\newcommand{\bmath}[1]{\mbox{$\mathbf{#1}$}}\newcommand{\q}{\bmath{q}}\newcommand{\y}{\bmath{y}}\newcommand{\x}{\bmath{x}}\newcommand{\de}{\bmathepgivjb}\newcommand{\ere}{\bmath{r}}\newcommand{\omegav}{\bmath{\omega}}\newcommand{\f}{\bmath{f}}\newcommand{\n}{\bmath{n}}\newcommand{\Pe}{\bmath{P}}\newcommand{\F}{\bmath{F}}\newcommand{\G}{\bmath{G}}\newcommand{\I}{\bmath{I}}\newcommand{\J}{\bmath{J}}\newcommand{\K}{\bmath{K}}\newcommand{\Ese}{\bmath{S}}\newcommand{\R}{\bmath{R}}\newcommand{\Ha}{\bmath{H}}\newcommand{\h}{\bmath{h}}\newcommand{\m}{\bmath{m}}\newcommand{\z}{\bmath{z}}\newcommand{\uve}{\bmath{v}}\newcommand{\V}{\bmath{V}}\newcommand{\undist}{\bmath{uds}}\newcommand{\dist}{\bmath{ds}}\newcommand{\amin}{\bmath{a}}\documentclass[a4paper,12pt]{article}\usepackage{times,epsfig}\setlength{\oddsidemargin}{-7mm}\setlength{\evensidemargin}{-7mm}\setlength{\topmargin}{-14mm}\setlength{\parindent}{0mm}\setlength{\parskip}{1mm}\setlength{\textwidth}{173mm}\setlength{\textheight}{244mm}\setlength{\unitlength}{1mm}%\input newcom.tex%\input symbols.tex\newcommand{\coursetitle}{SLAM Summer School 2006}\newcommand{\tutorialtitle}{Practical 2: SLAM using Monocular Vision}\newcommand{\docauthor}{Javier Civera, University of Zaragoza\\                        Andrew J. Davison, Imperial College London\\                        J.M.M Montiel, University of Zaragoza.}\newcommand{\docemails}{josemari@unizar.es, jcivera@unizar.es, ajd@doc.ic.ac.uk}\begin{document}\bibliographystyle{plain}\begin{center}\hrule\vspace{2mm}{\LARGE\bf \coursetitle}\vspace{2mm}{\Large\bf \tutorialtitle}\vspace{2mm}{\large \docauthor}\\{\large \texttt{\docemails}}\end{center}\hrule%\hrule\vspace{5mm}%\vspace{10mm}%\hrule%\vspace{2mm}\setcounter{page}{1}\setcounter{section}{0}\section{Objectives}\begin{enumerate}\item Understanding the characteristics of efficient (potentiallyreal-time) SLAM using a monocularcamera as the only sensor.\begin{enumerate}\item Map management.\item Feature initialization.\item Near and far features.\end{enumerate}\item Understanding the inverse depth parametrization of mapfeatures in monocular SLAM.\item Understanding the performancelimits of a constant velocity motion model for a camera when no odometry is available.\end{enumerate}\section{Exercise 1. Feature selection and matching.}One of the characteristics of vision-based SLAM is that there is toomuch information in an image sequence forcurrent computers to process in real-time. We therefore use heuristics to select which features toinclude in the map. The desirable properties of mapfeatures are:\begin{enumerate}\item Saliency: features have to be identified by distinct texture patches.\item A minimum number (e.g. 14) should  be visible in the image at all times --- when this is not the case, new map features are initialized.\item The features should be spread over the whole image.\end{enumerate}The goal of this exercise is to manually initialize featuresin order to meet the above criteria, and to understand better how anautomatic initialization algorithm should work. Run\texttt{mono\_slam.m}. With the user interface, you can addfeatures and perform step by step EKF SLAM:\begin{enumerate}\item In the first image add about ten salient features spread over the image.You can watch the movie \texttt{juslibol\_SLAM.mpg} (using forinstance \texttt{mpeg\_play} on a Unix workstation) as an example of howto select suitable features (but youcan of course select other ones). This movie shows the results ofapplying automatic feature selection.\item As the camera moves, some features will leave the field of view, and you willhave to add new ones in order to maintain around 14 visible map features.\end{enumerate}\section{Exercise 2. Near features and far features.}A camera is a bearing-only sensor. This means that the depth of afeature cannot be estimated with a single image measurement. Thedepth of the feature can be estimated only if the feature isobserved from different points of view (only if the cameratranslates enough to produce a significant parallax). So, it canbe distant features whose depth cannot be correctly estimated fora long time, or even never if the translation is never big enoughto produce parallax.The goal of this exercise is to observe the different estimationevolution for near and distant features and their influence in thecamera location estimate.\begin{enumerate}\item Open the video \texttt{parallax.mpg}.      Observe the different motion \emph{in the image} of different depth features.      Open the video \texttt{noparallax.mpg} and see now that the motion in the image       of different depth features is the same.\item Now analyse the video we are using for this practical (\texttt{juslibol.mpg}).Distinguish in this video parts with \emph{low parallax motion}.\item Run \texttt{mono\_slam.m}.\begin{itemize}\item  Observe what happens to the features in the 3D map(initialization value and covariance and value and covarianceafter some frames). Red dots are the estimated values and redlines limit the $95\%$ acceptance region.\item The code singles out features \#5 and \#15 and displays depthestimation and its $95\%$ acceptance region: [lower limit,estimation, upper limit]. When clicking, make sure that feature \#5corresponds to a near one (for example, on the car) and feature \#15corresponds to a far one (for example, the tree appearing on theleft).\item   Notice the difference between the evolution of near anddistant features.