Extrapolation and iteration for the problem of LFOV

Dr. Shuangren Zhao

Research Associate

Radiation Physics Department

Princess Margaret Hospital

What is LFOV and ROILFOV is “Limited field of view”

ROI is region of interest

Crop is the image inside the ROI

Crop outside ROI is the image outside the ROI

Projections we study are truncated

The Influence of truncated projections

Phantoms1. Shepp-Logan head phantom2. Body phantom

3. Modified Shepp-Logan head phantom4. Strong Modified Shepp-Logan head phantom5. further Modified Shepp-Logan head phantom 6. crops for the ROI

Truncated projectionsand their direct reconstruction

Extrapolations

Zero extrapolation *0 (a)Constant extrapolation *c (b)Linear extrapolation *(bx+c)Exponential extrapolation *exp(-x/αL)Exponential extrapolation *exp(-(x/αL)^2) (c)Cos extrapolation *cos(x)Quadratic extrapolation *(ax^2+bx+c) (d)Mixed extrapolation *(ax^2+bx+c)exp(-x/α) (e)Mixed extrapolation *(ax^2+bx+c)exp(-(x/α)^2) (f)Original projection without extrapolation (g)

Extrapolations for phantom 3

Extrapolations for phantom 3 and 4

Quadratic extrapolation(ax^2+bx+c) (d)

Projection should positive:

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Update from quadratic extrapolation to mixed extrapolation {exp(-x/aL)(ax^2+bx+c)}

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Different fits for the boundary values:1. The values of projections 2. The differential values of the projections

Update for fitting boundary values

Update for the mixed extrapolation of (ax2+bx+c)exp(-x/αL)

Update for the mixed extrapolation of (ax2+bx+c)exp(-x/αL)

The distances of reconstructed images to the image of phantom

ideal distance: reconstruction with non-truncated projections.

Reconstructions with different extrapolationsusing phantom 1

 

Reconstructions with different extrapolationsusing phantom 2

Reconstructions with different extrapolations using phantom 3

 

 

 

 

Reconstructions with different extrapolations using phantom 4

 

Reconstructions with different extrapolations using phantom 5

 

Iterative reconstruction algorithm:

Projections filter (for phantom 2)

Iterative reconstruction results for the phantom 1with exp(-(x/αL)^2) extrapolation α=0.5

Iterative reconstruction results for the phantom 2 with exp(-(x/αL)^2) extrapolation α =0.5

Iterative reconstruction results for the phantom 3 with exp(-(x/αL)^2) extrapolation α =0.5

Iteration results for phantom 5 with exp(-(x/αL)^2)(ax^2+bx+c) and exp(-(x/αL)^2)

extrapolation α=0.5

Further find the optimal parameters for for phantom 5

The stability of the parameters

Further find the optimal parameters for for phantom 4

Further find the optimal parameters for for phantom 5

The stability of the parameters

Find the optimal parameters for for phantom 4

Reconstruction with …menthod

Errors of iterative reconstruction without truncation

Phantom 5Iterative reconstructionwithout truncation

Crop of Phantom

iterative Reconstructionwith truncation

Reconstruction without truncation

Errors reconstruction without truncation

Errors of iterative reconstruction with truncation

Number of Projections=180

Distance=0.0253Distance=0.0221Distance=0.0348Distance=0

Reconstruction with …menthod

Phantom 5

Crop of Phantom

Reconstruction without truncation

Iterative reconstructionwithout truncation

Errors reconstruction without truncation

Errors of iterative reconstruction without truncation

iterative Reconstructionwith truncation

Errors of iterative reconstruction with truncation

Projections:360, 1st=mix 2, 2ed=exp 2, α1=0.65, α2=0.068,k=-1.04

Distance=0 Distance=0.0167 Distance=0.0145 Distance=0.0191

Contradiction

Our shield (extrapolation) is the best shield, it can resist all spears in the world.

Our spear (iteration) is the best spear, it can destroy all shields in the world.

Which one would you like to buy? The extrapolation or the iteration?