Loïc Mazo

Étienne Baudrier

Study on the digitization dual combinatorics and convex case

The action of a translation on a continuous object before its digitization generates several digitizations. The dual, introduced by the authors in a previous paper, stands for these digitizations in function of the translation parameters. This paper focuses on the combinatorics of the dual by making a link between the digitization number and the boundary curve, especially through its dual representation. The convex case is then studied and a few signicant examples are exhibited.

Introduction

For a given grid step and a given digitization method, a planar object produces several digitizations in function of its position on the grid. The object digital properties and digitally estimated characteristics depend on the obtained digitization. Thereby, this study of the digitization variability is an important issue in image analysis. This eld has been explored for some geometrical primitives. For instance, the set of straight segment digitizations in function of the segment slope and oset is known as the segment preimage and is used for digital straight segment recognition [START_REF] Dorst | Discrete Representation of Straight Lines[END_REF]. Several papers are also dedicated to the study of the generation and combinatorics of the disc digitization set in function of its radius and center position [START_REF] Heath-Brown | Lattice points in the sphere[END_REF][START_REF] Mazo | Lattice points in high-dimensional spheres[END_REF][START_REF] Zunic | On the number of ways to occupy n lattice points by balls in ddimensional space[END_REF][START_REF] Huxley | On the number of digitizations of a disc depending on its position[END_REF][START_REF] Zunic | On the number of digital discs[END_REF][START_REF] Nagy | An algorithm to nd the number of the digitizations of discs with a xed radius[END_REF]79] and the combinatorics of the strictly convex sets [START_REF] Huxley | The number of congurations in lattice point counting I[END_REF]. In the general case, the digitization set can be seen as the consequence of a group action on the object. A function, so-called dual, linking the group action and the produced digitization is used by the authors to study the digitization set up to a translation, for function graphs in [START_REF] Baudrier | Curve digitization variability[END_REF] and for planar object in [START_REF] Mazo | Object digitization up to a translation[END_REF].

In the latter case, the dual has been proved to be piecewise constant in function of the translation. This paper focuses on the dual combinatorics.

Two upper bounds are given for the number of digitizations of a planar object whose boundary is a Jordan curve. The rst one is expressed in terms of the number of grid cells crossed by the boundary and the second one in terms of the intersection number when plotting the boundary on the torus IR 2 /Z Z 2 . The latter bound is proved to be quadratic in the convex case. Some examples are provided in order to compare the two upper bounds both in the convex and the non-convex cases. A conclusion and some perspectives end the paper.

Background

Let us consider a connected compact set S in IR 2 whose boundary is a simple closed (Jordan) curve Γ . Thanks to the Jordan curve theorem, we may assume a continuous map f : IR 2 → IR such that Γ , resp. S, is implicitly dened by

Γ = {f (x) = 0 | x ∈ IR 2 } , resp. S = {x ∈ IR 2 | f (x) ≤ 0} .
We are interested in the variability of the Gauss digitization when the group of the translations acts on S, that is on the sets (u + S) ∩ Z Z 2 , u ∈ IR 2 . In this paper, we focus on the combinatorial aspects of this variance. Of course, the variance has to be understood up to integer translations. This is the reason why we dened in a previous paper [START_REF] Mazo | Object digitization up to a translation[END_REF] the dual by translation of the digitizations of S as a set-valued function ϕ S dened on the torus T = IR 2 /Z Z 2 which maps each point t ∈ T to the digitization, up to integer translations, of u + S where the vector u is any representative of t in IR 2 . Let us pick a representative in each class of T so as to form a connected set M called the structuring element. We note C the symmetric of M with respect to the origin: C = -M . The family of sets p + C, p ∈ Z Z 2 , is a tiling of IR 2 . For instance, we can take M = [0, 1) 2 , C = (-1, 0] 2 . From now on, to simplify the notations, we confuse t ∈ T with its representative in M and the subsets of Z Z 2 with their orbits for the action of the integer translations so we can write ϕ S (t) = (t + S) ∩ Z Z 2 . For any point or set X in IR 2 , we denote by proj(X) its projection on the quotient space T.

We dene the grid boundary B as the set of grid points that lie in the (morphological) dilation of the boundary Γ of S by the structuring element M :

B = (Γ ⊕ M ) ∩ Z Z 2 ,
where ⊕ denotes the Minkowski sum. The set B contains all the points of Z Z 2 whose membership to the digitization may change when the set S is shifted by a vector u ∈ M . Nevertheless, since IR 2 is connected, M is not open so there may exist some points in B ∩ S not liable to change, namely those points p in B for which p + C ⊆ S (see Fig. 1). That is why we have in fact to consider the toggling boundary B as the set of grid points whose membership eectively toggle for some translation by a vector u ∈ M :

B = B \ {p ∈ Z Z 2 | p + C ⊆ S} .
The set S ∩ Z Z 2 \ B of the grid points that are in any digitization of u + S, u ∈ M , is called the digitization core.

Finally, for any p ∈ Z Z 2 , we set and also intersects Z Z 2 \ S); blue disc: a point q ∈ B \ B (the blue square q + C intersects the boundary Γ but is included in S); green disc: a point r in the digitization core which is not in B (the green square r + C is included in S and does not intersect Γ ). (b) Black: a Jordan curve, red: the toggling boundary, blue: two points that lie in the grid boundary but that cannot toggle, green: the digitization core. so Γ p = ∅ if p / ∈ B, and we denote by 1l p the indicator function of the set Γ p , so 1l p is not constant i p ∈ B. Then,

Γ p = (-p) + Γ ∩ C , (a)
Γ p p r q (b)
ϕ S (t) = {p ∈ Z Z 2 | 1l p (-t) = 1} .
We proved in [START_REF] Mazo | Object digitization up to a translation[END_REF] that the plot of the curve Γ on the torus T, proj(Γ ), delineates regions on which the dual function ϕ S is constant. Property 1 ([11]). Let S be a compact subset of IR 2 whose boundary Γ is a Jordan curve. Then, the dual ϕ S is constant on the connected components of T \ proj(Γ ); when M = [0, 1) 2 , putting i = (1, 0), j = (0, 1), it holds for any p ∈ Z Z 2 \ Γ and any x, y ∈ [0, 1) that,

• 1l p (xi) = lim λ→1 1l p-j (xi + λj),

• 1l p (yj) = lim λ→1 1l p-i (yj + λi).

In the rest of the article, we use the notation E for the cardinal of a set E ( E ∈ IN ∪ {∞}), CC(A) for the family of the connected components of a subset A of IR 2 and I J for the disjoint union of the sets I and J.

In the following section, upper bounds are given for the number of digitizations up to a translation. The rst upper bound is naive. For the second one, the idea is to link the digitization number to the number of the intersection of the translated curves Γ p . Indeed, the dual induces a partition of the torus whose cells frontiers are arcs of the curve proj(Γ ). Then, we bound from above the partition size by counting the number of curve intersections in proj(Γ ).

Contribution

Bounding up by counting the crossed tiles

Since the grid boundary B contains any integer point whose value may change when shifting the set S, we have a rst, obvious, upper bound on the number of Gauss digitizations given by 2 B . As B is also the number of tiles p + C, p ∈ Z Z 2 , crossed by the frontier of S, we can state the following proposition.

Proposition 1. The number of Gauss digitizations, up to integer translations, is upper bounded by 2 a(Γ ) where a(Γ) is the number of tiles crossed by Γ . dual is divided into 8 regions).

Bounding up by counting the intersections

In Sec. 3.2, we assume a parametrization of Γ . It induces an order on the points of the curve Γ (for Γ is simple) that is used in the proof of the following proposition.

The dual ϕ S can be regarded as the projection, on the torus T of a nite labeled partition of the tile C (whose cells need not be connected). For the order by renement on the partitions, this partition is lower bounded by the inmum of the binary partitions P p associated to the indicator functions 1l p , p ∈ B (actually, proj(1l p ), p ∈ B). Indeed some cells of p∈B P p need to be merged whenever the corresponding digitizations are equivalent up to an integer translation. Then, the next proposition proposes an upper bound on the size of the partition p∈B P p and thereby an upper bound to the number of digitizations by inductively counting the intersections between the curves Γ p , p ∈ B. The idea is to count the partition cells created when adding a curve Γ p . To do this we count the intersection of Γ p with the already added curves.

Nevertheless, these intersections can be with one or more curves and can be a singleton, several points or an arc. Then, let us dene what means intersection in this context (the denition is illustrated in Fig. 3).

