On π -line reconstruction formulas in tomography Adel Faridani Joint work with Ryan Hass Department of Mathematics Oregon State University faridani@math.oregonstate.edu http://people.oregonstate.edu/ faridana/ January 14, 2014 On π -line reconstruction formulas in tomography – p. 1/59
Tomography with sources on a curve Data: Measurements of the divergent beam transform � ∞ D f ( y , θ ) = f ( y + t θ ) dt. 0 source curve. y ( s ) = On π -line reconstruction formulas in tomography – p. 2/59
Example 1: 2D fan-beam tomography y (s) Θ α R s y ( s ) = R (cos( s ) , sin( s )) , Df ( y ( s ) , Θ ( s, α )) = g ( s, α ) ( s, α ) = "curved detector coordinates". On π -line reconstruction formulas in tomography – p. 3/59
Example 2: 3D Helical Tomography w ¯ x x y ( s ) � R cos( s ) , R sin( s ) , P � Source Curve: y ( s ) = 2 πs Let S denote the interior of the helix cylinder. supp ( f ) ⊂ S . On π -line reconstruction formulas in tomography – p. 4/59
Example 2: 3D Helical Tomography w ¯ x x y ( s ) Which source positions are needed for reconstruction at a point x ? On π -line reconstruction formulas in tomography – p. 4/59
π -line and π -interval w y ( s t ) I π ( x ) x y ( s b ) y ( s ) A so-called π -line through x intersects the source curve twice within one turn. On π -line reconstruction formulas in tomography – p. 5/59
π -line and π -interval w y ( s t ) I π ( x ) x y ( s b ) y ( s ) A so-called π -line through x intersects the source curve twice within one turn. For the helix there is a unique π -line through x . On π -line reconstruction formulas in tomography – p. 5/59
π -line and π -interval w y ( s t ) I π ( x ) x y ( s b ) y ( s ) A π -line through x gives rise to the π -interval I π ( x ) = [ s b ( x ) , s t ( x )] . On π -line reconstruction formulas in tomography – p. 5/59
π -line and π -interval w y ( s t ) I π ( x ) x y ( s b ) y ( s ) A π -line through x gives rise to the π -interval I π ( x ) = [ s b ( x ) , s t ( x )] . Sources y ( s ) with s ∈ I π ( x ) lie on the green arc. On π -line reconstruction formulas in tomography – p. 5/59
π -line reconstruction formulas Definition 1 A π -line reconstruction formula uses for reconstruction at a point x only data from sources within the π -interval of x . w y ( s t ) I π ( x ) x y ( s b ) y ( s ) On π -line reconstruction formulas in tomography – p. 6/59
Example: Backprojection-filtration Define the Hilbert transform of f in direction θ ∈ S n − 1 as H θ f ( x ) = 1 � f ( x − t θ ) dt. π t R Then � − 1 1 ∂ � � ∂q D f ( y ( q ) , β ( s, x )) ds = H β ( s b ( x ) , x ) f ( x ) � 2 π | x − y ( s ) | � I π ( x ) q = s β ( s, x ) = unit vector pointing from y ( s ) to x . Right-hand side is Hilbert transform along the π -line of x . Originally due to Gel’fand and Graev (1991). Basis for backprojection-filtration algorithm (Zou and Pan (2004)). On π -line reconstruction formulas in tomography – p. 7/59
Example: Filtered backprojection � 2 π � f ( x ) = − 1 � 1 ∂ dγ ds � ∂q D f ( y ( q ) , Θ ( s, x , γ )) � 2 π 2 | x − y ( s ) | sin γ I π ( x ) 0 � q = s Θ ( s, x , γ ) = cos( γ ) β ( s, x ) + sin( γ ) β ⊥ ( s, x ) . β ( s, x ) = unit vector pointing from y ( s ) to x . (Katsevich 02, 04, Katsevich & Kapralov 07) On π -line reconstruction formulas in tomography – p. 8/59
Example: Filtered backprojection � 2 π � f ( x ) = − 1 � 1 ∂ dγ ds � ∂q D f ( y ( q ) , Θ ( s, x , γ )) � 2 π 2 | x − y ( s ) | sin γ I π ( x ) 0 � q = s Θ ( s, x , γ ) = cos( γ ) β ( s, x ) + sin( γ ) β ⊥ ( s, x ) . β ( s, x ) = unit vector pointing from y ( s ) to x . (Katsevich 02, 04, Katsevich & Kapralov 07) Both formulas hold in dimensions 2 and 3 for a large family of source curves. In dimension 3, β ⊥ has to be carefully chosen (Katsevich 02, 04). On π -line reconstruction formulas in tomography – p. 8/59
κ -Plane and Katsevich’s formula w ¯ x β ⊥ ( s, x ) x β ( s, x ) y ( s ) κ − plane � 2 π � f ( x ) = − 1 1 ∂ dγ ds � � ∂q D f ( y ( q ) , Θ ( s, x , γ )) � 2 π 2 | x − y ( s ) | sin γ � I π ( x ) 0 q = s Θ ( s, x , γ ) = cos( γ ) β ( s, x ) + sin( γ ) β ⊥ ( s, x ) . On π -line reconstruction formulas in tomography – p. 9/59
Characteristics of π -line formulas Flexibility in choosing π -lines in 2D. On π -line reconstruction formulas in tomography – p. 10/59
