Presentation on theme: "Data Acquisition, Representation and Reconstruction of medical images"— Presentation transcript:
1 Data Acquisition, Representation and Reconstruction of medical images Application of Advanced Spectral Methods
2 Acquisition Methods for medical images X-RaysComputer Tomography (CT or CAT)MRI (or NMR)PET / SPECT (Positron Emission Tomography, Single Photon Emission Computerized TomographyUltrasoundComputationalWhat about microscopic scanners!What about electron microscopes?What about voxelization / discretization?What about synthetic methods?Provide Links to the numerous online tutorials!
4 X-Rays - Physics cheap and relatively easy to use X-Rays are associated with inner shell electronsAs the electrons decelerate in the target through interaction, they emit electromagnetic radiation in the form of x-rays.patient is located between an x-ray source and a film -> radiographcheap and relatively easy to usepotentially damaging to biological tissue
5 X-RaysX-Rayssimilar to visible light, but higher energy!
6 X-Rays - Visibilitybones contain heavy atoms -> with many electrons, which act as an absorber of x-rayscommonly used to image gross bone structure and lungsexcellent for detecting foreign metal objectsmain disadvantage -> lack of anatomical structureall other tissue has very similar absorption coefficient for x-rays
7 X-Rays - Images X-Rays can be used in computerized tomography Add images!!!X-Rays can be used in computerized tomography
10 Non-Intrusive Medical Diagnosis based on Computerized Tomography Computer tomography CT(From Jain’s Fig.10.1)An X-ray CT scanning system
11 Non-Intrusive Medical Diagnosis based on Transmission Tomography Source and Detector are rotating around human’s body(From Bovik’s Handbook Fig )
12 Non-Intrusive Medical Diagnosis based on projections Observe a set of projections (integrations) along different angles of a cross-sectionEach projection itself loses the resolution of inner structureTypes of measurementstransmission (X-ray),emission, magnetic resonance (MRI)Want to recover inner structure from the projections“Computerized Tomography” (CT)
13 Non-Intrusive Medical Diagnosis based on Emission Tomography Emission tomography: ET measure emitted gamma rays by the decay of isotopes from radioactive nuclei of certain chemical compounds affixed to body parts.MRI: based on that protons possess a magnetic moment and spin.In magnetic field => align to parallel or antiparallel.Apply RF => align to antiparallel. Remove RF => absorbed energy is remitted and detected by Rfdetector.f(x,y) is 2D image as before
14 Radon Transform Principles A linear transform f(x,y) g(s,)Line integral or “ray-sum”Along a line inclined at angle from y-axis and s away from originFix to get a 1-D signal g(s)We have now a set of images g(s) which represent g(s,)(From Jain’s Fig.10.2)This is a transform from 2D to 2D spaces
15 Tomography and Reconstruction Lecture OverviewApplicationsBackground/history of tomographyRadon TransformFourier Slice TheoremFiltered Back ProjectionAlgebraic techniquesMeasurement of Projection dataExample of flame tomography
16 Applications & Types of Tomography Medical ApplicationsType of TomographyFull body scanX-rayRespiratory, digestive systems, brain scanningPET Positron Emission TomographyRespiratory, digestive systems.Radio-isotopesMammographyUltrasoundWhole BodyMagnetic Resonance (MRI, NMR)MRI and PET showing lesions in the brain.PET scan on the brain showing Parkinson’s Disease
17 Applications & Types of Tomography – non medical Non Medical ApplicationsType of TomographyOil Pipe FlowTurbine PlumesResistive/Capacitance TomographyFlame AnalysisOptical TomographyECT on industrial pipe flows
18 CT or CAT - Principles Computerized (Axial) Tomography Radon again! introduced in 1972 by Hounsfield and Cormacknatural progression from X-raysbased on the principle that a three-dimensional object can be reconstructed from its two dimensional projectionsbased on the Radon transform (a map from an n-dimensional space to an (n-1)-dimensional space)Radon again!From 2D to 3D !
