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Tomographic Image Reconstruction

Tomographic Image Reconstruction. Dr. Rajeev Srivastava. Overview. Image creation Image reconstruction Brute force Iterative techniques Backprojection Filtered backprojection. Image Creation. Tomogram image of a slice taken through a 3D volume. Projection

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Tomographic Image Reconstruction

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  1. Tomographic Image Reconstruction Dr. Rajeev Srivastava

  2. Overview • Image creation • Image reconstruction • Brute force • Iterative techniques • Backprojection • Filtered backprojection

  3. Image Creation • Tomogram • image of a slice taken through a 3D volume • Projection • Attenuation profile through the object • The projection function represents the summation of the attenuation coefficients along a given X-ray path

  4. Image Creation • Sinogram • 2D data set – result of stacking all the projections together • Radon transform • Transformation of a function (image) into the sinogram, p(r) • Computes projections of an image along specified directions

  5. Image Reconstruction • Process of estimating an image from a set of projections • Several algorithms exist to accomplish this task: • Brute force • Iterative techniques • Backprojection • Filtered backprojection

  6. Brute Force • projection set defines a system of simultaneous linear equations - can be solved using algorithms from linear algebra • not practical for real systems (can have hundreds of simultaneous equations for a single slice)

  7. Iterative Reconstruction • Known as algebraic reconstruction technique – ART, consists of three steps: • Make an initial guess at the solution • Compute projections based on the guess • Refine the guess based on the weighted difference between the actual projections and the desired projections • Original reconstruction method used in medical imaging • Works, but is slow and susceptible to noise

  8. Backprojection • Propagates sinogram back into the image space along the projection paths (inverse Radon transform) • Backprojection image is a blurred version of the original image • The projection theorem (central slice theorem) - provides an answer to inverse Radon transform problem • Set of 1D Fourier transform of the Radon transform of a function is the 2D Fourier transform of that function

  9. Fourier Reconstruction • Calculate the 1D Fourier transform of all projections [p(r) = P(k)] • Place P(k) on polar grid to get P(k,) • Resample in Cartesian space to get F(kx,ky) • Calculate the 2D inverse Fourier transform of F(kx,ky) to get f(x,y) – image • Resultant image is noisy

  10. Fourier Reconstruction

  11. Filtered Backprojection • Take projections - sinogram • Transform data to the frequency domain • Filter data • Inverse transform – smoothed sinogram • Backproject

  12. Filtered Backprojection

  13. Filtered Backprojection • ramp filter + nearest neighbor algorithm • ramp & Hamming filter + nearest neighbor algorithm • ramp filter + linear interpolation • ramp & Hamming filter + linear interpolation 2 1 4 3

  14. References • Image Processing: The Core of Nuclear Cardiology, Scott M. Leonard, MS, CNMT, Northwestern University, ppt presentation • Xiang Li , Jun Ni and Ge Wang, Parallel iterative cone beam CT image reconstruction on a PC cluster, Journal of X-Ray Science and Technology 13 (2005) 63–72 • HARISH P. HlRlYANNAlAH, X-ray Computed Tomography for Medical Imaging, IEEE SIGNAL PROCESSING MAGAZINE

  15. Thank you

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