From 9cb091d707d728eaa562d52213c6de236c0da18b Mon Sep 17 00:00:00 2001 From: toni Date: Fri, 29 Apr 2016 18:13:05 +0200 Subject: [PATCH] some minor changes and added smoothing transition section --- tex/bare_conf.dvi | Bin 57608 -> 64448 bytes tex/bare_conf.tex | 1 - tex/chapters/abstract.tex | 2 +- tex/chapters/conclusion.tex | 2 +- tex/chapters/filtering.tex | 2 +- tex/chapters/introduction.tex | 2 +- tex/chapters/relatedwork.tex | 2 ++ tex/chapters/smoothing.tex | 65 +++++++++++++++++++++++++++++++++- tex/misc/keywords.tex | 2 ++ 9 files changed, 72 insertions(+), 6 deletions(-) diff --git a/tex/bare_conf.dvi b/tex/bare_conf.dvi index 300aedfd720eaa4266be3927cf538b993d3891bd..026dca0b152e8a9d74ced4029db28a2eb78afd08 100644 GIT binary patch delta 9551 zcmb7K3wTu3wVpZWa>&e_aL!CZ=(Q68 zL8~+5u~)di_4W zGdX*owbx#It$+P%Z67}>9=;|v49&h{Lh<YljxQk|)!jQgG zyk5vtjTlCihWBe~u#oHx%9y}3e~fDp)L9oz(+ztCYssi|Y&2GlOUH_tnbl{YbS%86 zQemXOVg$VbgBg_?gw@6Zm2!YFIU_$b;4Lj%(c|KPw`JzxGcgj-G_SRe&Qzae0DMp^YQ?3>UT%>8RG-3(U^VO;u!WXG*yDTj&De`O=;E4S63ngWdJL8@#mD!`emB=Qrfu%fmT9q? zn4&`+rWg>+k0EKdo;a(h_^?CH=4b1c%e@Lqe0@{tZK15!WP{Zz_zX|k6tfP(F)lV) zj)6fqhUFC+j70v>$r)&Ybce%oiD=rAuysW7v?Wi$S6C+tLXZoa(v>1Jtt_36%qOeU z{4tLvdzs457@`7NRJuIg(7ZL9FE%MmF6aA-OP5Q1a2~i8+yj!Z0Uz{*B?4M3@$I-b z#rr2mKAX2V5(zhc^K5n4>=JrnpBiaQzij;uL?Y(R(v`L%Q=Hs5+aDDtM_d}5vEgFV zssPg}ShcJNRF_x5_{JP>K+(ze2P)WecikE4=MC>Ip0JN(WWB!lfG6B|ec6rhzIhNo z@ke2G@1$akbW6P7*1iilD&@kes;3g}9G7x`W+=2u%#eN6Ud1S6ve(cU;>D}DW67nB zB2&se0K@1C0uXz*ts6>Ch>CZYsYYD9d#TKHrCR6l#1A@#9l$iPYRxcII$SIGVnzV* z6b`e`=B*^O*-09lb zR>*2;qpUR$6;5Rz=*Ln`z=D5azDDS=AOjU&f5ek}mU@Fpe0 zWrUsUeR5U9?>$Of*tsESC}eDQVq%}{m`n3j2RVANZu@((hqsQH*v3N`&bA2n6BX+v z+o5nEepU|S!TaDC@HBE~Y@;Kf6RUnO@Fv!>sg!wV4tH*3AG|u9?e?XyU(A@D)tuvS za3o}ZoH0Ay0!4OZ<}~LPHf+`~vCGMi6Z2<1Axe#1PJWbl@80=0iGzpbu~83<6ARJ~ zCn_Hp;$&OL0Mj0vDKzIWVR5l|%OkxK;})BOb4%jwAKm4Yw&&b8f^{vcmYRDGJj>RW z?T6?)m&y{Y%$k-SwBDC3dlv7nFI&yuCu*0M2+kMSPI*2bPTY|5oER@~-D8a_t}H?9 z_3%};QW=z)_rf(E(d^Gd@)CQLmz`q&r#@s$J-22x-t1gz4aryX*fvj3sj=I{8g{^Q zFSN_3)M<@JRy@mXGsQRD`Ek2#23D;}vCWHB%WvA={n0MruqnuuY@^nf4fkH?aj|q? z+wid?B9TaA_x=4WGw$=Pfua-qCn36v_D8Y1{t3ITjS%`wNDQmi1hMzSpR#B4ono(D zU$fc99x-olR^qym=M)PDpG))(YND9^mvsH`L_cs%NeJov|AMBeAZ{mYD9v>nIJ(eXt`)P)d-hE5s;KH)R zS5KXEX1sjdvgqVncd{REjAm?H(_#Ek4a=P{<}$%sNtTq%zg}%-KR2hOnDPU2iipI^ z25*|rpEa9)khMMMrQKmZ_p@gnz}nL{=Ly1v$Jn5neu>r3KH(hPZ^VwUd6%ht_`vMh zKw^huA+lA&t~{QwB8@!?)+E07+zZYHIWO(vasZan&@n*sxHNQZh355Y4Tl}I6q2{-B{z4w;o!4T!hGW8O;ma1SAHq<YK?*Y@6PMR{h+BTgZ6RboS8e45nQBjX!(_`s;p3t_XH!_hXyh)XS_4Mp*rQcE#% zuEhWfW98s6QF@g;TTmDPuughg=?#w1k-%)IFP9C)9hFK?X-FH4=O7l)WuzLOS)|g_ zKFzIoi_Ae%>5cpSUX>#^c}mJOWO-#I+q-EzYkhUN^J)@Jd=`g{3NMaQ{c+(%Xze1Y z?19;ZObviY#3((qCqmGJ6+t7w+`0@%uO)qy0Qse#?;+9-+IDI#GKDpPS;$O zSzARg0QwUbvo3VX%SfPS*E9+%-`E$}ds}hD%wM|?2^3kGTLdZE&FvPlqlvkKx(+U6 zuVITa#f95=mJgd*1Lj!LB4*CEt(5T>?k38!aGMJULu5zwyFsfv(KAeH4jL5JNw&d~ zG^o&8Db%b6qT<0~Vju@g{EAC43|X%uTad*yKUivqQ8*-|7=@JgSr+qqlIGE(;>54z z_7#VjH;nO+KJhCYAKx=eu|p_n2+Hj$Xuj-+3IS9o018&#aQsv-X%k{sQSo`7RuLDU z_j2Y!h|<0gycqPK*mrAy?DKrhA{=2ymhW&@TT}Aa5Gth;S4oaV&%oYXK`;0f+yS1O zvR6|)aRu8gSIC^Gk?a_NEWi@t-#({5y)w7s)Egch+NR==Mn8h!y4d7t_**GBP+T1S zlYmB{&~UiBqej(&hQ;YP_l>b5^}Ry1pgpp;3TxTQmMfJqtUq4ogN{&AlZhZMJhg{QXy9gIn9ml|U_UJ}N9e6(G;i zEW3iuo5JFgwr6GzS=Hg0W}jQOv>{J-m_BnzUD8RiUpk2Z5+3R|+fPwIGN>3tT!Ks+ zN+E*=p>|8f>f92kV2n%-BfQ`5zD~|6hcSv29Ul$^6gcVAw+pE zE&T`Bc04V8k`eUzWL;g&p&s!