Firstly, strong tracking factor is introduced to modify priori covariance matrix to improve the accuracy of the eye tracking algorithm.Secondly, the finite-difference method is proposed to replace partial derivatives of nonlinear functions to eye tracking.From above deduction, the new strong finite-difference Kalman filter becomes very simple because of replacing partial derivatives calculation using finite-difference value.The last experiment results show that STFDEKF has more accuracy and covariance estimation, improves the robustness of target tracking, and can be used for the high nonlinear stochastic systems such as eye tracking.
tracking experiment is developed in platform of OPEN CV.Our system uses a ViewQuest VQ680 video camera to capture human images.The experiment is tested on a Pentium III 1.7G CPU with 128MB RAM.Eye tracking based on the proposed method can reach 10 frames per second.The format of input video is 352×288.Fig.1represents the eye tracking using STFDEKF algorithm.The Correct Rate of eye tracking is shown in

Table 1 .
Result of eye tracking using STFDEKF algorithmCorrect Rate of eye tracking is defined as in equation (25).

Table 2 .
Comparison of eye tracking algorithms www.intechopen.comInordertoqualitatively gauge performance and discuss resulting issues, we consider using the traditional measures of performance: the RMSE (Root Mean Square Error) and MSE (Mean Square Error).The simulation results of RMSE and MSE are depicted in Table3.

Table 3 .
RMSE and MSE of eye tracking filtering algorithmsThe results of above experiments indicate that the proposed method has better performance.So we can use STFDEK algorithm for eyes tracking.