From 8d29605121c6fbd23bb24d572ee6941ef7758068 Mon Sep 17 00:00:00 2001 From: Toni Date: Tue, 19 Apr 2016 07:45:23 +0200 Subject: [PATCH 1/2] added section recursive state estimation, and 3/4 of related work --- tex/chapters/relatedwork.tex | 14 +++++--- tex/chapters/system.tex | 63 +++++++++++++----------------------- tex/egbib.bib | 41 ++++++++++++----------- 3 files changed, 55 insertions(+), 63 deletions(-) diff --git a/tex/chapters/relatedwork.tex b/tex/chapters/relatedwork.tex index 07a643c..15b3284 100644 --- a/tex/chapters/relatedwork.tex +++ b/tex/chapters/relatedwork.tex @@ -18,16 +18,20 @@ On the other hand, fixed-interval smoothing requires all observations until time The origin of MC smoothing can be traced back to Genshiro Kitagawa. In his work \cite{kitagawa1996monte} he presented the simplest form of smoothing as an extension to the particle filter. This algorithm is often called the filter-smoother since it runs online and a smoothing is provided while filtering. -This approach can produce an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid \vec{o}_{1:t})$ with computational complexity of only $\mathcal{O}(N)$. -\commentByFrank{kleines n?} -However, it gives a poor representation of previous states \cite{Doucet11:ATO}. -\commentByFrank{wenn noch platz, einen satz mehr dazu warum es schlecht ist?} +This approach uses the particle filter steps to update weighted paths $\{(\vec{q}_{1:t}^i , w^i_t)\}^N_{i=1}$, producing an accurate approximation of the filtering posterior $p(\vec{q}_{t} \mid \vec{o}_{1:t})$ with a computational complexity of only $\mathcal{O}(N)$. +However, it gives a poor representation of previous states due a monotonic decrease of distinct particles caused by resampling of each weighted path \cite{Doucet11:ATO}. Based on this, more advanced methods like the forward-backward smoother \cite{doucet2000} and backward simulation \cite{Godsill04:MCS} were developed. Both methods are running backwards in time to reweight a set of particles recursively by using future observations. Algorithmic details will be shown in section \ref{sec:smoothing}. %wo werden diese eingesetzt, paar beispiele. offline, online -In recent years, smoothing gets attention mainly in the field of computer vision and ... Here, ... +In recent years, smoothing gets attention mainly in other areas as indoor localisation. +The early work of \cite{isard1998smoothing} demonstrates the possibilities of smoothing for visual tracking. +They used a combination of the CONDENSATION particle filter with a forward-backward smoother. +Based on this pioneering approach, many different solutions for visual and multi-target tracking have been developed \cite{Perez2004}. +For example, in \cite{Platzer:2008} a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. Or \cite{} + + Nevertheless, their are some promising approach for indoor localisation systems as well. For example ... diff --git a/tex/chapters/system.tex b/tex/chapters/system.tex index 9d0b113..865d2e4 100644 --- a/tex/chapters/system.tex +++ b/tex/chapters/system.tex @@ -1,11 +1,8 @@ \section{Recursive State Estimation} - \commentByFrank{schon mal kopiert, dass es da ist.} - \commentByFrank{die neue activity in die observation eingebaut} - \commentByFrank{magst du hier auch gleich smoothing ansprechen? denke es würde sinn machen weils ja zum kompletten systemablauf gehört und den hatten wir hier ja immer drin. oder was meinst du?