There are the NIST SP 800-90 series of test for randomness (with source), specifically:
Discrete Fourier Transform (Spectral) Test
Description: The focus of this test is the peak heights in the discrete Fast Fourier Transform. The purpose of this test is to detect periodic features (i.e., repetitive patterns that are near each other) in the tested sequence that would indicate a deviation from the assumption of randomness.
2nd Question:
Is that a better way to go for my problem? How can I use this for my purpose?
There's a StackOverflow Q&A on feature detection/extraction: Difference between Feature Detection and Descriptor Extraction and DSP.SE's Purpose of image feature detection and matching, while not calculating randomness (as the prior example does) it can classify the features in case there's something you want to exclude.
There's the power of Cloud Computing, for example: Google's Cloud Vision API, again this won't say if "it is random" but if it isn't random it will tell you what is in the photo. It's quick and using very few of your CPU cycles to obtain the answer.
Once you know 'what and where' you can subtract it from the image and test for randomness on the remainder.