Denition 1. Let

B = {b 1 , • • • , b i , • • • , b n }, n ≥ 1. Let m ∈ [2, n]. Then, inter γ,I = CC γ ∩ i∈I Γ bi \ j∈J Γ bj where γ ∈ CC(Γ bm ) and I J = [1, m -1] , # inter m = γ∈CC(Γ bm ) ∅⊂I⊆[1,m-1] w I inter γ,I
where w I = min( I , 2) .

Note that the components of all the inter γ,I are two by two disjoint and that # inter m may be innite.

The set inter γ,I stands for the intersection of the curve γ with the curves whose indexes are in a subset I, excluding any other curve Γ bi whose index is not in I. The necessity to have all the index subsets comes from the fact that intersections between multiple curves can occur.

Γ b2 (# inter 2 = inter Γ b 2 ,{1} = 4. (c) Brown: Γ b3 (# inter 3 = inter Γ b 3 ,{1} + inter Γ b 3 ,{2} +2 inter Γ b 3 ,{1,2} = 0+ 0+2 = 2. (d) Red: Γ b4 (# inter 4 = inter Γ b 4 ,{1} +2 inter Γ b 4 ,{1,2,3} = 2+2 = 4.
We are now able to state the proposition that relates the number of digitizations and the number of intersections # inter m .

Proposition 2. The size of the partition p∈B P p is upper bounded by

2 + n m=2 # inter m + CC(Γ bm ) .
Proof. The proposition is proved by induction. For m = 1, the result is obvious since, for any p ∈ B, P p is a binary partition. Let m > 1. We assume that the number of cells of the partition m-1 i=1 P bi of the tile C induces a partition Q of Γ bm , as a subset of C, with N m cells (these cells of Γ bm need not be connected). Then, the idea of the proof is to map each cell of Q to its supremum, for the order induced by the parametrization of Γ namely to the intersection, as dened in Def. 1 its supremum belongs to, or to the empty set when no such intersection exists. Then, though this mapping is not one-to-one, a careful examination of the dierent cases will permit to conclude that N m is upper bounded by # inter m + CC(Γ bm ) .

i=1 P bi is upper bounded by 2 + m-1 k=2 # inter k + CC(Γ b k ) .
Let γ be a connected component of Γ bm and s ∞ be the supremum of γ for the order induced by the parametrization of Γ . Let Q γ be the restriction of Q to γ. We set inter γ = ∅⊂I⊆[1,m-1] inter γ,I , and we assume the following facts that will be prove further:

a) each component in inter γ is included in a cell of Q γ ; b) if a cell of Q γ has a supremum s distinct from s ∞ ,
then s belongs to some component of inter γ ; c) if two cells of Q γ have their supremums in the same component K of inter γ , then these supremums are equal to the inmum of K. d) if three cells of Q γ have the same supremum s, then inter γ has innitely many components; e) if two cells of Q γ have the same supremum s and inter m is nite then either s = s ∞ and s ∞ belongs to some component of inter γ , or s belongs to at least two curves Γ bi and Γ bj where i, j < m.

Then, with these ve assumptions, we deduce the desired upper bound as follows.

We map each cell of Q γ to the component of inter γ its supremum belongs to, if any. The other cells are map on the empty set. We denote by ψ this mapping. Hence, from Fact b,the supremum of the preimages by ψ of ∅, if any, is s ∞ and so, s ∞ / ∈ inter γ if such preimages exist.

K 1 , K 2 of m-1 i=1 P bi such that K∩K 1 = ∅ and K ∩ K 2 = ∅. Let c 1 ∈ K ∩ K 1 and c 2 ∈ K ∩ K 2 . Since K 1 = K 2 , there exists i ∈ [1, m -1] such that 1l bi (c 1 ) = 1l bi (c 2 ). Then, the segment [c 1 , c 2 ]
in γ contains a point c 3 ∈ Γ bi which also belongs to K for K is connected. Then, on the one hand i ∈ I for c 3 ∈ Γ bi ∩ K and, on the other hand, c 1 , or c 2 , is in (1l bi = 0), that is i / ∈ I. Contradiction. b) Suppose that the supremum s of the cell K of Q γ is not in any component of inter m . Then, for any i ∈ [1, m -1], s / ∈ Γ bi . In other words, for any i ∈ [1, m -1], s is in the open set (1l bi = 0) or in the interior of (1l bi = 1). Thus, there exists an open neighborhood U of s in the tile C which does not intersect any curve Γ bi , 1 ≤ i < m. Thereby, U is included in a cell of the partition m-1 i=1 P bi . Since s = sup(K), there is no point t in γ ∩ U such that t > s. Thus, s = s ∞ . c) Let K 1 , K 2 be two cells of Q γ whose supremums are in K ∈ inter γ . Let K 3 be the cell of Q γ which includes this component (Fact a). If s is not the inmum of K then there exists an interval [u, s], with u < s, in K ⊆ K 3 (for K is connected) and s is not the supremum of both K 1 and K 2 . Contradiction. d) Let K 1 , K 2 , K 3 be three cells of Q γ that share the same supremum s. Since s belongs to at most one of these three cells, it is a limit point for the two others. For instance, assume that s is a limit point for K 1 and K

2 . Let k ∈ [1, m -1] such that 1l b k takes two distinct values on K 1 and K 2 . For instance, K 1 ⊆ (1l b k = 0) and K 2 ⊆ (1l b k = 1). Then, we can inductively build an innite sequence c 1 < c 2 < • • • < c i < • • • < s such that c 2i-1 ∈ K 1 and K 2i ∈ K 2 for any i ≥ 1. In particular, we have 1l b k (c 2i-1 ) = 0 and 1l b k (c 2i ) = 1 for any i ≥ 1. Then, c 2i-1 / ∈ Γ b k and [c 2i-1 , c 2 * i ] intersects Γ b k .
Therefore, inter γ contains innitely may components between c 1 and s. e) We assume # inter m < ∞. Let K 1 , K 2 be two cells of Q γ that share the same supremum s. This supremum cannot be a limit point both in K 1 and K 2 otherwise we could make the same reasoning as in the previous item and concludes that # inter m = ∞ which contradicts our assumption. So, s is an isolated point in one of the two cells, for instance K 1 (therefore, s ∈ K 1 , s / ∈ K 2 and s is a limit point in K 2 ). From Fact b, there exist a subset I of [1, m -1] and a component K in inter γ,I such that s ∈ K. As in the proof of Fact b, we derive that there exists an open neighborhood U of s in C which does not intersect any curve Γ bi , i / ∈ I. Suppose that I is a singleton, say I = {b 1 }. Then, taking a point in K 2 ∩ U (recall that s is a limit point of K 2 ), we found that necessarily 1l b1 = 0 on K 2 , while 1l b1 = 1 on K 1 , and 1l bi , i = 1, coincides on K and K 2 . If s = s ∞ , there exists a point u in γ ∩ U greater than s. This point is not in K 1 nor in K 2 for s = sup(K 1 ) = sup(K 2 ). Thereby, 1l b1 (u) = 0 and 1l b1 (u) = 1 which is absurd. Thus, either s ∞ = s ∈ K 1 or I is not a singleton.

When projecting the partition n i=1 P bi on the torus T, the cells that touch the boundary of C are identied two by two, which decreases the number of cells in the partition. Unfortunately, it is dicult to count these cells in the general case. Nevertheless, in the most simple case (each curve crosses the boundary of C in two points, all points and cells are distinct), the reader can easily check that the identication leads to remove the term n i=1 CC(Γ bi in Prop. 2.

The convex case

It seems plain that the structure of the dual should be simpler when the set S is convex compare to a set with a winding boundary. We could even hope that the digitizations coincide with the cells of the partition p∈B Γ p , after sewing the boundary of the tile C, and with the connected regions of the torus T delineated by the curve Γ . Figure 4 annihilates this hope by exhibiting a convex object and one of its digitizations whose inverse image by the dual is not connected. Nevertheless, in this section we show that the structure of the sets inter γ,I is simple when the set S is strictly convex and permits to obtain a quadratic bound for the complexity of the dual in terms of the grid boundary size (to be compare to the exponential bound of Prop. 1). Proposition 3. We assume a convex quadrilateral structuring element M . Let n = B . The number of Gauss digitizations of a strictly convex planar object up to a translation is upper bounded by 4n 2 + 4n -6 .

Proof. The proof invokes Lemma 3 which is stated and proved in the appendix.

