Efficient Multiscale and Multidirectional Representation of 3D Data using the 3D Discrete Shearlet Transform Bart Goossens, Hiêp Luong, Jan Aelterman, Aleksandra Pižurica and Wilfried Philips Ghent University, Dept. of Telecommunications and Information Processing, TELIN-IPI-IBBT, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium ABSTRACT In recent years, there has been a lot of interest in multiresolution representations that also perform a multidirectional anal- ysis. These representations often yield very sparse representation for multidimensional data. The shearlet representation, which has been derived within the framework of composite wavelets, can be extended quite trivially from 2D to 3D. However, the extension to 3D is not unique and consequently there are different implementations possible for the discrete transform. In this paper, we investigate the properties of two relevant designs having different 3D frequency tilings. We show that the first design has a redundancy factor of around 7, while in the second design the transform can attain a re- dundancy factor around 3.5, independent of the number of analysis directions. Due to the low redundancy, the 3D shearlet transform becomes a viable alternative to the 3D curvelet transform. Experimental results are provided to support these findings. Keywords: multiresolution transforms, wavelets, shearlets 1. INTRODUCTION Many applications require processing of large volumetric (3D) or even higher dimensional data sets in an effective manner and in a relatively short amount of time. Examples are in video processing, medical image processing (CT, MRI, DTI, ...), remote sensing, hyper spectral imaging. One of the well studied approaches in this area are multiscale/multiresolution so- lutions. This has led to various techniques to efficiently compress large data sets, multiresolution approaches to reconstruct CT/MRI images, signal/image denoising and restoration methods, techniques to detect object edges and salient features, ... Multiresolution transforms decompose an image in a natural way: the image is approximated by successively adding detail information to it in subsequent refinement steps. Such approach is effective as natural images (e.g., photographic images) are often low-pass in nature and because details are usually localized and clustered. The wavelet transform is a multiresolution transform that offers a good spatial localization of features and is excellent for signals with point-wise singularities. However, in higher dimensions, the transform can not efficiently represent other types of singularities (for example object edges in images): many non-zero coefficients are then needed to accurately represent these singularities. To overcome this limitation, there has been a lot of interest lately for transforms with a better directional selectivity, such as steerable pyramids, 1 dual-tree complex wavelets, 2 Marr-like wavelet pyramids, 3 2-D (log) Gabor transforms, 4,5 contourlets, 6 ridgelets, 7,8 curvelets 9 and surfacelets. 10 The shearlet transform 11–13 is a recent sibling in the family of multiresolution representations, and can be seen as an extension of the wavelet transform that combines multiresolution theory with geometric transforms (dilations and shear transforms). As we will demonstrate in this paper, this approach gives a lot of flexibility with respect to applications, while still having full control on the mathematical properties of the transform. For example, in 3D, the redundancy factor of the transform (i.e. the ratio of the number of transform coefficients and the number of input samples) can be made relatively low and independent of the number of analysis directions, while still offering the desired properties such as shift-invariance and energy preservation. This paper is organized as follows: in Section 2, we give background information on the topic of 3D shearlets. In Section 3 we explain two different designs for a 3D shearlet transform, both designs can be useful in certain applications and have certain advantages and disadvantages. A practical algorithm for computing these 3D shearlet transforms is explained in Section 4. Next, a comparison between the two 3D shearlet transform designs is given in Section 5. Finally, Section 6 concludes this paper. Further author information: (Send correspondence to B. Goossens) B. Goossens: E-mail: bart.goossens@telin.ugent.be, Telephone: +32 (0) 9 264 79 66