We study construction of structured regular quantizers for overcomplete expansions in RN. Our goal is to design structured quantizers which allow simple reconstruction algorithms with low complexity and which have good performance in terms of accuracy. Most related work to date in quantized redundant expansions has assumed that the same uniform scalar quantizer was used on all the expansion coefficients. Several approaches have been proposed to improve the reconstruction accuracy, with some of these methods having significant complexity. Instead, we consider the joint design of the overcomplete expansion and the scalar quantizers (allowing different step sizes) in such a way as to produce an equivalent vector quantizer (EVQ) with periodic structure. The construction of a periodic quantizer is based on lattices in RN and the concept of geometrically scaled- similar sublattices. The periodicity makes it possible to achieve good accuracy using simple reconstruction algorithms (e.g., linear reconstruction or a small lookup table).