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Computed Tomography Tomos = slice CT scan • Mathematical idea developed by Radon in 1917 • Cormack did the instrumentation research 1963 published it • A practical x-ray CT scanner was built by Hounsfield. When was the first computer introduced in laboratories? The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles Projection Radon transform Inverse back-projection is used to reconstruct the original image from the projected image CT images • Maps of relative linear attenuation of tissue • µ relative attenuation coefficient is expressed in Hounsfield units (HU) also known as CT numbers • HUx = 1000.(µx - µwater)/µwater • HUwater = 0 • HU depends on photon energy CT images • FOV (field of view) Diameter of the region being imaged (head 25 cm) • Voxel Volume element in the patient – Pixel area x slice thickness CT scan generations • 1st generation – Translate rotate, pencil beam • 2nd generation – Translate rotate, fan beam • 3rd generation – Rotate rotate, fan beam • 4th generation – Rotate, wide fan • 5th generation – Fixed array of detectors X-ray tube • High voltage xray tubes • For large focal spots (1mm) ->high power (60kW), smaller spots (0.5 mm) low power rating (below 25kW) • Copper and aluminum filters used for beam hardening effect • Collimators both in x ray tube and detector Detectors • Measure radiation through patient • High xray efficiency • Scintillation – Crystals produce visible range photons coupled with PMT • Xenon gas ionization detector – Gas chamber anode and cathode at potential. Used in 3rd gen., stable. BREAK CT Image Reconstruction CT • Please read Ch 13. • Homework is due 1 week from today at 12 pm. Tomographic reconstruction detectors = 0o The main idea detectors = 90o The main idea Reconstruct the image of a non uniform sample using its x-ray projection at different angles The Sinogram Detectors position Image reconstruction • Back projection • Filtered Back projection • Iterative methods (CH 22) Back-projection • Given a sample with 4 different spatial absorption properties A B D1= A+B=7 C D D2=C+D=7 =0o Back-projection A B C D = 90o D3= A+C=6 D4= B+D=8 Back-projection 9 A B C D 6 8 7 A+B=7 A+C=6 A+D=5 B+C=9 B+D=8 C+D=7 7 5 2 5 4 3 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 f x, y x cos y sin x'dxdy f p x ' 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 f x, y x cos y sin x'dxdy f p x ' p x ' f (x, y) p ( ) F cos sin 2D FT of an image at angle 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