Emission Discrete Tomography and
Optimization Problems
Attila Kuba
Department of Image Processing and Computer Graphics
University of Szeged
OUTLINE
Theoretical foundations of (Transmission) Discrete
Tomography (DT)
Existence
Uniqueness
Reconstruction
Theoretical foundations of Emission Discrete Tomography
(EDT)
Existence
Uniqueness
Reconstruction
Optimization in EDT, experiments
DISCRETE TOMOGRAPHY (DT)
Reconstruction of functions from their projections, when the
functions have known discrete range D = {d1,...dk}
BINARY TOMOGRAPHY
reconstruction of functions having binary range
2 1 1
4 1 1 1 1
3 1 1 1
4 1 1 1 1
1 1
3 4 3 2 1 1
sets (characteristic functions) binary matrices
BINARY MATRICES
F ( f ij ) mn , f ij 0,1
row sums:
r1 f11 f12 f1n n
r2 f21 f22 f2n ri f ij , i 1, , m
j 1
R
column sums:
m
s j f ij ,
fm1 fm2 fmn
rm j 1, , n
s1 s2 ... sn i 1
S
CLASSIFICATION OF THE PROJECTIONS
3 3 1 1 1 1 1 1 1
3 2 1 1 1 1 1 1
1 1 1 1 1 1 1
3 3 1 3 2 1
inconsistent unique non-unique
SWITCHING COMPONENT
1 1
configuration
1 1
2 1 1 2 1 1
4 1 1 1 1 4 1 1 1 1
3 1 1 1 3 1 1 1
4 1 1 1 1 4 1 1 1 1
1 1 1 1
3 4 3 2 1 1 3 4 3 2 1 1
It is necessary and sufficient for non-uniqueness.
CONSISTENCY
1 1 1 1
2
1 1 1 1 4 1 1 1 1
1 1 1 3 1 1 1
1 1 1 1 4 1 1 1 1
1 1 1
S* 5 4 3 2 0 0 3 4 3 2 1 1
k k
1 0
∑ s j ≥ ∑ s j’
*
1 1 1 1
j=1 j=1 3 1 1 1
1 1 1
1 1 1 1 4 1 1 1 1
k = 1,…,n 1 1 1 1 1 1 6 1 1 1 1 1 1
S’ 4 3 3 3 1 1
CONSISTENCY
k k
∑ sj* ≥ ∑ sj’, k = 1,…,n
j=1 j=1
it is a necessary and sufficient condition for the existence
Gale, 1957, Ryser 1957
example:
1 1 1 3
1 1 1 3
1 1
3 2 2 3 3 1
1 1 1 1
k = 2:
1 1 1 1 1
3+2 2 ?
ABSORBED PROJECTIONS
RECONSTRUCTION
optimization
cost function
Φ = ║Ag - y║2 + γ·║g║2
Metropolis algorithm (SA)
EXPERIMENTS
binary object (0 – black, 1 - white)
128×128
fan-beam projections
401 detectors/proj
stopping condition:
there was no accepted change in
the last 10.000 iterations
University of Szeged Zoltán Kiss,
Antal Nagy,
Lajos Rodek,
László Ruskó
NUMBER OF PROJECTIONS
32 166 s 619 s
16 176 s 307 s
8
126 s 90 s
10 % noise
DISTANCE CENTR. - DETECTOR
150 166 s 619 s
600 682 s 494 s
#proj. = 32
900 509 s 425s
10 % noise
ABSORPTION COEFFICIENT
0.005 166 s 619 s
0.009 626 s 652 s
#proj. = 32
0.03 626 s 652 s
10 % noise
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: direct@inf.u-szeged.hu
WORKSHOP ON DISCRETE TOMOGRAPHY
13-15 June, 2005
Graduate Center, City University of New York
Organisers:
Gabor T. Herman E-mail:gherman@gc.cuny.edu
Attila Kuba E-mail:kuba@inf.u-szeged.hu