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 obtained from sampling along a path in a given d-dimensional Hilbert space. We prove upper and lower bounds for the maximal Euclidean reconstruction error in terms of geometric quantities for the path. While the upper bounds are independent of the particular quantizer used, the lower bounds require quantizers that assume only integer multiples of a step-size δ (mid-tread). We calculate these bounds for various known frame families obtained from sampling and introduce new such paths, the so-called d-circles and semicircles frames. The latter give a slight improvement in the upper bound over the harmonic frames. The bounds we derive for N frame vectors in dimension d and quantization step-size δ are of the order δd3/2/N, with numerical constants that are comparable to that of coordinatewise application of the sigma–delta algorithm.