Efficient image representation using binary space partitioning trees

Abstract

Images are traditionally represented by a set of picture elements, or pixels, defined over a discrete two or three-dimensional lattice. Pixel-based representation of images, however, requires a large amount of storage memory and processing power when performing complex image operations. Several image processing, computer vision, and computer graphics applications have resorted to employing more efficient representations of images. Tree-based representations, such as quadtrees and octrees, have received a great deal of attention by researchers in many imaging, vision and graphics areas. Binary Space Partitioning (BSP) trees have been used in computer graphics as an efficient representation of objects with planar faces (i.e., polyhedra). In this dissertation we propose using BSP trees as an efficient representation of images for image compression and geometric transformation applications. The BSP tree approach partitions the image domain, in a recursive manner, using arbitrarily oriented lines. The result of this partitioning is a set of convex (unpartitioned) regions, known as the BSP tree cells. In order to generate an efficient segmentation of the image, the number of cells has to be as small as possible such that the image signal within each cell is smooth. Therefore, the most critical aspect of a BSP tree construction method is the criterion used for selecting the partitioning lines. Here, we introduce a boundary-based approach and an optimization method for selecting the lines of a BSP tree. Under the former approach, the partitionings are based on the boundaries of the objects within the image under consideration. By using an optimization method, the lines are selected based on a Least-Square-Error (LSE) criterion. As demonstrated in this work, both the boundarybased and LSE methods provide efficient segmentation when applied to complex images. In addition, we develop a multiresolution algorithm for constructing a multiresolution BSP tree. In this case, the coarse (high frequency) information of the image is represented by the nodes close to the root node, and the fine (low frequency) details are contained within the nodes close to the leaves of the BSP tree. Based on these BSP tree construction methods, a coding system is proposed for compressing still images. A hierarchical method for coding the partitioning lines of the tree is developed. Meanwhile, the image signal within the unpartitioned regions (cells) is coded using low order polynomials (zero and first order). Moreover, an optimum pruning algorithm is used to reduce the number of bits required for encoding the BSP tree representation while minimizing distortion. By employing this BSP tree-based compression method on typical images (of human faces), high compression ratios in the range of 50-100 were obtained. Unlike other tree-based representations (e.g. quadtrees), and due to the flexible representation of the BSP tree approach (namely, the use of arbitrarily oriented lines), line-preserving transformations can be applied very efficiently to BSP trees without the need for reconstructing the tree under consideration. In here, we show the result of performing affine and perspective projection transformations on the BSP tree representation of images. These geometric manipulations were achieved by performing parametric transformations on the partitioning lines of the tree.