Let N be the digitization number. Thanks to Prop. 2, we have

N ≤ 2 + n m=2 # inter m + CC(Γ bm ) . ( 1
)
From Lemma 3, we derive that, for any m ∈ triangle. The four colored regions correspond to the same digitization: the horizontal pair. Moreover, the two red regions belong to the same cell of the partition p∈B Γ p , the cell associated to the digitization bc (these two regions are separated by the unique point of the cell abcd). The orange region is the cell cd and the pink region is the cell ae. (c), (d) The congurations of the shifted triangle in the two red regions. Since they are disconnected, it is not possible to continuously move to one conguration to the other without hurting a black toggling point. Note that the triangle could be slightly inated so as to provide a strictly convex counterexample.

[1, n], γ ∈ CC(Γ bm ), i ∈ [1, m -1], I i inter γ,I ≤ 2 CC(Γ bi ) . (a) (b) (c) (d)
Then,

# inter m ≤ 2 m-1 k=1 CC(Γ b k ) .
Eventually, Eq (1) turns into

N ≤ 2 + 2 n m=2 m-1 k=1 CC(Γ b k ) + n m=2 CC(Γ bm ) ≤ 2 1 + n k=1 (n -k + 1) CC(Γ b k ) -CC(Γ b1 )
.

It can easily be seen (from Lemma 2 for instance) that, for any convex curve Γ and any convex polygon P with c edges, the number of connected arcs in the intersection Γ ∩ P is upper bounded by c. Then, if we assume that the tile C is a convex quadrilateral, we straightforwardly obtain the desired bound.

The term 4n 2 in Prop. 3 comes down to n 2 for suciently high resolutions because each curve Γ bi then have just one connected component instead of possibly 4 in the general case. Then, when S a disc of radius r, the result of Prop. 3 is close to the one in [START_REF] Huxley | Dierent digitisations of displaced discs[END_REF] which states that the number of digitizations of the disc is asymptotically 4πr 2 + O(r). As the ratio between the radius r and the size of the grid boundary is π/4 for the disc, our upper bound in function of the radius r of the disc is asymptotically equivalent to (16/π 2 )r 2 .

Conclusion

We present in this paper two upper bounds on the number of digitizations obtained from all the translated of a continuous object. The rst one is exponential in the number of toggling object-boundary pixels and a generic example reaching this bound is given. The second one is based on the passage from the dual connected-component count to the curve intersection count which, if the curve is parametrized, comes down to count some equation solutions. In the convex case, the latter upper bound is shown to yield a quadratic digitization number in term of the grid boundary size. An example of a convex object is given where the set of translation parameter classes corresponding to a given digitization is not connected.

The perspectives of this work are rst to explicit the second upper bound under assumptions less restrictive that the convexity, e.g. bounded curvature; then to study the combinatorics of the digitization under the rigid transformations and to propose an algorithm for the digitization generation.

A Convex sets

The proof of Lemma 3 relies on the two following lemmas about convex sets that seem obvious at the rst glance. Nevertheless, since we did not nd any result related to these lemmas in the literature, we provide our own justications of the two statements. Proof. Since [a, b] Γ , the line (ab) does not support S at any point (the notion of supporting line of a convex set is exposed for instance in [2]). So, there exists two supporting lines of S at a and b that cross the line (ab). Then, (ab Γ , from Lemma 1, (ab) ∩ Γ = {a, b}. Thereby, by connectivity, either σ 1 ⊂ H -or σ 1 ⊂ H + and σ 2 ⊂ H -or σ 2 ⊂ H + . Suppose for instance that σ 1 ⊂ H -and σ 2 ⊂ H -. Then, S, which is the connected subset of IR 2 bounded by σ 1 ∪σ 2 ∪{a, b} (citeJordan) is included in H -∪(ab) and, since (ab

) ∩ S is included in [a, b]. Let c ∈ [a, b] ∩ Γ , c = a.
)∩S = [a, b] from Lemma 1, [a, b] ⊂ Γ : contradiction.
Lemma 3 (Intersection of two segments of a convex curve). Let Γ be a Jordan curve whose interior is convex. Let Γ 1 and Γ 2 two disjoint (closed) segments/arcs of the curve Γ and τ a vector of IR 2 . Then the intersection of Γ 1 and τ + Γ 2 is composed of none, one, two points or a line segment. Proof. Let p, q be two distinct points in Γ 1 ∩ (τ + Γ 2 ) if such a pair exists. We denote by Σ 1 the open segment of Γ 1 between p and q. Alike, Σ 2 is the open segment of Γ 2 between -τ + p and -τ + q (see Fig. 5). We set Σ 1 = Σ 1 ∪ {p, q} and Σ 2 = Σ 2 ∪ {p, q}. Firstly, we prove that Σ 1 ∪ (τ + Σ 2 ) is a straight line segment whenever it contains more than two points. p q .

-τ + p

-τ + q Σ 1 Σ 2 τ + Σ 2 x H 1
case: Σ 1 ∪ (τ + Σ 2 ) ⊆ (pq). Then, since Σ 1 and Σ 2 are connected and Γ is simple, Σ 1 = Σ 2 = [p, q].
Second case: ∃x ∈ Σ 1 ∪(τ +Σ 2 ) \(pq). For instance, we assume x ∈ Σ 1 \(pq). By Lemma 2, Σ 1 = Γ ∩ H 1 , where H 1 is the open half-plane bounded by the line (pq) and containing x, and -τ + p is in IR 2 \ H 1 . Then, it can easily be seen that -τ + H 1 is the open half-plane bounded by the line joining -τ + p and -τ +q and including p. Thanks to Lemma 2, we derive that Σ 2 does not intersect -τ + H 1 . Thus, τ + Σ 2 does not intersect H 1 . In particular, (τ + Σ 2 ) ∩ Σ 1 = ∅.

This achieves the rst part of the proof. Now, let r be a point in Γ 1 ∩(τ +Γ 2 ) which is not in Σ 1 (if such a point exists). For instance, p belongs to the segment of Γ 1 between q and r. Then, the rst part of the proof, applied to the points q and r, implies that Γ 1 ∩ (τ + Γ 2 ) includes the segments [q, r].We straightforwardly concludes that either the intersection of Γ 1 and τ + Γ 2 is composed of at most two points or it is a line segment.

B Examples and counterexamples B.1 Building examples without proper congruent digitizations in the image of the dual

Let u and v be two vectors in [0, 1) 2 such that the sets u+S and v+S have distinct but congruent digitizations. Then, there exists an integer vector w, w = 0, such that (u + S)

∩ Z Z 2 = w + ((v + S) ∩ Z Z 2 ) = (w + v + S) ∩ Z Z 2 .
Let p be a point in the digitization core. Then, p ∈ (u + S) ∩ Z Z 2 and p ∈ (v + S) ∩ Z Z 2 . Therefore, w + p ∈ (u + S) ∩ Z Z 2 and -w + p ∈ (v + S) ∩ Z Z 2 , which can be rewritten as p ∈ ((-w + u) + S) ∩ Z Z 2 and p ∈ ((w + v) + S) ∩ Z Z 2 . Then, at least one of the vector w + v or -w + u has one of its coordinates which is negative. We derive that if there exists a point in the digitization core which is maximal in S for both coordinates then there is no proper congruent digitizations in the dual. An example with such a maximal point in the digitization core is provided in Fig. 6.

B.2 Building toric partitions in one-to-one correspondence with the power set of the toggling boundary

In this section, we exhibit a way to modify the boundary of the set S in order to ensure that any set in the power set of the toggling boundary is represented in the dual. To do so, we move along B, ordered in the same way as in Def. 1.

Then, a new boundary is built thanks to the approximations of the Hilbert lling curve: the segment of Γ intersecting the n-th cell of B is replaced by a n-th approximation of the Hilbert lling curve H n (extended at its extremities to ensure the continuity of the boundary): see Fig. 6. We consider the family of binary partitions P n of the unit square that comes with the curves H n . We claim that each curve H n+1 crosses each cell of the partition n i=1 P i so that the size of the nal torus partition is 2 N where N is the cardinal of the toggling boundary. To justify our claim, we divide the unit square in a family of 2 n × 2 n small squares (K n i,j ) 1≤i,jleq2 n (n ≥ 0) whose sizes are 1 2 n × 1 2 n . It can be seen that on the one hand, the Hilbert curve H n passes through the center of each of the squares K n i,j and, on the other hand, does not intersect any of the interior of the squares K n+1 i,j (H 0 is just the center of the unit square). Thereby, the partition n i=1 P i is coarser than the partition {K n+1 i,j | 1 ≤ i, j < n} (the boundaries of the squares K k i,j are assigned to the cells so as to coincide with P n ). Since H n+1 passes through the center of each of the squares K n+1 i,j , it passes in each cell of n i=1 P i which gives the claim. Thanks We thank Renan Lauretti for his idea to link the dual regions to the dual region border crossings in his study of the function graph dual.

