An image and surface representation based on regularization theory is introduced in this paper. This representation is based on a hybrid model derived from the physical membrane and plate models. The representation, called the lambda tau-representation, has two dimensions; one dimension represents smoothness or scale while the other represents the continuity of the image or surface. It contains images/surfaces sampled both in scale space and the weighted Sobolev space of continuous functions. Thus, this new representation can be viewed as an extension of the well-known scale space representation. We have experimentally shown that the proposed hybrid model results in improved results compared to the two extreme constituent models, i.e., the membrane and the plate models. Based on this hybrid model, a generalized edge detector (GED) which encompasses most of the well-known edge detectors under a common framework is developed. The existing edge detectors can be obtained from the generalized edge detector by simply specifying the values of two parameters, one of which controls the shape of the filter (tau) and the other controls the scale of the filter (lambda). By sweeping the values of these two parameters continuously, one can generate an edge representation in the lambda tau space, which is very useful for developing a goal-directed edge detection scheme for a specific task. The proposed representation and the edge detector have been evaluated qualitatively and quantitatively on several different types of image data such as intensity, range, and stereo images.