Here's one possibility: use \titlepagefootline before the frame containing the title page and \mainfootline after this frame. New commands \supervisor, \departament and \defense were also provided to add the information about the supervisor, the Departament, and the "Defense" text, respectively.
To get your University logo in the title page and in the inner frames, replace ctanlion with the name of the image file that contains the logo, in both places in which my example uses ctanlion. Since I didn't have any specific information about the dimensions of the logo, perhaps you'll need to modify some lengths:
\documentclass{beamer}
\usetheme{Madrid}
\def\supervisor#1{\gdef\insertsupervisor{#1}}
\def\departament#1{\gdef\insertdepartament{#1}}
\def\defensetext#1{\gdef\insertdefensetext{#1}}
\supervisor{}
\departament{}
\defensetext{}
\makeatletter
\defbeamertemplate*{title page}{mydefault}[1][]
{
\vbox{}
\vspace*{-8ex}
\begin{beamercolorbox}[wd=\paperwidth,sep=8pt,left,#1]{institute}
{\usebeamercolor[fg]{titlegraphic}\raisebox{-.3\height}{\inserttitlegraphic}}\quad
{\usebeamerfont{institute}\insertinstitute}
\end{beamercolorbox}
\vfill
\begin{centering}
\begin{beamercolorbox}[sep=8pt,center,#1]{title}
\usebeamerfont{title}\inserttitle\par%
\end{beamercolorbox}%
\vskip1em\par
\begin{beamercolorbox}[sep=8pt,center,#1]{date}
\usebeamerfont{author}\insertsupervisor\par\medskip
\insertauthor
\end{beamercolorbox}
\end{centering}
\vfill
\begin{beamercolorbox}[sep=8pt,center,#1]{date}
\usebeamerfont{date}\insertdate
\end{beamercolorbox}
}
\newcommand\mainfootline{%
\titlegraphic{\includegraphics[height=3ex]{ctanlion}}
\setbeamertemplate{footline}
{
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=4.25ex,dp=1ex,left]{author in head/foot}%
\raisebox{.8\baselineskip}{\parbox[t][4ex][t]{.333333\paperwidth}{\raggedright\hskip1ex\usebeamerfont{author in head/foot}%
\raisebox{-,5\height}{\inserttitlegraphic}\hskip2ex\insertshortinstitute}}
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=4.25ex,dp=1ex,center]{title in head/foot}%
\usebeamerfont{title in head/foot}\raisebox{.8\baselineskip}{\parbox[t][4ex][t]{.333333\paperwidth}{\centering\insertdefensetext\par\insertdepartament}}
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=4.25ex,dp=1ex,right]{date in head/foot}%
\raisebox{.8\baselineskip}{\parbox[t][4ex][t]{.333333\paperwidth}{\raggedleft\usebeamerfont{date in head/foot}\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}}}
\end{beamercolorbox}}%
\vskip0pt%
}%
}
\newcommand\titlepagefootline{%
\setbeamertemplate{footline}
{%
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=\paperwidth,ht=.25ex,dp=2ex,center]{institute}
\large\insertdepartament
\end{beamercolorbox}%
}%
}%
}
\makeatother
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{frametitle}[default][center]
\setbeamertemplate{title page}[mydefault]
\setbeamerfont{institute}{size=\LARGE}
\setbeamerfont{title}{size=\LARGE}
\setbeamerfont{author}{size=\large}
\setbeamerfont{institute}{size=\LARGE}
\title{The Title}
\institute{The Institute}
\supervisor{The Supervisor}
\author[Author]{The Author \\ 2876547}
\titlegraphic{\includegraphics[height=1cm]{ctanlion}}
\departament{The Departament}
\defensetext{Defense}
\date{}
\begin{document}
\titlepagefootline
\begin{frame}
\maketitle
\end{frame}
\mainfootline
\begin{frame}
\frametitle{Test Frame Title}
Test
\end{frame}
\end{document}

singaporetheme maybe closer to what you're looking for. Also see the Beamer solutions for examples of how to overlay institution logos etc. CTAN Beamer docs – oLas May 13 '13 at 21:35