New perspectives in the reconstruction of convex polyominoes from orthogonal projections

New perspectives in the reconstruction of convex polyominoes from orthogonal projections Paolo Dulio, Andrea Frosini 2, Simone Rinaldi 3, 4, Laurent Vuillon 5 Politecnico di Milano, 2 Università di Firenze, 3 Università di Siena, 4 Université Nice Sophia Antipolis, 5 Université Savoie Mont Blanc Reconstruction of convex polyominoes from orthogonal projections4-5-28 / 28

Pigami game Reconstruction of convex polyominoes from orthogonal projections4-5-28 2 / 28

Outlines Digital convex Polyominoes Reconstruction of convex polyominoes from orthogonal projections4-5-28 3 / 28

Outlines Digital convex Polyominoes 2 Combinatorics on words for discrete geometry Reconstruction of convex polyominoes from orthogonal projections4-5-28 3 / 28

Outlines Digital convex Polyominoes 2 Combinatorics on words for discrete geometry 3 Local modications Reconstruction of convex polyominoes from orthogonal projections4-5-28 3 / 28

Outlines Digital convex Polyominoes 2 Combinatorics on words for discrete geometry 3 Local modications 4 Conclusion and perspectives Reconstruction of convex polyominoes from orthogonal projections4-5-28 3 / 28

Polyomino Denition A polyomino is a (nite) connected set of points in the lattice Z 2. Reconstruction of convex polyominoes from orthogonal projections4-5-28 4 / 28

Polyomino Denition A polyomino is a (nite) connected set of points in the lattice Z 2. Denition A polyomino P is said digitally convex (DC) if its convex hull contains no integer point outside P Reconstruction of convex polyominoes from orthogonal projections4-5-28 4 / 28

(a) (b) (c) Reconstruction of convex polyominoes from orthogonal projections4-5-28 5 / 28

(a) (b) (c) (a) Non connected set, (b) A polyomino, (c) A DC polyomino. Reconstruction of convex polyominoes from orthogonal projections4-5-28 5 / 28

Border The border of a DC polyomino S, Bd(S), is the 4-connected path that follows clockwise the points of S that are 8-adjacent to some points not in S. Reconstruction of convex polyominoes from orthogonal projections4-5-28 6 / 28

Border The border of a DC polyomino S, Bd(S), is the 4-connected path that follows clockwise the points of S that are 8-adjacent to some points not in S. This path is a word in {,,, }, starting by convention from the leftmost lower point considered in the clockwise order. Reconstruction of convex polyominoes from orthogonal projections4-5-28 6 / 28

Example A DC polyomino S and its border. N E S W Reconstruction of convex polyominoes from orthogonal projections4-5-28 7 / 28

Example A DC polyomino S and its border. N E S W The points W, N, E and S are dened as in the gure. Reconstruction of convex polyominoes from orthogonal projections4-5-28 7 / 28

Example A DC polyomino S and its border. N E S W The points W, N, E and S are dened as in the gure. The word w A coding the WN path is w =. Reconstruction of convex polyominoes from orthogonal projections4-5-28 7 / 28

Reconstruction of convex polyominoes from orthogonal projections4-5-28 8 / 28

Theorem (B,L,P,R 2) A word w is WN -convex i its unique Lyndon factorization l n l n2 2...l n k k that all l i are Christoel words. is such Reconstruction of convex polyominoes from orthogonal projections4-5-28 8 / 28

Lyndon words Standard Lexicographic sequences was introduced by Roger Lyndon in 954. Reconstruction of convex polyominoes from orthogonal projections4-5-28 9 / 28

Lyndon words Standard Lexicographic sequences was introduced by Roger Lyndon in 954. Denition A w A + is a Lyndon word if it is the smallest between all its conjugates with respect to the lexicographic order. Reconstruction of convex polyominoes from orthogonal projections4-5-28 9 / 28

If w is a Lyndon word, ww is not. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

If w is a Lyndon word, ww is not. First Lyndon words on A :,,,,,,. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

If w is a Lyndon word, ww is not. First Lyndon words on A :,,,,,,. Theorem (Chen-Fox 954) Every non empty word w admits a unique factorization as a lexicographically decreasing sequence of Lyndon words. w = l n l n2 2 l n k, s.t l k > l l 2 > l l k where n i and l i are Lyndon words. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

