What are the meanings of the sentences where "Not that" is followed by an object-missing expression?

According to my observation, there are at least two types of using "Not that....". And my question is: what does "not that" mean in its second type of usage?

In the first usage, "not that" is followed by a structure-complete sentence or expression.

Example 1:

I've been teetering down my twittering and self editing my Facebooking lately. It's not that I don't have the urge to write things. It's just I have to fight the urge to write really inappropriate things that maybe my old High School English teacher doesn't really need or want to know.

Example 2:

It’s not that I don’t care about football. I just don’t care enough.

In the second usage, "not that" is followed by an object-missing expression.

Example 3 (from Fringe):

Olivia: Anything to do with metamorphic ability? Peter: Not that I can tell.

Here, tell is a transitive verb and its object is missing.

Example 4 (from Fringe):

Olivia: Were you involved in...? Walter: Not that I recall.

Here, recall is a transitive verb and its object is missing.

I understand that, in the first usage, "not that" sort of means "it doesn't mean...". But I'm not sure, in the second usage:

1. Why are objects of transitive verbs missing?
2. What does this type of "not that..." sentences mean here?

Solution 1:

The second usage is idiomatic, which you can translate roughly into "there's nothing that", or "there's not anything that":

Olivia: Anything to do with metamorphic ability?
Peter: There's not anything that I can tell (that has anything to do with metamorphic ability)

Olivia: Were you involved in...?
Walter: There's nothing that I recall (about being involved in ...)

It is essentially a way to say "I don't think so, but I could be wrong." There is nothing that you can think of that would make you wrong, but you know there could be other factors of which you are unaware, and so you hedge your bets by saying "not that I can [x]" instead of just straight-out saying "No."

Grammatically speaking, just as one person can say "where's John?" and the second can say "He's over there" and it's understood that "he" refers to John even though the second person never said John, it's understood that the object of the "Not that" verb is whatever the questioner was asking about.

Solution 2:

I'd say the easiest way to handle this kind of usage is to imagine the phrase "in any way", "in any respect", or "in any sense" immediately after the "not". Thus, Example 3 would read:

Not in any way that I can tell/recognize/discern

or, more fully,

It has nothing to do with metamorphic ability in any way that I can tell.

which could further be paraphrased as

It has no connection that I can discern with metamorphic ability.

Thus, the object of tell or recall in examples 3 and 4 can be considered to be way, although the "real" subject in example 3 is connection.