1. IMAGE, RADON, AND FOURIER SPACE
FILTERED BACK-PROJECTION FOR PLANAR PARALLEL PROJECTIONS
convolution
ramp filter
back-projection

2. FILTERED BACK-PROJECTION FOR FAN BEAM PLANAR DETECTOR CASE
P(x,y) P’ S R U y x a  
Fan beam projection on linear detector: pF(,a)
1) Weighing by cos and ramp filtering
2) Back-projection

3. CONE BEAM PROJECTIONS ON FLAT PANEL DETECTORS
RX SOURCE
Flat Panel Detector
RX CONE
z, b x y a   
VIRTUAL DETECTOR

4. 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. S
z, b a
Fan
inclined fan z
Sorgente
detector

5. virtual planar detector
FDK ALGORITHM
z, b x y a   
virtual planar detector
P(x,y) S P’
1) Weighing by coscos :
2) Row by row filtering with the ramp filter
3) Back-projection

6. FDK PROPERTIES
FDK algorithm is an approximate extension to the 3D cone beam case on planar detector of the 2D
Exact on the central plane, z=0, 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  FDK artifact
Satisfactory reconstructions were demonstrated even at fairly high copolar angles (40°-50°). Usually much lower copolar angles are exlored (10°) in the field of view, with higher precision.

7. Theory of 3D reconstruction from projections
RADON TRANSFORM IN 3D
The full Radon transform implies integration of volume over planes which are projected on a point located at the intercept of the normal line through the origin
integration plane
integration plane
Radon value x y z  
versor normal to the integration plane

8. CENTRAL SECTION THEOREM IN 3D – FULL RADON TRANSFORM
Full Radon transform, 2D projection of parallel planes on the orthogonal axis
Result:
The 1D Fourier transform of the projection axis t gives the 3D Fourier values on the corresponding axis t

9. THE FULL RADON TRANSFORM IS A SUFFICIENT DATA SET FOR 3D IMAGE RECONSTRUCTIONx y z t x y t z
f(x,y,z)
Rf(,,t)
F3Df(x, y, z)   .
F3D
F1D
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.

10. CENTRAL SECTION THEOREM IN 3D – PARTIAL RADON TRANSFORM
Partial Radon transform, p1D, r (s,t) = projection of lines parallel to a ray axis r on the orthogonal projection plane at coordinates (s,t)
The 2D Fourier transform of projected values on the projection plane (s,t) gives the 3D Fourier values on the corresponding plane frequency domain plane (s , t)

11. Deriving Full Radon (integrals on planes) Transform from Partial Radon Transform (line integrals)
F3D
projection plane x y z s r t t0 
integration plane z y x t s  r
Points on a projection plane represent 1D line integrals parallel to r. A further integration on a line t=t0 correspond to a 2D integration on a plane orthogonal to t and a distance t0 from the origin. Hence the Full Radon Transform sample in position (r=0, s=0, t=t0) .
Note that 1D projections are much more informative than 2D projections, due to lesser integration: a 1D projection on a plane fills an entire plane in the 3D Radon and Fourier spaces

12. 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 1 directions filling the Radon Space and the Fourier space is sufficient for the reconstruction.
Passing from parallel projections to cone beam projections an appropriate trajectory of the focal spot passing through 1 points can cover the entire 3D transform spaces if it satisfies proper conditions (see Tuy Smith sufficient condition in the next slide).

13. SUFFICIENT SET OF DATA IN 3D FOR CONE BEAM
Tuy-Smith sufficient condition (1985): A cone-beam projection permits to derive the integral of each plane passing through the source S. Hence, if the source in its trajectory encounters each plane through the object a sufficient set is obtained
Explanation: Any plane (,,t) through S is filled by X-rays and we know the integral (1D projection) along them; the 2D projection of the plane (i.e., integral over the plane) can be derived by summing up the 1D projection values giving a 3D Radon sample Rf(,,t). Note that 2 planes pass through a point S, which multiplied by  positions of S on the trajectory can fill all the 3D Radon space, if a proper trajectory is chosen. x y z t
corresponding point of Radon Transform Rf(,,t)   S
plane (,,t) touched by S
trajectory of S

14. ALMOST 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 y z
shadow zone x x

15. TRAJECTORIES SATISFYING TUY-SMITH CONDITION
helics (used in multislice “spiral” CT)
two non parallel circles (possibly used in C arm cone beam)
circle and line (just theoretical)