Non-uniqueness of π -lines in 2D In 2 dimensions we lack uniqueness of π -lines. Any line through x may be chosen as the π -line of x , denoted by L π ( x ) . y ( s b ( x )) x y ( s t ( x )) I π ( x ) may be chosen to correspond to either of the two arcs. On π -line reconstruction formulas in tomography – p. 11/59
Example: Orthogonal-long π -lines L π ( x ) is orthogonal to x and I π ( x ) = [ s b ( x ) , s t ( x )] corresponds to the longer arc. y ( s b ) x y ( s t ) Superior performance for R close to 1! On π -line reconstruction formulas in tomography – p. 12/59
Comparison for R=1.01 PI − M5 orthogonal − long PI − M4 orthogonal − long 1 1 R = 1.01, NSD = 1.53 0.5 0.5 0 0 − 0.5 − 0.5 − 1 − 1 − 1 0 1 − 1 0 1 Relerr = 0.094041 Relerr = 0.094041 2PI − M5 Shepp − Logan Std. Shepp − Logan 1 1 P = 333, Q = 318 0.5 0.5 0 0 − 0.5 − 0.5 − 1 − 1 − 1 0 1 − 1 0 1 Relerr = 0.15796 Relerr = 2.6397 On π -line reconstruction formulas in tomography – p. 13/59
Characteristics of π -line formulas Flexibility in choosing π -lines in 2D. On π -line reconstruction formulas in tomography – p. 14/59
Characteristics of π -line formulas Flexibility in choosing π -lines in 2D. Region of Backprojection not equal to S . RBP ( s ) = set of all points where data from source y ( s ) is used for reconstruction = { x : s ∈ I π ( x ) } . RBP(s) depends on the family of π -lines. On π -line reconstruction formulas in tomography – p. 14/59
Orthogonal-long π -lines y ( s b ) x y ( s t ) No two points have the same π -interval. The set RBP ( s ) and its boundary are not immediately obvious. On π -line reconstruction formulas in tomography – p. 15/59
RBP for orth.-long π -lines For orthogonal-long π -lines, RBP ( s ) contains all points outside the disk D ( s ) = { x : | x − y ( s ) / 2 | < | y ( s ) / 2 |} . y ( s ) D ( s ) Hass-F ., SIAM J. Imag Sci., (2012) On π -line reconstruction formulas in tomography – p. 16/59
Sources close to object The numerically most challenging parts of the reconstructions are those where data from an x-ray source contribute to the image at points very close to the source. But most of such points are not in the RBP for orthogonal long π -lines. So the most challenging parts are avoided! On π -line reconstruction formulas in tomography – p. 17/59
Characteristics of π -line formulas Flexibility in choosing π -lines in 2D. On π -line reconstruction formulas in tomography – p. 18/59
Characteristics of π -line formulas Flexibility in choosing π -lines in 2D. Region of Backprojection not equal to S . RBP ( s ) = set of all points where data from source y ( s ) is used for reconstruction = { x : s ∈ I π ( x ) } . RBP(s) depends on the family of π -lines. On π -line reconstruction formulas in tomography – p. 18/59
Characteristics of π -line formulas Flexibility in choosing π -lines in 2D. Region of Backprojection not equal to S . RBP ( s ) = set of all points where data from source y ( s ) is used for reconstruction = { x : s ∈ I π ( x ) } . RBP(s) depends on the family of π -lines. Comet tail artifacts. On π -line reconstruction formulas in tomography – p. 18/59
Comet tail artifacts Reconstructions from real data. The reconstruction from the π -line filtered backprojection formula (left) shows a large comet tail artifact that is not present in a standard reconstruction (right). On π -line reconstruction formulas in tomography – p. 19/59
Comet tail artifacts Reconstructions from real data. The reconstruction from the π -line filtered backprojection formula (left) shows a large comet tail artifact that is not present in a standard reconstruction (right). In this case most of the artifact is due to a previously undetected data misalignment in the fan angle. The π -line formula is much more sensitive to such misalignments. On π -line reconstruction formulas in tomography – p. 19/59
Finding the correct alignment TV-Norm -1 -0.5 0 0.5 1 shift The correct alignment (about 0 . 19 detector widths) corresponds here to a minimum of the total variation � |∇ f ( x ) | d x ( here of a subregion of the image). TV ( f ) = On π -line reconstruction formulas in tomography – p. 20/59
Reconstruction with corrected alignment The comet tail artifact is much reduced. On π -line reconstruction formulas in tomography – p. 21/59
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