Well, there are some rules for this, let me give a translation.
So, transitive verbs with negation are governed by genitive or accusative in following cases:
genitive (usually used)
- With strong negation with
ни particle - "Не люблю ни чрезмерной жары, ни чрезмерного холода".
- With partitive objects - Не cделал вычислений.
- After verbs
видеть, слышать, думать, хотеть, чувствовать, ждать
- With words used for describing abstract concepts (like time, honor and so one) - "Не теряет времени"
accusative (usually used)
- In cases when in English one most likely will use "the" - "Не прочёл книгу, которую ему рекомендовали" (he hadn't read the book, which…).
- With animate nouns and proper nouns.
- With double negations - Нельзя не заметить определённую логику в его доводах.
- With adverbs of extreme degree (like almost, nearly etc.) - Едва не пропустил урок.
- If there is a word (usually this is an adjective) which is logically connected with both the verb and and the corresponding noun. Here we need a bit of explanation. For example, in phrase "не считаю книгу интересной" you can just drop "книгу" and leave "не считаю интересной" - this is exactly what is meant by connection with both verb and noun.
- If negation is separated by some additional word (thus, if the verb is not negated directly) - "Не часто слушаю музыку".
- If it is subsequent infinitive which is the subject of negation, not a noun - Не хотел читать книгу.
If you are a sharp reader, you've noticed that I've a bit shortened the second list of rules. This is not because I'm lazy - well, not only because of this :)
Actually, the rules are getting quite vague, even self-repeating and, most important, all these rules are not always strict. You indeed can say "не читал этой книги" or "не читал эту книгу"; but at the same time there are cases when only one form is (de-facto) appropriate.
Your question belongs to that kind of questions which makes learning any language a difficult task, since even rules trying to describe something are quite fuzzy.
If you ask any native Russian speaker why he/she prefers one form to another, the less probabilistic scenario is that this native speaker will name you any of the rules described above.
But! Speaking of the verb "видеть" there's a little mnemonic trick you can use.
If you use "не видеть" in the meaning "не замечать" (do not notice) in colloquial speech you most probably will end with genitive form.