Diapositiva 1

1
IMAGE, RADON, AND FOURIER SPACE
FILTERED BACK-PROJECTION FOR PLANAR PARALLEL
PROJECTIONS
ramp filter
convolution
back-projection
2
FILTERED BACK-PROJECTION FOR FAN BEAM
Fan beam projection on linear detector pF(?,a)
1) Weighing by cos? and ramp filtering
2) Back-projection
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CONR BEAM PROJECTIONS ON FLAT PANEL DETECTORS
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RADON TRANSFORM IN 3D
The full Radon transform implies integration of
planes E which are projected on a point located
at the intercept of the normal line through the
origin
integration plane
versor normal to the integration plane
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CENTRAL SECTION THEOREM IN 3D FULL RADON
TRANSFORM
Full Radon transform, p2D projection of
parallel planes on the orthogonal axis The 1D
Fourier transform of the projection axis gives
the 3D Fourier values on the corresponding axis
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SUFFICIENT SET OF DATA IN 3D
A complete set of data is defined if all
integration planes through the object are
present. The subset of parallel planes defined by
the normal direction (?,?) fill the radial axis
with the same direction in F3D?f(x,y,z)?. So we
need ?2 directions times ? parallel shifts i.e.,
?3 planar integrations. These can be provided by
?2 planar projections.
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CENTRAL SECTION THEOREM IN 3D PARTIAL RADON
TRANSFORM
Partial Radon transform, p1D projection of
parallel lines on the orthogonal plane The 2D
Fourier transform of a projection plane gives the
3D Fourier values on the corresponding plane
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REDUNDANCY OF PROJECTIONS ON ALL PLANES
Any projection plain containing axis t can
furnish the integral of the integration plane
Et,t0 by integrating on the intersection line of
the two planes. Hence ? families of ? planes can
be used for filling the Radon space and the
Fourier space. Availability of all ?2 planes
gives redundancy.Each projection plane permits
to fill the corresponding plane in Fourier space.
Again ? planes are needed to fill this space.
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SUFFICIENT SET OF DATA IN 3D FOR CONE BEAM
A cone-beam projection permits to derive the
integral of each plane passing through the
source. Hence, if the source in its trajectory
encounters each plane through the object a
sufficient set is obtained. This is the Tuy-Smith
sufficient condition (1985). A circular
trajectory, most often used, satisfies this
condition only partially planes parallel to the
trajectory are never encountered. Hence, a torus
is filled in Radon space with a hole, called
shadow zone, close to the rotation axis z.
z
x
x
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TRAJECTORIES SATISFYING TUY-SMITH CONDITION

• helics (spiral)
• two non parallel circles
• circle and line

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FELDKAMP, DAVIS, KRESS (FDK) ALGORITHM (1984)
Approximated Filtered Back-Projection for
cone-beam and circular trajectory Satisfactory
approximation even with quite high copolar angles
(e.g., ?20) It reconstructs the volume crossed
by rays at any source position on the circles
hence a cilinder plus two cones.
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FDK ALGORITHM
1) Weighing by cos??cos?
2) Row by row filtering with the ramp filter
3) Back-projection
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FDK PROPERTIES

• Exact on the central plane, z0, where it
  coincides with the Fan Beam solution
• Exact for objects homogeneous along z, f(x,y,z)
  f(x,y).
• Integrals along z, ? f(x,y,z)dz, is preserved
• Integrals on moderately tilted lines preserved as
  well
• Main artifact blurring along z at high copolar
  angles