Initialize a repository for writeup

templates/baposter-template/examples/branch-and-bound/Makefile

@@ -0,0 +1,7 @@
+ALL: poster_portrait.pdf 
+
+%.pdf: %.tex Makefile
+	pdflatex --enable-write18 $< &&	pdflatex $< && pdflatex $<
+
+clean:
+	  rm -f *.aux	*.bbl	*.blg	*.log	poster_*.pdf 

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templates/baposter-template/examples/branch-and-bound/poster_portrait.aux

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+\relax 
+\bibstyle{ieee}
+\bibcite{amberg11:bnb}{1}
+\pgfsyspdfmark {pgfid1}{1757794}{774754}
+\newlabel{eqn:selection}{{3}{2}}
+\newlabel{eqn:cost}{{4}{2}}

templates/baposter-template/examples/branch-and-bound/poster_portrait.tex

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+\documentclass[portrait,final,a0paper,fontscale=0.277]{baposter}
+
+\usepackage{calc}
+\usepackage{graphicx}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{relsize}
+\usepackage{multirow}
+\usepackage{rotating}
+\usepackage{bm}
+\usepackage{enumitem}
+\usepackage{url}
+\usepackage{booktabs}
+
+\usepackage{graphicx}
+\usepackage{multicol}
+
+%\usepackage{times}
+%\usepackage{helvet}
+%\usepackage{bookman}
+\usepackage{palatino}
+
+\newcommand{\captionfont}{\footnotesize}
+
+\graphicspath{{images/}{../images/}}
+\usetikzlibrary{calc}
+
+
+\newcommand{\Matrix}[1]{\begin{bmatrix} #1 \end{bmatrix}}
+\newcommand{\Vector}[1]{\begin{pmatrix} #1 \end{pmatrix}}
+
+\newcommand*{\norm}[1]{\mathopen\| #1 \mathclose\|}% use instead of $\|x\|$
+\newcommand*{\abs}[1]{\mathopen| #1 \mathclose|}% use instead of $\|x\|$
+\newcommand*{\normLR}[1]{\left\| #1 \right\|}% use instead of $\|x\|$
+
+\newcommand*{\SET}[1]  {\ensuremath{\mathcal{#1}}}
+\newcommand*{\FUN}[1]  {\ensuremath{\mathcal{#1}}}
+\newcommand*{\MAT}[1]  {\ensuremath{\boldsymbol{#1}}}
+\newcommand*{\VEC}[1]  {\ensuremath{\boldsymbol{#1}}}
+\newcommand*{\CONST}[1]{\ensuremath{\mathit{#1}}}
+
+\DeclareMathOperator*{\argmax}{arg\,max}
+\DeclareMathOperator*{\diag}{diag}
+\DeclareMathOperator*{\argmin}{arg\,min}
+\DeclareMathOperator*{\vectorize}{vec}
+\DeclareMathOperator*{\reshape}{reshape}
+
+%\font\dsfnt=dsrom12
+
+\newcommand{\SNN}{\ensuremath{\mathbb N}}
+\newcommand{\SRR}{\ensuremath{\mathbb R}}
+\newcommand{\SZZ}{\ensuremath{\mathbb Z}}
+%-----------------------------------------------------------------------------
+% Matrices of the shape model
+\renewcommand{\a}{\VEC\alpha}
+\renewcommand{\v}{\VEC v}
+\renewcommand{\l}{\VEC l}
+\newcommand*{\m}{\VEC{\mu}}
+\newcommand*{\M}{\MAT{M}}
+\renewcommand*{\P}{\MAT{\Pi}}
+
+%\newcommand{\J}{\SET J}
+\newcommand{\J}{\SET{P}}
+\newcommand{\Active}{\mathcal{A}}
+\newcommand{\Selection}{\mathbf{S}}
+\newcommand{\AllSelections}{\mathfrak{S}}
+\newcommand{\Params}{\VEC\Theta}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%% Some math symbols used in the text
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Multicol Settings
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\setlength{\columnsep}{1.5em}
+\setlength{\columnseprule}{0mm}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Save space in lists. Use this after the opening of the list
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\compresslist}{%
+\setlength{\itemsep}{1pt}%
+\setlength{\parskip}{0pt}%
+\setlength{\parsep}{0pt}%
+}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Begin of Document
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Here starts the poster
+%%%---------------------------------------------------------------------------
+%%% Format it to your taste with the options
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Define some colors