Fig. 1 :

 1 Fig. 1: (a) Black: a Jordan curve Γ ; light blue: tiles z + C, z ∈ Z Z 2 ; red disc: a point p ∈ B (the red square p + C intersects Γ in brown: Γ p = (p + C) ∩ Γand also intersects Z Z 2 \ S); blue disc: a point q ∈ B \ B (the blue square q + C intersects the boundary Γ but is included in S); green disc: a point r in the digitization core which is not in B (the green square r + C is included in S and does not intersect Γ ). (b) Black: a Jordan curve, red: the toggling boundary,

  Generally, the digitization enumeration provided by Prop. 1 includes false positives and multiple counts. For instance, the grid boundary of a circle with diameter 1.7 involves 4 to 8 pixels, depending on the grid position, which gives an upper bound according to Prop. 1 equals to 16, while there exists only 8 digitizations (seeFig 2). Nevertheless, from any set S, it is possible to build a new set that avoid false positives (but not multiple counts) by replacing the initial boundary by a family of Hilbert curves (see Appendix B).

Fig. 2 :

 2 Fig. 2: (a) A circle with diameter 1.7 and the four pixels in the toggling boundary (which is equal to the grid boundary): a, b, c, d. (b) The (atten) dual of the closed disc bounded by the circle. Among the 16 = 2 4 potential digitizations provided by the 4-pixels a, b, c, d, three of them does not actually appear (∅, ac and bd) and some others appear multiple times (the four singular digitizations, which are obviously congruent, and the vertical and horizontal pairs which each appear twice. Eventually, there are only 8 digitizations (and actually the (sewed)

Fig. 3 :

 3 Fig. 3: (a) Blue: the curve Γ b1 . (b) Green: Γ b2 (# inter 2 = inter Γ b 2 ,{1} = 4. (c) Brown: Γ b3 (# inter 3 = inter Γ b 3 ,{1} + inter Γ b 3 ,{2} +2 inter Γ b 3 ,{1,2} = 0+ 0+2 = 2. (d) Red: Γ b4 (# inter 4 = inter Γ b 4 ,{1} +2 inter Γ b 4 ,{1,2,3} = 2+2 = 4.

  The cells of m-1 i=1 P bi which are included in one of the two cells of P bm , namely the sets (1l bm = 0) and (1l bm = 1), stay unchanged in the partition m i=1 P bi . Conversely, the cells of m-1 i=1 P bi that are intersected by both (1l bm = 0) and (1l bm = 1), which obviously are cells intersected by Γ bm , are divided in two new cells. Thereby, the number N m of new cells is upper bounded by N m , the number of cells in m-1 i=1 P bi intersected by Γ bm . Besides, the partition m-1

  If a component K of inter γ has exactly two preimages K 1 and K 2 by the mapping ψ, then from Fact c, we derive that the two cells share the same supremum and from Fact e, and the very denition of # inter m , we see that the weight of K in # inter m is 2, which corresponds to the number of its preimages, or s = s ∞ and the empty set has no preimage. If the empty set has two or more preimages by ψ, then # inter m = ∞ (Fact e). If a component K of inter γ has three or more preimages by ψ, these premimages share the same supremum (Fact c) and from d we derive that # inter m = ∞. We readily conclude that the number of cells of Q γ is upper bounded by 1 + ∅⊂I⊆[1,m-1] w I inter γ,I where w I = min(2, I ). By summing on all the connected components of Γ bm , we derive that N m , the number of cells of Q, is upper bounded by # inter m + CC(Γ bm ) which achieves the proof.Let us now prove the ve facts stated above. a) We demonstrate Fact a by contradiction. Suppose it exists K in inter γ,I , I ⊆ [1, m-1], and two distinct cells

Fig. 4 :

 4 Fig. 4: (a) A triangle and its toggling boundary. (b) The dual of the (lled)

Lemma 1 (

 1 Chords of convex sets). Let [a, b] be a chord of the boundary Γ of a closed convex set S. If [a, b] Γ , then (ab) ∩ S = [a, b] and (ab) ∩ Γ = {a, b}.

  Applying the rst part of the proof to the chord [a, c], we derive that (ac) ∩ S ⊆ [a, c] and, since b ∈ (ac) ∩ S, we conclude that b = c. Lemma 2 (Cuts of convex sets). Let a, b be two points of the boundary Γ of a closed convex set S. If [a, b] Γ , then the open curve segments of Γ , σ 1 , σ 2 , whose extremities are a and b are included in distinct open half-planes bounded by the line (ab).

  Proof. Let H -and H + be the two open half-planes bounded by (ab). Since[a, b]

Fig. 5 :

 5 Fig. 5: Light green: the arcs Γ 1 and Γ 2 .

First

Fig. 6 :

 6 Fig. 6: The boundary of the set S depicted in the gure is obtained by connecting n-th approximations of the Hilbert lling curve, 0 ≤ n ≤ 4. Red square and red disc: digitization core. Red circle: the ve points in the toggling boundary. The red disc is both in the grid boundary and in the digitization core. Furthermore, it is maximal for both coordinates which ensures that there is no proper congruent digitizations in the image of the dual. The set S has 2 5 digitizations.


David Guerrero
email: david.guerrero@ifsttar.fr

Hidekazu Itoh

Ports, regions and manufacturing systems: Automobile manufacturing in

Keywords: Port, Manufacturing, Automobile industry, Core-periphery model

The locations of seaports and manufacturing activities in Japan have changed considerably since World War II. Despite the geographic spread of economic activities over decades and the uneven development of ports, the cores of both systems have long remained in the same metropolitan areas. While co-location does not provide necessary or causal connection, strong a priori grounds can be offered to posit that a necessary relationship exists between the Japanese manufacturing system's geographic expansion and changes in the maritime transport network. A case study of automobile industry's recent development in the peripheral region of Kyushu identifies some drivers of these evolutions at manufacturing level. This demonstrates that, despite the development of high capacity transport infrastructure and manufacturing facilities in the Japanese periphery, the current manufacturing core is not yet threatened.

Introduction

This paper explores the relationships between regional manufacturing systems and seaports; the core/periphery approach is used to conceptualize their interactions.

Despite the potential understanding the driving forces of globalization, few works have analyzed the nexus between regional manufacturing and maritime transport systems [START_REF] Ducruet | Ports and the local embedding of commodity flows[END_REF][START_REF] Hall | Shifting proximities: The maritime ports sector in an era of global supply chains[END_REF]. However, transport improvements have played a major role in the historical extension of manufacturing systems. The Roman Empire's completion of a road network system or the development of steam engine's development in the nineteenth century are examples of advances in transportation that have set the pace of capitalist expansion, driving major shifts in both production and trade. The increase in transportation efficiency notably allowed large-scale manufacturing, implying further economic specialization in countries and regions, and favoring economies of agglomeration. Consequently, territories became increasingly interdependent. Among the different transport modes involved in the economic interaction between cities and regions, maritime has historically played a major role, enabling the emergence and development of a network of cities in the Mediterranean as early as the first millenium B.C. The region's success layed in such manufacturing activities as the production of textiles and olive oil, which were heavily dependent on overseas inputs [START_REF] Sherratt | The growth of the Mediterranean economy in the early first millennium BC[END_REF]. Today, 80% per cent of international merchandise trade, measured in tons-kilometers, is carried by sea and handled by seaports worldwide [START_REF]Review of Maritime Transport[END_REF]. Maritime transport's dominance is not limited to long-distance trade, as it also plays an important role in medium -and short-distance connections. Markets can only expand overseas if they benefit from frequent and reliable maritime transport services.

This paper aims to contribute to a better understanding of the interplay between maritime transport and regional manufacturing. Japan has been selected because its status as an insular country, providing relevant grounds to observe port-manufacturing interactions with less ground transport interference. Moreover, it is a rare country in which where detailed port traffic and employment data are available over several decades. The study focuses on the automobile industry in a peripheral Japanese region as an example to understand how the core and periphery interact. This industry is strongly reliant on frequent maritime transport services between its historical centers and new manufacturing areas. Specifically, we propose the case study of northern Kyushu because it has two important specificities: first, its location in southwestern Japan allows for the daily deliveries of automotive parts by either ship or truck from central Japan as well as low-cost suppliers in South Korea and China. Second, a relatively dense network of local suppliers has been progressively developed.

