Curvelets, Multiresolution Representation, and Scaling Laws Emmanuel J. Cand` es and David L. Donoho Department of Statistics Stanford University Stanford, CA 94305-4065, USA ABSTRACT Curvelets provide a new multiresolution representation with several features that set them apart from existing representations such as wavelets, multiwavelets, steerable pyramids, and so on. They are based on an anisotropic notion of scaling. The frame elements exhibit very high direction sensitivity and are highly anisotropic. In this paper we describe these properties and indicate why they may be important for both theory and applications. Keywords: Edges. Partitioning. Subband Filtering. Local Fourier Transform. Ridge functions. Ridgelets. Multiscale ridgelets. Pyramids. 1. INTRODUCTION Images have edges. Viewing an image as a mathematical object, one might think of an image as an otherwise smooth function with discontinuities along curves, a description which would not be suitable, however, for all kinds of images. Indeed, images of natural scenes are more than just smooth luminance surfaces separated by step discontinuities; for instance, we do not make any claims about the representation of textures, which are typically unsmooth. Our intention is rather to describe a class of images where edges are clearly the dominating features, e.g. cartoons or geometric images. 1.1. Our Viewpoint: Harmonic Analysis The curvelet construction 1,2 was originally developed for providing efficient representations of smooth objects with discontinuities along curves; the underlying motivation being to apply this construction to classical image processing problems such as: • Data compression. Compression of digitally acquired image data. • Image restoration, image reconstruction or edge-preserving regularization. Enhancement (noise re- moval) of digital images possibly obtained via indirect measurements as in tomography. 3 One of the main challenges here is to develop smoothing or reconstruction techniques that would smooth out the ‘flat’ part of the image without blurring the edges. Our viewpoint is that of modern harmonic analysis whose aim is to develop new representation systems. That is, one is searching for new collection of templates or “elementary forms” that will serve for both the analysis and the synthesis of an object under study. The most classical example is, of course, that of Further author information: Send correspondence to emmanuel@stat.stanford.edu