+
+\definecolor{lightorange}{rgb}{0.9,0.4,0}
+\definecolor{lightestorange}{rgb}{1,0.8,0.5}
+\definecolor{darkorange}{rgb}{0.2,0.1,0}
+
+\hyphenation{resolution occlusions}
+%%
+\begin{poster}%
+  % Poster Options
+  {
+  % Show grid to help with alignment
+  grid=false,
+  % Column spacing
+  colspacing=1em,
+  % Color style
+  bgColorOne=lightestorange,
+  bgColorTwo=white,
+  borderColor=darkorange,
+  headerColorOne=darkorange,
+  headerColorTwo=lightorange,
+  headerFontColor=white,
+  boxColorOne=lightestorange,
+  boxColorTwo=lightorange,
+  % Format of textbox
+  textborder=faded,
+  % Format of text header
+  eyecatcher=true,
+  headerborder=closed,
+  headerheight=0.1\textheight,
+%  textfont=\sc, An example of changing the text font
+  headershape=roundedright,
+  headershade=shadelr,
+  headerfont=\Large\bf\textsc, %Sans Serif
+  textfont={\setlength{\parindent}{1.5em}},
+  boxshade=plain,
+%  background=shade-tb,
+  background=plain,
+  linewidth=2pt
+  }
+  % Eye Catcher
+  {\includegraphics[height=7em]{images/search_tree_ex1-crop.pdf}} 
+  % Title
+  {\bf\textsc{Optimal Landmark Detection using Shape Models and Branch and Bound}\vspace{0.5em}}
+  % Authors
+  {\textsc{\{ Brian.Amberg and Thomas.Vetter \}@unibas.ch}}
+  % University logo
+  {% The makebox allows the title to flow into the logo, this is a hack because of the L shaped logo.
+    \includegraphics[height=9.0em]{images/logo}
+  }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Now define the boxes that make up the poster
+%%%---------------------------------------------------------------------------
+%%% Each box has a name and can be placed absolutely or relatively.
+%%% The only inconvenience is that you can only specify a relative position 
+%%% towards an already declared box. So if you have a box attached to the 
+%%% bottom, one to the top and a third one which should be in between, you 
+%%% have to specify the top and bottom boxes before you specify the middle 
+%%% box.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    %
+    % A coloured circle useful as a bullet with an adjustably strong filling
+    \newcommand{\colouredcircle}{%
+      \tikz{\useasboundingbox (-0.2em,-0.32em) rectangle(0.2em,0.32em); \draw[draw=black,fill=lightblue,line width=0.03em] (0,0) circle(0.18em);}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{Problem}{name=problem,column=0,row=0}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  Fitting statistical 2D and 3D shape models to images is necessary for a
+  variety of tasks, such as video editing and face recognition. Much progress
+  has been made on local fitting from an initial guess, but determining a close
+  enough initial guess is still an open problem. We propose a method to locate
+  fiducial points, which can then be used to initialize the fitting.
+ }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{Contributions}{name=contribution,column=0,below=problem}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  We overcome the inherent ambiguity in landmark detection by using global shape
+  information. We solve the  combinatorial problem of selecting out of a large
+  number of candidate landmark detections the configuration which is best
+  supported by a shape model. Our method, as opposed to previous approaches,
+  always finds the globally optimal configuration.
+
+  The algorithm can be applied to a very general class of shape models and is
+  independent of the underlying feature point detector. Its theoretic optimality
+  is shown, and it is evaluated on a large face dataset.
