1. Radon Transform, Ridgelets and theirApplications in Image ProcessingImage Analysis ProjectBy: Vanya V.ValindriaEng 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 isnot localProjection over all hyper planes is requiredfor 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 usedfor Inverse Radon Transform 9. Why Ridgelets? Weakness in wavelets only effective to adapt in point singularities Aridgelet iseffectiveforhigher 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 CWTWavelet in 2D is the tensor product of:LinePoint b, b1, b2 12. Ridgelets, Radon and WaveletsWavelets RidgeletsPoint Lines Radon!! 13. Relation of Ridgelets with Radon transform Original Image: fRadon Domain: Rf Radon Transform Ridgelets Domain: Df 14. Fourier Slice Theorem Links Radon and Fourier transformF-1F (Rf) = F(u,v)f(x,y) Base for reconstruction 15. Relation between Transform 2D Fourier DomainRadonDomainRidgeletDomain 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 400x 30 50 20100 10150 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 RidgeletsTransform Line DetectionOriginal Image From wavelet From ridgeletcomponentcoefficients 24. Conclusion Radon transform is the key method for tomographic imaging Waveletscan 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 ProcessingMagazine, 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: TechnicalUniversity; 1996. Ph.D. Thesis. Farrokhi, F.R. Wavelet-Based Multiresolution Local Tomography. 2007. IEEE Transcations onImage Processing, Vol.6 No.10. Candes, E., Donoho, D.L.: Ridgelets: A Key to Higher-Dimensional Intermittency? .1999.Phil. Trans. R. Soc. Lond. A, 24952509.Zhao, S. Welland, G. Wavelet Sampling and Localization Schemes for the Radon Transformin 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. IEEETransactions on Image Processing 1:1628.Hasegawa,M. A Ridgelet Representation of Semantic Objects Using WatershedSegmentation. 2004. International Symposium on Communication and InformationTechnologies, Japan.