1. Póth Miklós Polytechnical Engineering College, Subotica
Branch and Bound Method for Discrete Tomography Reconstruction of hv-convex Binary Matrices
Póth Miklós
Polytechnical Engineering College, Subotica

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. Examples
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4. Examples
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5. Classification
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6. Branch and Bound (B&B) Global Optimization Method.
For problems with finite, but very large number of solutions.
For problems where no polynomial algorithm is known.
Can be slow (exponential growth with problem size).
In some cases faster convergence.
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7. Branch and Bound (B&B) The search space is very large.
B&B works with the divide and conquer principle: the search space is subdivided into smaller subregions (branching).
Strength of B&B: when bounds on a large subregion show that it contains only inferior solutions, and so the entire subregion can be discarded without further examination.
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8. 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
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9. Convexity
h-convex
v-convex
hv-convex
no-convex
By introducing hv-convexity we exclude the switching components.
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10. Explanation of the method
We have n-k+1 possibilities to place k black
pixels on n positions (in a convex way).
R=[ ]
C=[ ]
Three possibilities to place two 1’s on four positions
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11. Tree development
The tree after two levels of development
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12. Edge cutting The four digit number at the nodes shows
the current column sum.
Does any of the four digits exceed the
given column sum? If so, we cut that edge.
On the previous picture the leftmost and
rightmost nodes fail to satisfy this condition
(2210 vs and 0122 vs. 1331).
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13. Edge cutting
Since we aim to reconstruct a hv-convex matrix, we know that in any column of the matrix after a 1-to-0 transition no 1 can follow.
To track the occurance of 1-to-0 transitions, we put a marker (dot) on the digit where the transition occurred.
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14. Tree analysis
The number of levels of the tree equals the number of rows of the matrix to be reconstructed (assuming that a solution exists).
The number of solutions equals the number of leaves.
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15. Performance and speed-up
Let’s consider the matrix with the following
row and column sum
C=[ ]
R=[ ]
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16. Performance and speed-up
After a 90° CW rotation of the above matrix,
R and C becomes:
R=[ ]
C=[ ]
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17. Row sum maximum start
Matrix rotation does not necessarily help to increase the computational speed.
Very often, the majority of the 1’s is placed in the middle of the matrix.
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18. Row sum maximum start R=[2 2 6 8 7 3 2 1] C=[2 3 3 6 7 6 3 1]
In this case we’ll have at least 42 nodes on
level 2.
Possible solution: to start from the row with
most ones.
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19. Developing the tree from row with most 1’s
Row sum maximum start
Developing the tree from row with most 1’s
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20. Conclusion
We showed a new method for reconstructing binary matrices from its orthogonal projections.
This method finds all the possible results that satisfy the row sum and column sum constraint and the hv-convexity property.
So before reconstruction, it is suggested to analyze the matrix first, and according to the elements of R and C to decide which method to use.
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21. Thank you for your attention. Questions?
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