-m4Bf9$YP4ygIZVddjgfH;nLEZW@%4Teq=ng>mdRt zTQFFb5tY`}>&nWY3PEvc-J=0r!BM*r+wB@sLgV;^LNJC*1wins9$wk0zLCjD(e1M* zN5R5k=D9^lbc%|Z`)Mr-z!1T6g+;0`=|64KW`{EHi6+`Ixl8C(F!xF~ghhpw12~pI z12h+?yjnSsBEsJ8cbq=A^g2%1Ivi@^;BZ*A!rrXU4W-Dpz$z+joeVotdA#*5qJgb_ zt-88l$;G`TD6MPK!u!Z`9FBuL22sG?eDT{hw}S21m`m?tmf@}30gE4Y;G`%7l5}z^ zon=VaF`JKhgD|?V;~t_hu1XDTBIZ`AVSd7DTV>fkT;%dt>nhfU0M$Hx9KYM%g^WZv z$x8=hij%6W95U&V8R%Fbt17Woyvpxnn^45;D$TO+)bLT=ucH7~QK_TnGO4OsL3b!X zdfS~4L`d!CB(6uL^y9&51h%T9bpWG+rlr5lX-#d+vL>Qfm#N(VuuuOJfj!*RaxtJp zWJ`$Sjn(eJW}{;6c*u#1xuyRs>#=gAb!0@FY1nl?nU<(lO5y*9{3K%Lcr~l6hUN>= zC_>l46h8M_jfd=4LTlg{QM{;B-8ez&+{A^?edHi>Wu5pR=LSThkSf5TumNf%8aax) zFosaVc_odH%Aaxrz|Rfz(uyKru_%rf5!tGjDixWZrRqoQSZy!}8-P5VY+k|fIn#+W zTWKNp^H)>T!u|hNS^&QIDJ}f{yJ;c&J81y`*u7z9e|AXg6XG8syS(vH%LppHS|_HY z244JHl@f1_M8%(`__$&$u2PdwotSk!SlWW6^>0GR1acc^| zyTAbeic9%7RI7i8yn}-P3vMZHozOBG){zQ^!8#UAv{H@iQ)03aLLRB+#fqH7TcUjR z5-rAt7Z;C+n0YUKRf$e4oEwcOpX>&V+3s!LxV;ocYyN``<-7-MX8?Ll`$0yb4tODY zz{%z`5zyPGnNst!O305)RHN#eKbb;!<|vFx9+dydbOl-llMMG`R?^Kzn3AV=dxFNf zt~RB2cO|)movCdj89^6S23G43S%db+YZQYGg!hsQP^+1D5@NN+1R`fs0T#X3%+x6C z|1YDE-K!1kLu!bnyE~}YzSR*^ikrl{Limk8F3hn5Z9+<*jk7t}-5e7ME2CoUtPGt! 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In contrast to normal filtering procedures like particle filtering, smoothing methods are able to incorporate future measurements instead of just using current and past data. This enables many possibilities for further improving the position estimation. -Both smoothing techniques are deployed as fixed-lag and fixed-interval smoother and two novel approaches for incorporating them easily within our localisation system are presented. +Both smoothing techniques are deployed as fixed-lag and fixed-interval smoother and a novel approach for incorporating them easily within our localisation system is presented. All this is evaluated on four floors within our faculty building. The results show that smoothing methods offer a great tool for improving the localisation results. Especially fixed-lag smoothing provides a great runtime support by reducing timely errors and improving the overall estimation with affordable costs. diff --git a/tex/chapters/conclusion.tex b/tex/chapters/conclusion.tex index 7bae3ee..159ecc8 100644 --- a/tex/chapters/conclusion.tex +++ b/tex/chapters/conclusion.tex @@ -1,3 +1,3 @@ \section{Conclusion} - +map information into smoothing. better way and faster then just dijkstra. compensate big jumps caused by wifi. better method for estimation and drawing of particles in backward simulation. more advanced smoothing transition. not used evaluating using the observations, but using the given information for more advanced approaches. diff --git a/tex/chapters/filtering.tex b/tex/chapters/filtering.tex index 5b5ddf3..1463605 100644 --- a/tex/chapters/filtering.tex +++ b/tex/chapters/filtering.tex @@ -75,7 +75,7 @@ \section{Transition} - +\label{sec:transition} The