Date post: | 27-Dec-2015 |
Category: |
Documents |
Upload: | spencer-lindsey |
View: | 216 times |
Download: | 0 times |
1
Discrete Tomography and Its Applications in Discrete Tomography and Its Applications in Medical ImagingMedical Imaging**
Attila Kuba
Department of Image Processing and Computer Graphics
University of Szeged
*This presentation is based on the joint paper
G.T. Herman, A. Kuba, Discrete Tomography in Medical Imaging,
Proc. of IEEE 91, 1612-1626 (2003).
2
OUTLINEOUTLINE
What is Discrete Tomography (DT) ? Application in Medical Imaging
AngiographySPECTPETEM
Discussion
3
DISCRETE TOMOGRAPHY (DT)
Reconstruction of functions from their projections, when the functions have known discrete range D = {d1,...dk}
Example: CT image of homogeneous object object – made of wood,projections – X-ray images,function – absorption coefficient,
D = {dair,dwood:}: absorption coefficients
4
WHY DISCRETE TOMOGRAPHY ?
use the fact that the range of the function to be reconstructed is discrete and known
Consequence:
in DT we need a few (e.g., 2-10) projections,
(in CT we need a few hundred projections)
5
WHY DISCRETE TOMOGRAPHY?
# projs. FBP Discretized FBP DT method
8
12
16
6
THE RECONSTRUCTION PROBLEM
y = A f ,
where
f: vector representing the object,
y: vector of measurements,
A: matrix describing the projections
f1 f2
f3 f4
y1 = 1·f1 + 1·f2 + 0·f3 + 0·f4
y2 = 2·f1 + 0·f2 + 0·f3 + 2·f4
7
RECONSTRUCTION METHOD
cost function to be minimized, e.g.
C(f) = || Af - y ||2
Task: find f such that C(f) is minimal
solution by optimization (e.g., simulated annealing)
8
APPLICATIONS
HRTEM (High Resolution Transmission
Emission Microscopy)
NDT(Non-destructive testing)
9
DT IN MEDICAL IMAGING
but the human body is not a homogeneous object,
it cannot even be considered to contain just a few homogenous regions
DT can be applied to medical imaging in special circumstances,
e.g. contrast material is injected (angiography), then
two regions:
contrast enhanced organ and
surrounding tissues
10
OBJECT TO BE RECONSTRUCTED
object – function f(x)
2D , 3D, … - two, three, … variables
range: D = {d1, d2, …, dc} known
binary object/image/function D = {0,1} c = 2
11
OBJECT TO BE RECONSTRUCTED
a priori knowledge:
F : class of functions f (having discrete, known range) that
may describe an object in that application area
examples:
convex sets,
functions having constant values on closed 3D regions with triangulated boundary surfaces,
12
PROJECTIONS
projections – integrals (e.g., along straight lines)
SgdxxfSfS
Pf(x)
S
g(S)
13
PROJECTIONS
binary object (a (0,1)-matix),
its horizontal and vertical projections (row and column sums)
14
PROJECTIONS
other kinds of projections
fan-beam
cone-beam (3D)
strips
15
PROJECTIONS
if f defined on a discrete domain (e.g., digital image on pixels/voxels):
f1
fI
gj
Jjgfai
jiij ,...,1,
gAf
linear equation system
projection data y ≈ g, (an approximation to g)
available from measurements,
Af ≈ y
16
RECONSTRUCTION PROBLEM
Let F be a class of functions having discrete, known range D
Given: The projections data y(S)
Task: Find a function f in F such that
[P f](S) ≈ y(S)
in the case of CT the class F is more general, e.g., D = [0,+∞)
17
RECONSTRUCTION METHODS
Heuristic
Discretization of classical reconstruction methods
Optimization
18
HEURISTIC RECONSTRUCTION METHODS
19
DISCRETIZATION OF CLASSICAL RECONSTRUCTION METHODS
20
RECONSTRUCTION METHODSBASED ON OPTIMIZATION
Af = g ≈ y
big size
under determined (not enough data)
contradiction (measurement errors, no solution)
instead of exact solution minimization of a cost function:
C = ║Af - y║2 + Φ(f)
how far is an f
from the measurements
how undesirable
is a solution f
21
REGULARIZATION
C = ║Af - y║2 + γ·║f║2
different selections of C
γ regularization parameter
C = ║Af - y║2 + γ·║f – f(0)║2
f(0) a given prototype (a similar object)
C = ║Af - y║2 + γ·∑ici·fi
ci weight of cost of the position i
22
OPTIMIZATION METHODS
ICM (Iterated Conditional Modes)
a local descent-based method
SA (Simulated Annealing)
initialization: let f be in Finteration: change f to reach less cost C
23
COMPARISON OF DT IMAGES
%100'
'2
%100'
'2
%100'
ii
ii
ii
ii
i
ii
ff
ffVE
ff
ffSE
f
ffRrelative mean error
shape error
volume error
24
ANGIOGRAPHY
digital subtraction angiography
object with two homogeneous regions: contrast medium having known absorption value (μcontrast), background (μ=0)
it can be reduced to a binary object (μ = 0,1)
a small number of projections can be taken, e.g., 2
biplane angiography
25
ANGIOGRAPHY OF CARDIAC VENTRICLES
Chang, Chow 1973
clay model of a dog’s heart
two X-ray projections
cross-sections: convex, symmetric polygons
heuristic reconstruction method
26
ANGIOGRAPHY OF CARDIAC VENTRICLES
Onnasch, Heintzen 1976
heart ventricles
two X-ray projections
prior information: similar neighbor slices, similar to 3D model
heuristic reconstruction method
enddiastolic RV cast
27
Onnasch, Prause, 1999
ANGIOGRAPHY OF CARDIAC VENTRICLES
28
Onnasch, Prause, 1999
ANGIOGRAPHY OF CARDIAC VENTRICLES
29Onnasch, Prause, 1999
ANGIOGRAPHY OF CARDIAC VENTRICLES
30
ANGIOGRAPHY OF CORONARY ARTERIES
coronary arterial segments from two projectionsReiber, 1982
31
Human iliac bifurcation, elliptic-based model
Pellot, 1994
ANGIOGRAPHY OF CORONARY ARTERIES
32
PET
Bayesian reconstruction and use of anatomical a priori informationBowsher, 1996
33
SPECT
MAP reconstructions of the bolus boundary surfaceCunningham, Hanson, Battle, 1998
34
ET
Phantom FBP DT
Chan, Herman, Levitan, 1998
35
SPECT
MAP reconstructions of the bolus boundary surfaceCunningham, Hanson, Battle, 1998
36
SPECT
Tomography reconstruction using free-form deformation modelsFFDs reconstructions for different levels of noise
Battle, Bizais, Le Rest, Turzo, 1999
37
SPECT
Tomography reconstruction using free-form deformation modelsFFDs reconstructions for different levels of noise
Battle, 1999
38
PET
Edge-preserving tomography reconstruction with nonlocal regularization.Emission phantom, FBP, Huber penalty, proposed penalty
Yu, 2002
39
DIscrete REConstruction Tomography
software tool for
generating/reading projections
reconstructing discrete objects
displaying discrete objects (2D/3D)
available via Internet
http://www.inf.u-szeged.hu/~direct/
it is under development
E-mail: [email protected]
40
DISCUSSIONDISCUSSION
If the object to be reconstructed consists of known materials, then DT can be applied.
DT offers new theory and techniques for reconstructing images from less number of projections.
Further experiments are necessary (including e.g. fan-beam projections, scattering)