Image Representations Using Multiscale Diierential Operators

Diierential operators have been widely used for multiscale geometric descriptions of images. The eecient computation of these diierential operators is always desirable. Moreover, it has not been clear whether such representations are invertible. For certain applications, it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it. In this paper, such problems are studied. We consider multiscale diierential representations of images using diierent types of operators such as the directional derivative operators and Laplacian operators. In particular, we provide a general approach to represent images by their multiscale and multi-directional derivative components. For practical implementation, eecient pyramid-like algorithms are derived using spline technique for both the decomposition and reconstruction of images. It is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales; therefore, these representations can be used for edge based image processing purposes. Furthermore, the intrinsic relationships of the proposed representations with the compact wavelet models, and some classical multiscale approaches are also elucidated in the paper.