Book of Abstracts - Extract th Annual Meeting. March 23-27, 2015 Lecce, Italy. jahrestagung.gamm-ev.de

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1 GESELLSCHAFT für ANGEWANDTE MATHEMATIK und MECHANIK e.v. INTERNATIONAL ASSOCIATION of APPLIED MATHEMATICS and MECHANICS 86 th Annual Meeting of the International Association of Applied Mathematics and Mechanics March 23-27, 20 Lecce, Italy Book of Abstracts - Extract 20 jahrestagung.gamm-ev.de

2 Sun day 22 Time Scientific Program - Timetable Monday 23 Tuesday 24 Wednesday 25 Thursday 26 Friday 27 9: 10: ( in parallel) Moritz Diehl von Mises prize lecture ( in parallel) (14 in parallel) 11: 12: Registration Thomas Böhlke General Assembly Ferdinando Auricchio (11 in parallel) 13: Opening Univ. Chorus Performance Lunch Lunch Lunch Closing 14: : Prandtl Lecture Keith Moffatt Giovanni Galdi Enrique Zuazua Nikolaus Adams ( in parallel) Daniel Kressner Stanislaw Stupkiewicz Registration pre- opening 16: 17: 18: 19: 20: 21: Minisymposia & Young Reseachers' Minisymposia (10 in parallel) Opening reception at Castle of Charles V Poster session (14 in parallel) Public lecture Francesco D'Andria ( in parallel) Conference dinner at Hotel Tiziano Poster session ( in parallel)

3 GAMM 20 Università del Salento Table of contents YRMS4: Co-/Sparsity, Inverse Problems and Compressive Imaging 4 Recovering overcomplete sparse representations from structured sensing Needell - Krahmer - Ward Computational Aspects of Sparse Recovery Tillmann Joint reconstruction and segmentation from sparse Radon data Frikel - Storath - Weinmann - Unser Empirical phase transitions in sparsity-regularized X-ray CT Jørgensen Cosparse models and recovery algorithms for inverse problems in acoustics and electro-encephalography Bertin - Kitic - Albera - Gribonval

4 YRMS4 YRMS4: Co-/Sparsity, Inverse Problems and Compressive Imaging At the core of many inverse and imaging problems are signal models which in addition to stabilizing the solutions of these problems are also capable of dealing with undersampling. In this context, a very popular model is the sparse synthesis model. Here, the assumption is that the signals can be expressed as a linear combination of a few signal components from a dictionary. In the last decade, there has been a lot of theoretical and practical work dealing with this model in the field of compressive sensing. Despite their success, practical applications often do not meet the theoretical conditions that compressive sensing relies on. Hence, their success is yet to be mathematically grounded. The goal of this mini-symposium is to draw attention to the theoretical gap between theory and practice, and to give young researchers working in inverse problems, imaging and compressed sensing the opportunity to present their new results. GAMM 20 4

5 YRMS4 Monday, March 23 16:-17:10 (Giotto Room) Needell Recovering overcomplete sparse representations from structured sensing Felix Krahmer 1, Deanna Needell 2, Rachel Ward 3 1 Inst. Numerische und Angewandte Mathematik, Universität Göttingen 2 Claremont McKenna College 3 University of Texas at Austin In many signal processing applications, one wishes to acquire images that are approximately sparse in transform domains such as wavelets using frequency domain samples. Often the quality of the sparsity based model significantly improves when one considers redundant representation systems such as wavelet frames. To date, compressed sensing with redundant representation systems has, however, only been studied for measurement systems that have certain concentration properties, which is not the case for frequency domain samples. In this talk, we close this gap, providing more general reconstruction guarantees for signals that are sparse with respect to redundant systems. GAMM 20 5

