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Yes, but of my sending the picture is not a constituent; of goes with approve to form a transitive verb. The gerund clause my sending the picture is the direct object of approve of.
My sending the picture is grammatical, and falutes slightly higher, but with pronouns there are many idioms. Gerund clauses have two complementizers: the normal Acc-ing complementizer (without him telling me), and the Poss-ing complementizer (without his telling me). Both are correct, both are common, but Acc-ing is somewhat more common in practice.
Acc
-ing complementizer (without him telling me), and thePoss
-ing complementizer (without his telling me). Both are correct, both are common, but Acc-ing is somewhat more common in practice.