We give a characterization, in terms of computational complexity, of the family \recU \ of the unary picture languages that are tiling recognizable. We introduce quasi-unary strings to represent unary pictures and we prove that any unary picture language $L$ is in \recU \ \iff the set of all quasi-unary strings encoding the elements of $L$ is recognizable by a one-tape nondeterministic Turing machine that is space and head-reversal linearly bounded. In particular, the result implies that the family of binary string languages corresponding to tiling-recognizable square languages lies between \text{\scshape NTime}$(2^n)$ and \text{\scshape NTime}$(4^n)$. %This also provides a tool for the construction of nontiling-recognizable unary languages. This also implies the existence of a nontiling-recognizable unary square language that corresponds to a binary string language recognizable in nondeterministic time $O(4^n \log n)$.