Christoel path Let a and b be two relatively prime natural numbers. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

Christoel path Let a and b be two relatively prime natural numbers. Figure: The Christoel path of the line segment joining O(, ) to (b, a). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

Christoel path Let a and b be two relatively prime natural numbers. Figure: The Christoel path of the line segment joining O(, ) to (b, a). The closest path to the line segment. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

Christoel path Let a and b be two relatively prime natural numbers. Figure: The Christoel path of the line segment joining O(, ) to (b, a). The closest path to the line segment. 2 There are no points of Z Z between the path and line segment. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 / 28

Christoel word Horizontal step. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Christoel word Horizontal step. Vertical step. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Christoel word Horizontal step. Vertical step. The Christoel word of slope a b denoted w = C( a b ) where a b = w w. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Christoel word Horizontal step. Vertical step. The Christoel word of slope a b denoted w = C( a b ) where a b = w w. Property Christoel words are of the form: w, where w is a palindrome. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Particular points (8,5) P w 2 O(,) w Q P the closest point to the line segment on the Christoel path. (Standard factorization) Reconstruction of convex polyominoes from orthogonal projections 4-5-28 3 / 28

Particular points (8,5) P w 2 O(,) w Q P the closest point to the line segment on the Christoel path. (Standard factorization) 2 Q the furthest point from the line segment on the Christoel path. (Palindromic factorization) Reconstruction of convex polyominoes from orthogonal projections 4-5-28 3 / 28

Particular points (8,5) P w 2 O(,) w Q P the closest point to the line segment on the Christoel path. (Standard factorization) 2 Q the furthest point from the line segment on the Christoel path. (Palindromic factorization) The uniqueness of these points is due to Borel, Laubie (BL993). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 3 / 28

Example Consider the following WN-convex path v =. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 4 / 28

Example Consider the following WN-convex path v =. The Lyndon factorization of v is: Reconstruction of convex polyominoes from orthogonal projections 4-5-28 4 / 28

Example Consider the following WN-convex path v =. The Lyndon factorization of v is: v = () () () 2 () (). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 4 / 28

Example Consider the following WN-convex path v =. The Lyndon factorization of v is: v = () () () 2 () (). where,,, and are all Christoel words. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 4 / 28

Example Consider the following WN-convex path v =. The Lyndon factorization of v is: v = () () () 2 () (). where,,, and are all Christoel words. The Christoel words are arranged in a decreasing order of slopes. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 4 / 28

Example Consider the following WN-convex path v =. The Lyndon factorization of v is: v = () () () 2 () (). where,,, and are all Christoel words. The Christoel words are arranged in a decreasing order of slopes. > 2 > > 3 > Reconstruction of convex polyominoes from orthogonal projections 4-5-28 4 / 28

Reconstruction w4 w3 w2 w Reconstruction of convex polyominoes from orthogonal projections 4-5-28 5 / 28

Reconstruction w4 w3 w2 w w, w 2, w 3 and w 4 are C.W. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 5 / 28

Reconstruction w4 w3 w2 w w, w 2, w 3 and w 4 are C.W. Where to add a single point in order to preserve the convexity? Reconstruction of convex polyominoes from orthogonal projections 4-5-28 5 / 28

Reconstruction w4 w3 w2 w w, w 2, w 3 and w 4 are C.W. Where to add a single point in order to preserve the convexity? 2 Can we add more than one point? Reconstruction of convex polyominoes from orthogonal projections 4-5-28 5 / 28

Minimum Denition Let w be a Christoel word. We dene min(w) to be the length of the prex needed to reach the minimal point Q of w. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 6 / 28

Minimum Denition Let w be a Christoel word. We dene min(w) to be the length of the prex needed to reach the minimal point Q of w. Remark Since Q is unique hence the minimum value is unique too. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 6 / 28

Minimum Denition Let w be a Christoel word. We dene min(w) to be the length of the prex needed to reach the minimal point Q of w. Remark Since Q is unique hence the minimum value is unique too. 2 If k = min(w), then w[k] =, and w[k + ] =. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 6 / 28

Split operator Let w be a Christoel word with k = min(w) and w = l. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 7 / 28

Split operator Let w be a Christoel word with k = min(w) and w = l. The point of the path at position k is the furthest from the line segment and the unique that respects: Reconstruction of convex polyominoes from orthogonal projections 4-5-28 7 / 28

Split operator Let w be a Christoel word with k = min(w) and w = l. The point of the path at position k is the furthest from the line segment and the unique that respects: Proposition The words u = w[, k ] and u 2 = w[k +, l], are two Christoel words. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 7 / 28

Split operator Let w be a Christoel word with k = min(w) and w = l. The point of the path at position k is the furthest from the line segment and the unique that respects: Proposition The words u = w[, k ] and u 2 = w[k +, l], are two Christoel words. We have: Slope(u ) > Slope(u 2 ). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 7 / 28

Example (8,5) P w 2 O(,) w Q Reconstruction of convex polyominoes from orthogonal projections 4-5-28 8 / 28

Example (8,5) P w 2 O(,) w Q C( 5 8 ) = (w, w 2 ), Reconstruction of convex polyominoes from orthogonal projections 4-5-28 8 / 28

Example (8,5) P w 2 O(,) w Q C( 5 8 ) = (w, w 2 ), 2 k = min(w) = OQ = = 5, Reconstruction of convex polyominoes from orthogonal projections 4-5-28 8 / 28

Example (8,5) P w 2 O(,) w Q C( 5 8 ) = (w, w 2 ), 2 k = min(w) = OQ = = 5, 3 u = C( 2 3 ) =, u 2 = C( 3 5 ) = are Christoel words with 2 3 > 3 5. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 8 / 28

Adding one point Reconstruction of convex polyominoes from orthogonal projections 4-5-28 9 / 28

Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Let ρ(w ) = 3 5 > ρ(w 2) = 2. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Let ρ(w ) = 3 5 > ρ(w 2) = 2. The split(w ) = u v gives ρ(u ) = 2 3 and ρ(v ) = 2. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Let ρ(w ) = 3 5 > ρ(w 2) = 2. The split(w ) = u v gives ρ(u ) = 2 3 and ρ(v ) = 2. u v w 2 u w 3 Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Let ρ(w ) = 3 5 > ρ(w 2) = 2. The split(w ) = u v gives ρ(u ) = 2 3 and ρ(v ) = 2. u v The order of the slopes is: w 2 u w 3 2 3 > 2 < 2. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Let ρ(w ) = 3 5 > ρ(w 2) = 2. The split(w ) = u v gives ρ(u ) = 2 3 and ρ(v ) = 2. u v The order of the slopes is: w 2 u w 3 2 3 > 2 < 2. Join v with w 2 to form one line segment. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Let ρ(w ) = 3 5 > ρ(w 2) = 2. The split(w ) = u v gives ρ(u ) = 2 3 and ρ(v ) = 2. u v The order of the slopes is: w 2 u w 3 2 3 > 2 < 2. Join v with w 2 to form one line segment. This concatenation is not exactly the same new Christoel word w 3. v.w 2 =. w 3 =. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 2 / 28

Adding two points Reconstruction of convex polyominoes from orthogonal projections 4-5-28 22 / 28

First case: Let w and w 2 be two consecutive Christoel words in the same octant of a W N path of a Polyomino where split(w ) = u v and split(w 2 ) = u 2 v 2. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 23 / 28

First case: Let w and w 2 be two consecutive Christoel words in the same octant of a W N path of a Polyomino where split(w ) = u v and split(w 2 ) = u 2 v 2. u = c d v = w b a u 2= b a v = 2 w 2 c d Reconstruction of convex polyominoes from orthogonal projections 4-5-28 23 / 28

First case: Let w and w 2 be two consecutive Christoel words in the same octant of a W N path of a Polyomino where split(w ) = u v and split(w 2 ) = u 2 v 2. u = c d v = w b a u 2= b a v = 2 w 2 c d Theorem (P.Dulio et al.) If slope(v ) > slope(w 2 ) and slope(w ) > slope(u 2 ), then it holds slope(v ) > slope(u 2 ). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 23 / 28

