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Radon Transform, Ridgelets and their Applications in Image Processing

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- 1. Radon Transform, Ridgelets and their Applications in Image Processing Image Analysis Project By: Vanya V.Valindria Eng Wei Yong -VIBOT 4-
- 2. Outline Radon Transform Wavelets and Radon Transform Ridgelets Transform Applications in Image Processing
- 3. Radon Transform Computes the projection of an image matrix along specific axes
- 4. Radon Transform
- 5. Sinograms
- 6. Sinograms
- 7. Wavelets and Radon In even dimension Radon Transform is not localProjection over all hyper planes is required for reconstruction of image In odd dimension only hyperplane that is in the neighborhood of x is requiredLess radiation exposed to patient is desired Hence, application of wavelet theory to RT
- 8. Wavelets and Radon The expansion of Radon using wavelets analysis: The wavelets coefficients can be used for Inverse Radon Transform
- 9. Why Ridgelets? Weakness in wavelets only effective to adapt in point singularities A ridgelet is effective for higher dimensional singularities (line, curve, etc.) Next generation of wavelets
- 10. What is Ridgelets?
- 11. Ridgelets and WaveletsContinuous Ridgelet ContinuousTransform Wavelets Transform For a 2D Separable CWT Wavelet in 2D is the tensor product of: Line Point b, θ b1, b2
- 12. Ridgelets, Radon and Wavelets Wavelets Ridgelets Point Lines Radon!!
- 13. Relation of Ridgelets with Radon transform Original Image: f Radon Domain: Rf Radon Transform Ridgelets Domain: Df
- 14. Fourier Slice Theorem Links Radon and Fourier transform F-1 F (Rf) = F(u,v) f(x,y) Base for reconstruction
- 15. Relation between Transform 2D Fourier Domain Radon Domain Ridgelet Domain
- 16. Application of Radon Transform CT (Computed Tomography)-Scan
- 17. Application of Radon Transform CT-Scan Acquisition
- 18. Application of Radon Transform CT – Tomography Methods
- 19. Application of CT Technique in MATLABOriginal Image Sinogram -150 60 -100 50 Radon -50 Transform 40 0 x 30 50 20 100 10 150 0 0 20 40 60 80 100 120 140 160
- 20. Application of CT - FBPReconstruction in MATLAB
- 21. CT Reconstruction fromWavelets Coefficients Wavelets Coefficients Reconstruction from wavelet coefficients
- 22. Application of Radon Transform
- 23. Application of Ridgelets Transform• Line Detection Original Image From wavelet From ridgelet component coefficients
- 24. Conclusion Radon transform is the key method for tomographic imaging Wavelets can be applied for Radon localization and inverse Radon transform Ridgelets can be derived from Radon and wavelets transform Radon transform and Ridgelets have wide applications in image processing
- 25. References Berenstein, C. Radon Transforms, Wavelets, and Applications. Technical Research Report: Engineering Research Center Program the University of Maryland. Hiriyannaiah, H. P. X-ray computed tomography for medical imaging. IEEE Signal Processing Magazine, March 1997: 42-58. Chen, G.Y. Image Denoising with Complex Ridgelets. 2007. Pattern Recognition 40, pp.578- 585. Carre, P., Andres.Eric. Discrete Analytical Ridgelet Transform. 2004. Signal Processing 84, pp.2165 – 2173. Toft, Peter. The Radon Transform: Theory and Implementation. Denmark: Technical University; 1996. Ph.D. Thesis. Farrokhi, F.R. Wavelet-Based Multiresolution Local Tomography. 2007. IEEE Transcations on Image Processing, Vol.6 No.10. Candes, E., Donoho, D.L.: Ridgelets: A Key to Higher-Dimensional Intermittency? .1999. Phil. Trans. R. Soc. Lond. A, 2495–2509. Zhao, S. Welland, G. Wavelet Sampling and Localization Schemes for the Radon Transform in Two Dimensions. 1997. Journal in Applied Mathematics, Vol.57, No.6 pp.1749 – 1762. Do MN, Vetterli M. The finite ridgelet transform for image representation. 2003. IEEE Transactions on Image Processing 1:16–28. Hasegawa,M. A Ridgelet Representation of Semantic Objects Using Watershed Segmentation. 2004. International Symposium on Communication and Information Technologies, Japan.

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