Introduction to Medical Imaging. Cone-Beam CT. Introduction. Available cone-beam reconstruction methods: Our discussion:

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1 Intoduction Intoduction to Medical Imaging Cone-Beam CT Klaus Muelle Available cone-beam econstuction methods: exact appoximate Ou discussion: exact (now) appoximate (next) The Radon tansfom and its invese ae impotant mechanisms to undestand cone-beam CT Compute Science Depatment Stony Book Univesity Cone-Beam Tansfom 2D Radon Tansfom

2 3D Radon Tansfom Fouie-Slice Theoem in 2D Fouie-Slice Theoem in 3D Exact Reconstuction in 2D and 3D In 2D: use 2D invesion fomula: the filteed backpojection pocedue we have seen a spatial technique, only pefoming filteing in the fequency domain (in a pola gid) but may also intepolate the pola gid in the fequency domain and invet the esulting catesian lattice employ linogam techniques fo the latte (see late) In 3D: use 3D invesion fomula: not nealy as staightfowad than 2D invesion full fequency-space methods also exist moe details next (on all)

3 Exact Invesion Fomula The basic 3D invesion filteed backpojection fomula, due to Nattee (1986): θ is the angle, a unit vecto on a unit sphee x, ρ ae object and Radon space coodinates, esp.: ρ = x θ involves a 2 nd deivative of the 3D Radon tansfom the second deivative opeato can be teated as a convolution kenel Some manipulations can educe the second deivative to a fist deivative, along with convolution opeatos see, fo example, this fomula: Gangeat s Algoithm Phase 1: fom cone-beam data to deivatives of Radon data Phase 2: fom deivatives of Radon data to econstucted 3D object Thee ae many ways to achieve Phase 2 diect, O(N 5 ) a two-step pocedue, O(N 4 ) [Ma et al, 1981] a Fouie method, O(N 3 log N), [Axelsson/Danielsson, 1994] a divide-and-conque stategy, O(N 3 log N) [Basu/Besle, 2002] we shall discuss the fist thee hee many diffeent vaiants have been poposed - fo example: Kudo/Saito (1990), Smith (1985) But fist let us see how Radon data ae geneated fom conebeam data Tansfoming Cone-Beam to Radon Data Tansfoming Cone-Beam to Radon Data fom Axelsson/Danielsson Stategy: weigh detecto data with a facto 1/SA integate along all intesections (lines) between the detecto plane and the equied Radon planes - thee ae N 2 such lines (N lines and N otations) take the deivative in the s-diection (in the detecto plane pependicula to t) weight the 2D data set esulting fom a single souce position by the facto SC / cos 2 β The ode of these opeations can be switched since they ae all linea (Gangeat swapped the ode of opeation 2 and 3)

4 Radon Data to Object: Diect Method Radon Data to Object: Two-Step Method Thee ae O(N 3 ) data points in Radon (deivative) space Each is due to a plane integal The diect method simply insets the plane data into the object space, one by one this is basically the expansion of a point into a plane, defined by (θ, ρ) this gives ise to an O(N 5 ) algoithm fom Axelsson/Danielsson Radon Data to Object: Two-Step Method Radon Data to Object: Fouie Space Appoach Each vetical plane holds all Radon points due to plane integals of pependiculaly intesecting planes filteed backpojection educes the plane integals to line integals, confined to hoizontal planes The hoizontal planes ae then econstucted with anothe filteed backpojection Each such opeation is O(N 3 ) and thee ae O(N) of them, esulting in a complexity of O(N 4 ) fom Axelsson/Danielsson

5 Radon Data to Object: Fouie Space Appoach Long Object Poblem Takes advantage of the O(N log N) complexity of the FFT at vaious steps It also uses linogams [Edholm/Heman, 1987] to educe 2D intepolation to 1D intepolation The complexity is then O(N 3 log N) Tuy's Sufficiency Condition Concept of PI-Lines

6 Examples of Complete Tajectoies Cicula Souce Path A pominent example of an incomplete tajectoy 3D Radon Data Acquied by a Cicula Tajectoy Challenges in Cone-Beam Reconstuction The naive application of the 3D Radon invesion fomula is pohibitive due to long object poblem enomous computational expense Simplifications have to found to end up in an efficient and numeically stable econstuction algoithm pefeably in a shift-invaiant 1D-filteed backpojection algoithm Utilization of edundant data is obscue. Ideally edundancy in collected Radon planes has to be consideed. Howeve, this appoach is suboptimal because: it is quite complicated undeestimates the edundancy of data typically in cone beam, the data ae highly edundant in appoximation

7 Tansmission CT Popula Appoximation A typical econstuction algoithm is Filteed Backpojection X-ay souce attenuating object detecto Feldkamp-Davis-Kess (FDK) Cone-beam econstuction Pojection filteing FFT multiply by amp invese FFT pe-weighting Backpojections Post-weighting FDK: Filteing FDK: Backpojection Y ϕ Y ϕ Y v Y v Z ϕ Z ϕ x ϕ x ϕ X v X v Y s Y s Z v Z v filteed pojection data cicula pe-weighting D ˆ D P ( Y, Z ) = P ( Y, Z )** g ( Y ) D + Y + Z d Z s X s pojection data S amp filte d D ˆ ( ) ˆ y z P = P ( Y ( ), Z( )), Y ( ) = D, Z( ) = D d + x d + x Z s S X s voxel pojection mapping pojection coodinates of mapped voxel

8 FDK: Accumulation, Depth-Weighting Y ϕ Y v Z ϕ x ϕ X v Z v Y s econstucted voxel d D 2π 2 1 d f ( ) = Pˆ ( ) d 4 π ( d + x ) Z s S X s accumulation fo all pojections depth-weighting