The definite article is justified in phrases like this because we are talking about a specific kind of manipulation: the of part specifies what kind. The definite article is justified, but not required, in phrases like this. It is justified in this particular sentence, but the definite article is not always justified in "[the] noun1 of noun2" phrases: there are exceptions.
It will generally be OK in sentences where noun1 defines an action (for example words ending in -tion) and noun2 defines what is acted upon.
The definite article can also used when noun1 is an abstract noun and noun2 is something having that quality "the courage of heroes" and "the spirit of Christmas". It does not, however, work in all situations like this, for example "the peace of mind".
This NGram graph shows that, for manipulation, it's OK with or without the definite article. As you can see from the graph, fashions change with time, and at the moment, without the definite article appears to be ahead, but I wouldn't read much into that, because generally shorter ngrams occur more frequently.
If you look instead at "by the abuse of power", the version with the definite article is ahead.
Looking at your second sentence, the definite article would be justified if, rather than the compound noun "picture manipulation", you used "manipulation of pictures":
The format allows [the] fast manipulation of pictures.