Fast Piecewise Constant Approximation of Images

In this work, we present a Least-Square-Error (LSE), recursive method for generating piecewise-constant approximations of images. The method is developed using an optimization approach to minimize a cost function. The cost function, proposed here, is based on segmenting the image, recursively, using Binary Space Partitionings (BSPs) of the image domain. We derive a LSE necessary condition for the optimum piecewise-constant approximation, and use this condition to develop an algorithm for generating the LSE, BSP-based approximation. The proposed algorithm provides a significant reduction in the computational expense when compared with a brute force method. As shown in the paper, the LSE algorithm generates efficient segmentations of simple as well as complex images. This shows the potential of the LSE approximation approach for image coding applications. Moreover, the BSP-based segmentation provides a very simple (yet flexible) description of the regions resulting from the partitioning. This makes the proposed approximation method useful for performing image affine transformations (e.g., rotation and scaling) which are common in computer graphics applications.