This research studies these interactions by building on the frameworks of the core-periphery model and global commodity chains, which are presented and criticized in Section 2. The area of study is defined in Section 3, and long term ) locational manufacturing and seaport trends are presented in section 4. We investigate the recent changes in manufacturing and port systems (1990-2010) and their geographical expansion in Section 5 and in section 6 we rely on a case of automobile manufacturing in northern Kyushu to explain the evolving relationships between port and manufacturing systems. Finally, Section 7 presents a discussion of our results and conclusions.

Core-periphery and commodity chains: Conceptualizing the regional shifts in manufacturing and seaports

Wallerstein's world-systems analysis provides a compelling framework to analyze the evolving locations of ports and manufacturing activities, as it aims to understand how the dynamic of capital accumulation contributes to shape the world as a core-periphery structure that relies on two basic dichotomies [START_REF] Wallerstein | A world-system perspective on the social sciences[END_REF]. The first is a class dichotomy, in which ruling groups' control depends on their capacity to make decisions regarding the nature and quantity of the production of goods-via property rights, accumulated capital, control over technology and so on. The second dichotomy, which is particularly relevant to our study, is the spatial hierarchy of economic specialization, or core versus periphery. This involves an appropriation of surplus from the producers of low-profit goods, by the producers of highprofit goods in a purported "unequal exchange". Spatial hierarchies change to respond to the pressures of cyclical economic crises within the system, but without significantly changing class hierarchies [START_REF] Wallerstein | A world-system perspective on the social sciences[END_REF]. The core and periphery can then be considered as only temporary outcomes of the capitalist system. Core processes imply high wages, high technology and high profit input; periphery processes imply the opposite. A transnational firm's managing of a commodity chain is an example of a core mechanism. These processes tend to spatially concentrate and segregate over time, reflecting the evolution of market power, entry barriers, and forms of chain governance [START_REF] Brown | World city networks and global commodity chains: Towards a world-systems' integration[END_REF]. This dy-namic produces places in which core processes are dominant and places where periphery processes are dominant, or the "core" and "periphery" places, respectively. Naturally, core places also host peripheral processes; some core processes could eventually occur in a periphery.

Wallerstein's model [START_REF] Wallerstein | The capitalist world-economy[END_REF][START_REF] Wallerstein | World-Systems Analysis: An Introduction[END_REF] provides a rather sophisticated approach to the relationships between core and periphery, rather than suggesting a dual world. This offers valuable insight to understand the hierarchical relationships within urban systems [START_REF] Flint | Political geography: World-economy, Nation-state, and Locality[END_REF][START_REF] Beaverstock | World-city network: A new metageography?[END_REF], and explains the concentration of decisionmaking activities in a few large urban areas. This configuration leads to income disparities between the core (some large urban areas) and the periphery (the rest) [START_REF] Abdel-Rahman | Toward a generalequilibrium theory of a core-periphery system of cities[END_REF]. The commodity chain concept is used to explain how value is transferred from the periphery to the core, maintaining or deepening the differences between both. Hopkins & Wallerstein [START_REF] Hopkins | Commodity chains in the world-economy prior to 1800[END_REF] define it as a "network of labor and production processes whose end result is a finished commodity". Further developed by Gereffi & Korzeniewicz [START_REF] Gereffi | Commodity Chains and Global Capitalism[END_REF], the (global) commodity chain is considered a system of value creation employed by firms and other agents, in which market power asymmetries lead to unequal value distributions. Participants along the chain are unequally able to appropriate rents and barriers to entry exist. Lead firms act as chain drivers, commanding the coordination of the whole commodity chain, by controlling the other firms in the chain [START_REF] Gereffi | Beyond the producer-driven/buyer-driven dichotomy the evolution of global value chains in the Internet era[END_REF]. The lead firms' superior profitability is a result of their capacity to generate different types of rents, using scarce assets (i.e. infrastructure, machinery, brands, marketing, etc.). These assets lead to the creation of barriers to entry and result in different types of high rents [START_REF] Gereffi | Beyond the producer-driven/buyer-driven dichotomy the evolution of global value chains in the Internet era[END_REF], which allow the firms ensuring core processes to be relatively insulated from capitalist competition. Conversely, firms realizing peripheral processes would not have the power to contest the organizational leadership, and are more exposed to competitive pressures [START_REF] Gereffi | Commodity Chains and Global Capitalism[END_REF]. Despite its utility in describing the relationships between firms, the global commodity chain approach has been criticized for simply assuming the power differential between firms implied in a chain, without providing a more general explanation of how these differences have been created [START_REF] Taylor | Rethinking the global production of uneven development[END_REF][START_REF] Starosta | Global commodity chains and the Marxian law of value[END_REF]. According to these critics, chains should be further conceived as moments in a global circuit of capital [START_REF] Taylor | Rethinking the global production of uneven development[END_REF]. The connection between commodity chains and the general dynamics of capitalism would be partially reached using the Marxian law of value [START_REF] Starosta | Global commodity chains and the Marxian law of value[END_REF]. Indeed, the formation of commodity chains is therefore the concrete form taken by the competition among normal or average capitals over extra surplus value that escapes the hands of small capitals. (Starosta, 2010:451) The trend of outsourcing manufacturing activities can then be regarded as a method for large firms to multiply the sources of extra surplus value released by small capitals. The lead firm is in the best position to coordinate the entire chain and exert control on the other firms to capture the "extraordinary profits flowing out of small capitals" [START_REF] Starosta | Global commodity chains and the Marxian law of value[END_REF].

The automobile industry is one of the most extensively researched commodity chain [START_REF] Gereffi | A commodity chains framework for analyzing global industries[END_REF][START_REF] Barnes | The German connection: Shifting hegemony in the political economy of the South African automotive industry[END_REF][START_REF] Kaplinsky | Globalisation and unequalisation: What can be learned from value chain analysis[END_REF][START_REF] Humphrey | Globalization and supply chain networks: the auto industry in Brazil and India[END_REF], and illustrates how profits freed by small capitals are captured by normal capitals. Three main players exist within this particular chain: car manufacturers, first-tier suppliers, and lower-tier suppliers. Automobile chain is controlled by automobile manufacturers, which typically play a central role. They are large firms with highly automated labor processes. The chain is highly hierarchical with multilayered production systems involving thousands of firms.

The automobile value chain's situation has changed in the 1990s; the hierarchical restructuring into tiers became even more pronounced [START_REF] Gereffi | A commodity chains framework for analyzing global industries[END_REF], following a decrease in the number of suppliers at all levels, with each manufacturer relying on a small group of first-tier suppliers. These large suppliers not only operate large plants, but they have also assumed over many of the functions previously centralized by car manufacturers, and are often responsible for the design, manufacturing, and delivery of complete modules to automobile assembly plants. The latter have become responsible for selecting lower-tier suppliers and coordinating the automobile supply chain's core segments at a global level. According to Gereffi, top first-tier suppliers are "challenging the assemblers to control over the key high value activities in automobile production" [START_REF] Gereffi | A commodity chains framework for analyzing global industries[END_REF] (p.5). As many of the leading auto suppliers manufacture parts in the periphery, this could provide a chance for its firms to move up in the industry. However, not all the major first-tier suppliers succeed in capturing higher value in the chain, and some are experiencing difficulties [START_REF] Frigant | Winners and losers in the auto parts industry: Trajectories followed by the main first tier suppliers over the past decade[END_REF]. Further, some lower-tier suppliers can eventually capture higher value from the chain. One example involves steel producers, where the invention of new processes generates opportunities for product innovation and the creation of higher value steels through forming and shaping new and existing materials [START_REF] Hudson | Critical political economy and material transformation[END_REF]. Finally the core/periphery structures can vary considerably between regions, despite the general principles of commodity chains' organization. Maritime transport plays an important role in accessibility in remote or insular areas, and its improvement can eventually accelarete the rearrangement of core/periphery structures.

Area of study and data collection

Japan presents a noteworthy geographical context to study the links between seaports and manufacturing. The country's manufacturing system is highly reliant on seaports due to both its insularity and a lack of natural resources. Moreover, the latter handles the bulk of Japanese international trade (99% in tons, and 75% in value in 2010). Many studies have been devoted to the regional shifts of manufacturing [START_REF] Fujita | Regional growth in postwar Japan[END_REF][START_REF] Akita | Regional income inequality in the postwar Japan[END_REF][START_REF] Shioji | Composition effect of migration and regional growth in Japan[END_REF] and to the evolution of its seaports [START_REF] Itoh | Efficiency changes at major container ports in Japan: A window application of Data Envelopment Analysis[END_REF][START_REF] Rimmer | Japanese seaports: Economic development and state intervention[END_REF] in such a context, but few of the former analyze both dimensions simultaneously.