\item Observe the camera location uncertaintyevolution (Use the axes limit controls in the user interface).\item Observe what happen to features and camera location uncertaintiesduring the low parallax motion part discussed in point 2. Noticethe difference between this part of the 3D map (constructed withlow parallax information) and the high parallax parts.\end{itemize}\end{enumerate}\section{Exercise 3. Inverse depth parameterization.}Initializing a feature in monocular SLAM is a challenging issue,because the depth uncertainty is not well modelled by a Gaussian.This problem is overcome using inverse depth instead of theclassical $XYZ$ representation.\begin{figure}\centering\includegraphics[width=0.5\columnwidth]{FeatureObservationAndParameterization.eps}\caption{Feature parameterization and measurement equation.}\label{fig_feat_par}\end{figure}Matlab code of the practical is coded in inverse depth so statevector is:\begin{equation}\x=\left(\x_v^\top, \y_1^\top, \y_2^\top, \ldots\y_n^\top\right)^\top.\end{equation}it is composed of:\begin{enumerate}\item 13 components that correspond to the location and velocity ofthe camera:\begin{equation}\x_v=\left(\begin{array}{c}\ere^{WC}\\\q^{WC}\\\uve^{W}\\\omegav^{W}\end{array}\right).\end{equation}\item The rest of the components are features. Each feature is represented by 6 parameters, that are the position of the camera the first time the feature was seen $x_i,  y_i,  z_i$, the ray coded with azimuth-elevation angles ($\theta,\phi$), in absolute reference, and the inverse depth, $\rho$, of the feature along the ray:\begin{equation}\y_i=\left(\begin{array}{cccccc}x_i & y_i & z_i & \theta_i &\phi_i & \rho_i\end{array}\right)^\top\end{equation}So the transformation from the inverse depth coding to theEucleidean coding in the absolute frame is:\begin{equation}\left(\begin{array}{c}x\\y\\z\end{array}\right)=\left(\begin{array}{c}x_i\\y_i\\z_i\end{array}\right)+                    \frac{1}{\rho_i}\m\left(\theta_i,                    \phi_i\right).\end{equation}where:\begin{equation}\m=\left(\begin{array}{ccc}\cos\phi_i \sin\theta_i&                     -\sin\phi_i&                     \cos\phi_i \cos\theta_i\end{array}\right)^\top                     \label{eq-m}\end{equation}\end{enumerate}The goal of this exercise is to understand the inverse depthparameterisation.\begin{enumerate}\item The code stores partial information about features \#5 and\#15 in the file \texttt{history.mat}:\begin{itemize}\item \texttt{feature5History} is a 6 row matrix, Each column containing the feature \#5location coded in inverse depth at step $k$.\item \texttt{rhoHistory\_5} is a row vector containing the inverse depth estimationhistory for feature 5.\item \texttt{rhoHistory\_15} is a row vector containing the inverse depth estimationhistory for feature 15.\item \texttt{rhoStdHistory\_5} is a row vector containing the inverse depthstandard deviation history for feature 5.\item \texttt{rhoStdHistory\_15} is a row vector containing the inverse depth standard deviationhistory for feature 15.\end{itemize}\item Compute the $XYZ$ Euclidean location for feature \#5 afterprocessing all images.\item Do a graph with the value of the inverse depth and the  $95\%$ acceptance region history for both features \#5 and \#15.  Use the matlab functions \texttt{open}, \texttt{figure}, \texttt{hold}, and \texttt{plot}.  Comment the difference between the two graphs.\item After processing the whole sequence, what is the estimation and theacceptance region \emph{expressed in depth} for both features?Think about a feature at infinity, what inverse depth would have?Try to see in the previous graphs when the infinity is included inboth features estimation, that is, the feature can be at any depthin the ray.\end{enumerate}\section{Exercise 4. Constant velocity motion model (optional)}Monocular SLAM uses camera as the unique sensor, without anyodometry input. It is used a constant velocity model so thesystems needs as input both the camera frame rate and the maximumexpected angular and linear acceleration.For a given camera acceleration, the frame rate defines theacceptance region for the point matching, so for any acceleration,the acceptance regions can be kept low if a frame rate high enoughis used.The goal of this exercise is to analyse the effect of the linearacceleration, the angular acceleration and the frame rate.\begin{enumerate}\item The initial tuning is $6\frac{m}{s^2}$ and $6\frac{\mbox{rad}}{s^2}$.\item Increase the angular acceleration only (for instance, doublethe value) and analyse the effect on the search region. Find thisparameter in the file \texttt{mono\_slam.m}, its name is\texttt{sigma\_alphaNoise}.\item Increase the linear acceleration only (for instance, doublethe value), analyse the effect and compare with the previouspoint. The name of this parameter is \texttt{sigma\_aNoise}.\item Reduce the frame rate processing and see the effect:\begin{enumerate}\item 1 out of 2 images.\item 1 out of 4 images.\end{enumerate}The processing time has increased or decreased as the result ofprocessing less images? Can you explain this?Clue: Find the variable step and the code associated with thisvariable. You will also have to modify the variable\texttt{deltat}, that codes the time between frames.\end{enumerate}\nocite{Hartley2004,Davison2003,Montiel2006RSS,Bar-Shalom-88}\bibliographystyle{ieee}\bibliography{IEEEabrv,practical}\end{document}