Tomographic Image Reconstruction

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Tomographic Image Reconstruction - PowerPoint PPT Presentation

Tomographic Image Reconstruction. Miljenko Markovic. 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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Presentation Transcript

Tomographic Image Reconstruction

Miljenko Markovic

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
• Attenuation profile through the object
• The projection function represents the summation of the attenuation coefficients along a given X-ray path
Image Creation
• Sinogram
• 2D data set – result of stacking all the projections together
• Transformation of a function (image) into the sinogram, p(r)
• Computes projections of an image along specified directions
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
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)
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
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
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
Filtered Backprojection
• Take projections - sinogram
• Transform data to the frequency domain
• Filter data
• Inverse transform – smoothed sinogram
• Backproject
Filtered Backprojection
• ramp filter + nearest neighbor algorithm
• ramp & Hamming filter + nearest neighbor algorithm
• ramp filter + linear interpolation
• ramp & Hamming filter + linear interpolation

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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