The introduction of vanishing moment recovery (VMR) functions in our recent work (also called “fundamental functions” in an independent paper by Daubechies, Han, Ron, and Shen) modifies the so-called “unitary extension principle” to allow the construction of compactly supported affine frames with any desirable order of vanishing moments up to the order of polynomial reproduction of the given associated compactly supported scaling function. The objective of this paper is to unify and extend certain tight-frame results in the two papers mentioned above, with primary focus on the investigation of tight frame generators with minimum supports. In particular, a computational scheme to be described as an algorithm is developed for constructing such minimum-supported tight frame generators. An example is included as an illustration of this algorithm.