A new class of fast shape-adaptive orthogonal transforms and their application to region-based image compression

Region-based approaches to image and video compression have been very actively explored in the last few years. It is widely expected that they will result in rate/quality gains and expanded functionalities. In such approaches, one of the essential problems is the representation of luminance and color in arbitrarily shaped regions. For rectangular blocks extracted from natural images, the discrete cosine transform (DCT) has been found to perform close to the eigentransform. Although for arbitrarily shaped regions orthogonalization-based procedures have been shown to perform very well, their computational complexity and memory requirements are prohibitive for today's technology. Therefore, other approaches are presently investigated, and particular attention is paid to low implementation complexity. In this paper, we propose a new class of orthogonal transforms that self-adapt to arbitrary shapes. The new algorithms are derived from flow graphs of standard fast transform algorithms by a suitable modification of certain butterfly operators. First, we show how to derive a shape-adaptive transform from the discrete Walsh-Hadamard transform (DWHT) flow graph. Then, we discuss modifications needed to arrive at a DCT-based shape adaptive transform. We give implementation details of this transform, and compare its computational complexity with several well-known approaches. We also evaluate the energy compaction performance of the new transform for both synthetic and natural data. We conclude that the proposed DCT-based shape-adaptive transform gives a very beneficial compaction/complexity ratio compared to other well-known approaches. The complexity of the new method does not exceed the complexity of two nonadaptive DCT's on a circumscribing rectangle, and therefore, unlike other tested methods with comparable energy compaction, it is suitable for large regions. This property should prove very valuable in the future when true region-based image/video compression methods are developed.