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Regularization For Inverting The Radon Transform With Wedge Consideration I. Aganj 1, A. Bartesaghi 2, M. Borgnia 2, H.Y. Liao 3, G. Sapiro 1, S. Subramaniam.

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Presentation on theme: "Regularization For Inverting The Radon Transform With Wedge Consideration I. Aganj 1, A. Bartesaghi 2, M. Borgnia 2, H.Y. Liao 3, G. Sapiro 1, S. Subramaniam."— Presentation transcript:

1 Regularization For Inverting The Radon Transform With Wedge Consideration I. Aganj 1, A. Bartesaghi 2, M. Borgnia 2, H.Y. Liao 3, G. Sapiro 1, S. Subramaniam 2 1.Department of Electrical Engineering, University of Minnesota 2.Center for Cancer Research, National Institutes of Health 3.Institute for Mathematics and Its Applications, University of Minnesota

2 Radon Transform

3

4 Missing Wedge Problem

5

6 Reconstruction R:Radon Transformf: Reconstructed Image w:Projections

7 Reconstruction R:Radon Transformf: Reconstructed Image w:Projections

8 Reconstruction R:Radon Transformf: Reconstructed Image w:ProjectionsP(f): Penalty Function

9 Total Variation Low TVHigh TV Introduced in: L. Rudin, S. Osher, and C Fatemi “Nonlinear total variation. based noise removal algorithms” Physica D, vol. 60, pp. 259-–268, 1992.

10 Reconstruction by Total Variation Minimization R:Radon Transformf: Reconstructed Image w:Projections fidelitypenalty

11 Reconstruction by Total Variation Minimization Original image TV minimization Weighted Back Projection Phantom image originally from: A.H. Delaney and Y. Bresler, “Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,” IEEE. Trans. Imag. Proc., vol. 7, pp. 204-221, 1998.

12 Anisotropic Regularization Isotropic Total Variation: Anisotropic Total Variation:

13 Reconstruction by Anisotropic Regularization R:Radon Transformf: Reconstructed Image w:Projections

14 Reconstruction by Anisotropic Regularization This is the optimum direction, choosing more than one direction is redundant! R:Radon Transformf: Reconstructed Image w:Projections

15 Reconstruction by Anisotropic Regularization TV minimization Weighted Back Projection Anisotropic Regularization Original image

16 Results on Real Data (Bdellovibrio Bacterium) ARTSIRTAnisotropic TV

17 THANK YOU!

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