We identify where core and peripheral processes are dominant by using the typology of Japanese prefectures established by Fujita & Tabuchi [START_REF] Fujita | Regional growth in postwar Japan[END_REF] (Table 1). Decision-making functions, such as those associated to headquarters (HQs), and such knowledge intensive activities as research and development (R&D), have been considered as core processes. These often imply high wages, technology and profit inputs and are located at the top of the urban hierarchy. Manufacturing functions (MNF) and regional branches have been associated with peripheral processes, which are located in the rest of the country and overseas (Table 2).

Level Name of the Urban Area

Functions* We analyze the changes in ports and manufacturing systems by considering the three largest metropolitan areas (MAs) located along the Japanese manufacturing belt (as defined by the Japanese Ministry of Economy, Trade and Industry, Figure 1). Additionally, this research focuses on northern Kyushu, as it is an emerging manufacturing region. The Tokyo MA is constructed by the Saitama, Chiba, Tokyo, and Kanagawa prefectures; the Nagoya MA is constructed by the Gifu, Aichi and Mie prefectures; the Osaka MA is constructed by the Kyoto, Osaka, Hyogo and Nara prefectures; and finally, the Kyushu MA is constructed by the Fukuoka, Saga, Nagasaki, Kumamoto and Oita prefectures.

A combination of data analysis, literature review, and case study research was used in this work. The study of the joint evolution of manufacturing and ports at the prefecture level has been permitted bu a comparison of employment data (as per the Census conducted by the Statistics bureau of Japan every five years) and port throughput (Port Statistics Yearbook, published annually by the Ministry of Land, Infrastructure, Transport, and Tourism). systems at the level of prefectures has been allowed by the comparison of data on employment (Census, carried out by the Statistics bureau of Japan every five years) and port throughput (Port Statistics Yearbook, published every year by the Ministry of Land, Infrastructure, Transport and Tourism). As our focus is on manufacturing activities, we only considered flows of manufacturing products and employment in manufacturing firms. Unfortunately, the level of data aggregation did not allow us to distinguish the employment and cargo flows related to core and peripheral processes. This obstacle has been partially overcome by a case study based on automobile manufacturing. Given the specificity topic's specificity and the lack of data, a series of 20 semi-structured, in-depth interviews were conducted with automobile manufacturers and logistics providers in primary manufacturing regions (the Nagoya, Tokyo, and Kyushu MAs) between May and September 2015. The interviewees essentially held senior positions at logistics or production management departments. . 4. Regional growth in Post-War Japan (1945-1990)

Urban

The manufacturing and seaport systems' current core/periphery structures cannot be understood without referencing the historical process that produced them. The Japanese manufacturing core prior to World War II (WWII) was essentially structured around the Tokyo and Osaka Metropolitan Areas (MAs), and the textile industry was its most prominent (Figure 2). Osaka was slightly more important than Tokyo at the time as a manufacturing center: its success was built on textile and the related trade. Local merchants and traders ensured the link between domestic manufacturers and foreign suppliers and buyers. Further, the domestic industry's inputs and outputs were handled bu ports in the Osaka and Tokyo MAs. The need for proximity between trade, manufacturing, and transport functions led to the use of ports directly located at the core (Figure 3).

The Korean War (1950)(1951)(1952)(1953) in the aftermath of WWII, acted as a catalyst for Japanese economic revitalization, and aroused manufacturing activities specializing in military and consumable goods. Shortly thereafter (1960), an ambitious infrastructure investment policy was implemented: the "double-income plan". This aimed to support the manufacturing growth by ensuring production by ensuring high-capacity transport connections between Osaka, Tokyo, and the primary peripheral MAs. The manufacturing system's core was still shared by Osaka and Tokyo MAs during this rapid economic growth period, but large manufacturing centers emerged at the periphery: Nagoya, Hiroshima, Okayama, and Fukuoka. Heavy industry, such as shipbuilding, became increasingly important. The seaport hierarchy remained unchanged despite peripheral port expansion with Osaka and Tokyo MAs at the forefront.

The rapid growth period abruptly ended with the first oil shock in 1973. The foreign currency exchange system's transition around the same time, from fixed to floating rates, induced the appreciation of the Japanese yen against the United States' dollar. The combination of high oil prices, the yen's higher value and an intensified competition with newly industrialized economies in East Asia provoked the Japanese economy's second major structural change. This involved a change from heavy industry to knowledgeintensive activities, such as automobile and electronic manufacturing. Japanese manufacturing firms then created overseas production facilities to remain competitive in international markets. This shift involved the emergence of a new regional system, the center of which is exclusively the Tokyo MA, which concentrated most emerging multinational firms' headquarters. The port system became increasingly dominated by the Osaka MA (Kobe), a pioneer of containerization in Asia [START_REF] Guerrero | The waves of containerization: Shifts in global maritime transportation[END_REF], that still concentrates most of its functions on trade and shipping companies. The manufacturing system's international expansion decreased the need for proximity between manufacturing, trade and seaport functions. The most labor-intensive processes, such as largescale manufacturing, are increasingly transferred to the periphery with the manufacturing system's core specializing in advanced services.

The post-bubble context (1990-2010)

The Japanese economy was substantially transformed in the 1980s and 1990s. A new wave of internationalization of manufacturing occurred, primarily in Southeast Asia, South Korea, and China. Large container hubs emerged in neighboring countries (South Korea, Taiwan, and Hong-Kong), diverting many deep-sea container services from Japanese ports, which were increasingly feederized, and became peripheral in a regional hierarchy dominated by foreign ports [START_REF] Comtois | The evolution of containerization in East Asia[END_REF][START_REF] Fleming | World container port rankings[END_REF][START_REF] Frémont | The emergence of a Mega-port: From the global to the local, The case of Busan[END_REF]. Although not the main cause of the port's decline, the 1995 Kobe earthquake acted as a catalyst for a collapse [START_REF] Itoh | Efficiency changes at major container ports in Japan: A window application of Data Envelopment Analysis[END_REF]. The port function became increasingly disconnected from the manufacturing system's core.

The impacts of these transformations have been particularly dramatic within the manufacturing sector, as between 1990 and 2010 employment decreased from 14.5 to 9.5 million. Aside from the economic factors at work (i.e., increased productivity, manufacturing firms' outsourcing of services, internationalization, etc.) which are not on the within this paper's scope, it is important to underline the uneven geographical impact of these transformations. Manufacturing employment decreased everywhere, but the periphery (the Nagoya MA and the rest of the periphery) resisted better than the manufacturing core (the Tokyo and Osaka MAs) (Figure 4 2 ).

The observation of port traffic during the same period provides an alternative perspective on the geographical shifts in manufacturing. The volumes of manufactured goods handled by seaports increased both for imports (+51%, from 51.4 Mt to 104.8 Mt) and exports (+23%, from 134.3 Mt to 174.3Mt), between 1990 and 2010. However the geographical dynamics of imports and exports significantly differ. Regarding imports, the seaports of Tokyo, and to a lesser extent Kyushu MAs, increase their shares on import flows to the detriment of the Osaka and Nagoya MAs, and especially after 2005. Fostered by its large local consumer market's demand, the Tokyo MA strengthened its gateway position for imports. The opposite occurred in the Osaka MA, where the local demand for imports weakened (Figure 5).

The geographical distribution of export flows approximates that of manufacturing employment, as a substantial share of the manufacturing outputs 2 Detailed data on seaports and employment was not available at an international level. Graphic schemes have been elaborated as complements to clarify the geographical shifts of port and manufacturing systems in the international periphery. is shipped overseas. The Tokyo and Osaka MAs lost ground during the 1990-2005 in favor of the rest of the country, and particularly of Nagoya and Kyushu MAs (Figure 6). The ties between decision-making and manufacturing activities loosened as production moved from the core to peripheral areas, including overseas.

