Xu Dan, Do Minh N
IEEE Trans Image Process. 2006 Oct;15(10):3225-30. doi: 10.1109/tip.2006.877479.
Adaptive multiscale representations via quadtree splitting and two-dimensional (2-D) wavelet packets, which amount to space and frequency decompositions, respectively, are powerful concepts that have been widely used in applications. These schemes are direct extensions of their one-dimensional counterparts, in particular, by coupling of the two dimensions and restricting to only one possible further partition of each block into four subblocks. In this paper, we consider more flexible schemes that exploit more variations of multidimensional data structure. In the meantime, we restrict to tree-based decompositions that are amenable to fast algorithms and have low indexing cost. Examples of these decomposition schemes are anisotropic wavelet packets, dyadic rectangular tilings, separate dimension decompositions, and general rectangular tilings. We compute the numbers of possible decompositions for each of these schemes. We also give bounds for some of these numbers. These results show that the new rectangular tiling schemes lead to much larger sets of 2-D space and frequency decompositions than the commonly-used quadtree-based schemes, therefore bearing the potential to obtain better representation for a given image.
通过四叉树分裂和二维（2-D）小波包实现的自适应多尺度表示，分别相当于空间和频率分解，是已在应用中广泛使用的强大概念。这些方案是其一维对应方案的直接扩展，特别是通过二维的耦合，并限制每个块仅进一步划分为四个子块的一种可能方式。在本文中，我们考虑更灵活的方案，这些方案利用多维数据结构的更多变化。同时，我们限制在适合快速算法且索引成本低的基于树的分解。这些分解方案的示例包括各向异性小波包、二进矩形平铺、单独维度分解和一般矩形平铺。我们计算这些方案中每种方案的可能分解数。我们还给出其中一些数的界限。这些结果表明，新的矩形平铺方案比常用的基于四叉树的方案导致二维空间和频率分解的集合大得多，因此有可能为给定图像获得更好的表示。
