Nonparametric fixed-interval smoothing of nonlinear vector-valued measurements

This paper addresses the problem of estimating a smooth vector-valued function given noisy nonlinear vector-valued measurements of that function. We present a nonparametric optimality criterion for this estimation problem, and develop a computationally eficient iterative algorithm for its solution. The new criterion is the natural generalization of our earlier work on vector splines with linear measurement models. The new algorithm provides an alternative to the extended Kalman filter, as it does not require a parametric state-space model. We also present an automatic procedure that uses the measurements to determine how much to smooth. The algorithm demonstrates subpixel estimation accuracy on two examples: the estimation of a curved edge in a noisy image, and a biomedical image-processing application. I . INTRODUCTION HIS paper considers the problem of estimating a smooth T vector-valued function from noisy measurements observed through a nonlinear mapping. We assume the following nonlinear measurement model: y, = h,(x , ) + E,, II = 1, * * * , N where E,,, y, E &”, x, E @”” and h,: 6iM + BL”. We assume the additive measurement errors are independent between samples and are normally distributed with mean zero. Without loss of generality, we assume the covariance matrix of E, is a2Z, where u2 may be unknown.’ The states { x, } are (possibly unequally spaced) samples of a smooth vector-valued function g : where “ ’ ” denotes matrix transposition. The goal is to estimate g from the measurements { y, } : = I . The prevalent approach to this estimation problem is the extended Kalman filter (EKF) [l] . The EKF hinges on an assumption that the states adhere to a parametric Gauss-Markov state-space model. However. in applications such as the edgeestimation example given in Section VI, the parameters required by the EKF formulation (state evolution matrices and Manuscript received September 25, 1989; revised May 10, 1990. This work was supported in part by the National Institute of Health under Contract N01-HV-38045 and Grants R01-HL-39045, HL-39297, HL-34962, and HL-39478, by the National Science Foundation under Contract ECS8213959, and by GE Medical Systems Group under Contract 22-84. The author was with the Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305. He is now with the Division of Nuclear Medicine, University of Michigan Medical Center, Ann Arbor, MI 48109. IEEE Log Number 90423 1 1 . ‘If the measurement error has the (positive definite) covariance matrix U*&,, then we can premultiply y. and h, by E;’/*. Singular covariances may be the result of linearly dependent measurements, indicating that other constraints should be incorporated. covariances) are unknown and are difficult to determine. Furthermore, the state-space formulae imply the a priori variance of the function varies with t . Although it is natural for tracking applications, where one is given a starting state that evolves with increasing uncertainty over time, this variation is counterintuitive for off-line applications such as image processing, where t often represents space rather than time. For example, when detecting and estimating an edge in an image, the a priori variance of the position of the edge (the uncertainty before actually seeing the image) is the same throughout the image. Despite these objectives to parametric methods, we must use our a priori knowledge of the smoothness of the underlying functions if we are to obtain accurate estimates. This necessity has motivated nonparametric approaches to smoothing [2], [3], and is the basis for the new algorithm presented in this paper. In [4], we presented a computationally efficient algorithm for nonparametric smoothing for the special case when h, is linear, and we presented the rationale behind ‘‘penalized likehood” estimation. Here, just as in the linear case, we must compromise between the agreement with the data and the smoothness of the estimated functions. Thus we propose the following optimality criterion: