Back Projection Reconstruction for CT, MRI and Nuclear Medicine
Back Projection Reconstruction for CT, MRI and Nuclear Medicine F33AB5.
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Transcript of Back Projection Reconstruction for CT, MRI and Nuclear Medicine F33AB5.
• Introduction• Coordinate systems• Crude BPR• Iterative reconstruction• Fourier Transforms• Central Section Theorem• Direct Fourier Reconstruction• Filtered Reconstruction
Crude back projection
• Add up the effect of spreading each projection back across the image space.
• This assumes equal probability that the object contributing to a point on the projection lay at any point along the ray producing that point.
• This results in a blurred image.
Solutions
• Two competitive techniques– Iterative reconstruction
• better where signal to noise ratio is poor
– Filtered BPR • faster
• Explained by Brooks and di Chiro in Phys. Med. Biol. 21(5) 689-732 1976.
Coordinate system
• Data collected as series of – parallel rays, at
position r, – across projection
at angle .
• This is repeated for various angles of .
Detec
tor,
trans
lated
X-ra
y tu
be, t
rans
late
d
X-ray beam
Sample
r
s
Attenuation of ray along a projection
• Attenuation occurs exponentially in tissue.
(x) is the attenuation coefficient at position x along the ray path.
dxxeII
)(
0
Definition of a projection• Attenuation of a ray at position r, on the
projection at angle , is given by a line integral.
• s is distance along the ray, at position r across the projection at angle .ds s)(r, =
ds )(x, =
IIln = ) ,p(r 0
yDet
ecto
r, tra
nslat
ed
X-ra
y tu
be, t
rans
late
d
X-ray beam
Sample
r
s
Coordinate systems• (x,y) and (r,s) describe the distribution of
attenuation coefficients in 2 coordinate systems related by .
• where i =1..M for M different projection orientations• angular increment is = /M.
x
y r (along projection)
S (along ray path)
tube
detectorii yxr sincos
Crude back projection• Simply sum effects of back-
projected rays from each projection, at each point in the image. ) ,p(r = y)(x, i
M
1=i
*
) ,ysin+(xcos p = y)(x, iii
M
1=i
*
Crude back projection
• After crude back projection, the resulting image, *(x,y), is convolution of the object ((x,y)) with a 1/r function.
Convolution
• Mathematical description of smearing. • Imagine moving a camera during an
exposure. Every point on the object would now be represented by a series of points on the film: the image has been convolved with a function related to the motion of the camera
Iterative Technique
• Guess at a simulated object on a PxQ grid (j, where j=1PxQ),
• Use this to produce simulated projections
• Compare simulated projections to measured projections
• Systematically vary simulated object until new simulated projections look like the measured ones.
• For your scanner calculate jj(r,i), the path length through the jth voxel for the ray at (r,i)
j need only be estimated once at the start of the reconstruction,
j is zero for most pixels for a given ray in a projection
1 2 3 4
16
5 6 7 8
1211109
13 14 15
j=02=0.17=1.2
• The simulated projections are given by:
j is mean simulated attenuation coefficient in the jth voxel.
jij
QP
ji r ),( = ) ,(r
1
1 2 38 9 4
7 6 5
1415
6
16 17 2
6
2118
27
6
27
6
27
6
Object and projections
15 15 15
1019
109
1513
First ‘guess’
From Physics of Medical Imaging by Webb
To solve• Analytically, construct P x Q simultaneous
equations putting (r,i) equal to the measured projections, p(r,i):
•– this produces a huge number of equations – image noise means that the solution is not exact and the
problem is 'ill posed’
• Instead iterate: modify j until (r,i) looks like the real projection p(r,i).
jj
ji i
)(r, = )(r, p
Iterating• Initially estimate j by projecting data
in projection at = 0 into rows, or even simply by making whole image grey.
• Calculate (r,i) for each i in turn.
• For each value of r and , calculate the difference between (r,) and p(r,).
• Modify i by sharing difference equally between all pixels contributing to ray.
21/3
71/3
61/3
22/3
72/3
62/3
16
5
16 17 12
1015
122/3
Next iteration
27
6
27
6
27
6
15 15 15
1513
First ‘guess’
1 2 38 9 4
7 6 5
Object
1 2 38 9 4
7 6 5
1415
6
16 17 2
6
2118
1019
10
Fourier Transforms
• Imagine a note played by a flute.
• It contains a mixture of many frequency sound waves (different pitched sounds)
• Record the sound (to get a signal that varies in time)
• Fourier Transforming this signal will give the frequencies contained in the sound (spectrum)
Time Frequency
Fourier transforms of images
• A diffraction pattern is the Fourier transform of the slit giving rise to it
kkxx
kkyy yy
xx
FTFT
Central Section theorem• The 1D Fourier transform of a
projection through an object is the same as a particular line through 2DFT of the object.
• This particular line lies along the conjugate of the r axis of the relevant projection.
kkxx
kkyyyy
xx
FTFT
Projection
Direct Fourier Reconstruction
• Fourier Transform of each projection can be used to fill Fourier space description of object.
y
x
ky
kx
Fp(r,1)
Fp(r,2)
InverseFourierTransform
Direct Fourier Reconstruction
• BUT this fills in Fourier space with more data near the centre.
• Must interpolate data in Fourier space back to rectangular grid before inverse Fourier transform, which is slow.
Relationship between object and crude BPR
results• Crude back projection from above:
• Defining inverse transform of projection as:
• then
, d ) ,ysin+p(xcos = y) ,(x o
*
dk,e )(k, F = ) ,p(r ikr2p
-
ddk |k|
[k] e ) ,(k F = y)(x, )ysin+ik(xcos2
p
-o
*
• The right hand side has been multiplied and divided by k so that it has the form of a 2DFT in polar coordinates – k conjugate to r k conjugate to r
– the integrating factor is kdrd dxdy
ddk |k|
[k] e ) ,(k F = y)(x, )ysin+ik(xcos2
p
-o
*
• Crude back projected image is same as the true image, except Fourier amplitudes have been multiplied by (magnitude of spatial frequency)-1
.
– Physically because of spherical sampling. – Mathematically because of changes in
coordimates.
k
)(k,F = )k,k(Fp
yx*
Filtered BPR
• Multiplying 2 functions together is equivalent to convolving the Fourier Transforms of the functions.
• Fourier transform of (1/k) is (1/r)• Multiplying FT of image with 1/k is
same as convolving real image with 1/r
• ie BPR has effect we supposed.