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This paper explains the development of a highly efficient progressive 3-D mesh geometry coder based on the region adaptive transform in the spectral mesh compression method. A hierarchical set partitioning technique, originally proposed for the efficient compression of wavelet transform coefficients in high performance wavelet based image coding methods, is proposed for the efficient compression of the coefficients of this transform. Experiments confirm that the proposed coder employing such a region adaptive transform has a high compression performance rarely achieved by other state of the art 3-D mesh geometry compression algorithms. A new, high performance fixed spectral basis method is also proposed for reducing the computational complexity of the transform. Many-to-one mappings are employed to relate the coded irregular mesh region to a regular mesh whose basis is used. To prevent loss of compression performance due to the low-pass nature of such mappings, transitions are made from transform based coding to spatial coding on a per region basis at high coding rates. Experimental results show the performance advantage of the newly proposed fixed spectral basis method over the original fixed spectral basis method in the literature that employs one-to-one mappings.

Constrained storage vector quantization, (CSVQ), introduced by Chan and Gersho (1990, 1991) allows for the stagewise design of balanced tree-structured residual vector quantization codebooks with low encoding and storage complexities. On the other hand, it has been established by Makhoul et al. (1985), Riskin et al. (1991), and by Mahesh et al. (see IEEE Trans. Inform. Theory, vol.41, p.917-30, 1995) that variable-length tree-structured vector quantizer (VLTSVQ) yields better coding performance than a balanced tree-structured vector quantizer and may even outperform a full-search vector quantizer due to the nonuniform distribution of rate among the subsets of its input space. The variable-length constrained storage tree-structured vector quantization (VLCS-TSVQ) algorithm presented in this paper utilizes the codebook sharing by multiple vector sources concept as in CSVQ to greedily grow an unbalanced tree structured residual vector quantizer with constrained storage. It is demonstrated by simulations on test sets from various synthetic one dimensional (1-D) sources and real-world images that the performance of VLCS-TSVQ, whose codebook storage complexity varies linearly with rate, can come very close to the performance of greedy growth VLTSVQ of Riskin et al. and Mahesh et al. The dramatically reduced size of the overall codebook allows the transmission of the code vector probabilities as side information for source adaptive entropy coding.