Critical Parameter Values and Reconstruction Properties of Discrete Tomography : Application to Experimental Fluid Dynamics

• Autorzy: Petra S. , Schnörr C. , Schröder A.
• Języki publikacji: EN

Abstrakty

EN

We analyze representative ill-posed scenarios of tomographic PIV (particle image velocimetry) with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine: (i) the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.

Słowa kluczowe

EN

compressed sensing; large deviation; limited angle tomography; sparsity; tail bound; TomoPIV; underdetermined nonnegative linear systems

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2013
• Tom: Vol. 125, nr 3/4
• Strony: 285--312
• Opis fizyczny: Bibliogr. 19 poz., fot., rys., wykr.

Twórcy

autor

Petra S.

• Image and Pattern Analysis Group, University of Heidelberg, Speyerer Str. 6, 69115 Heidelberg, Germany

autor

Schnörr C.

• Image and Pattern Analysis Group, University of Heidelberg

autor

Schröder A.

• Institute of Aerodynamics and Flow Technology, German Aerospace Center, Bunsenstr. 10, 37073 Göttingen, Germany

Bibliografia

• [1] Azuma, K.: Weighted Sums of Certain Dependent Random Variables, Tohoku Math. J., 19(3), 1967, 357367.
• [2] Berinde, R., Indyk, P.: Sparse Recovery Using Sparse Random Matrices, 2008, MIT-CSAIL Technical Report.
• [3] Candes, E. J., Romberg, J., Tao, T.: Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information, IEEE Trans. Inf. Theor., 52(2), 2006, 489-509.
• [4] DasGupta, A.: Asymptotic Theory of Statistics and Probability, Springer, 2008.
• [5] Donoho, D.: Compressed Sensing, IEEE Trans. Inf. Theor.I, 52(4), 2006, 1289-1306.
• [6] Donoho, D., Tanner, J.: Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming, Proc. National Academy of Sciences, 102(27), 2005, 9446-9451.
• [7] Donoho, D., Tanner, J.: Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications, Discrete Comput. Geom., 43(3), 2010, 522-541.
• [8] Donoho, D., Tanner, J.: Precise Undersampling Theorems, Proceedings of the IEEE, 98(6), 2010, 913-924, ISSN 0018-9219.
• [9] Elsinga, G., Scarano, F., Wieneke, B., van Oudheusden, B.: Tomographic Particle Image Velocimetry, Exp. Fluids, 41, 2007, 933-947.
• [10] G.T. Herman, G., Kuba, A.: Discrete Tomography: Foundations, Algorithms and Applications, Birkhauser, 1999.
• [11] Khajehnejad, M., Dimakis, A., Xu, W., Hassibi, B.: Sparse Recovery of Positive Signals with Minimal Expansion, IEEE Trans. Sig. Proc., 59, 2011, 196-208.
• [12] Mangasarian, O., Recht, B.: Probability of Unique Integer Solution to a System of Linear Equations, Eur. J Oper. Res., 214(1), 2011, 27-30.
• [13] Petra, S., Schnorr, C.: TomoPIV meets Compressed Sensing, Pure Math. Appl., 20(1-2), 2009,49-76.
• [14] Petra, S., Schnorr, C.: Average Case Recovery Analysis of Tomographic Compressive Sensing, arXiv:1208.5894v2 [math.NA], August 30 2012.
• [15] Petra, S., Schroder, A., Schnorr, C.: 3D Tomography from Few Projections in Experimental Fluid Mechanics, in: Imaging Measurement Methods for Flow Analysis (W. Nitsche, C. Dobriloff, Eds.), vol. 106 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer, 2009, 63-72.
• [16] Vlasenko, A., Schnorr, C.: Variational Approaches for Model-Based PIV and Visual Fluid Analysis, in: Imaging Measurement Methods for Flow Analysis (W. Nitsche, C. Dobriloff, Eds.), vol. 106 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer, 2009, 247-256.
• [17] Wang, M., Xu, W., Tang, A.: A Unique ”Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices, IEEE Trans. Sig. Proc., 59(3), 2011, 1007-1016.
• [18] Wendel, J.: A Problem in Geometric Probability, Math. Scand., 11, 1962, 109-111.
• [19] Xu, W., Hassibi, B.: Efficient Compressive Sensing with Deterministic Guarantees Using Expander Graphs, Information Theory Workshop, 2007. ITW ’07. IEEE, 2007.