)sin,cos(),()(),( 21 tytxDtD yxfFtfRF

dyxttgtfRyxf P 2

0)sincos,()(),(),(

ramp filter

ttj

ttDP de

πFtg t

2

1)( 1

1

convolution

back-projection

IMAGE, RADON, AND FOURIER SPACE

FILTERED BACK-PROJECTION FOR PLANAR PARALLEL PROJECTIONS

Fan beam projection on linear detector: pF(,a)

1) Weighing by cos and ramp filtering

2) Back-projection

)a(gcos)a,(p)a(gaR

R)a,(p)a,(p PFP

22

FF

d)),y,x(a,(p),y,x(U

R)y,x(f F2

0 2

2

FFBP

sinycosxR),y,x(U,

sinycosxR

cosysinxR),y,x(a

R

aarctansinRsinRt

,R

aarctan

P(x,y)

P’

SR

U

y

x

a

FILTERED BACK-PROJECTION FOR FAN BEAM PLANAR DETECTOR CASE

RX SOURCE

Flat Panel Detector

RX CONE

CONE BEAM PROJECTIONS ON FLAT PANEL DETECTORS

22

22

aR

barctan

R

aarctan

tanaRb

tanRa

z, b

x

ya

VIRTUAL DETECTO

R

z

Sorgente

detector

S

z, b

a

Fan

inclined fan

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.

z, b

x

ya

virtual planar

detector

P(x,y)

S

P’

1) Weighing by coscos :

)b,a,(pcoscos)b,a,(pbaR

R FF

222

2) Row by row filtering with the ramp filter

)a(g)b,a,(pcoscos)b,a,(p PFF

3) Back-projection

dzyxbyxap

yxU

Rzyxf F

FDK )),,,(),,,(,(),,(

),,(2

0 2

2

sinycosxR),y,x(U

sinycosxR

Rz),z,y,x(b,

sinycosxR

cosysinxR),y,x(a

FDK ALGORITHM

FDK algorithm is an approximate extension to the 3D cone beam case on planar detector of the 2D

1) Exact on the central plane, z=0, where it coincides with the Fan Beam solution

2) Exact for objects homogeneous along z, f(x,y,z) = f(x,y).

3) Integrals along z, f(x,y,z)dz, is preserved

4) 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.

FDK PROPERTIES

x̂t

integration plane

Radon value

x

y

z

T

tzyxxtrE ),,(,ˆ:

0,ˆ

xd)tˆx()x(f)t,,(fR

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

T)cos,sinsin,cos(sin),(ˆ

integration plane

versor normal to the integration plane

)(),,( ),(2 tptf DR

Full Radon transform, 2D projection of parallel planes on the orthogonal axis

)()(),,(

|),,(),0,0(),,(

212

00

)(3

tDDt

tsrtD

tpFdtedrdszyxf

dtdsdrezyxfzyxfF

t

s

r

tsr

CENTRAL SECTION THEOREM IN 3D – FULL RADON TRANSFORM

)(),,( ),(2 tptf DR

Result:

The 1D Fourier transform of the projection axis t gives the 3D Fourier values on the corresponding axis t

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 F3Df(x,y,z). So we need 2 directions times parallel shifts; i.e., 3 planar integrations. These can be provided by 2 planar projections.

x x

y y

z z

t

x

y

t

z

f(x,y,z) Rf(,,t) F3Df(x, y, z)

.

F3D

F1D

THE FULL RADON TRANSFORM IS A SUFFICIENT DATA SET FOR 3D IMAGE RECONSTRUCTION

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)

CENTRAL SECTION THEOREM IN 3D – PARTIAL RADON TRANSFORM

),(),(),,(

|),,(),,0(),,(

,12)(2

0)(2

3

tsrDDts

tsrtsD

tspFdtdsedrzyxf

dtdsdrezyxfzyxfF

ts

r

tsr

)(),( 0),(,20,1

0

ttpdsttsp D

tt

rD

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

F3D

projection plane

x

y

z

s

r

t

t0

integration plane

)s,r(E0t,t̂

)t,s(E 0,r̂

Deriving Full Radon (integrals on planes) Transform from Partial Radon Transform (line integrals)

1. 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.

2. 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).

REDUNDANCY OF PROJECTIONS ON ALL PLANES

SUFFICIENT SET OF DATA IN 3D FOR CONE BEAMTuy-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

xy

z

shadow zone

x

x

z

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.

1. helics (used in multislice “spiral” CT)

2. two non parallel circles (possibly used in C arm cone beam)

3. circle and line (just theoretical)

TRAJECTORIES SATISFYING TUY-SMITH CONDITION