distribution $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is sampled via random walks on a graph $G=(V,E)$, which is generated from the buildings floorplan \todo{FUSION2016}. $p(\mStateVec_{t} \mid \mStateVec_{t-1})$ is determined by walking along adjacent edges $\mEdgeAB$ connecting diff --git a/tex/chapters/introduction.tex b/tex/chapters/introduction.tex index 973cead..b9e3f3e 100644 --- a/tex/chapters/introduction.tex +++ b/tex/chapters/introduction.tex @@ -80,7 +80,7 @@ Since the problem of navigation, especially the representation of complex moveme Namely, forward-backward smoothing \cite{doucet2000} and backward simulation \cite{Godsill04:MCS}. Within this work, we investigate the benefits and drawbacks of those techniques using our indoor localisation system presented in \cite{Ebner-16}. -We provide both, fixed-lag and fixed-interval smoothing as well as two novel approaches for incorporating them easily within the localisation procedure. +We provide both, fixed-lag and fixed-interval smoothing as well as a novel approach for incorporating them easily within the localisation procedure. The main goal is to solve above mentioned problems and to investigate new possibilities for even more advanced systems. diff --git a/tex/chapters/relatedwork.tex b/tex/chapters/relatedwork.tex index 451d807..2f3016a 100644 --- a/tex/chapters/relatedwork.tex +++ b/tex/chapters/relatedwork.tex @@ -6,6 +6,8 @@ Sequential MC filter, like aforementioned particle filter, use all observations $\mObsVec_{1:t}$ until the current time $t$ for computing an estimation of the state $\mStateVec_t$. In a Bayesian setting, this can be formalized as the computation of the posterior distribution $p(\mStateVec_t \mid \mObsVec_{1:t})$ using a sample of $N$ independent random variables, $\vec{X}^i_{t} \sim (\mStateVec_t \mid \mObsVec_{1:t})$ for $i = 1,...,N$ for approximation. Due to importance sampling, a weight $W^i_t$ is assigned to each sample $\vec{X}^i_{t}$. +In context of particle filtering $\{\vec{X}^i_{1:t}, W^i_{1:t} \}_{i=1}^N$ is a weighted set of samples, also called particles. +Therefore a particle is a representation of one possible system state $\mStateVec$. By considering a situation given all observations $\vec{o}_{1:T}$ until a time step $T$, where $t \ll T$, standard filtering methods are not able to make use of this additional data for computing $p(\mStateVec_t \mid \mObsVec_{1:T})$. This problem can be solved with a smoothing algorithm. diff --git a/tex/chapters/smoothing.tex b/tex/chapters/smoothing.tex index aac81d2..52e77a9 100644 --- a/tex/chapters/smoothing.tex +++ b/tex/chapters/smoothing.tex @@ -6,7 +6,7 @@ As mentioned before, those algorithm are able to compute probability distributio %Especially fixed-lag smoothing is very promising in context of pedestrian localisation. In the following we discuss the algorithmic details of the forward-backward smoother and the backward simulation. -Further, two novel approaches for incorporating them into the localisation system are shown. +Further, a novel approaches for incorporating them into