| dc.description.abstract | Quantization is an important part of signal processing. Several issues influence the performance
of a quantization algorithm. One is the “basis” we choose to represent the signal, another is how
we quantize the “basis” coefficients. We desire to explore these two things in this thesis. In the first chapter, we review frame theory and fusion frame theory. In the second chapter, we introduce a
popular quantization algorithm known as Sigma-Delta quantization and show how to apply it
to finite frames. Then we give the definition and properties of Sobolev duals which are optimized
duals associated to Sigma-Delta quantization. The contraction Sobolev duals depends on the frame, and in chapter 3, we prove that for any finite unit-norm frame, the best error bound that can be achieved
from the reconstruction with Sobolev duals in rth Sigma-Delta quantization is equal to order O(N^-r), where the error bound can be related to both operator norm and Frobenius norm. In the final chapter, we develop Sigma-Delta quantization for fusion frames. We construct stable first-order and high-order Sigma-Delta algorithms for quantizing fusion frame projections of f onto W_n, where W_n is an M_n dimension subspace of R^d. Our stable 1st-order quantizer uses only log2(Mn+1) bits per subspace. Besides, we give an algorithm to calculate the Kashin representations for fusion frames to improve the performance of the high-order Sigma-Delta quantization algorithm. Then by defining the left inverse and the canonical left inverse for fusion frames, we prove the property that the canonical left inverse has the minimal operator norm and Frobenius norm. Based on this property, we give the idea of Sobolev left inverses for fusion
frames and prove it leads to minimal squared error. | |
