CT Image Reconstruction

CT Please read Ch 13. Homework is due 1 week from today at 4 pm.

Tomographic reconstruction  = 0 o detectors

The main idea  = 20 o detectors

Reconstruct the image of a non uniform sample using its x-ray projection at different angles The main idea  = 90 o detectors

The Sinogram Detectors position Projection angle    

Image reconstruction Back projection Filtered Back projection Iterative methods (CH 22)

Back-projection Given a sample with 4 different spatial absorption properties AB CD D1= A+B=7 D2=C+D=7  =0 o

AB CD D3= A+C=6  = 90 o D4= B+D=8 Back-projection

AB CD A+B=7 A+C=6 A+D=5 B+C=9 B+D=8 C+D=7 Back-projection 25 43

Real back-projection In a real CT we have at least 512 x 512 values to reconstruct We don’t know where one absorber ends where the next begins ~ 800,000 projections

Back projection The projection of a function is the radon transform of that function

Projections Are periodic in +/-  The radon transform of an image produces a sinogram

Central Slice Theorem Relates the 1 D Fourier transform of a projection of an object –F(p(x’)) at a given angle  To a line through the center of the 2D Fourier transform of the object at a given angle 

Central Slice Theorem

Why is it important? If you compute the 1D Fourier transform of all the projection (at all angles f) you can “fill” the 2 D Fourier transform of the object. The object can then be reconstructed by a simple 2D Fourier transform.

FILTERED back-projection If only the 2D inverse Fourier transform is computed you will obtain a “blurry” image. (it is intrinsic in inverse Radon) The blur is eliminated by deconvolution In filtered back projection a RAMP filter is used to filter the data

Homework Prove the center slice theorem. Use imrotate

Imaging in Matlab An image is a 2D matrix of numbers imread - reads an image file imwrite - writes an image to file