Computer Tomography: How t Works Medical maging and Pattern Recognition Lecture 7 Computed Tomography Oleh Tretiak Only one plane is illuminated. Source-subject motion provides added information. 2 How it Works First CT Scanner Original CT Scanner Head only One minute scanning time Two sections Ten minutes to compute images Extremely successful! 3 4 Before and After CT Contemporary CT 5 6
Fan-Beam Computer Tomography Contemporary Spiral Scanner Configuration: 4 slices per rotation maximum Other options are 32 slices or 6 slices 4 mm axial distance scanned in one rotation.4 sec per rotation 6 kw generator 7 8 Example: Head Example: Head Bleeding due to injury Can cause brain injury if not treated Blood between brain and dura, easy to treat FRONTAL CONTUSON WTH SUBARACHNOD HEMORRHAGE 9 Chest Study Abdomen Pneumothorax (air between lung and chest) Also note the bilateral lower lobe consolidation of lungs, right being greater than left. There is a chest tube within the right hemithorax. Appendicitis (arrow) Contrast agents in stomach and in blood 2 2
Mathematics of Computed Tomography Model for measurements Direct problem nverse problem Algorithm for computed tomography Direct Problem µ µ µ µ Beam with intensity µ µ µ µ enters body with varying a = e µ t, b = e µ 2 t a, attenuation c = e µ 3 t b, = e µ 4 t c Each layer has thickness t = e µ 4 t e µ 3t e µ 2t e µ t = e (µ +µ 2 +µ 3 +µ )t 4 ln( / ) = (µ + µ 2 + µ 3 + µ 4 )t 3 4 ntegral Equation Radon Transform (y) y y f(x,y) x (y) = exp( µ(x, y)dx) ln( / (y)) = µ(x, y)dx ln( / (t,)) = t x (t, ) µ(t cos lsin,t sin + lcos)dl g(t,) = t x (t, ) g(t, ) f (t cos lsin,t sin + lcos)dl 5 6 nverse Radon Transform Example Given: X-ray transmission measurements (t, ). Find µ(x, y) Given: g(t, ). Find f(x, y) Method: (a) convolution g (t,) = h(t )g(,)d (b) backprojection f (x, y) = g (x cos + y sin,)d 7.5 -.5 - - f(x, y).2.8.6.4.2 -.2 -.4 -.6 - -.5.5 Lines: g(t, ), same for all Dots: g (t, ), after convolution 8 3
Backprojection More Backprojection.5.5.5.5.5.5.5.5 -.5 - - -.5 - - -.5 - - -.5 - - -.5 - - -.5 - - One, two, and four angles of backprojection 8, 5, and 3 angle backprojection 9 2 Pictures Backprojection at 4, 6, and angles f(x, y) g(t, ) Theta horizontal 2 22 History More History 97, Joachim Radon Solved formal inverse problem. nterest in theory of integration and geometry 958, Simeon Tetelbaum of KP publishes a paper about X-ray tomography. Publishes valid inverse problem solution. 963 John Cormack publishes theoretical and experimental results. Experiment with cylindrical objects 972 Godfrey Hounsfield develops CT scanner 979 Hounsfield and Cormack receive Nobel prize in Medicine 23 24 4
Example of Contrast More Contrast Operations 2 bit image, full contrast range. Window for low densities Window for high densities 25 26 3-D mages Bronchoscopy Spiral scan procedures produce sets of sectional images suitable for 3-D imaging Resectioning: Compute new section plane Projection: Compute sums along rays Rendering: Segment image and show surface. Path View 27 28 Colonoscopy Medical Practice Path View n the fall of 23 Siemens became the first CT supplier ever to receive clearance from the FDA for a computer-aided technique of identifying nodules, that is, possible tumors, in the lung. CT is also used for the diagnosis of colon cancer: A virtual flight through the human colon can detect even the smallest polyps. f these are removed in time, an outbreak of colon cancer can very probably be prevented. 29 3 5
Comparison Left: A polyp seen with optical endoscopy. Right: View in virtual endoscopy. Summary Computer tomography became successful because it showed soft tissue dierences that could not be seen on X- rays. Evolution of high-speed (spiral scan) machines came about through improvements in X-ray detectors This has led to 3-D imaging methods Surgery planning Virtual endoscopy 3 32 6