Is the Synthesis Matrix Invertible?
In [5] there is no explicit statement that the synthesis matrix [P Q] is invertible. The reader may wonder if there can be a case where the analysis matrix cannot be produced by inversion. Without diving too far into the details of [5], it is relevant [5] shows [P Q] to be of the following block form: In this form, O are the rows corresponding to the vertices already a part of the lower level of subdivision, N are the rows corresponding to the vertices created by the splits, and a is a special “magic” matrix derived in [5]. Although not explicitly clear, a section in [5] regarding the whether O is invertible for primal schemes (Loop is a primal scheme) implies that O is invertible for Loop. The text in [5] states that most primal schemes have invertible O, with Catmull-Clark as the only noted exception. Therefore we aim to show here only that if O is invertible, then so is [P Q]. It can be shown easily through block matrix multiplication that the inverse of [P Q] as given above is as f