the localisation system is shown. \subsection{Forward-backward Smoother} @@ -80,4 +80,67 @@ Therefore, \cite{Godsill04:MCS} presented the backward simulation (BS). Where a This method can be seen in algorithm \ref{alg:backwardSimulation} in pseudo-algorithmic form. Again, a particle filter is performed at first and then the smoothing procedure gets applied. Here, $\tilde{\vec{q}}_t$ is a random sample drawn approximately from $p(\vec{q}_{t} \mid \tilde{\vec{q}}_{t+1}, \vec{o}_{1:T})$. Therefore $\tilde{\vec{q}}_{1:T} = (\tilde{\vec{q}}_{1}, \tilde{\vec{q}}_{2}, ...,\tilde{\vec{q}}_{T})$ is one particular sample realization from $p(\vec{q}_{1:T} \mid \vec{o}_{1:T})$. Further independent realizations are obtained by repeating the algorithm until the desired number $N_{\text{sample}}$ is reached. The computational complexity for one particular realization is $\mathcal{O}(N)$. However, the computations are then repeated for each realization drawn \cite{Godsill04:MCS}. \subsection{Transition for Smoothing} +As seen above, both algorithms are reweighting particles based on a state transition model. +Unlike the transition presented in section \ref{sec:transition}, it is not possible to just draw a set of new samples. +Here, $p(\vec{q}_{t+1} \mid \vec{q}_{t})$ needs to provide the probability of the \textit{known} future state $\vec{q}_{t+1}$ under the condition of the current state $\vec{q}_{t}$. +In case of indoor localisation using particle filtering, it is necessary to not only provide the probability of moving to a particle's position under the condition of its ancestor, but also of all other particles at time $t$. +The smoothing transition model therefore calculates the probability of being in a state $\vec{q}_{t+1}$ in regard to previous states and the pedestrian's walking behaviour. +This means that a state $\vec{q}_t$ gets rewarded with a high probability, if it is a proper ancestor (realistic previous position) of a future state $\vec{q}_{t+1}$. +%observations von barometer und turn sind ziemlich genau. +%of course, instead of the line of side one could choose to calculate the the shortest path. however, this requires a trombendes calculation time and is therefore not further discussed within this work. + +By writing +\begin{equation} +p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{step}} = \mathcal{N}(\Delta d_t \mid \mu_{\text{step}}, \sigma_{\gDist}^2) +\label{eq:smoothingTransDistance} +\end{equation} +we receive a statement about how likely it is to cover a distance $\Delta d_t$ between two states $\vec{q}_{t+1}$ and $\vec{q}_{t}$. +In the easiest case, $\Delta d_t$ is the linear distance between two states. +Of course, based on the graph structure, one could calculate the shortest path between both and summarize the respective edge lengths. +However, this requires tremendous calculation time for negligible improvements. +Therefore this is not further discussed within this work. +The average step length $\mu_{\text{step}}$ is based on the pedestrian's walking speed and $\sigma_{\gDist}^2$ denotes the step length's variance. +Both values are