The Reconstruction of Some 3D Convex Polyominoes from Orthogonal - - PowerPoint PPT Presentation

The Reconstruction of Some 3D Convex Polyominoes from Orthogonal Projections

Maciej G˛ ebala Institute of Mathematics Wrocław University of Technology email: mgc@im.pwr.wroc.pl

SOFSEM’02 – p.1

Discrete Tomography

Basic Problem: Reconstruction of finite point sets that are acces- sible only through some of their discrete X-rays.

SOFSEM’02 – p.2

2D Convex Polyomino

Set of cells S in a m × n grid with properties: S is connected, S in each row and each column is connected.

SOFSEM’02 – p.3

3D Convex Polyomino

Set of cells S in n × n × n grid with properties: S is connected, each two-dimensional orthogonal section

• f the grid contains 2D convex polyomino.

SOFSEM’02 – p.4

Orthogonal Projections

Orthogonal projections - the number of cells in each bar of the grid that belongs to S. Top projections PT

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 2 2 2 2 2 2 3 2 1 2 3 3 2 1 2 2 2 2 3 4 3 1 4 2 1 2 3 1

Front projections PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 4 1 2 2 4 4 2 2 4 3 1 1 4 5 2 1 1 3 1 2 3 3 2 2 2 1

Side Projections PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 2 1 4 4 2 2 3 4 2 2 1 2 5 4 1 1 2 1 3 1 2 3 3 1 2 2

SOFSEM’02 – p.5

Reconstruction Problem

Given three assigned matrices: PF, PS, PT ∈ {0, . . . , n}n×n, we examine whether there exists at least one convex polyomino S with adequate orthogonal projections.

SOFSEM’02 – p.6

Full 3D Convex Polyomino

3D Convex Polyomino with at least one matrix

• f projections does not contain zeros (full

matrix). We determine that PT is a full matrix. PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.7

Our Result

• Theorem. The problem of the reconstruction
• f full convex 3D polyominoes from orthogonal

projections has the complexity O(n7 log n).

SOFSEM’02 – p.8

Properties of 2D Convex Polyom.

auxiliary values Dց

j , U ց j , Dւ j , U ւ j

(easy computable) ∀j∈{1,...,n} U ց

j

≤ Dց

j

∧ U ւ

j

≤ Dւ

j .

∀j∈{1,...,n−1} Dց

j + 1 ≥ U ց j+1

∧ Dւ

j+1 + 1 ≥ U ւ j .

• Lemma. Let cells [1, p1] and [n, p2] belong to the convex

polyomino S. Then

• 1. if p1 ≤ p2 then cells [1, p1], . . . , [Dց

p1, p1], and

[U ց

j , j], . . . , [Dց j , j]

∀j∈{p1+1,...,p2−1}, and [U ց

p2 , p2], . . . , [n, p2] also belong to the S, or

• 2. if p1 ≥ p2 then cells [U ւ

p2 , p2], . . . , [n, p2], and

[U ւ

j , j], . . . , [Dւ j , j]

∀j∈{p2+1,...,p1−1}, and [Dւ

p1, p1], . . . , [1, p1] also belong to the S.

SOFSEM’02 – p.9

Example of Properties

1 2 3 4 6 8 8 7 7 7 7 7 7 7 6 8 8 5 3 1 1 3 6 10 16 24 32 39 46 53 60 67 74 81 87 95 103 108 111 112 2 3 5 7 9 10 7 7 5 5 5 5 7 8 8 7 5 3 2 2 2 5 10 17 26 36 41 50 55 60 65 70 77 85 93 100 105 108 110 112 U D vj hi Vj Hi

SOFSEM’02 – p.10

Algorithm

• 1. Compute start positions

(a) select one cell in each corner vertical bar (b) from Lemma compute positions of cells belonging to S in both lateral slices of the grid (c) from Lemma compute positions of cells belonging to S in vertical slices

• rthogonal to slices from (b)
• 2. Perform Filling Procedure

SOFSEM’02 – p.11

Example of Reconstruction

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Example of Initial Positions

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Example of Using the Lemma

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Example of Using the Lemma

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Example of Using the Lemma

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Example of Start Positions

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Example of 3D Convex Polyomino

PT

1 1 2 3 2 2 1 1 2 3 2 1 2 1 2 3 2 2 2 2 1 3 2 2 3 3 4 2 2 1 2 3 3 3 3 2 2 2 4 3 1 1 1 1 1 2 3 1 1

PF

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 5 4 3 2 1 3 4 5 4 3 3 2 2 4 5 2 4 4 4 4 2 2 2 4 4 3 1 3 2 1

PS

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 1 2 3 4 5 3 3 4 5 4 3 4 4 2 5 4 2 2 2 2 4 4 3 4 4 2 1 2 3 1

SOFSEM’02 – p.12

Filling Procedure - properties

It returns fail if 3D convex polyomino with computing start positions and adequate matrices of projections does not exist. Otherwise it returns 3D convex polyomino. It works correctly if start positions contain at least one cell of polyomino in each vertical bar.

SOFSEM’02 – p.13

Filling Procedure - the main idea

h = 10 - the value of the projection for the bar 1 1 1 1 p ˜ p ˜ l l r ˜ r ˜ q q

• peration ⊕ - the integration of the block of 1’s

1 1 1 1 1 1 p ˜ p ˜ l l r ˜ r ˜ q q

• peration ⊖ - the integration of the two final blocks of 0’s

0 0 0 0 1 1 1 1 1 1 0 0 h h

• peration ⊗ - the expansion of the block of 1’s

0 0 0 0 1 1 1 1 1 1 1 0 0 h h

• peration ⊙ - the expansion of the two final blocks of 0’s

0 0 0 0 1 1 1 1 1 1 1 0 0 0 p ˜ p ˜ l l r ˜ r ˜ q q

SOFSEM’02 – p.14

Complexity

n4 – the number of different initial positions; O(n2) – the cost of computing of the start positions; O(n3 log n) – the cost of the Filling Procedure. Complexity of the Algorithm – O(n4 · n3 log n).

SOFSEM’02 – p.15