Reduce constraints?? Reconstruction of convex polyominoes from orthogonal projections 4-5-28 24 / 28

Second case: Letρ(w ) = 3 4 > ρ(w 2) = 5 7. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 25 / 28

Second case: Letρ(w ) = 3 4 > ρ(w 2) = 5 7. The split(w ) = u v and split(w 2 ) = u 2 v 2 give 5, 9 26, 3 4 and 2 3. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 25 / 28

Second case: Letρ(w ) = 3 4 > ρ(w 2) = 5 7. The split(w ) = u v and split(w 2 ) = u 2 v 2 give 5, 9 26, 3 4 and 2 3. v 2 u w3 Reconstruction of convex polyominoes from orthogonal projections 4-5-28 25 / 28

Second case: Letρ(w ) = 3 4 > ρ(w 2) = 5 7. The split(w ) = u v and split(w 2 ) = u 2 v 2 give 5, 9 26, 3 4 and 2 3. v 2 u w3 We have: v =, u 2 = with ρ(v ) < ρ(u 2 ). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 25 / 28

Second case: Letρ(w ) = 3 4 > ρ(w 2) = 5 7. The split(w ) = u v and split(w 2 ) = u 2 v 2 give 5, 9 26, 3 4 and 2 3. v 2 u w3 We have: v =, u 2 = with ρ(v ) < ρ(u 2 ). v.u 2 =. w 3 =. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 25 / 28

Second case: Letρ(w ) = 3 4 > ρ(w 2) = 5 7. The split(w ) = u v and split(w 2 ) = u 2 v 2 give 5, 9 26, 3 4 and 2 3. v 2 u w3 We have: v =, u 2 = with ρ(v ) < ρ(u 2 ). v.u 2 =. w 3 =. The decreasing order of the slopes obtained is: 5, 5, 5, 2 3. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 25 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. The split(w ) = u v and split(w 2 ) = u 2 v 2,give 2 3, 2, 4 7 and 53 93. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. The split(w ) = u v and split(w 2 ) = u 2 v 2,give 2 3, 2, 4 7 and 53 93. The slope ρ(v ) < ρ(u 2 ), with v = and u 2 =. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. The split(w ) = u v and split(w 2 ) = u 2 v 2,give 2 3, 2, 4 7 and 53 93. The slope ρ(v ) < ρ(u 2 ), with v = and u 2 =. u 2 v w Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. The split(w ) = u v and split(w 2 ) = u 2 v 2,give 2 3, 2, 4 7 and 53 93. The slope ρ(v ) < ρ(u 2 ), with v = and u 2 =. u 2 v w By concatenating these two words, we get w 3 = of slope 5 9 = 2 4 7. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. The split(w ) = u v and split(w 2 ) = u 2 v 2,give 2 3, 2, 4 7 and 53 93. The slope ρ(v ) < ρ(u 2 ), with v = and u 2 =. u 2 v w By concatenating these two words, we get w 3 = of slope 5 9 = 2 4 7. Still having ρ(u ) > ρ(w 3 ) < ρ(v 2 ). Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Third case: Let ρ(w ) = 3 5 > ρ(w 2) = 57. The split(w ) = u v and split(w 2 ) = u 2 v 2,give 2 3, 2, 4 7 and 53 93. The slope ρ(v ) < ρ(u 2 ), with v = and u 2 =. u 2 v w By concatenating these two words, we get w 3 = of slope 5 9 = 2 4 7. Still having ρ(u ) > ρ(w 3 ) < ρ(v 2 ). Another point has to be added, in order to maintain the convexity. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 26 / 28

Final Aims Extend the work in the case of two words in dierent octants. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 27 / 28

Final Aims Extend the work in the case of two words in dierent octants. 2 Produce a nal reconstruction algorithm that extends the strategy of Barcucci et al. for HV convex polyominoes by preserving step by step digital convexity. Reconstruction of convex polyominoes from orthogonal projections 4-5-28 27 / 28

Final Aims Extend the work in the case of two words in dierent octants. 2 Produce a nal reconstruction algorithm that extends the strategy of Barcucci et al. for HV convex polyominoes by preserving step by step digital convexity. Thank You Reconstruction of convex polyominoes from orthogonal projections 4-5-28 27 / 28