Back Projection Reconstruction

for CT, MRI and Nuclear Medicine

F33AB5

CT collects Projections

• Introduction• Coordinate systems• Crude BPR• Iterative reconstruction• Fourier Transforms• Central Section Theorem• Direct Fourier Reconstruction• Filtered Reconstruction

To produce an image the projections are back projected

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.

Crude v filtered BPR

90

360

Crude BPR Filtered BPR

Sinograms

r

r

Stack up projections

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.

Filtered BPR

• Therefore there are two possible approaches to deblurring the crude BPR images:

• Deconvolve multiplying by f (1/f x f = 1) in Fourier domain.

• Convolve with Radon filter in the image domain, to overcome effect of being filtered with 1/r by crude BPR.