Image representations using multiscale differential operators

Differential operators have been widely used for multiscale geometric descriptions of images. The efficient computation of these differential 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 differential representations of images using different 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 multidirectional derivative components. For practical implementation, efficient pyramid-like algorithms are derived using the spline technique for both the the composition 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.