} - - We consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem. - Using a recursive Bayes filter that satisfies the Markov property, the posterior distribution at time $t$ can be written as +As mentioned before, most smoothing methods require a preceding filtering. +In our previous work \cite{Ebner-16}, we consider indoor localisation as a time-sequential, non-linear and non-Gaussian state estimation problem. +Therefore, a Bayes filter that satisfies the Markov property is used to calculate the posterior, which is given by % \begin{equation} \arraycolsep=1.2pt @@ -13,40 +10,33 @@ &p(\mStateVec_{t} \mid \mObsVec_{1:t}) \propto\\ &\underbrace{p(\mObsVec_{t} \mid \mStateVec_{t})}_{\text{evaluation}} \int \underbrace{p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})}_{\text{transition}} - \underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}} \enspace, + \underbrace{p(\mStateVec_{t-1} \mid \mObsVec_{1:t-1})d\vec{q}_{t-1}}_{\text{recursion}} \enspace. \end{array} \label{equ:bayesInt} \end{equation} % - where $\mObsVec_{1:t} = \mObsVec_{1}, \mObsVec_{1}, ..., \mObsVec_{t}$ is a series of observations up to time $t$. - The hidden state $\mStateVec$ is given by +Here, the previous observation $\mObsVec_{t-1}$ is included into the state transition \cite{Koeping14-PSA}. +For approximating eq. \eqref{equ:bayesInt} by means of MC methods, the transition is used as proposal distribution, also known as CONDENSATION algorithm \cite{isard1998smoothing}. + +In context of indoor localisation, the hidden state $\mStateVec$ is defined as follows: \begin{equation} \mStateVec = (x, y, z, \mStateHeading, \mStatePressure),\enskip x, y, z, \mStateHeading, \mStatePressure \in \R \enspace, \end{equation} % - where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the - relative atmospheric pressure prediction in hectopascal (hPa). - \commentByFrank{hier einfach kuerzen und aufs fusion paper verweisen? auch wenn das noch ned durch ist?} - The recursive part of the density estimation contains all information up to time $t-1$. - Furthermore, the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ models the pedestrian's movement as described in section \ref{sec:trans}. - %It should be noted, that we also include the current observation $\mObsVec_{t}$ in it. - As proven in \cite{Koeping14-PSA}, we may include the observation $\mObsVec_{t-1}$ into the state transition. - - Containing all relevant sensor measurements to evaluate the current state, the observation vector is defined as follows: +where $x, y, z$ represent the position in 3D space, $\mStateHeading$ the user's heading and $\mStatePressure$ the relative atmospheric pressure prediction in hectopascal (hPa). Further, the observation is given by % \begin{equation} - \mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsSteps, \mObsPressure, \mObsActivity) \enspace, + \mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsSteps, \mObsPressure, x) \enspace, \end{equation} % - where $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{}) - and \docIBeacon{}s, respectively. $\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number - of steps detected for the pedestrian. $\mObsPressure$ is the relative barometric pressure with respect to a fixed reference. - Finally, $\mObsActivity$ contains the activity, currently estimated for the pedestrian, which is one of: unknown, standing, walking or - walking stairs. - %For further information on how to incorporate such highly different sensor types, - %one should refer to the process of probabilistic sensor fusion \cite{Khaleghi2013}. - By assuming statistical independence of all sensors, the probability density of the state evaluation is given by +covering all relevant sensor measurements. +Here, $\mRssiVec_\text{wifi}$ and $\mRssiVec_\text{ib}$ contain the measurements of all nearby \docAP{}s (\docAPshort{}) and \docIBeacon{}s, respectively. +$\mObsHeading$ and $\mObsSteps$ describe the relative angular change and the number of steps detected for the pedestrian. +$\mObsPressure$ is the relative barometric pressure with respect to a fixed reference. +Finally, $x$ contains the activity, currently estimated for the pedestrian, which is one of: unknown, standing, walking or walking stairs. + +The probability