6 YRMS4 Monday, March 23 17:10-17: (Giotto Room) Tillmann Computational Aspects of Sparse Recovery Andreas M. Tillmann Technische Universität Braunschweig The fundamental problem of sparse recovery from underdetermined linear measurements reads min x 0 such that Ax b δ, (1) where δ 0 is an estimate of measurement noise, x 0 denotes the number of nonzeros in x (i.e., the support size) and is some norm (often the l 2 -norm). This problem is well-known to be NP-hard in the strong sense and also difficult to approximate, even in the noiseless case δ = 0. In this talk, we discuss a selection of aspects from the sparse recovery context, with a focus on computational matters: Replacing the discrete objective by the l 1 -norm has become a standard approach to obtain sparse solutions, backed by empirical success as well as theoretical guarantees for when this strategy in fact solves (1). For the noiseless basis pursuit model min x 1 such that Ax = b, (2) in the presence of a sparse solution, a simple "heuristic optimality check" (HOC) procedure can improve both solution speed and quality of many different problem-specific algorithms [1]. We extend the HOC idea to several noise-aware l 1 -minimization models (for both the synthesis and analysis approach) and provide preliminary numerical results that demonstrate the potential effectiveness of HOC schemes in this context. It is well-known that basis pursuit (2) can be written as a linear program. We show that the converse result is also true (i.e., every LP can be transformed into a basis pursuit problem via a polynomial reduction) and sketch possible applications and complexity-theoretical implications. In general, verifying that a relaxation or heuristic such as basis pursuit indeed provides the sparsest possible solution is often as hard as solving (1) itself, since evaluating the strongest known conditions that guarantee recovery success for heuristics is also NP-hard [2, 3]. On the one hand, this gives rise to the question whether there are other polynomially solvable special cases of (1) besides those identified by the known recovery conditions. On the other, it motivates tackling (1) directly by exact methods from discrete and/or combinatorial optimization. In fact, there seem to be only very few attempts to solve (1) directly. Continuing the work from [4], we investigate branch-and-cut methods for binary set-covering integer program reformulations of the sparse recovery task. In this context, to make use of the observation that (measurement) equations with few nonzero coefficients may provide stronger information regarding the optimal support, we encounter another interesting combinatorial problem named matrix sparsification, which can also be seen as a special dictionary learning problem [5]. We briefly discuss work-in-progress on the branch-and-cut approach and the connection to matrix sparsification. References [1] D. A. Lorenz, M. E. Pfetsch, A. M. Tillmann. Solving basis pursuit: Subgradient algorithm, heuristic optimality check, and solver comparison. ACM T. Math. Software, to appear (2014). [2] A. M. Tillmann. Computational aspects of compressed sensing. Doctoral dissertation, TU Darmstadt, [3] A. M. Tillmann, M. E. Pfetsch. The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing. IEEE Trans. Inf. Theory 60(2) (2014), [4] S. Jokar, M. E. Pfetsch. Exact and approximate sparse solutions of underdetermined linear equations. SIAM J. Sci. Comput. 31(1) (2008), [5] A. M. Tillmann. On the computational intractability of exact and approximate dictionary learning. IEEE Signal Process. Lett. 22(1) (20), 49. GAMM 20 6

7 YRMS4 Monday, March 23 17:-17:50 (Giotto Room) Frikel Joint reconstruction and segmentation from sparse Radon data Martin Storath 1, Andreas Weinmann 2, Jürgen Frikel 2, and Michael Unser 1 1 Biomedical Imaging Group, EPFL Lausanne, Switzerland 2 Helmholtz Center Munich, Germany In many tomographic imaging setups the reconstruction and segmentation are classically performed in two separate steps. In such cases, the performance of the segmentation step greatly depends on the quality of the reconstruction and, hence, often yields unsatisfactory results. In particular, if only a few noisy projections are available, such strategies lead to particularly poor performance. To overcome this, combined reconstruction and segmentation (one-step) strategies have shown to yield promising results. In this talk, we present an approach for joint reconstruction and segmentation using the Potts model, cf. [1]. More specifically, we propose a new algorithmic approach to the non-smooth and nonconvex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider sparse angle data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from 7 angular views only. As further applications, we also consider reconstructions from spherical Radon data which arises in photoacoustic tomography. References [1] M. Storath, A. Weinmann, J. Frikel, and M. Unser. Joint image reconstruction and segmentation using the Potts model. To appear in Inverse Problems, Preprint at arxiv: GAMM 20 7

