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ECE 5810: Computed Imaging Systems Week 10: Image Reconstruction from Projections : 

ECE 5810: Computed Imaging Systems Week 10: Image Reconstruction from Projections © K. Dobson, 2008 Reference: Chapters 12 & 13, The Essential Physics of Medical Imaging, Bushberg Computed Tomography, Kalender, Verlag, 2000. Chapter 12, Intermediate Physics for Medicine and Biology, 3rd Ed., Hobbie. Chapter 3, Principles of Computerized Tomographic Imaging, Kak and Slaney, IEEE Press Medical Physics and Biomedical Engineering, Brown, et al, IoP Publishing.

Image Reconstruction: 

Image Reconstruction 2-D Fourier transform review Backprojection Filtered Backprojection

1) Basics – Fourier Transform of an image f(x,y): 

1) Basics – Fourier Transform of an image f(x,y) Transformed image is described in k-space, where kx= 2π/x, ky=2π/y are spatial frequencies Ref: Hobbie

1) What does the DFT image represent ?: 

1) What does the DFT image represent ?

2) Fourier Slice Theorem – MATLAB example: 

2) Fourier Slice Theorem – MATLAB example

3) Image Reconstruction : projection data: 

3) Image Reconstruction : projection data

3) Backprojection - Linear single backprojection: 

3) Backprojection - Linear single backprojection

3) Backprojection – two linear projections: 

3) Backprojection – two linear projections

3) Backprojection – Multiple linear projections: 

3) Backprojection – Multiple linear projections

3) Image Reconstruction – concept of backprojection: 

3) Image Reconstruction – concept of backprojection

3) Relating θ to x-y coordinate back-projection: 

3) Relating θ to x-y coordinate back-projection

3) The Radon Transform – Kak and Slaney: 

3) The Radon Transform – Kak and Slaney Ref: Kak and Slaney

3) Fourier Slice Theorem – Kak and Slaney: 

3) Fourier Slice Theorem – Kak and Slaney Ref: Kak and Slaney

3) Fourier Slice Theorem – Kak and Slaney: 

3) Fourier Slice Theorem – Kak and Slaney

2) Fourier Slice Theorem – what does the DFT image represent ?: 

2) Fourier Slice Theorem – what does the DFT image represent ?

2) Fourier Slice Theorem – what does the DFT image represent ?: 

2) Fourier Slice Theorem – what does the DFT image represent ?

2) Fourier Slice Theorem – what does the DFT image represent ?: 

2) Fourier Slice Theorem – what does the DFT image represent ?

MATLAB implementation of a test phantom: 

MATLAB implementation of a test phantom

MATLAB implementation of the Radon transform (i.e. projection data): 

MATLAB implementation of the Radon transform (i.e. projection data)

MATLAB implementation of the Inverse Radon transform (i.e. image reconstruction from projection data) : 

MATLAB implementation of the Inverse Radon transform (i.e. image reconstruction from projection data)

MATLAB implementation of the Inverse Radon transform (i.e. image reconstruction from projection data): 

MATLAB implementation of the Inverse Radon transform (i.e. image reconstruction from projection data)

Effect of projection number: 

Effect of projection number

Effect of noise in the projection data : 

Effect of noise in the projection data sd=0 sd=0.05 sd=0.10 sd=0.20 sd=0.50 imshow(theta,xp,RN1,[],'notruesize'), colormap(jet), colorbar; P = phantom('Modified Shepp-Logan',200); [RP,xp] = radon(P,theta); RN1=RP.*(1+sd*randn(size(RP)));