density of the state evaluation is given by % \begin{equation} %\begin{split} @@ -54,23 +44,16 @@ p(\vec{o}_t \mid \vec{q}_t)_\text{baro} \,p(\vec{o}_t \mid \vec{q}_t)_\text{ib} \,p(\vec{o}_t \mid \vec{q}_t)_\text{wifi} - \enspace. + \enspace %\end{split} \label{eq:evalBayes} \end{equation} % - Here, every single component refers to a probabilistic sensor model. - The barometer information is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$, - whereby absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{ib}$ for - \docIBeacon{}s and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}. - - %It is well known that finding analytic solutions for densities is very difficult and only possible in rare cases. - %Therefore, numerical solutions like Gaussian filters or the broad class of Monte Carlo methods are deployed \cite{sarkka2013bayesian}. - Since we assume indoor localisation to be a time-sequential, non-linear and non-Gaussian process, - a particle filter is chosen as approximation of the posterior distribution. - \commentByFrank{smoothing?} - %Within this work the state transition $p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$ is used as proposal distribution, - %also known as CONDENSATION algorithm \cite{Isard98:CCD}. +and therefore similar to \cite{Ebner-16}. +Here, we assume a statistical independence of all sensors and every single component refers to a probabilistic sensor model. +The barometer information is evaluated using $p(\vec{o}_t \mid \vec{q}_t)_\text{baro}$, whereby absolute position information is given by $p(\vec{o}_t \mid \vec{q}_t)_\text{ib}$ for \docIBeacon{}s and by $p(\vec{o}_t \mid \vec{q}_t)_\text{wifi}$ for \docWIFI{}. + + diff --git a/tex/egbib.bib b/tex/egbib.bib index aaaf1bd..7f9f2eb 100644 --- a/tex/egbib.bib +++ b/tex/egbib.bib @@ -975,16 +975,6 @@ ISSN={0162-8828},} year={2006}, } -@misc{VAP2, - title={IEEE P802.11 Wireless LANs - Virtual Access Points}, - author={Aboba, Bernard}, - year={2003}, - REMurl={http://aboba.drizzlehosting.com/IEEE/11-03-154r1-I-Virtual-Access-Points.doc}, - note={\url{http://aboba.drizzlehosting.com/IEEE/11-03-154r1-I-Virtual-Access-Points.doc}}, - note={zuletzt abgerufen am 28.11.2013}, - pages={13}, -} - % reference points to reset errors, automatic floorplan generation, backend-phase @inproceedings{crowdinside, author = {Alzantot, Moustafa and Youssef, Moustafa}, @@ -2082,16 +2072,15 @@ year = {2014} @incollection{Platzer:2008, year={2008}, isbn={978-3-540-78639-9}, - booktitle={Bildverarbeitung für die Medizin 2008}, + booktitle={Bildverarbeitung f\"ur die Medizin 2008}, series={Informatik Aktuell}, - editor={Tolxdorff, Thomas and Braun, Jürgen and Deserno, ThomasM. and Horsch, Alexander and Handels, Heinz and Meinzer, Hans-Peter}, + editor={Tolxdorff, Thomas and Braun, J\"urgen and Deserno, Thomas M. and Horsch, Alexander and Handels, Heinz and Meinzer, Hans-Peter}, doi={10.1007/978-3-540-78640-5_58}, - title={3D Blood Flow Reconstruction from 2D Angiograms}, - url={http://dx.doi.org/10.1007/978-3-540-78640-5_58}, + title={{3D Blood Flow Reconstruction from 2D Angiograms}}, publisher={Springer Berlin Heidelberg}, author={Platzer, Esther-S. and Deinzer, Frank and Paulus, Dietrich and Denzler, Joachim}, - pages={288-292}, - language={English} + pages={288--292}, + language={English}, } @article{haugh2004monte, @@ -2273,12 +2262,12 @@ IGNOREmonth={Apr}, } @incollection{isard1998smoothing, - title={A Smoothing Filter for Condensation}, + title={{A Smoothing Filter for Condensation}}, author={Isard, Michael and Blake, Andrew}, booktitle={Computer Vision—ECCV'98}, pages={767--781}, year={1998}, - publisher={Springer} + publisher={Springer}, } @inproceedings{klaas2006fast, @@ -2717,5 +2706,21 @@ volume = {13}, year = {1967} } +@article{Perez2004, +author = {P{\'{e}}rez, Patrick and Vermaak, Jaco and Blake, Andrew}, +doi = {10.1109/JPROC.2003.823147}, +isbn = {0018-9219}, +issn = {00189219}, +journal = {Proceedings of the IEEE}, +keywords = {Color,Data fusion,Motion,Particle filters,Sound,Visual tracking}, +IGNOREmonth = {mar}, +number = {3}, +pages = {495--513}, +shorttitle = {Proceedings of the IEEE}, +title = {{Data Fusion for Visual Tracking with Particles}}, +url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1271403}, +volume = {92}, +year = {2004} +} From 033466111a76075ce16a094075e43e9675c5ecf0 Mon Sep 17 00:00:00 2001 From: toni Date: Tue, 19 Apr 2016 18:28:30 +0200 Subject: [PATCH 2/2] first draft related work --- tex/bare_conf.dvi | Bin 12772 -> 38596 bytes tex/chapters/experiments.txt | 111 +++++++++++++++++++++++++++++++++++ tex/chapters/relatedwork.tex | 31 ++++++++-- tex/chapters/smoothing.tex | 8 +++ tex/egbib.bib | 33 ++++++++++- 5 files changed, 176 insertions(+), 7 deletions(-) create mode 100644 tex/chapters/experiments.txt diff --git a/tex/bare_conf.dvi b/tex/bare_conf.dvi index 0133dd42d90eb0b14f81139ae23b7cb8e10d4b90..0c34719071b7ec6232802ca40020e9440e59fbf3 100644 GIT binary patch literal 38596 zcmdUY3wRaPwSPD>C%nR|ARv`+Im~-t4i&)w$^9bTA%!XYwf+~Odb?$zx#dn z(yuCWX7=p8_Im!-+H2ko4ZGro2c|8^!yoO{lc(33Uz(oCq+2rS>E)$mvr9_PDJh#b zzie*#oUh@IVRYNE{v5%dHz&0iEoL%hZ8jIqEtzG-i^|HZ_^!}*FW=U&>!Z&;b5}=) 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zL!Z7C&W_-79Rz|cH=7|zF&!5GO8Q-zYSZtulLiuDc0`zlpv}S<%pGYUJQ~g{-rQ;i z+ZaOJ%bDmqhn%P1U9r5Ok%-Z&6^)1Mzt*+t%O~wXOM3f`&YWE|lVXQ=z4`m8O8o7` zV({>7HG%kx!?%59;>SJBS{~OJa4a(R&xhBQyYBh2u_nOo<(FHmVuraD<>i+s8cnGH zk@|*(5fp+o8SKPl@N$3W&<`&aUCV}44=}zxfG9%HaP*R(l}8tH)d2>=z4VW even better + path1_shortest + cnt(171) min(64.679) max(2185.99) range(2121.31) med(639.352) avg(746.264) stdDev(462.957) // similar: much better + + path2_simple + cnt(193) min(31.4961) max(1596.12) range(1564.62) med(671.196) avg(714.763) stdDev(447.28) + path2_multi + cnt(193) min(2.47806) max(1447.66) range(1445.18) med(440.189) avg(500.696) stdDev(381.872) + path2_shortest + cnt(193) min(13.2449) max(1453.03) range(1439.78) med(449.287) avg(532.14) stdDev(402.977) + + path3_simple + cnt(158) min(36.0375) max(1863.92) range(1827.88) med(615.248) avg(678.632) stdDev(428.826) // short, stairs only, bad baromter, nothing to optimize + path3_multi + cnt(158) min(26.7108) max(1610.37) range(1583.66) med(562.175) avg(582.494) stdDev(382.947) // only slightly better + path3_shortest + cnt(158) min(39.0438) max(1778.07) range(1739.03) med(587.44) avg(648.729) stdDev(374.347) // only slightly better + + path4_simple + cnt(293) min(32.1262) max(1875.25) range(1843.13) med(889.001) avg(836.58) stdDev(484.112) + path4_multi + cnt(293) min(17.1514) max(1686.81) range(1669.66) med(620.385) avg(702.897) stdDev(446.311) + path4_shortest + cnt(293) min(17.6632) max(1740.36) range(1722.7) med(732.178) avg(778.587) stdDev(438.974) // die treppe am ende ist problematisch diff --git a/tex/chapters/relatedwork.tex b/tex/chapters/relatedwork.tex index 15b3284..18e7ce9 100644 --- a/tex/chapters/relatedwork.tex +++ b/tex/chapters/relatedwork.tex @@ -29,14 +29,33 @@ In recent years, smoothing gets attention mainly in other areas as indoor locali The early work of \cite{isard1998smoothing} demonstrates the possibilities of smoothing for visual tracking. They used a combination of the CONDENSATION particle filter with a forward-backward smoother. Based on this pioneering approach, many different solutions for visual and multi-target tracking have been developed \cite{Perez2004}. -For example, in \cite{Platzer:2008} a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. Or \cite{} - - - -Nevertheless, their are some promising approach for indoor localisation systems as well. For example ... +For example, in \cite{Platzer:2008} a particle smoother is used to reduce multimodalities in a blood flow simulation