Vertical alignment of subfigures LATEX

Another solution (which works with the subcaption package is

\begin{figure}[p]
        \centering
        \begin{subfigure}{.49\linewidth}
            \centering
            \caption{Large Picture}
            \includegraphics{LARGEPIC}
        \end{subfigure}
        \hfill
        \begin{subfigure}{.49\linewidth}
            \centering
            \caption{SMALL PIC}
            \includegraphics{small picture}
            \begin{minipage}{.1cm}
            \vfill
            \end{minipage}
        \end{subfigure} 
        \caption{Two pictures}
\end{figure}

The \vfill alone does not work, that's why it is put into the minipage

My method is using square minipage which centers its contents:

\begin{figure}
\subfloat[Figure a]{%
\begin{minipage}[c][1\width]{0.5\textwidth}%
\includegraphics[clip,width=1\textwidth]{figurea}%
\end{minipage}}\subfloat[Figure b]{\centering{}%
\begin{minipage}[c][1\width]{0.5\textwidth}%
\begin{center}
\includegraphics[clip,width=0.6\textwidth]{figureb}
\par\end{center}%
\end{minipage}}
\caption{main caption}
\end{figure}

This code was generated by LyX, however, so it's a bit ugly.

You can also use \raisebox{x}{\includegraphics[...]{...}} where x is negative to shift it downwards and positive to shift upwards.

Edit: it seems like subfig package has problems, particularly when working with hyperref. In that case, try subcaption package as mentioned in this answer.

If you use subfig package, you can do this easily. The solution is in section 5.4 of the manual:

\newsavebox{\tempbox}
\begin{figure}[H]
\sbox{\tempbox}{\includegraphics[width=0.28\textwidth]{sparsesmall}}
\subfloat[sparse($\mathbf{A}$)]{\usebox{\tempbox}\label{sparse}}%
\qquad
\subfloat[full($\mathbf{A}$)]{\vbox to \ht\tempbox{%
  \vfil
  \includegraphics[width=0.55\textwidth]{fullsmall}
  \vfil}\label{full}}%
  \caption{Representation of $\mathbf{A}$ in MATLAB}\label{schematic}
\end{figure}

I haven't tested it, and there may be typos, but it should work.