We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a/spl grave/ trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.

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The curvelet transform for image denoising

Global observations of nonlinear mesoscale eddies

List of Contributors. 1. The Development of Automated Diffraction in Scanning and Transmission Electron Microscopy D.J. Dingley. 2. Theoretical Framework for Electron Backscatter Diffraction V. Randle. 3. Representation of Texture in Orientation Space K. Rajan. 4. Rodriques-Frank Representations of Crystallographic Texture K. Rajan. 5. Fundamentals of Automated EBSD S.I. Wright. 6. Studies on the Accuracy of Electron Backscatter Diffraction Measurements M.C. Demirel, B.S. El-Dasher, B.L. Adams, A.D. Rollett. 7. Phase Identification Using Electron Backscatter Diffraction in the Scanning Electron Microscope J.R. Michael. 8. Three-Dimensional Orientation Imaging D.J. Jensen. 9. Automated Electron Backscatter Diffraction: Present State and Prospects R.A. Schwarzer. 10. EBSD: Buying a Systems A. Eades. 11. Hardware and Software Optimization for Orientation Mapping and Phase Identification P.P. Camus. 12. An Automated EBSD Acquisition and Processing System P. Rolland, K.G. Dicks. 13. Advanced Software Capabilities for Automated EBSD S.I. Wright, D.P. Field, D.J. Dingley. 14. Strategies for Analysis of EBSD Datasets W.E. King, J.S. Stolken, M. Kumar, A.J. Schwartz. 15. Structure-Property Relations: EBSD-Based Materials-Sensitive Design B.L. Adams, B.L. Henrie, L.L. Howell, R.J. Balling. 16. Use of EBSD Data in Mesoscale Numerical Analyses R. Becker, H. Weiland. 17. Characterization of Deformed Microstructures D.P. Field, H. Weiland. 18. Anisotropic Plasticity Modeling Incorporating EBSD Characterization of Tantalum and Zirconium J.F. Bingert, G.C. Kaschner, T.A. Mason, P.J. Maudlin, G.T. Gray III. 19. Measuring Strains Using Electron Backscatter Diffraction A.J. Wilkinson. 20. Mapping Residual Plastic Strain in Materials Using Electron Backscatter Diffraction E.M. Lehockey, Yang-Pi Lin, O.E. Lepik. 21.EBSD Contra TEM Characterization of a Deformed Aluminum Single Crystal Xiaoxu Huang, D.J. Jensen. 22. Continuous Recrystallization and Grain Boundaries in a Superplastic Aluminum Alloy T.R. McNelley. 23. Analysis of Facets and Other Surfaces Using Electron Backscatter Diffraction V. Randle. 24. EBSD of Ceramic Materials J.K. Farrer, J.R. Michael, C.B. Carter. 25. Grain Boundary Character Based Design of Polycrystalline High Temperature Superconducting Wires A. Goyal. Index.

Electron Backscatter Diffraction in Materials Science

3D electron microscopy in the physical sciences: the development of Z-contrast and EFTEM tomography

Diffusion MRI of Complex Neural Architecture

Providing basic information about the properties of radon transform, this book contains examples and documents a wide variety of applications. It offers guidance to literature related to transform, and is aimed at those with a basic undergraduate background in mathematics.

Stanley Roderick Deans

Papers

The Radon Transform and Some of Its Applications