Adjust logo height to match \frametitle box height

Question

I wanted to add a logo at the top right corner of my frames so I edited the frametitle with \addtobeamertemplate{frametitle} and added the logo using a TikZ \node (as explained in this answer), like this:

\addtobeamertemplate{frametitle}{}{%
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=north east,yshift=2pt] at (current page.north east) {\includegraphics[height=0.8cm]{example-image}};
\end{tikzpicture}}

I would like to adjust the height of the logo so that it matches exactly the height of the frametitle box.

Currently I can only do this by trial and error, changing the image height and the yshift value of the \node. This makes getting the desired result tricky and very time consuming.

As far as I know, the frametitle box is created with the beamercolorbox environment (e.g. here and here). I would like to know if there is a command that gets the height of the frametitle box, so that I can assign it to the logo height like this:

\includegraphics[height=\frametitleheight]{example-image}

---

Here is my MWE:

\documentclass{beamer}
\usetheme{Madrid}

\usepackage{tikz}

\addtobeamertemplate{frametitle}{}{%
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=north east,yshift=2pt] at (current page.north east) {\includegraphics[height=0.8cm]{example-image}};
\end{tikzpicture}}

\begin{document}

\begin{frame}
\frametitle{Frame title}
Hello world!
\end{frame}

\end{document}

Answer 1

You can measure the height of the frametitle colour box with the help of the \settoheight macro. Basically you could duplicate the content of the frametitle there, but to avoid the duplication of the content, just save the frametitle color box in a \savebox and re-use this savebox both in the frametitle template and in measuring the height.

\documentclass{beamer}
\usetheme{Madrid}

\usepackage{tikz}

\makeatletter
\setbeamertemplate{frametitle}{%
  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
  \@tempdima=\textwidth%
  \advance\@tempdima by\beamer@leftmargin%
  \advance\@tempdima by\beamer@rightmargin%
  \newsavebox{\mybox}% <- NEW
  \sbox{\mybox}{% <- NEW
      \begin{beamercolorbox}[sep=0.3cm,left,wd=\the\@tempdima]{frametitle}
        \usebeamerfont{frametitle}%
        \vbox{}\vskip-1ex%
        \if@tempswa\else\csname beamer@fteleft\endcsname\fi%
        \strut\insertframetitle\strut\par%
        {%
          \ifx\insertframesubtitle\@empty%
          \else%
          {\usebeamerfont{framesubtitle}\usebeamercolor[fg]{framesubtitle}\insertframesubtitle\strut\par}%
          \fi
        }%
        \vskip-1ex%
        \if@tempswa\else\vskip-.3cm\fi% set inside beamercolorbox... evil here...
      \end{beamercolorbox}%
     }% <- NEW
     \usebox{\mybox}% <- NEW
     \newlength{\frametitelheight}% <- NEW
     \settoheight{\frametitelheight}{\usebox{\mybox}}% <- NEW
}
\makeatother

\addtobeamertemplate{frametitle}{}{%
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=north east,yshift=5pt] at (current page.north east) {\includegraphics[height=\frametitelheight]{example-image}};
\end{tikzpicture}}

\begin{document}

\begin{frame}
\frametitle{Frame title}
Hello world!
\end{frame}

\end{document}

Answer 2

I managed to fix several bugs in samcarters solution:

1. resolved the mybox error mentioned in the comments
2. added the height of the header to match for themes like Berlin or Darmstadt
3. Fixed the yshift mentioned in the comments

\documentclass{beamer}
\usetheme{Darmstadt}
\usepackage{tikz}

\makeatletter
\newlength{\frametitleheight}% <- NEW
\newsavebox{\beamer@titlebox}% <- NEW

\setbeamertemplate{frametitle}{%
  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
  \@tempdima=\textwidth%
  \advance\@tempdima by\beamer@leftmargin%
  \advance\@tempdima by\beamer@rightmargin%

  \sbox{\beamer@titlebox}{% <- NEW
      \begin{beamercolorbox}[sep=0.3cm,left,wd=\the\@tempdima]{frametitle}
        \usebeamerfont{frametitle}%
        \vbox{}\vskip-1ex%
        \if@tempswa\else\csname beamer@fteleft\endcsname\fi%
        \strut\insertframetitle\strut\par%
        {%
          \ifx\insertframesubtitle\@empty%
          \else%
          {\usebeamerfont{framesubtitle}\usebeamercolor[fg]{framesubtitle}\insertframesubtitle\strut\par}%
          \fi
        }%
        \vskip-1ex%
        \if@tempswa\else\vskip-.3cm\fi% set inside beamercolorbox... evil here...
      \end{beamercolorbox}%
     }% <- NEW
     \usebox{\beamer@titlebox}% <- NEW
\settoheight{\frametitleheight}{\usebox{\beamer@titlebox}}% <- NEW
\addtolength{\frametitleheight}{\headheight}
}
\makeatother

\addtobeamertemplate{frametitle}{}{%
\begin{tikzpicture}[remember picture,overlay]
\node[inner sep=0, outer sep=0, anchor=north east] at (current page.north east) {\includegraphics[height=\frametitleheight, keepaspectratio]{example-image}};
\end{tikzpicture}
}

\begin{document}

\begin{frame}{Frame Title}
Hello world!
\end{frame}

\begin{frame}
Hello world!
\end{frame}

\begin{frame}{Frame Title}
Hello world!
\end{frame}

\end{document}

---

If you only want to match the headline, for instance when some frames have no title, but only the headline, use instead

\addtobeamertemplate{headline}{}{%
\begin{tikzpicture}[remember picture,overlay]
\node[inner sep=0, outer sep=0, anchor=north east] at (current page.north east) {\includegraphics[height=\headheight, keepaspectratio]{example-image}};
\end{tikzpicture}
}