Diapositiva 1

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BASIC TOMOGRAPHY PRINCIPLES

• TOMOGRAPHY AS AN ALGEBRAIC PROBLEM DISCRETE
  PIXEL N VS DISCRETE PROJECTION N
• ALGEBRAIC RECONSTRUCTION TECHNIQUE (ART) AND
  EXPECTATION MAXIMIZATION (EM) ITERATIVE SOLUTIONS
• EXTENSIONS CONSIDERING DETAILED SENSITIVITY
  FUNCTIONS OF DETECTORS (RESOLUTION RECOVERY) AND
  REGULARIZING BASIS FUNCTIONS
• TOMOGRAPHY AS A REGULARIZATION PROBLEM THE
  MOORE-PENROSE CONTINUOUS SOLUTION FROM DISCRETE
  MEASUREMENTS
• TOMOGRAPHY AS AN ANALYTICAL PROBLEM THE
  CONTINUOUS RECONSTRUCTION FROM CONTINUOUS
  MEASUREMENTS INVERSE RADON TRANSFORM AND FILTERED
  BACK-PROJECTION
• THE SAMPLED APPROXIMATION (SAMPLING PROBLEMS)
• THE 3D EXTENSION (FDK ALGORITHM)

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BOTH TRANSMISSION AND EMISSION TOMOGRAPHYDERIVE
FROM PROJECTIONS
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TOMOGRAPHY AS NUMERICAL PROBLEM
(DISCRETE-DISCRETE)
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SYSTEM MATRIX AND ALGEBRAIC PROBLEM
array of I projection measurements pi
array gray value in J pixels (voxels) fj
system matrix (I x J)

• I order N2 (e.g., 256 x 256) or N3 in a volume
• J order N2 (e.g., 360 x 256 samples) or N3 in a
  volume
• dimension of W order N4 or N6 in a volume

J reconstructed values must satisfy (approximate)
I measurement constrains each defined by a
projection ray
a ray touches order N pixels (voxels) out of N2
(N3) the other weights are null hence the system
matrix W is largely sparse
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TYPICAL ITERATIVE SOLUTION (X-ray)ALGEBRAIC
RECONSTRUCTION TECHNIQUE (ART)
additive error
simulation of projectionat step (i-1)
additive correction
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MAX LIKELYHOOD BY EXPECTATION MAXIMIZATION
(EM)FOR EMISSION TOMOGRAPHY WITH POISSON NOISE
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GENERALIZATION TO A LINEAR SPACE VARIANT PROBLEM
integration kernelsensitivity function of the
i-th detector
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GENERALIZATION OF THE ALGEBRAIC
PROBLEMSENSITIVITY FUNCTIONS AND BASIS FUNCTIONS

• Approximation of f(x,y,z) as a linear combination
  of J basis functions bj(x,y,z) with unknown
  weights

• System matrix coefficients computation

• Measurement constrains

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MOORE-PENROSE SOLUTION OF THE DISCRETE TO
CONTINUOUS PROBLEM SENSITIVITY FUNCTIONS AS
BASIS FUNCTIONS (NATURAL PIXELS)

• Problem of Moore-Penrose least squares solution

• Solution approximation of f(x,y,z) as a linear
  combination of the I sensitivity functions
  hj(x,y,z) with unknown weights

• System matrix coefficients computation

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THE ANALYTICAL CONTINUOUS TO CONTINUOUS
PROBLEMTHE RADON TRANSFORM AND THE SINOGRAM
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PROJECTION AND BACK-PROJECTIONPROVIDES A BLURRED
SOLUTION
back-projection operator
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IMAGE, RADON, AND FOURIER SPACETHE CENTRAL
SECTION THEOREM
FILTERED BACK-PROJECTION FOR PLANAR PARALLEL
PROJECTIONS
ramp filter
convolution
back-projection
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IMAGE RECONSTRUCTION FROM PROJECTIONSTHE INVERSE
RADON TRANSFORM
ramp filter
ramp filtering
back-projection
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FILTERED BACK PROJECTIONA LESS FORMAL
DEMONSTRATION

• Projection/back-projection is a Linear Space
  Invariant (LSI) operation (x,y)?(x,y)
• It is described by a Point Spread Function (PSF,
  impulse response) expected to be bell-shaped and
  low pass filtering
• We need the inverse filter to correct
  (Back-Projection Filter)
• The PSF with finite projections is a star of
  bladesthe densities of which is 1/?
• With infinite projections the PSF 1/?
• Its FT (Modulation Transfer Function, MTF) has
  the same shape 1/O FTblade orthogonal blade
  FTstar-of-blades star-of-blades
• The inverse filter MTF-1 is an upside-down cone O
• The Central Section of the 2D cone is the 1D ramp
  filter ? FBP

(da F.Rocca, 1998)
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SAMPLED SOLUTION
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POLAR SAMPLING OF PROJECTIONS VS ORTHOGONAL
SAMPLING OF RECONSTRUCTED IMAGE
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BACK TO THE ALGEBRAIC PROBLEMWHAT CAN REALLY
GIVE US ART FROM LIMITED VIEW ANGLES?
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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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FULL 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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Deriving Full Radon (integrals on planes)
Transform from Partial Radon Transform (line
integrals)
Points on a projection plane represent line
integrals. Hence, selecting a line on the
projection plane and integrating provides the
integral on the plane orthogonal to the
projection plane (parallel to the rays) passing
through this integration line. This is a point of
the Full Radon Transform.
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REDUNDANCY OF PROJECTIONS ON ALL PLANES

• Projections planes are ?2 (azimuth and polar
  angle).In each plane ?2 integration lines can be
  defined.Hence ?4 values are found i.e., each
  Radon Transform point is found in ? ways.Indeed,
  the same can be computed on the family of ?
  projection planes containing axis t and fill the
  corresponding axis in the Full Radon Transform
  and in the 3D Fourier space.
• Equivalently, the Central Section Th. (in the
  Partial Radom Transform version), says that by
  transforming a planar projection an entire plane
  (?2 points) is filled in the 3D Fourier
  space.Given the ?2 planar projections we obtain
  a redundancy of ?3 over ?4 , again.
• In conclusion, a set of ? directions filling the
  Radon Space and the Fourier space is sufficient
  for the reconstruction.
• Passing from parallel projections to cone beam
  projections an appropriate trajectiory of the
  focal spot passing through ? points has also to
  cover the entire 3D transform spaces.

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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

• helices (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 cylinder 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

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CONCLUSION

• THE BASES OF TOMOGRAPHY PRESENT INTERESTING SETS
  OF BOTH NUMERICAL AND ANALYTICAL PROBLEMS
  MATCHED WITH THE PECULIAR PHYSICAL STRUCTURE OF
  SCANNERS AND THE RELEVANT TECHNOLOGICAL SOLUTIONS
• A VIEW INTEGRATING THE DIFFERENT APPROACHES
  GIVES A CONSIDERABLY LARGER FLEXIBILITY AND MAY
  PROVIDE ORIGINAL SOLUTIONS

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