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# Reconstruction from projections - TUT · PDF fileImage reconstruction ... The tomographic data...

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• Reconstruction from projections

Sakari Alenius

5.2.2003

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 1

Outline

Image reconstruction

Fourier slice theorem

Iterative reconstruction

Penalized iterative reconstruction

Conclusion

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 2

Image reconstruction

The image is not seen by the scanner, but its projection profiles are measured.

Measured PET sinogram data

-

Reconstructed image

Figure 1: Image reconstruction from projections (negative images)

The image is estimated computationally (inverse problem).

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 3

Different tomographic modalities reflect different things.

Figure 2: X-ray based CT(anatomical)

Figure 3: PET (functional)

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 4

The tomographic data acquisition is conventionally modeled by the Radontransform (Johann Radon, 1917).

Radon transform collects line integrals across the object at different angles.

m(t, ) , R{f} =

f(x, y)(x cos + y sin t) dx dy . (1)

Note: Radon m(t, ) is not a polar coordinate representation.

Measured data are collected as a sinogram matrix.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 5

Figure 4: Projection data collected as a sinogram (Radon transform of the unknownobject).

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 6

Fourier slice theorem

projection

1D FT

object

x

y

u

v1D FT ofanotherprojections

2D IFT

ts

Figure 5: The Fourier slice theorem: The 1D FT of a projection taken at angle equalsthe central radial slice at angle of the 2D FT of the original object.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 7

Figure 6: 2 projections reconstructed using simple direct Fourier method.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 8

Figure 7: 4 projections reconstructed using simple direct Fourier method.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 9

Figure 8: 8 projections reconstructed using simple direct Fourier method.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 10

Figure 9: 128 projection views reconstructed using simple direct Fourier method.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 11

Figure 10: The density difference btw. direct Fourier and FT of the original image.

The Fourier space filled in is most dense at and near the zero frequency.

Compensation by the distance from the center || Ramp-shaped filter.

Interpolation errors in the corners (high frequencies!) make direct application ofFourier slice theorem difficult not used in practice

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 12

The inverse 2D FT expressed using the polar coordinates and in the frequencyspace (u = cos , v = sin ) is

f(x, y) = F12 {F (u, v)} =

F (u, v) ej2(xu+yv) du dv

= 2

0

0

F ( cos , sin ) ej2(x cos +y sin ) u vu

v

=

d d

=

0

[

M(, )|| ej2(x cos +y sin ) d]d , (2)

where M(, ) is the 1D FT of the measure projection profile m(t, ). The multiplication by || serves as a ramp filter applied to each projection profile in

the frequency space.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 13

Figure 11: Common window functions used with ramp filter.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 14

(2) gives an algorithm to reconstruct the image f(x, y) from its projections m(t, )as the Filtered Back Projection, FBP: Set f(x, y) = 0,x, y. For each projectionprofile:

? Take 1D FT: m(t, ) M(, )? Apply the frequency domain filter (e.g. ramp + Hann)? Take inverse FT: ||M(, ) m(t, )? Back project (smear) filtered profile m over the image at the given angle

In the discrete implementation of FBP, the integrals are replaced by finitesummations and FFTs can be used.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 15

Figure 12: FBP with 2 projections (with noise)

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 16

Figure 13: FBP with 4 projections

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 17

Figure 14: FBP with 8 projections

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 18

Figure 15: FBP with 128 projections

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 19

Figure 16: Number of projection angles: 4, 8, 16, 32 views

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 20

Iterative reconstruction Iterative reconstruction: the reconstructed image is a solution of a maximization of

an objective function.

Starting from an initial guess, the image is updated iteratively so that it matchesbetter the measured projections.

Maximum likelihood expectation maximization (MLEM) searches for an image thatmakes the measured data most likely to occur (argmaxp(n|))

k+1b =

kbd pdb

d

nd pdbb kb pdb

(3)

k is the kth iteration emission image, b is the pixel index, d is the sinogram binindex, pdb is a system matrix element, and n is the measured sinogram.

With noisy data, the reconstructed image is also noisy.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 21

ART (algebraic reconstruction technique): The update is additive. The idea is tosolve a set of linear equations

n = A (4)

(n sinogram, A system matrix, image) and update the image according to thedifference btw. calculated and measured projections.

Other similar methods: SIRT, SART.

Due to noise, no unique solution exists.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 22

Figure 17: Simple example of MLEM iterations

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 23

Figure 18: Intermediate images and reprojections of MLEM iterations

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 24

Figure 19: True, FBP, MLEM (with noise)

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 25

Penalized iterative reconstruction

Ill-posed to well-posed: The image is required to fit with measured data, and also beconsistent with additional regularizing criteria that are set independent on the data.

The one step late (OSL) algorithm uses the current image k when calculating thevalue of the derivative of the energy function U().

k+1b =

kb

d pdb + b

U (, b) |=k

d

nd pdbb kb pdb

cLkb

= kb cP kb c

Lkb (5)

k is the kth iteration emission image, b is the pixel index, d is the sinogram binindex, pdb is a system matrix element, and n is the measured sinogram.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 26

The current image estimate k is updated using two factors: cP kb changes thepixel value such that prior assumptions are better met, and c

Lkb for better data

fitting (MLEM).

The penalty can be restricted to only non-monotonic structures in a neighborhoodby comparing the pixel against the local median.

Using this constraint in the term U() of Eq. (5), the penalty factor cP kb for MRP(median root prior) is

cP kb =

1d pdb +

kbMbMb

(6)

The penalty reference Mb = Med(k; b) is estimated as the median of the pixels ina neighborhood Nb

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 27

Figure 20: Flowchart of penalized MLEM reconstruction

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 28

Figure 21: Large number of iterations makes MLEM-image more noisy. MRP is morerobust.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 29

Figure 22: Transmission scans are used in PET for attenuation compensation.

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 30

Conclusion

FBP works if data are not very noisy and if the measurement can be accuratelymodeled as a Radon transform.

Noise regularization is frequency selective (cutoff & window) trade-off btw.resolution and noise.

Iterative methods maximize an objective function A probabilistic noise modelcan be used.

Quantitative accuracy is important in some studies (PET) noise reductionshould be unbiased.

Noise regularization: early stopping, priors & penalties.

MRP is effective, but some theoretical aspects are lost (no proof of convergence).

• Sakari Alenius 5.2.2003 7103100 Laaketieteelliset kuvausmenetelmat 31

Noise reduction directly from the sinogram is a demanding task. Some progress donealready.

? Kak, Slaney: Principles of Computerized Tomographic Imaginghttp://www.slaney.org/pct/

? Lets play PET http://laxmi.nuc.ucla.edu:8000/lpp/? Bruyant: Analytic and Iterative Reconstruction Algorithms in SPECThttp://www.snm.org/education/pq1002.html,http://jnm.snmjourn

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