There are already many image decoders for the NOAA APT weather satellites available, with the most common and feature rich program being the abandoned freeware "WXtoIMG".
However many people may not know how simple the APT signal DSP code is. Over on his blog post Dmitrii Eliuseev explains how only 50 lines of Python code are required to decode an image from the received APT audio. Dmitrii's post shows how a simple Hilbert transform is used on the APT audio which is essentially the entire decoding step. This is then followed by a for loop that calculates the pixel luminosity from the decoded data, and plots it onto an image file.
Of course the image is only grayscale (or in Dmitrii's case he decided to use greenscale), but adding false color and various other image enhancements are just standard image processing techniques.
Dmitrii concludes with the following:
Interesting to mention, that there are not so many operational radio communication systems in the world, the signal of which can be decoded using 20 lines of code. The NOAA satellites are about 20 years old, and when they finally will retire, the new ones will most likely be digital and format will be much more complex (the new Russian Meteor-M2 satellite is already transmitting digital data at 137 MHz). So those who want to try something simple to decode can be advised to hurry up.
