Frame Paths and Error Bounds for Sigma-delta Quantization

We study the performance of finite frames for the encoding of vectors by applying first-order sigma-delta quantization to the frame coefficients. Our discussion is restricted to families of uniform tight frames, given by N vectors in d-dimensional Hilbert spaces, and mostly concerns the use of quantizers that assume only integer multiples of a step-size δ (mid-tread). We prove upper and lower bounds for the maximal reconstruction error in terms of geometric quantities of a path interpolating the sequence of frame vectors. We calculate these bounds for various known frame families and introduce the so-called d-circles and semicircles frames. The latter give a slight improvement in the upper bound over the harmonic frames. We prove that the maximal error for all of these families is asymptotically of the order δd/N , with numerical constants that are comparable to that of coordinatewise application of the sigma-delta algorithm.