8 YRMS4 Monday, March 23 17:50-18:10 (Giotto Room) Jørgensen Empirical phase transitions in sparsity-regularized X-ray CT Jakob S. Jørgensen Technical University of Denmark, Denmark Sparsity regularization in X-ray computed tomography (CT) has shown large potential for accurate reconstruction from reduced data, see e.g. [1, 2], leading to substantially reduced patient x-ray exposure in medical imaging and shorter scan times in materials science. One driving factor for research in sparsity-regularized methods has been developments in compressed sensing (CS), connecting the possible undersampling level to the image sparsity level. CS offers guarantees of accurate reconstruction of sparse images from reduced data under suitable assumptions on the measurement matrix. However, existing CS guarantees, e.g. based on the restricted isometry property (RIP) do not apply to the structured and sparse measurement matrices of X-ray CT, while other coherence-based guarantees lead to essentially useless bounds. The empirical success of sparsity regularization for X-ray CT therefore remains unexplained from a theoretical viewpoint. Using empirical phase diagrams we study image recoverability from X-ray CT data as function of image sparsity and undersampling level. We consider sparsity in the image and gradient domains and reconstruction by 1-norm and TV regularization. We demonstrate empirically that X-ray CT exhibits a pronounced relation between image sparsity and the possible undersampling level as well as sharp phase transitions for certain image classes. We demonstrate that recovery performance of X-ray CT is almost comparable with the standard CS Gaussian matrix ensemble, which suggests a similarity with CS. However, we also highlight some important differences, for example, that structure in the non-zero locations of the image and not just sparsity level affects recoverability in X-ray CT. Furthermore we investigate the phase-transition behavior theoretically for a simple class of images consisting of sparse superpositions of radial basis functions (RBFs). RBFs are an often-used alternative to pixels and voxels for image representation in X-ray CT and commonly referred to as blobs. The use of RBFs in X-ray CT image reconstruction is motivated by a reduction of model errors arising from anisotropies of the rectangular pixels and voxels. The rotational symmetry of RBFs implies a regularity that can be exploited to analyze the setup from a CS perspective. The analysis exploits non-negativity of the sampling matrix and the signal and utilizes new mathematical tools from CS theory in the form of expander graphs. This approach has recently been used to derive average-case recovery guarantees for certain restricted discrete tomography setups [6]. Here, we address how to generalize the approach to the more complicated scanning setups used in standard (non-discrete) tomography, for example parallel and fan-beam geometries. We compare the theoretical recovery guarantees with empirical phase diagrams. References [1] E.Y. Sidky, X. Pan. Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Phys. Med. Biol. 53 (2008), [2] J. Bian, J.H. Siewerdsen, X. Han, E.Y. Sidky, J.L. Prince, C.A. Pelizzari, X. Pan. Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT. Phys. Med. Biol. 55 (2010), [3] J.S. Jørgensen, E.Y. Sidky. How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT. Submitted (2014). [4] J.S. Jørgensen, C. Kruschel, D.A. Lorenz. Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT. Inverse Probl. Sci. Eng. (2014), 23 pp. DOI: / [5] J.S. Jørgensen, E.Y. Sidky, P.C. Hansen, X. Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Submitted (2014). Available from: [6] S. Petra, C. Schnörr: Average case recovery analysis of tomographic compressed sensing. Linear Algebra Appl. 441 (2014), GAMM 20 8

9 YRMS4 Monday, March 23 18:10-18: (Giotto Room) Bertin Cosparse models and recovery algorithms for inverse problems in acoustics and electro-encephalography Srđan Kitić 1, Laurent Albera 1,2,3, Nancy Bertin 4,1, Rémi Gribonval 1 1 Inria, Centre Inria Rennes Bretagne Atlantique 2 Inserm UMR LTSI - Université Rennes 1 4 IRISA - CNRS UMR 6074 Sparse data models are powerful tools for solving ill-posed inverse problems. In their most general formulation, linear inverse problems can be expressed as the problem of recovering a signal x R n from measurements y R m : y Mx, (1) where M R m n is a transfer matrix modeling the signal acquisition process. In most cases, this problem is ill-posed, and solving it implies the use of additional knowledge in the problem formulation, in order to regularize it. Sparse data models have emerged as a pervasive tool to perform such regularization. The now ubiquitous sparse synthesis model states that the signal (x R n ) is constructed by a linear combination of a few column vectors (atoms) taken from a dictionary D R n d, such that x = Ds, (2) where the coefficient vector s contains very few non-zero elements and is called a sparse representation of x. A counterpart of sparse synthesis is the sparse analysis or cosparse data model, which has gained attraction more recently [1]. This model assumes that there exist an analysis operator A R b n (b n), such that the following analysis representation: z = Ax (3) of the signal x is sparse. A signal x whose analysis representation z = Ax R b contains l zero elements is said to be l-cosparse. The analysis and the synthesis sparse models are nominally equivalent in only one special case: when A = D 1, where the analysis operator and the dictionary are square, non-singular matrices [2]. In this presentation, we will show two settings where knowledge at hand is encoded through known Partial Differential Equations (PDEs), governing the underlying physical phenomena, which can be used to design the dictionary D (in the sparse synthesis case) or the analysis operator A (in the cosparse analysis model). We will present the resulting regularization framework based on the sparse analysis models for two problems governed by linear partial differential equations: acoustic pressure source localization [4], governed by the wave equation, and brain electrical current source localization [3], which can be described by the Poisson equation. The cosparse model can very naturally incorporate this physical knowledge in an analysis operator A resulting from a direct discretization of the underlying PDE. When solved by specially tailored optimization algorithm based on Alternating Direction Method of Multipliers, the underlying inverse problems can be solved efficiently and with much better scalability than their synthesis-based counterpart formulations. References [1] S. Nam, M.E. Davies, M. Elad, R. Gribonval. The cosparse analysis model and algorithms. Applied and Computational Harmonic Analysis 34 (2013). [2] M. Elad, P. Milanfar, R. Rubinstein. Analysis versus synthesis in signal priors. Inverse problems 23 (2007). [3] L. Albera, S. Kitić, N. Bertin, G. Puy, R. Gribonval. Brain source localization using a physics-driven structured cosparse representation of EEG signals. MLSP (2014). [4] S. Kitić, N. Bertin, R. Gribonval. Hearing behind walls: localizing sources in the room next door with cosparsity. ICASSP (2014). GAMM 20 9