Mini Workshop on Applied Harmonic Analysis and Tomography


Time: August 20, 2013, 13:00-16:40
Location: DTU Compute, Building/Room 324/020.
Registration: Participation is free, and there is no registration. All are welcome.
Organizers: Kim Knudsen (SC) and Jakob Lemvig (MAT).


Schedule


Time
Speaker and Institution
Title
13:00-13:50
Morten Nielsen, Aalborg University
On approximation with redundant dictionaries
13:50-14:40
Jürgen Frikel, Technische Universität München
Curvelet sparse regularization for x-ray computed tomography
14:40-15:00
Coffee Break Coffee Break
15:00-15:50
Jan Hesthaven, Ecole Polytechnique Fédérale de Lausanne Compressed Sensing and its Application to MRI and fMRI
15:50-16:40
Ole Christensen, Technical University of Denmark A preliminary report on integral representations of higher dimensional functions


Abstracts:
  • Morten Nielsen, Aalborg Univeristy
On approximation with redundant dictionaries
Abstract. Data approximation using sparse linear expansions from overcomplete dictionaries has become a central theme in signal and image processing with applications ranging from data acquisition (compressed sensing) to denoising and compression.
For a given dictionary, we can also study best m-term approximation rates for any specific function. Interestingly, the notions of sparse expansions and certain asymptotic approximation rates are closely linked in the case of nice non-redundant dictionaries (e.g., an orthonormal basis in a Hilbert space.)
In this talk, I will explore the link between sparse expansions from an overcomplete dictionary and asymptotic approximation rates. Redundancy complicates the analysis, and we show that the close link between the two notions fails in general. However, using a probabilistic approach, we show that the close link is retained for 'many' redundant dictionaries in a finite dimensional setting.

  • Jürgen Frikel, Technische Universität München
Curvelet sparse regularization for x-ray computed tomography
Abstract. The reconstruction of tomographic slices from x-ray CT data is an interesting and challenging problem in medical imaging. In particular, there are many applications, such as digital breast tomosynthesis, dental tomography, electron microscopy etc., where the data is available at a limited angular range only. In this case the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this talk, we introduce a reconstruction framework which is based based on curvelet expansion. The curvelet formulation allows preservation of edges as well as an exact analytic representation of the system matrix. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will present the practical relevance of these results.
  • Jan Hesthaven, Ecole Polytechnique Fédérale de Lausanne
Compressed Sensing and its Application to MRI and fMRI
Abstract. While MRI (Magnetic Resonance Imaging) is used routinely for medical imaging and diagnosis, the use of functional MRI (fMRI) remains a topic of active research, partly because the need for temporal resolution and the ability to collect data is severely constrained by physiological limits. To improve the temporal resolution it is therefore necessary to be smarter with the data that can be collected as there are limitations on how much data can be collected. In this talk we shall first give an overview of MRI and fMRI technology to better understand the limitations and continue by introducing ideas of compressed sensing to improve performance. While this improves performance considerably, it remains insufficient to reach our goals and we propose a new approach in which prior edge information from a T1-weighted image is combined with compressed sensing. We shall illustrate the performance of these algorithms and demonstrate the ability to reach a temporal resolution close to the limit set by signal frequencies in the human brain. We conclude the talk by discussing a number of open questions and challenges.
  • Ole Christensen, Technical University of Denmark
A preliminary report on integral representations of higher dimensional functions
Abstract. Grafakos and Sansing have shown how to obtain directionally sensitive time-frequency decompositions in L2(Rn) based on Gabor systems in L2(R); the key tool is the "ridge idea", which lifts a function of one variable into a function of several variables. We generalize their result by showing that similar results hold starting with general frames for L2(R), both in the setting of discrete frames and continuous frames. This allows to apply the theory for several other classes of frames, e.g., wavelet frames and shift-invariant systems.