I have used the GIMP in the past for studying and editing images such as for resizing, clipping, rotation, color curves, hue, saturation, noise reduction, sharpening, format conversion, studying the histogram, adding text, etc. It is true, among the FOSS Software I have used GIMP stands out as a tool packed with features. It used to be slow to startup loading all the data options and the plugins, but in the latest version it seems to be slow only the first time around. The GIMP and its source code are available to download and install on Windows, Linux and Mac OS.
I have been using Openoffice for almost a decade now. I have it installed on all the three laptops we have at home and the laptop I have at work also uses the same. It is very reliable and does almost everything that you can do with Microsoft Office. Openoffice Writer can read the .docx file type, but cannot write in .docx format. Openoffice Calc can read Excel worksheets and write as well. Some of my worksheets are secured with a password and the security functionality transfers as well. Openoffice is currently pushing for its use in education which will help in extending its deployment and reach.
The best part about Octave is you can load almost all the Matlab programs you wrote in the past and have it run as is. Octave includes an API to the Matlab MEX interface as well, although it is not as straightforward to use. You get several packages along with the Octave distribution, such as finite element analysis and image manipulation. In my last job, the only reason we chose to buy Matlab license is for the support.