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