+  }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\headerbox{Results}{name=results,column=1,span=2,row=0}{
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  {
+\smaller\centering
+\begin{tabular}{@{}rccccccc@{}}
+\begin{sideways}\makebox[0pt][c]{Success}\end{sideways} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_fa_success_1.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_fb_success_1.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_ql_success_1.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_qr_success_1.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_hl_success_1.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_hr_success_1.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_rc_success_1.pdf}} \\
+&
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_fa_success_2.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_fb_success_2.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_ql_success_2.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_qr_success_2.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_hl_success_2.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_hr_success_2.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_rc_success_2.pdf}} \\
+\midrule
+\begin{sideways}\makebox[0pt][c]{Failure}\end{sideways} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_fa_fail.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_fb_fail.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_ql_fail.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_qr_fail.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_hl_fail.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_hr_fail.pdf}} &
+\parbox[c]{0.11\linewidth}{\includegraphics[width=\linewidth]{images/l_rc_fail.pdf}} 
+\end{tabular}
+  }\\[-1em]
+      \begin{multicols}{2}
+Some randomly chosen images from the color feret database for each
+pose, and the detected landmark positions. The first two rows are success
+cases, the last row shows a failure case. 
+      \end{multicols}
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{Representation}{name=representation,column=2,below=results}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\centering\includegraphics[width=\linewidth]{images/representation.pdf}
+Subsets of solutions are encoded as the Kartesian product of subsets of landmark candidates per fiducial point. 
+  }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\headerbox{Scaling Behaviour}{name=scaling,column=2,below=representation}{%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \smaller%
+  \centering{Runtime as a function of the number of false positives}\\[0em]%
+  \centering{{\includegraphics[width=0.9\linewidth]{images/typical_random_no_noise-crop.pdf}}}\\[0em]%
+  \centering{Runtime as a function of detection accuracy}\\[0em]%
+  \centering{{\includegraphics[width=0.9\linewidth]{images/typical_random_add_noise-crop.pdf}}}\\[0em]%
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{References}{name=references,column=0,above=bottom}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    \smaller
+    \bibliographystyle{ieee}
+    \renewcommand{\section}[2]{\vskip 0.05em}
+      \begin{thebibliography}{1}\itemsep=-0.01em
+      \setlength{\baselineskip}{0.4em}
+      \bibitem{amberg11:bnb}
+        B.~Amberg, T. Vetter.
+        \newblock {O}ptimal {L}andmark {D}etection using {S}hape {M}odels and {B}ranch and {B}ound
+        \newblock In {\em ICCV '11}
+      \end{thebibliography}
+   \vspace{0.3em}
+  }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{Source Code}{name=source,column=2,above=bottom}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \noindent
+  \begin{minipage}{\linewidth}
+  \begin{minipage}{0.75\linewidth}
+    \indent{}The source code is available at \\
+    \url{http://www.cs.unibas.ch/personen/amberg_brian/bnb/}
+  \end{minipage}\hfill%
+  \begin{minipage}{0.23\linewidth}
+  \hfill\includegraphics[width=\linewidth]{chart}
+  \end{minipage}
+  \end{minipage}
+  }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{Formulation}{name=formulation,column=0,below=contribution,above=references}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The solution is constrained by a shape model
+\begin{align}
+  M(\Params) &= (m_1(\Params), \dots, m_N(\Params))\\
+   m_i &: \SRR^{N_{\Params}}\to\SRR^2\nonumber
+\end{align}
+mapping model parameters~$\Params$ to image positions $m_i(\Params)$.
+For each fiducial point $m_i$ a set of candidate positions 
+\begin{align}
+ L_i &= \{\l_i^1, \l_i^2, \dots\} & \l_i^j \in \SRR^2
+\end{align}
+is detected in the image.
