1. Póth Miklós Subotica Tech
Particle Swarm Optimization Approach to Discrete Tomography Reconstruction Problems of Binary Matrices
Póth Miklós
Subotica Tech
SISY 2014

2. Discrete Tomography
Reconstruction of a discrete set from its projections (2-10, in our case 2).
Binary Tomography: the matrix has only binary values.
Trivial necessary condition for existence of the solution: rowsum(15) = colsum(15).
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3. Discrete tomography
Since very large number of solution can exist, we usually limit the search to matrices with certain properties (convexity properties, number of connected com-ponents).
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4. Convexity
The convexity property means that the series of 1’s is not broken:
Horizontal convexity (h-convexity)
Vertical convexity (v-convexity)
Horizontal and vertical convexity (hv-convexity)
No convexity
h-convex
v-convex
hv-convex
no-convex
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5. What is Computed Tomography?
Computed Tomography (CT) is a computer aided imaging technique performed by illuminating the object of interest with radiation and measuring the attenuation.
Commonly, X-ray emitters and detectors are used to collect the attenuation data. That data is then processed to reconstruct 2D or 3D images of the regions of interests in a non-invasive manner, using a machine like the one on the right.
CT is widely used in the medical field in the diagnosis of cancers, disease, and to recover muscular-skeletal information in the human body. It can also be useful in determining dosimetry for radiation treatments.
CT Scanner
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6. Tomography Scanners
CT Machines are basically a motorized table with an array of
cathode ray X-ray tubes positioned around the patient’s body in a radial
fashion. With the aid of computers, image data can be acquired and
processed to reconstruct images of the underlying information.
[2]
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Check out some very impressive examples of CT at GE’s website below
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7. Computerized Tomographyx y s u σ
f (x,y)
X-rays
Tomography is an inverse problem where the content of the matrix is not known.
Reconstruct f (x,y) from its projections.
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8. Nonogram
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9. The algorithm
First the binary matrix is reconstructed using Ryser’s algorithm (1957)
Hv-convex binary matrix with two connected components.
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10. Basic operation - switching
By changing the position of ones the row sum and the column sum of the matrix does not change.
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11. The PSO – generating the population (scrambling)
Since the particle swarm optimization method is a population based global optimization technique it is needed to generate the population for the process of optimization.
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12. Evaluation of the quality of population members
The following formula was used:
C4O=2
C4=5
C8=3
C8O=2
HVR=2
HVC=3
Q=sqrt(23)
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13. The “winner”
The matrix with the lowest Q value is the “winner” matrix.
In the following step all other matrices will tend to look like as much as possible the winner of the previous step.
The changes in the matrices are done through switching.
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14. The PSO process
Swarm optimization process was first introduced by Kennedy and Eberhard (1995) and it has its origins in two distinguished concepts:
- the intelligence of swarms is based on examining swarms of different species such as birds, bees or fishes.
- evolutionary algorithms.
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15. The PSO process
The PSO algorithm consists of only three very simple steps:
- Determinig the fitness (quality) of each member of the swarm.
- Updating the individual and global best fitnesses and positions.
- Updating the velocity and position of each individual.
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16. The PSO process
The speed of each particle is updated according to the following formula:
the velocity of particle i at time t
the position of particle i at time t
the position of the best particle at time t
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17. The PSO process
the velocity of each particle in the next time step depends on three components:
is the inertial component. This component is responsible for the moving of the particle in the direction it originally had.
is the cognitive component, it serves as the memory of the particle, it influences the particle to return to positions where it had high fitness values.
is the social component, its task is to lead the particle towards the best position that the swarm found so far.
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18. Stopping condition
The PSO process can terminate either when a perfect solution is found or when the solution meets some other criteria. When the user is satisfied with a non-perfect solution, he can choose from one of the following criteria:
- fixed number of iterations performed.
- no improvement of the average fitness in the last 10 generations.
- no improvement of the best fitness in the last 10 generations.
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19. Experimental results MATRIX SIZE MIN MAX AVG 20X20 7.4 182.7 105.0
MINIMAL, MAXIMAL AND AVERAGE EXECUTION TIMES
IN SECONDS FOR MATRIX SIZES 20X20, 30X30 AND 50X50
MATRIX SIZE
MIN
MAX
AVG
20X20
7.4
182.7
105.0
30X30
15.1
405.2
244.3
50X50
44.3
INF
532.2
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20. NUMBER OF SWITCHING STEPS TO GET
Experimental results
NUMBER OF SWITCHING STEPS TO GET
THE FIRST SOLUTION
MATRIX SIZE
MIN
MAX
AVG
20X20 34
605
403
30X30 71
1143
766
50X50
206
INF
1674
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21. Thank you for your attention. Questions?
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