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# DFG Project: Geometric Multiscale Analysis

Duration: | September 2010 - August 2012 |

Project Heads: | Gitta Kutyniok |

Researchers: | Xiaosheng Zhuang |

Support: | DFG KU 1446/14 |

# Project Description

**Introduction**

Multivariate problems in applied mathematics are typically governed by anisotropic phenomena such as singularities concentrated on lower dimensional embedded manifolds or edges in digital images. Wavelets – a representation system that can be generated by dilations and translations of a single function – are nowadays considered to be indispensable mutliscale efficient/sparse encoding systems for a wide range of wide range of theoretically to practically oriented tasks. Two of the main reasons are their optimal sparse representations for 1-D smooth functions except finite number of discontinuities and their fast implementation for digital data on a rectangular grid based on a so-called multiresolution analysis.

However, wavelets do not perform equally well in higher dimensions due to the fact that they are designed to encode solely isotropic phenomena. Therefore, there have been many interesting attempts to find directional multiscale representation systems for efficiently encoding anisotropic features in 2-D settings. The most promising one is the system of shearlets, which are generated by parabolic scaling, shearing, and translation of some particular functions. The parabolic scaling and shear operators enable efficiently detection of directional features for high-dimensional data.

Directional representation systems have so far been employed in diverse applications such as geophysics for recovery of missing data or sparse approximation of seismic data, etc. However, the true goal is to provide methodologies to efficiently encode higher dimensional signals. The crucial dimension for this long-term goal is 3-D, since this is the first time that anisotropic features occur in different dimensions. Another reason for the urgency of develop methodologies for efficient analysis of 3-D signals arises from emerging new technologies, e.g., biological imaging, where new sensor types for the analysis of cells on a molecular level such as Confocal Microscopy or for the analysis of larger scale biological data produce 3-D signals in which tubes and sheets form 1-D and 2-D geometric features. Thus, there is a pressing need to derive a fundamental mathematical understanding of efficient encoding of anisotropic phenomena in 3D and provide fast implementation of those methods.

**Research Programe**

The main objective in this project is the development of multiscale representation systems possessing multiresoluton analysis for efficient decomposition and analysis of geometric features such as singularities or more complex objects in 3-D signals in the continuous or digital setting. The focus will be two-fold: we intend to provide a thorough mathematical analysis of key properties of these systems such as their sparsity and robustness as well as to investigate theoretical and computational aspects of problem complexes such as efficient data acquisition, missing data, and geometric separation. A further goal is to provide a webpage with a publicly available software-package implementing the developed methodologies.

Our first sub goal is the development and analysis of representation systems, which in a first step are inspired by shearlets as the most advanced directional multiscale representation system in the 2-D setting:

(I) Theoretical Analysis of Multiscale Representation Systems for 3-D Signals

a. Development of shearlet-like systems and variants of those with continuous and discrete parameters.

b. Analysis of wavefront set detection for distributions on 3-D as well as of sparsity properties.

c. Introduction of associated Multiresolution Analysis.

d. Investigate the extendibility of these systems to match geometric features of a more complex structure, for instance, by considering hybrid systems between directional representation systems and Gabor systems.

A second task complex then consists of the understanding of computational issues multiscale transforms of 3-D signals as well as of developing a digital theory.

(II) Computational Aspects of the Associated Transforms

a. Development of an associated theory for the systems developed in (I) now for digital 3D data.

b. Efficient implementation of the associated transforms.

c. Theoretical and experimental study of applying compressed sensing technique for transform coefficient acquisition.

d. Provision of a publicly available, well-tested software-package containing the documented codes.

After thoroughly studyin different types of multiscale representation systems and providing efficient implementations, the third task complex shall consist of the application of those systems to a selected variety of model problemsmotivated by seismic, astronomical, and biological applications:

(III) Application of the Developed methodologies

a. Inpainting problemsusing l_1 minimization techniques.

b. Geometric separation for 3-D signals containing morphologically distinct features using phase space heurstics.

# Project Output

**Publications**King, E. J., Kutyniok, G., and Zhuang, X. (2012) Analysis of inpainting via Clustered Sparsity and Microlocal Analysis,

*Journal of Mathematical Imaging and Vision, Submitted*

Kutyniok, G., Shahram, M., and Zhuang, X. (2011) ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm.* SIAM Journal on Imaging Sciences, Submitted.*

Bodmann, B. G., Kutyniok, G., and Zhuang, X. (2011) Coarse quantization with the fast digital shearlet transform. *Wavelet XI, San Diego, CA, SPIE Proc. (8318)*.

King, E. J., Kutyniok, G., and Zhuang, X. (2011) Analysis of data separation and recovery problems using clustered sparsity. *Wavelet XI, San Diego, CA, SPIE Proc. (8318). *

Donoho, D. L., Kutyniok, G., Shahram, M., and Zhuang, X. (2011) A rational design of a digital shearlet transform.* The 9th International Conference on Sampling Theory and Applications, Singapore (SampTA'11).*

**Software-Package**

ShearLab

**Cooperations**

Prof. Dr. Bernhard G. Bodmann

Prof. Dr. David D. Donoho

Dr. Emily J. King

Dr. Morteza Shahram