Overall, most labor-intensive activities moved to the periphery, either overseas or within Japan. The Tokyo MA remains a national gateway, as is the largest Japanese consumer market, but only plays a minor role in eastern-Asia. Most regional hubs3 are located overseas, mainly in South Korea (Busan) and Taiwan (Kaohsiung). Outbound international trade has been directly shipped from manufacturing regions (e.g., the Nagoya and Kyushu MAs) (Figure 7). 6. An exploration of the links between seaport and manufacturing systems: The case of the automobile industry in Kyushu

The recent development of automobile manufacturing in Japan's peripheral regions suitably illustrates the interdependence between the seaport and manufacturing systems. The automobile industry is doubly noteworthy for our study. On the one hand, it is by far Japan's most important industry, accounting for 3.6% of total employment4 [START_REF]The Motor Industry of Japan[END_REF], and generating substantial transport volumes. On the other hand, its organization, implies frequent deliveries between suppliers and assemblers, and makes it particularly sensitive to the quality of both transport services and infrastructure. This is particularly true for maritime transport within Japan, which is used for daily part deliveries from suppliers located in the historical cores of automobile industry in the Nagoya and Tokyo MAs, to assembly plants located in the peripheral regions.

As aforementioned (Figure 2), the Japanese manufacturing system's international expansion began in the early 1970s, after the Nixon shock. Automobile manufacturers followed the same trend, creating overseas assembly plants to supply their foreign markets. Twenty years later, in the early 1990s, Japanese car manufacturers' overseas production is higher than the domestic production. Currently, three out of five Japanese cars sold worldwide are produced outside Japan (Table 3). However, despite its high internationalization, domestic production remains crucial, with almost 10 million cars, nearly half of which is exported (4.85 million).

Long before the Japanese automobile firms' internationalization, most of them built keiretsu systems, in which very tight supplier relationships are developed. This form of vertical integration particularly common in Japan, involves capital partnership, the inclusion of representatives on the board, technical guidance, and financing, from a parent company to its subsidiary companies. Some suppliers in this context, usually located close to assembly plants, provide parts requested by car manufacturers on a just-in-time (JIT) basis. However, this is far from the case for every component, and different sourcing strategies are adopted depending on such factors as the minimum efficiency scale required for production, part diversity, and bulkiness among others [START_REF] Kaneko | The logistics of just-in-time between parts suppliers and car assemblers in Japan[END_REF]. Moreover, car manufacturers in the recent decades have largely adopted global purchasing strategies to save costs, often implementing long distance sourcing and substantial stocks [START_REF] Guerrero | The role of seaport-based logistics platforms in the automotive supply chain[END_REF].

Northern Kyushu provides a noteworthy illustration on how car manufacturers use ports to build more resilient supply chains. Japanese car manufacturers considerably expanded their production capacities outside their traditional core in the early 1990s. The reasons for this shift are threefold. The first reason is cost-related, labor costs in norther Kyushu are about 20% lower than in the traditional automobile industry cores in the Nagoya and Tokyo MAs. Moreover, land acquisition costs for industrial activities have increased considerably in the traditional cores. Second, northern Kyushu has higher accessibility to rapidly growing Asian consumer markets for new vehicles. Finally, frequent parts deliveries can be organized from low cost suppliers in South Korea and China to the northern Kyushu's assembly plants. For example, Nissan, uses high-speed ferry services between Busan (South Korea), Shimonoseki and Hakata (Japan) [START_REF] Fujiwara | Seamless Logistics Create a new Era of East Asia[END_REF]. Semi-trailers with both South Korean and Japanese license plates are used to convey the parts daily between suppliers and assembling plants at the northern Kyushu (Figure 8).

Sea, rail, and air transport are used to source parts from other regions of Japan, in addition to overseas services. Kitakyushu's container terminal expansion in 2005 and airport creation in 2006 were strongly motivated by the needs of automobile manufacturers. The air terminal, which is the first to be open 24 hours a day in northern Kyushu, is connected through a large-capacity highway to the manufacturing plants in Kokura (Toyota) and Kanda (Toyota and Nissan). Automobile manufacturers require both lean and robust supply chain networks to quickly respond to shifts in new vehicle demand, and to reduce their exposure to exchange rate fluctuations. This need for robust and resilient supply chain networks increases in northern Kyushu, as most production is destined to foreign markets, which often implies less stable demand than in domestic markets.

Although the automobile industry has existed in northern Kyushu since the 1970s, production substantially increased in quantity and importance after the 1990s, with the creation many manufacturing and parts supply plants (Table 4). However, local content ratios for the relatively recent northern Kyushu plants are lower (51% in 2006) than those from the traditional regions in the Nagoya and Tokyo MAs (about 84% in 2006) [START_REF] Yusuf | Growing Industrial Clusters in Asia: Serendipity and Science[END_REF]. This is because a majority of part suppliers have remained located in the automobile industry's traditional cores. Moreover, Kyushu's local content ratio could be overestimated, as approximately 50% of the parts delivered by local suppliers are manufactured in other regions of Japan [START_REF] Imbres | Japan's "Car Island": Evolution of Northern Kyushu Industrial Cluster in Light of the Changing Japanese Automotive Industry[END_REF][START_REF] Kido | Changes of the agglomeration process in newlydeveloped automobile industrial regions (in Japanese)[END_REF]. Northern Kyushu's production growth has outstripped the national production of new vehicles since the 1990s; the region accounted for 14.9% of domestic production in 2012 (Figure 9), and is the third largest automobile manufacturing region after the Nagoya and Tokyo MAs. The northern Kyushu's growth is particularly striking when considering the national automobile industry's sluggish development over the last two decades. These differences can be captured by comparing the employment trends in the transport and machinery sector, which is a slightly larger category than the singular automobile industry. Employment in northern Kyushu increased between 1990 and 2010 by 37% (from 39,000 to 63,000), but decreased at national level by 9% (from 1.11 to 1.02 million).

The automobile industry in northern Kyushu strongly relies on maritime transport, both for inbound and outbound logistics. Inbound transport machinery volumes in 2010 represented 18% (10.2 Mt) of domestic inbound volumes and 9% (1.0 Mt) internationally. Substantial domestic inbound flows involve parts from suppliers located in the Nagoya and Tokyo MAs. The transport machinery sector accounts for 19% (12.5 Mt) of domestic outbound volumes, and 6% (5.4 Mt) internationally. The main domestic destination of the northern Kyushu's shipments is the Nagoya MA, as finished cars and auto parts sent there are eventually redistributed overseas.

The ports in northern Kyushu handle most of the inbound flows of parts This research raises the question that, in an advanced economic context in which transport infrastructure is well-developed, can manufacturing firms from the peripheral areas improve their positions in the automobile value chain? Local suppliers in northern Kyushu, supported by universities and the local government, seem to be struggling to upgrade their positions. For instance, only a few local firms have succeeded in becoming first-tier suppliers for several models built in northern Kyushu. According to different works in this field, suppliers located close to headquarters and R&D centers, where the new models are designed, would have a significant advantage over suppliers located elsewhere [START_REF] Yusuf | Growing Industrial Clusters in Asia: Serendipity and Science[END_REF][START_REF] Humphrey | Globalization and supply chain networks: the auto industry in Brazil and India[END_REF]. The latter are hindered in their quest to improve their positions, as the first stages of development of new models requires frequent exchanges between car manufacturers and first-tier suppliers. Despite first-class transport connections to the Japanese automobile industry's core in the Nagoya and Tokyo MAs, suppliers of northern Kyushu would travel too far to compete with them. Another major challenge for part suppliers in northern Kyushu will be to address the increasing competition with low cost suppliers in South Korea and China. Nissan's CEO, Carlos Ghosn, has recently expressed his willingness to reduce Kyushu plants' sourcing cost by increasing the share of imported parts from low-cost suppliers in South Korea and China [START_REF] Unnamed | Kyushu industry faces a new challenge[END_REF]. Paradoxically, improving the northern Kyushu's maritime transport connections, heavily funded by the public sector, could imply the decline of local suppliers.

Conclusion

This paper presented an overview of the evolution of Japanese manufacturing and seaport systems since World War II, with a special focus on the post-bubble era . First, we illustrated that until the mid-1990s primary seaports have always been located in core regions. Japanese seaports have since been downgraded, primarily because of container hub development in neighboring countries, such as Busan in South Korea, Kaohsiung in Taiwan and Hong-Kong, and a substantial shift in Japanese manufacturing to emerging economies in east and southeast Asia. We then demonstrated that while decision-making functions have been increasingly concentrated in the core (Tokyo MA during the post-bubble era ), manufacturing functions have trickled down the urban hierarchy in Japan and overseas.

