Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames

Gitta Kutyniok, Jakob Lemvig, Wang-Q Lim
2012 SIAM Journal on Mathematical Analysis  
We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized 3D cartoon-like images. This function class will have two smoothness parameters: one parameter β controlling classical smoothness and one parameter α controlling anisotropic smoothness. The class then consists of piecewise C β -smooth functions with discontinuities on a piecewise C α
more » ... h surface. We introduce a pyramid-adapted, hybrid shearlet system for the 3D setting and construct frames for L 2 (R 3 ) with this particular shearlet structure. For the smoothness range 1 < α ≤ β ≤ 2 we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of nonlinear N -term approximations. Introduction. Recent advances in modern technology have created a new world of huge, multidimensional data. In biomedical imaging, seismic imaging, astronomical imaging, computer vision, and video processing, the capabilities of modern computers and high-precision measuring devices have generated 2D, 3D, and even higher-dimensional data sets of sizes that were infeasible just a few years ago. The need to efficiently handle such diverse types and huge amounts of data has initiated an intense study in developing efficient multivariate encoding methodologies in the applied harmonic analysis research community. In neuroimaging, e.g., fluorescence microscopy scans of living cells, the discontinuity curves and surfaces of the data are important specific features since one often wants to distinguish between the image "objects" and the "background," e.g., to distinguish actin filaments in eukaryotic cells; that is, it is important to precisely capture the edges of these 1D and 2D structures. This specific application is an illustration that important classes of multivariate problems are governed by anisotropic features. The anisotropic structures can be distinguished by location and orientation or direction, which indicates that our way of analyzing and representing the data should capture not only location, but also directional information. This is exactly the idea behind so-called directional representation systems which by now are well developed and understood for the 2D setting. Since much of the data acquired in, e.g., neuroimaging, are truly 3D, analyzing such data should be performed by 3D directional representation systems. Hence, in this paper, we therefore aim for the 3D setting. * 1.1. Dimension three. In the present paper we will consider sparse approximations of cartoon-like images using shearlets in dimension three. The step from the 1D setting to the 2D setting is necessary for the appearance of anisotropic features at all. When further passing from the 2D setting to the 3D setting, the complexity of anisotropic structures changes significantly. In two dimensions one "only" has to handle one type of anisotropic features, namely, curves, whereas in three dimensions one has to handle two geometrically very different anisotropic structures: curves as 1D features and surfaces as 2D anisotropic features. Moreover, the analysis of sparse approximations in dimension two depends heavily on reducing the analysis to affine subspaces of R 2 . Clearly, these subspaces always have dimension and codimension one in two dimensions. In dimension three, however, we have subspaces of codimension one and two, and one therefore needs to perform the analysis on subspaces of the "correct" codimension. Therefore, the 3D analysis requires fundamental new ideas. Finally, we remark that even though the present paper only deals with the construction of shearlet frames for L 2 (R 3 ) and sparse approximations of such, it also illustrates how many of the problems that arise when passing to higher dimensions can be handled. Hence, once it is known how to handle anisotropic features of different dimensions in three dimensions, the step from three to four dimensions can be dealt with in a similar way as also the extension to even higher dimensions. Therefore the extension of the presented result in L 2 (R 3 ) to higher dimensions L 2 (R n ) should be, if not straightforward, then at least be achievable by the methodologies developed. Modeling anisotropic features. The class of 2D cartoon-like images consists, as mentioned above, of piecewise C 2 -smooth functions with discontinuities on a piecewise C 2 -smooth curve, and this class has been investigated in a number of recent publications. The obvious extension to the 3D setting is to consider functions of three variables being piecewise C 2 -smooth function with discontinuities on a piecewise C 2smooth surface. In some applications the C 2 -smoothness requirement is too strict, and we will, therefore, go one step further and consider a larger class of images also containing less regular images. The generalized class of cartoon-like images in three dimensions considered in this paper consists of 3D piecewise C β -smooth functions with discontinuities on a piecewise C α surface for α ∈ (1, 2]. Clearly, this model provides us with two new smoothness parameters: β being a classical smoothness parameter and α being an anisotropic smoothness parameter; see Figure 1 .1 for an illustration. This image class is unfortunately not a linear space as traditional smoothness spaces, e.g., Hölder, Besov, or Sobolev spaces, but it allows one to study the quality of the performance of representation systems with respect to capturing anisotropic features, something that is not possible with traditional smoothness spaces.
doi:10.1137/110844726 fatcat:f4n3bktlsrfx3pp4u7asrcphqi