chosen depending on the activity $x$ recognized at time $t$. +For example $\mu_{\text{step}}$ gets smaller while a pedestrian is walking upstairs, then just walking straight. +This requires to extend the smoothing transition by the current observation $\mObsVec_t$. +Since $\mStateVec$ is hidden and the Markov property is satisfied, we are able to do so. + + +The heading information is incorporated using +\begin{equation} +p(\mStateVec_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{turn}} = \mathcal{N}(\Delta\alpha_t \mid \mObsHeading, \sigma^2_{\text{turn}})\enspace , +\label{eq:transHeadingSmoothing} +\end{equation} +where $\Delta\alpha_t$ is the absolute angle between $\vec{q}_{t+1}$ and $\vec{q}_{t}$ in the range of $[0, \pi]$. +The relative angular change $\mObsHeading$ is then used to receive a statement about how likely it is to walk in that particular direction. +Again the normal distribution of \refeq{eq:transHeadingSmoothing} does not integrate to $1.0$. Therefore the same assumption as in \refeq{eq:transHeading} has to be made. + + +To further improve the results, especially in 3D environments, the vertical (non-absolute) distance $\Delta z$ between two successive states is used as follows: +\begin{equation} +p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{baro}} = \mathcal{N}(\Delta z \mid \mu_z, \sigma^2_{z}) \enspace . +\label{eq:smoothingTransPressure} +\end{equation} +This assigns a low probability to false detected or misguided floor changes. +Similar to \refeq{eq:smoothingTransDistance} we set $\mu_z$ and $\sigma^2_{z}$ based on the activity recognized at time $t$. +Therefore, $\mu_z$ is the expected change in $z$-direction between two time steps. +This means, if the pedestrian is walking alongside a corridor, we set $\mu_z = 0$. +In contrast, $\mu_z$ is positive while walking downstairs or otherwise negative for moving upstairs. +The size of $\mu_z$ and also $\mu_{\text{step}}$ could be a predefined value or set dynamically based on the measured vertical and linear acceleration. + +Looking at \refeq{eq:smoothingTransDistance} to \refeq{eq:smoothingTransPressure}, obvious similarities to a sensor fusion process can be seen. By assuming statistical independence between those three, the probability density of the smoothing transition is given by +\begin{equation} + \arraycolsep=1.2pt + \begin{array}{ll} + p(\vec{q}_{t+1} \mid \vec{q}_t) = + &p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{step}}\\ + &p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{turn}}\\ + &p(\vec{q}_{t+1} \mid \vec{q}_t, \mObsVec_t)_{\text{baro}} + \end{array} +\enspace . +\end{equation} +% +It is important to notice, that all particles at each time step $t$ of the forward filtering need to be saved. +Therefore, the memory requirement increasing proportional to the processing time. + diff --git a/tex/misc/keywords.tex b/tex/misc/keywords.tex index 2d928f2..c253a71 100644 --- a/tex/misc/keywords.tex +++ b/tex/misc/keywords.tex @@ -29,3 +29,5 @@ \newcommand{\docsRSSI}{RSSI} \newcommand{\docDSimplex}{downhill-simplex} + +\DeclareMathOperator{\atan}{atan2}