+The task is to assign to every model vertex one of the candidate positions
+such that the shape model can be best fit to the selection $\Selection{}$, written as a tuple 
+\begin{align}
+  \Selection &=(j_1, j_2, \dots, j_N) & j_i &\in \SNN,\label{eqn:selection}
+\end{align}
+where $j_i$ is the index of a candidate of landmark $i$. 
+
+So we minimize the distance between the shape model and the image landmarks:
+\begin{align}
+  \Selection^* &= \argmin_{\Selection=(j_1, \dots, j_N)} f(\Selection)\nonumber\\
+  f(\Selection) &= \min_{\Params} \sum_i \rho\left( \normLR{ m_i(\Params) - \l_i^{j_i} }\right)\quad.\label{eqn:cost}
+\end{align}
+Where $\rho: \SRR\to\SRR$ is a robust function, allowing us to handle missing
+detections, and points which are invisible due to occlusion.
+  }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \headerbox{Splitting Strategy}{name=strategy,column=1,above=bottom}{
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+{\smaller\centering{Runtime as a function of the splitting strategy}\\[-0.5em]
+\includegraphics[width=0.95\linewidth]{images/typical_random_splitting_strategies.pdf}}
+Different splitting strategies result in vastly different performance.
+Note that  `split into equal sized problems' is one of the worst strategies for
+branch and bound.
+  }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\headerbox{Solution}{name=solution,column=1,row=0,below=results,above=strategy}{
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  This discrete optimization is solved by Branch and Bound, which is a method to minimize a function over a set. It requires us to (1) efficiently specify solution subsets, (2) determine a lower bound on the minimal cost of the solutions within a subset, and (3) specify a strategy to split a solution subset into two new subsets.
+%\begin{enumerate}[itemsep=2pt,parsep=0pt]
+%  \item Start with the set of all elements $\SET Q=\{ \AllSelections \}$
+%  \item \textbf{Repeat}:
+%  \begin{enumerate}[itemsep=2pt,parsep=0pt,topsep=2pt]
+%  \item  Take the minimal subset\vspace{-0.7em}
+%    \begin{align}
+%      \J_i &\leftarrow \argmin_{\J_i \in \SET Q} g(\J_i)\\
+%                  \SET Q   &\leftarrow \SET Q \setminus \{ \J_i \}\nonumber
+%    \end{align}
+%  \item
+%    \textbf{Return} $\Selection$ \textbf{if} $\J_i=\{\Selection\}$ is a single element.
+%  \item Split $\J_i$ into \vspace{-0.7em}
+%    \begin{align}
+%      \J_i^1 &\subset \J_i, \J_i^2 \subset \J_i\\\quad\text{ s.t. }\J_i &= \J_i^1 \cup \J_i^2.\nonumber
+%    \end{align}
+%  \item Add the new subsets to the candidates\vspace{-0.7em}
+%    \begin{align}
+%      \SET Q &\leftarrow \SET Q \cup \{ \J_i^1, \J_i^2 \}\nonumber
+%    \end{align}
+%  \end{enumerate}
+%\end{enumerate}
+
+  The ingredients in our case are:
+ \begin{enumerate}
+ \item Solution subsets are created by taking subsets of landmark
+ candidates, and considering the Kartesian product of all selected landmark
+ candidates
+ \item We bound the cost for such a solution set by taking for each
+ landmark the minimal distance to the convex hull of the selected candidates
+% \begin{align}
+%   g(\J) &= \min_{\Params} \sum_i \rho\left( d_{\text{convex hull}}(\l_i^{\J_i}, m_i(\Params)) \right)\\
+%         &< \min_{\Params} \min %\nonumber\\
+%   d_{\text{convex hull}}(\l_i^\J, \VEC x) &=  \min_{c \in \text{convex hull}(\l_i^{\J})}\normLR{ x - c }.
+% \end{align}
+ \item We found that splitting landmark candidates such that the convex hull of the resulting two landmark candidates are as distant as possible is most effective.
+ \end{enumerate}
+  }
+
+\end{poster}
+
+\end{document}
+