# BME 412 Biomedical Signal Processing Class 1

Document Sample

```					  BMME 560 & BME 590I
Medical Imaging: X-ray, CT, and
Nuclear Methods
Tomography Part 4
Today
• Tomography
– Other geometries
•   Fan-beam variants
•   Cone-beam
•   Pinhole
•   Ring
– Other reconstruction methods
• Matrix-based
• Iterative reconstruction
Tomographic Reconstruction
t
• The problem
y
p(t,q)
f(x,y)       t
s

q

x

Given p(t,q) for 0<q<p
Find f(x,y)
Other Geometries
• We have discussed only parallel (Radon)
projections so far.

• There are other projection geometries that are
more practical in certain situations.
–   Fan-beam
–   Cone-beam
–   Pinhole
–   Ring
Fan-beam Reconstruction
y                 y
f(x,y)            f(x,y)

x                 x

t
Fan-beam Variants
• There are a few variations on fan-beam
– Asymmetric fan
– Varifocal fan
– Multi-slice fan-beam
Asymmetric Fan
y                      y
f(x,y)                 f(x,y)

x                      x

Asymmetric               Symmetric
Varifocal fan-beam
y                y
f(x,y)           f(x,y)

x                x

Varifocal
3D Imaging
• So far, all of our treatments have been with
taking 1D projections of a 2D object.
• Traditional CT would take one slice at a time
and step the patient through the CT to build up
the 3D image from individual 2D images.
• Modern CTs take more than one slice at a
time.
Multi-slice Fan-beam: 3D
y                    y
f(x,y,z)             f(x,y,z)

z               x

Stack up a
few
detectors

z                       t
Multi-slice Fan-beam: 3D
• As long as the oblique angle is not too severe,
can use fan-beam reconstruction without much
distortion.

• As the oblique angle becomes more
significant, distortions become more
significant.

• This becomes a cone-beam.
Cone-beam Imaging

Issues:
•Data sufficiency
•Inexact reconstruction methods
•Field of view
Cone-beam Data Sufficiency
Consider a point in
the focal plane

To this point, projections look like a fan beam.
Cone-beam Data Sufficiency
Consider a point
above or below the
focal plane

To this point, projections look like a tilted fan beam.
Tilted fan beam is insufficient to completely reconstruct a 3D object.
Cone-beam Data Sufficiency
Consider a point off
axis and above or
below the focal
plane

To this point, projections look like tilted fan
beams at multiple oblique angles.
Data Insufficiency Artifact
Circular-orbit cone-beam

Single            Dual              Triple
Sufficient Data Orbits

Require complex motion of
camera and/or subject

Difficult to reconstruct:
•Can be done iteratively
•Some analytical methods exist
Cone-Beam Tomography
Options for reconstruction:
• Feldkamp algorithm: Like FBP
• Fourier rebinning: Estimate parallel-beam
from cone-beam
• Iterative algorithms
Feldkamp Algorithm
Same as fan-beam filtered backprojection, but backproject
along tilted fans.
Pinhole Tomography
• In SPECT, we can create a pinhole projection.

Subject
Only the photons that
pass through the        Detector
pinhole are detected.
Ring System
• Exclusive to PET

Detection
occurs between                 Every pair creates
pairs of                       a “line of
detectors                      response” (LOR)
Ring System
• 2D PET                   y

Each line of
response
represents a (t,q)                 q
pair.

x
t

How can you reconstruct this?
Is it sufficient?
Ring System
• 3D PET
Now, lines of
response are both
within a slice
(“directs”) and
across slices
(“obliques”).

How could you
reconstruct this?
Other Approaches to Reconstruction
• FBP develops problems when:
– Data is inconsistent
– Geometries do not translate well to parallel-beam
(especially 3D geometries: ring, cone-beam,
pinhole)
– Data is incomplete

• More general methods perform better in these
situations
Matrix Methods
• The forward problem is reformulated as a
matrix equation:
g  Hf
A vector of all image-
A vector of all                                 space pixels.
projection values
over t and q.           A matrix defining the
contribution of each
image-space pixel to
each projection bin.

Now, the problem is: Given g and H, solve for f.
Matrix Methods
• Solving the problem may be as simple as
inverting the matrix:
ˆ
f  H 1g

• If H represents the Radon projection process,
then H-1 would represent FBP.
• However, we can compute an H for virtually
any projection geometry and/or situation.
Matrix Methods
• Two problems:
– The matrix is BIG (For CT, 106 x 106?)
– For many geometries, the inverse does not exist or
is poorly conditioned
• Inversion is subject to big numerical errors, especially
with noise.

• Iterative methods can be used to approximate
the solution to the matrix equation.
Iterative Reconstruction
Essentially, an iterative approach to the matrix inversion.

Estimated
Current             Project
Projections
Estimate
Measured
Projections
Update
Compare

Image-Space                           Projection-Space
Backproject
Error                                    Error
ART Reconstruction
• A classic iterative reconstruction method:
Algebraic Reconstruction Technique

• Illustrate with this old example:
1    2     3

3    4     7

4    6
ART Reconstruction
along one direction
0    0     0   3

0    0     0   7

4    6
ART Reconstruction
• Compute errors and backproject errors

3   3
0 0      3    3
2   2
7   7
0 0      7    7
2  2

4   6
ART Reconstruction
• Project the result at another angle

3   3
3
2   2
7   7
7
2   2

5    5

4    6
ART Reconstruction
• Backproject those errors and update

3 1   3 1
            3
2 2   2 2
7 1   7 1
            7
2 2   2 2

-1    1

4     6
ART Reconstruction
• Then, start over again if necessary

1   2        3      3

3   4        7      7

4    6   This case has converged after
one pass through all of the data,
but that does not always happen,
especially with noisy data.
ART Reconstruction
• Sensitive to noise and the result may depend
on the order of angles
• Can be done mulitplicatively (MART)
• Important for historical perspective; variations
of it are used in research.
Iterative Reconstruction
Essentially, an iterative approach to the matrix inversion.

Estimated
Current             Project
Projections
Estimate
Measured
Projections
Update
Compare

Image-Space                           Projection-Space
Backproject
Error                                    Error
Statistical-based Iterative Algorithms
• More mathematically rigorous than ART
• Can model any geometry and any physical
• Better at limiting noise than FBP and ART
• Widely used in SPECT and PET
– OS-EM: Ordered-subset expectation maximization
• Algorithms exist for transmission CT, but are
not widely used.
Iterative Reconstruction
Essentially, an iterative approach to the matrix inversion.

Estimated
Current             Project
Projections
Estimate
Measured
Projections
Update
Compare

Image-Space                           Projection-Space
Backproject
Error                                    Error
Statistical-based Iterative Algorithms