Stable Edge-Adaptive Multiscale Decompositions Using Updated Normal Offsets

The normal offset decomposition is an adaptive (nonlinear) multiscale data transform, constructed in a way similar to the lifting scheme for wavelet transforms. The main difference lies in the fact that the detail coefficients are not wavelet coefficients corresponding to a wavelet basis function. Wavelet coefficients carry detail information situated at the location of the basis function. Besides detail information, normal offset coefficients also contain directional/geometrical information, i.e., information on where to locate the details. The normal offset decomposition is a jump-adaptive transform leading to a sparse representation of edges in images and sharp reconstructions of these edges. A major issue in this paper is the design of a new class of wavelet transforms that can be extended to normal offset decompositions. The combination of normal offsets and a stable lifting scheme is nontrivial. A necessary condition for an -stable lifted wavelet transform is that the wavelet basis functions have a vanishing integral (i.e., at least one vanishing moment). Although the normal offset decomposition is not a basis transform in the strict sense, a similar criterion for stability can be established and instability is equally a problem in a naive implementation. This paper constructs an ldquoupdatedrdquo normal offset decomposition that is stable and therefore appropriate for use in applications that sensitive to the numerical condition, such as applications involving (heavy) noise. The new scheme combines numerical stability, fast computation, and sharp edge representations.