19 CT or CAT - Methodsmeasures the attenuation of X-rays from many different anglesa computer reconstructs the organ under study in a series of cross sections or planescombine X-ray pictures from various angles to reconstruct 3D structures
20 The History of CATJohan Radon (1917) showed how a reconstruction from projections was possible.Cormack (1963,1964) introduced Fourier transforms into the reconstruction algorithms.Hounsfield (1972) invented the X-ray Computer scanner for medical work, (which Cormack and Hounsfield shared a Nobel prize).EMI Ltd (1971) announced development of the EMI scanner which combined X-ray measurements and sophisticated algorithms solved by digital computers.
24 Example of Simple Backprojection Reconstruction Given are sums, we have to reconstruct values of pixels A, B, C and D
25 Image Reconstruction: ART or Algebraic Reconstruction Technique
26 CT - Reconstruction: ART or Algebraic Reconstruction Technique METHOD 1: Algebraic Reconstruction Techniqueiterative techniqueattributed to GordonInitial GuessReconstructed modelBack- ProjectionProjectionActual Data Slices26
27 CT - Reconstruction: FBP Filtered Back Propagation METHOD 2 : Filtered Back Projectioncommon methoduses Radon transform and Fourier Slice TheoremyF(u,v)f(x,y)Gf(r)xsugf(s)fSpatial DomainFrequency Domain
28 COMPARISON : CT - FBP vs. ART Algebraic Reconstruction TechniqueStill slowbetter quality for fewer projectionsbetter quality for non-uniform project.“guided” reconstruct. (initial guess!)Filtered Back ProjectionComputationally cheapClinically usually 500 projections per sliceproblematic for noisy projections28
29 Fourier Slice Theorem and FFT review Patient’s body is described by spatial distribution of attenuation coefficient
30 Properties of attenuation coefficient Our transform: f(x,y) p(r,)
31 attenuation coefficient is used in CT_number of various tissues These numbers are represented in HU = Hounsfield UnitsCT_number uses attenuation coefficients
52 The Lung and The CTs 52 [LUNG] 1.Either of the pair of organs occupying the cavity of the thorax that effect the aeration of the blood.2.Balloon-like structures in the chest that bring oxygen into the body and expel carbon dioxide from the body[TYPES]1.Small Cell Lung Cancer (SCLC) - 20% of all lung cancers2.Non Small Cell Lung Cancer (NSCLC) - 80% of all lung cancer[Risks]In the United States alone, it is estimated that 154,900 died from lung cancer in In comparison,is estimated that 126,800 people died from colon, breast and prostate cancer combined, in 2002.[LUNG CANCER]Lung Cancer happens when cells in the lung begin to grow out of control and can than invade nearby tissues or spread throughout the body; Large collections of this out of control tissues are called tumors.52
53 We want to reconstruct shape of the lungs Starting PointBorder DetectionAt the moment two approaches are available.Left the algorithm developed at PisaRight the algorithm developed at Lecce53
54 Image Interpolation - Theory [IDEA]In order to provide a richer environment we are thinking of using interpolation methods that will generate “artificial images” thus revealing hidden information.[RADON RECONSTRUCTION]Radon reconstruction is the technique in which the object is reconstructed from its projections. This reconstruction method is based on approximating the inverse Radon Transform.[RADON Transform]The 2-D Radon transform is the mathematical relationship which maps the spatial domain (x,y) to the Radon domain (p,phi). The Radon transform consists of taking a line integral along a line (ray) which passes through the object space. The radon transform is expressed mathematically as:[FILTERED BACK PROJECTION - INVERSE R.T.]It is an approximation of the Inverse Radon Transform.[The principle] Several x-ray images of a real-world volume are acquired[The Data] X-ray images (projections) of known orientation, given by data samples.[The Goal] Reconstruct a numeric representation of the volume from these samples.[The Mean] Obtain each voxel value from its pooled trace on the several projections.[Resampling] At this point one can obtain the “artificial slices”[Reslicing] An advantage of the volume reconstruction is the capability of obtaining new perpendicular slices on the original ones.54
56 Image Interpolation - Graphical Representation (II) 56
57 Line Integrals and Projections We review the principleDiscuss various geometriesShow the use of filtering
58 Line Integrals and Projections The function P = Radon transformobject function f(x,y).The function is known as the Radon transform of the function f(x,y).
