SBU CSE 591 - Final Project Suggestion - Fast Cone-Beam CT

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CSE 591: GPU ProgrammingFinal Project Suggestion: Fast Cone-Beam CTKl M llKlaus MuellerComputer Science DepartmentStony Brook UniversityOutlinePi CTPrimer on CT• radiography• principles• helical 3D CT• cone-beam 3D CTFinal project taskFinal project task• cone-beam CTX-Ray DiscoveryDi d b Wilh l Rö t i 1895Discovered by Wilhelm Röntgen in 1895• accidentally, when performing experiments with cathode tubes and fluorescent screens • the “light” even illuminated the screen when the tube was placed into a box• he called this new type of radiation X-rays (X for unknown)th Xldt lth h llkid f t il tdiff t•these X-rays could travel through all kinds of materials, at different material-specific attenuations???screentube in boxX Ray Interaction With MatterThree types of interaction with matter:Three types of interaction with matter:• photo-electric absorption: absorption of a photon by an atom and release of an electron along the same direction (which is soon absorbed)absorbed)• Compton scattering: only partial absorption of photon energy. The hhdii(l )dl lphoton electronphotons changes direction (at lower energy) and an electron also gets released (which is soon absorbed).photonphoton• Pair production: when photon energy > 1.02 MeV, an electron-positron pair may form. Soon, the positron annihilates with another lt T ht f fliit itditi ( diphotonelectronelectron. Two photons form, flying in two opposite directions (used in nuclear imaging) annihilationphotonelectron-positron pairphoton pairCase Studies (1)Double mandibular fracturewith strong displacement tothe left.Solitary humeral bone cystknown as ”fallen leaf sign”Case Studies (2)Radiographicchestimagesho ingmltiplelngRadiographicchestimageshowingmultiplelungmetastasesCase Studies (3)Dense opacity withspicular borders in the leftbreasthichs ggestsaPostoperative fluoroscopic controlof bone fixation with plate andscre safteracompletefract rebreast,whichsuggestsamalignant lesionscrewsafteracompletefractureof the humerusCT OverviewSiScanning:rotate source-detector pair around the patientdatareconstruction routinesinogram: a line for every anglereconstructed cross-sectional sliceSinogramSt ki ll j ti (li i t l ) i ldStacking all projections (line integrals) yields the sinogram, a 2D dataset p(r,θ)T ill t t i i bj t th t iTo illustrate, imagine an object that is a single point:•it then describes a sinusoid in p(r,θ):projections point object sinogramReconstruction: ConceptGi th i(θ)tt thbjtGiven the sinogram p(r,θ) we want to recover the object described in (x,y) coordinatesRecall the early axial tomography methodRecall the early axial tomography method• basically it worked by subsequently “smearing” the acquired p(r,θ) across a film plate•for a simple point we would get:•for a simple point we would get:This is called backprojection:π∫0(, ) {(, )} ( cos sin , )bxy Bpr px y dθθθθθ==⋅+⋅∫Backprojection: IllustrationT d t d th bl i d thÆthFiThe Fourier Slice TheoremTo understand the blurring we need more theory Æthe Fourier Slice Theorem or Central Slice Theorem•it states that the Fourier transform P(θ,k) of a projection p(r,θ) is a line across the origin of the Fourier transform F(kx,ky) of function f(x,y)lidA possible reconstruction procedure would then:polar gridpp• calculate the 1D FT of all projections p(rm,θm), which gives rise to F(kx,ky) sampled on a polar grid (see figure)•resample the polar grid into a cartesian grid (using interpolation)resample the polar grid into a cartesian grid (using interpolation)• perform inverse 2D FT to obtain the desired f(x,y) on a cartesian grid However, there are two important observations:• interpolation in the frequency domain leads to artifacts • at the FT periphery the spectrum is only sparsely sampledFiltered Backprojection: Concept TtfthilitifthtbtiTo account for the implications of these two observations, we modify the reconstruction procedure as follows:•filter the projections to compensate for the blurring• perform the interpolation in the spatial domain via backprojectionÆ hence the name Filtered BackprojectionFiltering--what follows is a more practical explanation (forFiltering --what follows is a more practical explanation (for formal proof see the book):•we need a way to equalize the contributions of all frequencies in the FT’s polar gridFT s polar grid• this can be done by multiplying each P(θ,k) by a ramp function Æ this way the magnitudes of the existing higher-frequency samples in eachrampthe existing higherfrequency samples in each projection are scaled up to compensate for their lower amount• the ramp is the appropriate scaling function since the sample density decreases linearly towards the FT’s periphery Filtered Backprojection: Equation and Result1D F iramp-filtering1D Fouriertransform of p(r,θ)Æ P(k,θ)20(, ) ( (, ) ) ikrfxy Pk k e dk dππθθ∞−∞=⋅⋅∫∫inverse 1D Fourier transform Æ p(r,θ)backprojection for all anglesRecall the previous (blurred)backprojection for all anglesRecall the previous (blurred) backprojection illustration•now using the filtered projections:not filtered filteredFiltered Backprojection: Illustration Backprojection: Practical ConsiderationsAf i i f ti l fthi thA few issues remain for practical use of this theory:• we only have a finite set of M projections and a discrete array of Npixels (xi, yj)M1(, ) {(, )} ( cos sin , )Mijnm i mjmmmbx y Bpr px yθθθθ===⋅+⋅∑pixelray• to reconstruct a pixel (xi, yj) there may not be a ray p(rn,θn) (detector sample) in the projection setpixelythe projection set Æ this requires interpolation (usually linear interpolation is used)detector samples•the reconstructions obtained with the simplebackprojectionappearinterpolation•the reconstructions obtained with the simple backprojectionappear blurred (see previous slides)Image RepresentationWk tht di ti i ti fi lWe know that a discrete image is a matrix of pixels• do keep this in mind, however:an image is NOT a matrixan image is NOT a matrix of solid squaresth h i l i Dirather, each pixel is a Dirac impulse, with the pixel’s value as its heightSo, why do we not see isolated dots on the


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