A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Partially Linear Functions <s> Addresses real-time implementation of the simultaneous localization and map-building (SLAM) algorithm. It presents optimal algorithms that consider the special form of the matrices and a new compressed filler that can significantly reduce the computation requirements when working in local areas or with high frequency external sensors. It is shown that by extending the standard Kalman filter models the information gained in a local area can be maintained with a cost /spl sim/O(N/sub a//sup 2/), where N/sub a/ is the number of landmarks in the local area, and then transferred to the overall map in only one iteration at full SLAM computational cost. Additional simplifications are also presented that are very close to optimal when an appropriate map representation is used. Finally the algorithms are validated with experimental results obtained with a standard vehicle running in a completely unstructured outdoor environment. <s> BIB001 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Partially Linear Functions <s> The Unscented Kalman Filter oflers sign$- cant improvements in the estimation of non-linear discrete- time models in comparison to the Extended Kalman Fil- ter 1121. In this paper we use a technique introduced by Casella and Robert (2), known as Rao-Blackwellisation, to calculate the tractable integrations that are found in the Unscented Kalman Filter: We show that this leads to a re- duction in the quasi-Monte Carlo variance, and a decrease in the computational complexity by considering a common tracking problem. <s> BIB002 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Partially Linear Functions <s> Introduction.- Literature Survey: Autonomous Compliant Motion.- Literature Survey: Bayesian Probability Theory.- Kalman Filters for Nonlinear Systems.- The Non-Minimal State Kalman Filter.- Contact Modelling.- Geometrical Parameter Estimation and CF Recognition.- Experiment: A Cube-In-Corner Assembly.- Task Planning with Active Sensing.- General Conclusions. <s> BIB003 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Partially Linear Functions <s> A method for performing unscented Kalman filtering with a reduced number of sigma points is proposed. The procedure is applicable when either the process or measurement equations are partially linear in the sense that only a subset of the elements of the state vector undergo a nonlinear transformation. It is shown that for such models second-order accuracy in the moments required for the unscented Kalman filter recursion can be obtained using a number of sigma points determined by the number of nonlinearly transformed elements rather than the dimension of the state vector. A procedure for computing the sigma points is developed. An application of the proposed method to smoothed target state estimation from bearings measurements is presented. <s> BIB004 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Partially Linear Functions <s> In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that in a stationary state it has a covariance function which corresponds to a process with similar smoothness properties as the first-order smoothness Tikhonov regularization. The proposed method is computationally more lightweight than the method where the regularization is augmented as a measurement. The method was tested in the case of the inverse problem of DOT. A solid phantom with optical properties similar to tissue was made, incorporating two moving parts that simulate two different physiological processes: a localized change in absorption and a surrounding rotating two-part shell which simulates slow oscillations in the tissue background physiology. A sequence of measurements of the phantom was made and the reconstruction of the image sequence was computed using this method. It allows the recovery of the full time series of images from relatively slow measurements with one source active at a time. In practice, this allows instruments with a larger dynamic range to be applied to the imaging of functional phenomena using DOT. <s> BIB005
In BIB002 , algorithms for treating different setups of UKF were presented. The algorithms considered different implementations when the state transition model or measurement model has one of the following three forms The algorithms improve the computation efficiency when either or both of the state and measurement models belong to the first or second group. If both are in the first group the updates can be computed exactly using the KF. If the function belongs to the second group then (10)-(12) become and the approximation can be computed for the dimension of the state instead of the dimension of the augmented state. This optimization is widely used among different filters and is often considered to be the standard model. In BIB004 the function is split into a nonlinear g 1 (·) part that depends only on a part of the state z n and linear parts The UKF is used for computing the expected values of the nonlinear part and the correlation between nonlinear and linear parts. We generalize the above models to functions of form: where rank T equals the number of rows in T . To reduce the required resources in computation T is chosen to have minimal number of rows and g(·) to have minimal number of elements. Expectations (10)- (12) are computed for T z, which should have a smaller dimension than z. For computing the update, an algorithm that approximates expectations (10)- (12) is required. These are usually computed in the update stage of a KFE. The cross correlation matrix P zg(z) BIB003 is not needed in the normal state propagation, but the algorithm from the update stage can be used for this. The transformed augmented state is denoted When expectations µ gn(z) (10), P gn(z)gn(z) BIB001 , and Pz gn(z) BIB005 are known, the expectations for (28) are where This is based on the fact that the cross term Pz gn(z) describes the linear dependency of function g n (z) andz as in (20): and the term P zgn(z) is Solving J from (34) and substituting it into (35) we get  . In the update, the matrix with cross terms is required for state variables only. This can be extracted by when the state variables appear first in the augmented state. The algorithm for state transition is given in Algorithm 1 and an algorithm for state update is given in Algorithm 2. Use of these algorithms is beneficial when the dimension ofz is smaller than z and the matrix inverse in (33) is applied in a small dimension. Algorithms 1 and 2 are given in a general form and for applications they can be optimized further by considering the structures of matrices. Examples of exploiting the partially linear state are given in sections 7.1 and 7.2. ALGORITHM 1: State transition using a partially linear measurement Input: xt−1 ∼ N(µx t−1 , Px t−1 x t−1 ) // State estimate from previous time step εq ∼ N(µε q , Pε q εq ), // State transition noise Px t−1 εq // cross covariance between state and state transition noise f (x, εq) = f (z) = Ag (T z) + Hz, // State transition function of given form Output: , and Pz g(z) using a KFE Update using a partially linear measurement  Pεyεy // Covariance matrix of the augmented state , and Pz g(z) using a KFE