59 Various types of beams can be used Fan BeamsParallel BeamsA fan beam projection is taken if the rays meet in one locationParallel beams projections are taken by measuring a set of parallel rays for a number of different anglesVarious types of beams can be used
60 Line Integrals and Projections A projection is formed by combining a set of line integrals.Here the simplest projection, a collection of parallel ray integrals i.e constant θ, is shown.Notation for calculations in these projections
61 Line Integrals and Projections A simple diagram showing the fan beam projection
63 Fourier Slice TheoremThe Fourier slice theorem is derived by taking the one-dimensional Fourier transform of a parallel projection and noting that it is equal to a slice of the two-dimensional Fourier transform of the original object.It follows that given the projection data, it should then be possible to estimate the object by simply performing the 2D inverse Fourier transform.Start by defining the 2D Fourier transform of the object function asFor simplicity θ=0 which leads to v=0Define the projection at angle θ = Pθ(t)Define its transform byAs the phase factor is no-longer dependent on y, the integral can be split.
64 Fourier Slice TheoremAs the phase factor is no-longer dependent on y, the integral can be split.The part in brackets is the equation for a projection along lines of constant xSubstituting inThus the following relationship between the vertical projection and the 2D transform of the object function:
65 Fourier Slice Theorem Stanley and Kak Full details of derivation, not for now.
67 The Fourier Slice Theorem The Fourier Slice theorem relates the Fourier transform of the object along a radial line.tFourier transformvuθSpace DomainFrequency Domain
68 The Fourier Slice Theorem The Fourier Slice theorem relates the Fourier transform of the object along a radial line.Collection of projections of an object at a number of anglestvuFourier transformvuθFor the reconstruction to be made it is common to determine the values onto a square grid by linear interpolation from the radial points.But for high frequencies the points are further apart resulting in image degradation.Space DomainFrequency Domain
85 CT or CAT - Advantages significantly more data is collected superior to single X-ray scansfar easier to separate soft tissues other than bone from one another (e.g. liver, kidney)data exist in digital form -> can be analyzed quantitativelyadds enormously to the diagnostic informationused in many large hospitals and medical centers throughout the world
86 CT or CAT - Disadvantages significantly more data is collectedsoft tissue X-ray absorption still relatively similarstill a health riskMRI is used for a detailed imaging of anatomy – no Xrays involved.
87 Nuclear Magnetic Resonance (NMR) Magnetic Resonance Imaging (MRI)
88 MRINuclear Magnetic Resonance (NMR) (or Magnetic Resonance Imaging - MRI)most detailed anatomical informationhigh-energy radiation is not used, i.e. this is “safe method”based on the principle of nuclear resonance(medicine) uses resonance properties of protons
89 Magnetic Resonance Imaging MRI - polarized all atoms (core) with an odd number of protons have a ‘spin’, which leads to a magnetic behaviorHydrogen (H) - very common in human body + very well magnetizingStimulate to form a macroscopically measurable magnetic field
90 MRI - Signal to Noise Ratio proton density pictures - measures H MRI is good for tissues, but not for bonesignal recorded in Frequency domain!!Noise - the more protons per volume unit, the more accurate the measurements - better signal to noise ratio (SNR) through decreased resolution
91 PET/SPECTPositron Emission Tomography Single Photon Emission Computerized Tomography
92 PET/SPECTPositron Emission Tomography Single Photon Emission Computerized Tomographyrecent techniqueinvolves the emission of particles of antimatter by compounds injected into the body being scannedfollow the movements of the injected compound and its metabolismreconstruction techniques similar to CT - Filter Back Projection & iterative schemes
94 Ultrasoundthe use of high-frequency sound (ultrasonic) waves to produce images of structures within the human bodyabove the range of sound audible to humans (typically above 1MHz)piezoelectric crystal creates sound wavesaimed at a specific area of the bodychange in tissue density reflects wavesechoes are recorded
95 Ultrasound (2)Delay of reflected signal and amplitude determines the position of the tissuestill images or a moving picture of the inside of the bodythere are no known examples of tissue damage from conventional ultrasound imagingcommonly used to examine fetuses in utero in order to ascertain size, position, or abnormalitiesalso for heart, liver, kidneys, gallbladder, breast, eye, and major blood vessels
96 Ultrasound (3) by far least expensive very safe very noisy 1D, 2D, 3D scannersirregular sampling - reconstruction problems