How to set an image in a header that adjusts to the edges of the page?

Question

I'm struggling to add an image to the headers of an A4 paper document. Additionnally, my document is twoside. That's the way I'd like it to look :

Basically the logo with the name and the triangles are one single image, spanning the full width of an A4 page (I've taken it from a Word template), and adapted to displaying on odd pages. I designed a second image for even pages.

I tried to put them in a fancyhdr style, using the command \fancyhead[\includegraphics[width=\paperwidth]{even.png}]{\includegraphics[width=\paperwidth]{odd.png}}. The thing is the image doesn't adjust to the exact edges of the page. I've tried different solutions : using \setlength commands to modify the margins; using \fancyheadoffset command to change the width of the header; splitting the image into three smaller images (triangles, inverted triangles and logo) that I could send into the corners using \fancyhead's R and L options ... None of them would work. So I figured I needed your help.

So is it even possible to set an image in the header of a document to give the output above ?

Here's a MWE :

\documentclass[a4paper,twoside]{report}

\usepackage[T1]{fontenc}
\usepackage[UTF8]{inputenc}
\usepackage{graphicx, fancyhdr, lipsum}
\usepackage[top=2.0cm, bottom=2.5cm, inner=2.5cm, outer=2.5cm, marginparwidth=0pt]{geometry}
\pagestyle{fancy}%
\fancyhf{}%
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}

% These lines are my attempts at changing the layout
%\setlength\hoffset{0pt}
%\setlength\oddsidemargin{1cm}
%\setlength\voffset{0pt}
\setlength\headheight{85.0pt}
\addtolength{\textheight}{-85.0pt}
%\setlength\headwidth{\paperwidth}
%\fancyheadoffset{150pt}
%---

\fancyhead[RE, LO]{\includegraphics{logo.png}}
\fancyhead[LE]{\includegraphics{even.png}}
\fancyhead[RO]{\includegraphics{odd.png}} %

\begin{document}

\lipsum[1-15]

\end{document}

Answer 1

Here is a suggestion using scrlayer-scrpage instead fancyhdr. Then you can define two new layers: one for odd and one for even pages. These layers can be added to the pagestyle scrheadings. So the images will not be on chapter pages because they are using the page style plain.

\documentclass[a4paper,twoside]{report}
\usepackage[T1]{fontenc}
\usepackage[UTF8]{inputenc}
\usepackage{graphicx}
\usepackage[margin=2.5cm, top=5cm, marginparwidth=0pt]{geometry}

\usepackage{mwe}% dummy text and example images

\newsavebox\topodd
\newsavebox\topeven
\savebox\topodd{\includegraphics[height=85pt,width=\paperwidth]{example-image-A}}
\savebox\topeven{\includegraphics[height=85pt,width=\paperwidth]{example-image-B}}

\usepackage[automark]{scrlayer-scrpage}
\clearpairofpagestyles
\cfoot*{\pagemark}% page number in footer
\DeclareNewLayer[
  background,
  topmargin,
  oddpage,
  height=85pt,
  contents={\usebox\topodd}
]{top.odd}
\DeclareNewLayer[
  clone=top.odd,
  evenpage,
  contents={\usebox\topeven}
]{top.even}
\AddLayersToPageStyle{scrheadings}{top.odd,top.even}

\begin{document}
\tableofcontents
\Blinddocument
\end{document}

Note that I have enlarged the top margin to 5cm.