for human vessels. Or \cite{Hu2014} uses a smoother to overcoming the problem of particle impoverishment while predicting the Remaining Useful Life (RUL) of equipment (e.g. a Lithium-ion battery). %smoothing im bezug auf indoor -Smoothing solutions in indoor localisation werden bisher nicht wirklich behandelt. das liegt hauptsächlich daran das es sehr langsam ist \cite{}. es gibt ansätze von ... und ... diese benutzen blah und blah. wir machen das genauso/besser. +Nevertheless, their are some promising approaches for indoor localisation systems as well. +For example \cite{Nurminen2014} deployed a fixed-interval forward-backward smoother to improve the position estimation for non-real-time applications. +They combined Wi-Fi, step and turn detection, a simple line-of-sight model for floor plan restrictions and the barometric change within a particle filter. +The state transition samples a new state based on the heading change, altitude change and a fixed step length. +The experiments of \cite{Nurminen2014} clearly emphasize the benefits of smoothing techniques. The estimation error could be decreased significantly. +However, a fixed-lag smoother was treated only in theory. + +In the work of \cite{Paul2009} both fixed-interval and fixed-lag smoothing were presented. +They implemented Wi-Fi, binary infra-red motion sensors, binary foot-switches and a potential field for floor plan restrictions. +Those sensors were incorporated using a sigma-point Kalman filter in combination with a forward-backward smoother. +It was also proven by \cite{Paul2009}, that the fixed-lag smoother is slightly less accurate then the fixed-interval smoother. +As one would expect from the theoretical foundation. +Unfortunately, even a sigma-point Kalman filters is after all just a linearisation and therefore not as flexible and suited for the complex problem of indoor localisation as a non-linear estimator like a particle filter. +\commentByToni{Kann das jemand nochmal verifizieren? Das mit dem Kalman Filter. Danke.} +Additionally, the Wi-Fi RSSI model requires known calibration points and is deployed using a remarkable number of access points for very small spaces. +In our opinion this is not practical and we would further recommend adding a PDR-based transition instead of a random one. + +In contrast, the here presented approach is able to use two different smoothing algorithm, both implemented as fixed-interval and fixed-lag versions. +Further, our localisation system presented in \cite{Ebner-16} enables us to walk stairs and therefore going into the third dimension. +Therefore, a regularly tessellated graph is utilized to avoid walls, detecting doors and recognizing stairs. +Within this work, this is additionally supported by a simple classification that detects the activities unknown, standing, walking and walking stairs. +Finally, we incorporate prior navigation knowledge by using syntactically calculated realistic human walking paths \cite{Ebner-16}. +This method makes use of the given destination and thereby provides a more targeted movement. + diff --git a/tex/chapters/smoothing.tex b/tex/chapters/smoothing.tex index 4b81998..c97668d 100644 --- a/tex/chapters/smoothing.tex +++ b/tex/chapters/smoothing.tex @@ -2,3 +2,11 @@ \label{sec:smoothing} Consider a situation given all observations until a time step T... + + + + + +%komplexität eingehen +The reason for not behandeln liegt ... +However, \cite{} and \cite{} have proven this wrong and reduced the complexity of different smoothing methods. diff --git a/tex/egbib.bib b/tex/egbib.bib index 7f9f2eb..80bbf64 100644 --- a/tex/egbib.bib +++ b/tex/egbib.bib @@ -2718,9 +2718,40 @@ number = {3}, pages = {495--513}, shorttitle = {Proceedings of the IEEE}, title = {{Data Fusion for Visual Tracking with Particles}}, -url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1271403}, volume = {92}, year = {2004} } +@article{Hu2014, +abstract = {This work addresses the problem of predicting the Remaining Useful Life (RUL) of components for which a mathematical model describing the component degradation is available, but the values of the model parameters are not known and the observations of degradation trajectories in similar components are unavailable. The proposed approach solves this problem by using a Particle Filtering (PF) technique combined with a kernel smoothing (KS) method. This PF-KS method can simultaneously estimate the degradation state and the unknown parameters in the degradation model, while significantly overcoming the problem of particle impoverishment. Based on the updated degradation model (where the unknown parameters are replaced by the estimated ones), the RUL prediction is then performed by simulating future particles evolutions. A numerical application regarding prognostics for Lithium-ion batteries is considered. Various performance indicators measuring precision, accuracy, steadiness and risk of the obtained RUL predictions are computed. The obtained results show that the proposed PF-KS method can provide more satisfactory results than the traditional PF methods.}, +author = {Hu, Yang and Baraldi, Piero and {Di Maio}, Francesco and Zio, Enrico}, +doi = {10.1016/j.ress.2014.10.003}, +file = {:home/toni/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Hu et al. - 2015 - A particle filtering and kernel smoothing-based approach for new design component prognostics.pdf:pdf}, +issn = {09518320}, +journal = {Reliability Engineering and System Safety}, +keywords = {Battery,Kernel smoothing,Parameter estimation,Particle filtering,Prognostics,Remaining useful life}, +IGNOREmonth = {feb}, +pages = {19--31}, +title = {{A Particle filtering and Kernel Smoothing-Based Approach for New Design Component Prognostics}}, +volume = {134}, +year = {2014} +} + +@article{Paul2009, +abstract = {Solutions for indoor tracking and localization have become more critical with recent advancement in context and location-aware technologies. The accuracy of explicit positioning sensors such as global positioning system (GPS) is often limited for indoor environments. In this paper, we evaluate the feasibility of building an indoor location tracking system that is cost effective for large scale deployments, can operate over existing Wi-Fi networks, and can provide flexibility to accommodate new sensor observations as they become available. This paper proposes a sigma-point Kalman smoother (SPKS)-based location and tracking algorithm as a superior alternative for indoor positioning. The proposed SPKS fuses a dynamic model of human walking with a number of low-cost sensor observations to track 2-D position and velocity. Available sensors include Wi-Fi received signal strength indication (RSSI), binary infra-red (IR) motion sensors, and binary foot-switches. Wi-Fi signal strength is measured using a receiver tag developed by Ekahau, Inc. The performance of the proposed algorithm is compared with a commercially available positioning engine, also developed by Ekahau, Inc. The superior accuracy of our approach over a number of trials is demonstrated.}, +author = {Paul, Anindya S. and Wan, Eric A.}, +doi = {10.1109/JSTSP.2009.2032309}, +isbn = {1932-4553}, +issn = {19324553}, +journal = {IEEE Journal on Selected Topics in Signal Processing}, +keywords = {Bayesian inference,Indoor tracking,Received signal strength indication (RSSI)-based localization,Sigma-point Kalman filter,Sigma-point Kalman smoother,State estimation}, +IGNOREmonth = {oct}, +number = {5}, +pages = {860--873}, +shorttitle = {IEEE Journal of Selected Topics in Signal Processi}, +title = {{RSSI-Based Indoor Localization and Tracking using Sigma-Point Kalman Smoothers}}, +volume = {3}, +year = {2009} +} +