Compressing as well as the best tiling of an image

We investigate the task of compressing an image by using different probability models for compressing different regions of the image. We introduce a class of probability models for images, the k-rectangular tilings of an image, that is formed by partitioning the image into k rectangular regions and generating the coefficients within each region by using a probability model selected from a finite class of N probability models. For an image of size n×n, we give a sequential probability assignment algorithm that codes the image with a code length which is within O(k log Nn/k) of the code length produced by the best probability model in the class. The algorithm has a computational complexity of O(Nn³). An interesting subclass of the class of k-rectangular tilings is the class of tilings using rectangles whose widths are powers of two. This class is far more flexible than quadtrees and yet has a sequential probability assignment algorithm that produces a code length that is within O(k log Nn/k) of the best model in the class with a computational complexity of O(Nn² log n) (similar to the computational complexity of sequential probability assignment using quadtrees)