1Lucas Parra, CCNYCity College of New Yor kBME I5000: Biomedical ImagingLecture 4Computed TomographyLucas C. Parra, [email protected] some slides inspired by lecture notes of Andreas H. Hilscher at Columbia University.Blackboard: http://cityonline.ccny.cuny.edu/2Lucas Parra, CCNYCity College of New Yor kSchedule1. Introduction, Spatial Resolution, Intensity Resolution, Noise2. X-Ray Imaging, Mammography, Angiography, Fluoroscopy3. Intensity manipulations: Contrast Enhancement, Histogram Equalization4. Computed Tomography5. Image Reconstruction, Radon Transform, Filtered Back Projection6. Positron Emission Tomography7. Maximum Likelihood Reconstruction8. Magnetic Resonance Imaging9. Fourier reconstruction, k-space, frequency and phase encoding 10. Optical imaging, Fluorescence, Microscopy, Confocal Imaging11. Enhancement: Point Spread Function, Filtering, Sharpening, Wiener filter12. Segmentation: Thresholding, Matched filter, Morphological operations13. Pattern Recognition: Feature extraction, PCA, Wavelets14. Pattern Recognition: Bayesian Inference, Linear classification3Lucas Parra, CCNYCity College of New Yor k Biomedical ImagingImaging Modality Year Inventor WavelengthEnergyPhysical principleX-Ray1895Röntgen(Nobel 191)3-100 keV Measures variable tissueabsorption of X-RaysSingle PhotonEmission Comp.Tomography(SPECT) 1963Kuhl, Edwards 150 keV Radioactive decay.Measures variableconcentration of radioactiveagent.Positron EmissionTomography (PET)1953Brownell,Sweet150 keV SPECT with improved SNRdue to increased number ofuseful events.Computed AxialTomography(CAT) 1972Hounsfield,Cormack(Nobel 1979)keV Multiple axial X-Rayviews to obtain 3D volumeof absorption.Magnetic ResonanceImaging (MRI)1973Lauterbur,Mansfield(Nobel 2003)GHz Space and tissue dependentresonance frequency of kernspin in variable magneticfield. Ultrasound 1940-1955many MHz Measures echo of sound attissue boundaries.CT: Computed Tomography = CAT: Computed Axial Tomography4Lucas Parra, CCNYCity College of New Yor kCT - Origine●Mathematical basis developed by Radon (1917)●Idea popularized by Cormack (1963)●First practical x-ray CT scanner by Hounsfield (1971)5Lucas Parra, CCNYCity College of New Yor kCT – then and now1971 2000Original axial CT image from the dedicated Siretom CT scanner. Ability to see the soft tissue structures of the brain, including the black ventricles for the first time.128x128 pixel1-4 hours acquisition time1-5 days computationAxial CT image of a normal brain using a state-of-the-art CT system. 512 x 512 pixel 0.35 sec acquisition time1 sec computation6Lucas Parra, CCNYCity College of New Yor kCT - Imaging PrincipleComputed Axial Tomography: Multiple x-ray projections are acquired around the object and a 2D image is computed from those projections.Idea: Reconstruct 2D attenuation distribution µ(x,y) from multiple 1D x-ray projections g( ) taken at different angles φ .g x =∫dy  x , y x , yxy7Lucas Parra, CCNYCity College of New Yor kCT - Imaging PrinciplexyComputed Axial Tomography: Multiple x-ray projections are acquired around the object and a 2D image is computed from those projections.Idea: Reconstruct 2D attenuation distribution µ(x,y) from multiple 1D x-ray projections g( ) taken at different angles φ .8Lucas Parra, CCNYCity College of New Yor kCT - Imaging PrinciplexyComputed Axial Tomography: Multiple x-ray projections are acquired around the object and a 2D image is computed from those projections.Idea: Reconstruct 2D attenuation distribution µ(x,y) from multiple 1D x-ray projections g( ) taken at different angles φ .9Lucas Parra, CCNYCity College of New Yor kCT - Imaging PrinciplexyComputed Axial Tomography: Multiple x-ray projections are acquired around the object and a 2D image is computed from those projections.Idea: Reconstruct 2D attenuation distribution µ(x,y) from multiple 1D x-ray projections g( ) taken at different angles φ .10Lucas Parra, CCNYCity College of New Yor kCT - Imaging PrinciplexyComputed Axial Tomography: Multiple x-ray projections are acquired around the object and a 2D image is computed from those projections.Idea: Reconstruct 2D attenuation distribution µ(x,y) from multiple 1D x-ray projections g( ) taken at different angles φ .11Lucas Parra, CCNYCity College of New Yor kCT - Imaging PrinciplexyComputed Axial Tomography: Multiple x-ray projections are acquired around the object and a 2D image is computed from those projections.Idea: Reconstruct 2D attenuation distribution µ(x,y) from multiple 1D x-ray projections g( ) taken at different angles φ .12Lucas Parra, CCNYCity College of New Yor kCT – Simple Inversion ExampleGiven the observed detector values how can one compute the unknown attenuation coefficients?g(1,1) = µ(1,1) + µ(1,2)g(1,2) = µ(2,1) + µ(2,2)g(2,1) = (µ(1,1) + µ(1,2) + µ(2,2))/2g(2,2) = (µ(1,1) + µ(2,1) + µ(2,2))/2 x , yg r ,44105.5? ???g=M M =[1 1 0 00 0 1 10.5 0.5 0 0.50.5 0 0.5 0.5], =[1,11,22,12,2], g=[g 1,1g 1,2g 2,1g 2,2]13Lucas Parra, CCNYCity College of New Yor kCT – Inversion Simple ExampleGiven the observed detector values how can one compute the unknown attenuation coefficients?Answer: linear inversion!g=M =M−1g44105.51 346M−1=[0 −1 0 21 1 0 −21 1 −2 0−1 0 2 0], =[1364], g=[41045.5]14Lucas Parra, CCNYCity College of New Yor kCT – CT numberHounsfield Units or “CT number” are units for attenuation coefficient relative to watter attenuation at µwater at 70keV.Tissue CT number (HU)Bone 1000Liver 40 ... 60White matter (brain) 46Grey matter (brain) 43Blood 40Muscle 10 – 40Kidney 30Cerebrospinal fluid 15Water 0Fat –50 ... –100Air –1000HU =1000−waterwater15Lucas Parra, CCNYCity College of New Yor kCT – 1st Generation●EMI Mark I (Hounsfield): parallel-beam scanner (highly collimated beam) → excellent scatter rejection, now outdated.●180o-240o rotation angle in steps of 1o●Used for the head●5-min scan time, 20 min reconstruction●Original resolution: 80x80 pixels (ea. 3X3 mm2), 13 mm slice.Translation & Rotation16Lucas Parra, CCNYCity College of New Yor kCT – 2nd Generation●Hybrid system: Fan beam, linear detector array (~30 detectors)●Translation and rotation●Reduced number of view angles → scan time ~ 30 s●Slightly more complicated reconstruction algorithms because of fan-beam projection●Non-parallel rays