We have concluded that unlike the previous periods, when decision-making in manufacturing and seaport functions were concentrated at the core, a growing disconnect exists between both: decision-making remains at the core, but primary seaports have been located in neighboring countries since the mid-1990s. However, even if the seaports at the core (e.g., the Tokyo MA) are not the container network's primary nodes, they still dominate the national hierarchy and non-containerized shipping networks, and particularly in the case of imports. We have also concluded that the development of containerized hub-and-spokes networks has contributed to the marginalization of Japanese ports. However it is unclear if this is a real obstacle to the manufacturing system's development. Recent works indicate that most cargo carried by sea container is not particularly time-sensitive [START_REF] Mckinnon | Freight transport deceleration: Its possible contribution to the decarbonisation of logistics[END_REF]. Therefore, a few days' lengthening of the maritime transport induced by the transshipment of cargo in foreign countries does not seem to be critical for most Japanese-based firms. This is particularly the case when we know that when order and door-to-door lead times are considered.

The transport machinery industry follows a slightly different pattern than other manufacturing industries: the centrifugal movement of assembling plants is less powerful than in other such sectors as electronics, and many manufacturing plants remain at the traditional manufacturing regions. The core still plays an important role in this case, not only in decision-making functions but also in the mass production of parts and vehicles. However, efficient maritime transport has allowed for the relocation of non-negligible automobile production to peripheral areas, in which production costs are generally lower and car manufacturers can benefit from an existing supplier network, primarily inherited from electronic industry.

Improvements in ports and maritime transportation surely do not comprehensively capture changes in the manufacturing system, but are strongly suggestive of the type of explanation needed. Our study also indicated it is necessary to conduct a detailed activity-based analysis (in contrast to the traditional industrial sector-based analysis) combined with urban and transport systems approaches in the seaport/manufacturing systems, which are being experienced in both developed and developing countries worldwide. Although this paper has focused only on Japan, we hope to conduct a more comprehensive study of regional transformations in comparison with several closely related countries (including both developed and developing countries) using a similar approach in the future.

Figure 1 :

 1 Figure 1: Composition of the Japanese manufacturing belt (left) and of the main metropolitan areas (MAs) used in this study (right). Source: Authors (2016), based on the categories of Fujita & Tabuchi[START_REF] Fujita | Regional growth in postwar Japan[END_REF]

Figure 2 :

 2 Figure 2: Leading industries and main events in manufacturing and port systems. Source: Authors (2016), based on Fujita & Tabuchi [12].

Figure 3 :

 3 Figure 3: Scheme representing the changes in manufacturing and seaport systems since WWII. Source: Authors (2016).

Figure 4 :

 4 Figure 4: Graph (left) representing the evolution of Japanese MAs' contribution to total manufacturing employment (shares for the national total, noted in number of employees). The scheme (right) represents the evolution of manufacturing employments in Japanese firms at the international level. Source: Authors (2016); Data: JP Census.

Figure 5 :

 5 Figure 5: Graph (left) representing the evolution of Japanese MAs' contribution to total manufacturing cargo imports (in tons). The schemes (right) represent the evolution of manufacturing cargo imports on an international level. Source: Authors (2016); Data: MLIT.

Figure 6 :

 6 Figure 6: Graph (left) representing the evolution of Japanese MAs' contribution to total manufacturing cargo exports (in tons). The scheme (right) represent the evolution of Japanese firms' manufacturing cargo exports on an international level. Source: Authors (2016); Data: MLIT.

Figure 7 :

 7 Figure 7: Scheme representing the current organization of the Japanese seaport and manufacturing system. Source: Authors (2016).

Figure 8 :

 8 Figure 8: Automobile industry in northern Kyushu. Source: Authors (2016)

Figure 9 :

 9 Figure 9: Trends in the production of finished cars in northern Kyushu. Source: Fukuoka Prefecture

Figure 10 :

 10 Figure 10: Ports mentioned in Table 5

Table 1 :

 1 Organization of Japanese manufacturing firms by urban hierarchy level. Source: Authors (2016), based on the categories of Fujita & Tabuchi[START_REF] Fujita | Regional growth in postwar Japan[END_REF] *Mentioned in order of importance.

	1	Tokyo MA	HQs, R&D, MNF
	1.5	Osaka MA	HQs, R&D, MNF
	2	Nago., Fuku., Sapp., Hiro., Send. MNF, HQs, R&D, Branches
	3	Other provincial centers	MNF, Branches
	Others Rest of Japan	MNF

Table 2 :

 2 Detailed organization of Japanese manufacturing firms by urban hierarchy level.

	Level	HQ R&D Branch Factory
	1 Tokyo MA	*** ***	*	*
	1.5 Osaka MA	**	*	*	*
	2 (ex. Fukuoka) *	*	**	**
	3 (ex. Okayama) *		***	**
	Others			**	***
	Overseas 1	**	*	**	*
	Overseas Others				***

Source: Authors (2016), based on the categories of Fujita & Tabuchi

[START_REF] Fujita | Regional growth in postwar Japan[END_REF]

Table 3 :

 3 Finished car production (in million units) by Japanese firms, by region. Source: Authors (2016); Data: Nissan (2012)

Table 4 :

 4 Characteristics of Japanese car manufacturers' plants in northern Kyushu. Source: Fukuoka Prefecture, Ishiro & Mokudai (2013). *Number of parts. **Share of the vehicle's value produced in northern Kyushu.

Corresponding author at Université Paris-Est, Ifsttar, Ame-Splott, 14-20 Boulevard Newton, Cité Descartes, Champs-sur-Marne, 77447 Marne la Vallée Cedex

2, France. Tel.: +33 1 81 66 87 90.

In the context of the rapid increase in container ships' sizes, the major Japanese ports are increasingly served via foreign hubs, and particularly in the connections with Europe and the Middle-East. Comparatively, Japanese ports are well-positioned for trade with North America, as transshipment in South Korea or Taiwan implies considerably longer transit times.

Including automobile production, materials and equipment supply, and sales and services.

Roll-on/roll-off (ro-ro) ships are vessels designed to carry wheeled cargo, such as automobiles or trucks, driven on and off the ship on their own wheels.

[START_REF] Comtois | The evolution of containerization in East Asia[END_REF] Most of these mega-ships were ordered in the mid 2000s when global car sales were booming and shipping companies raced to keep up with the demand from car manufacturers.


Corrections to "A Global High-Gain Finite-Time Observer"

Tomas Ménard, Emmanuel Moulay and Wilfrid Perruquetti

Abstract-This note fix the proof of Theorem 2 in the article [START_REF] Ménard | A global high-gain finite-time observer[END_REF].

Equations from the original paper will be denoted with a star (for example (1 * )) whereas equations of the present corrected paper will be denoted without a star (for example [START_REF] Munkres | Topology[END_REF]).

I. THE ERROR

The function Ṽα used in the proof of Theorem 2 in [START_REF] Ménard | A global high-gain finite-time observer[END_REF], and derived from Theorem 10 in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], is not C 1 with respect to (e, α). Indeed, one has

∂ e k 1 α k-1 q ∂e k = 1 α k q e k 1 α k q -1 . (1)
Hence, when α → 1, 1 α k q → 1 and when one of the component of e goes to zero, the limit lim (α,e)→(1,e0)

1 α k q e k 1 α k q -1
does not exist. Thus the function Ṽα cannot be used as a candidate Lyapunov function.

II. THE FIX

Let us first recall Theorem 2 from [START_REF] Ménard | A global high-gain finite-time observer[END_REF].

Theorem 1. Let us consider system (3 * ) with a bounded input u. Then there exists θ * ≥ 1 such that for all θ > θ * there exists > 0 such that system (3 * ) admits the following global finite-time observer:

       ẋ1 = x2 + k 1 ( e 1 α1 + ρe 1 ) + m j=1 g 1,j (x 1 )u j . . . ẋn = k n ( e 1
αn + ρe 1 ) + ϕ(x) + m j=1 g n,j (x)u j for all α ∈]1 -, 1[, where e 1 = x 1 -x1 , the powers α i are defined by (5 * ), the gains k i by (6 * ) and ρ = n 2 θ where S 1 is defined by (8 * ). In addition, the settling time of the error dynamics is bounded by T 1 (e 0 ) + T 2 (e 0 ) (with e 0 = x 0 -x0 ), where T 1 , T 2 are respectively given by (18 * ) and (6).

The statement of Theorem 2 in [START_REF] Ménard | A global high-gain finite-time observer[END_REF] remains correct, except for the settling time which has to be corrected.

We can define the function V 1 (e) = e T S ∞ (1)e, for e ∈ R n , where S ∞ (1) is the solution of (7 * ) for θ = 1. This choice corresponds to the linear case, that is α = 1. Proceeding as in [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Yu | Adaptive finite-time consensus in multi-agent networks[END_REF], one can construct a candidate Lyapunov function with properties stated next.

Proposition 1. Let a ∈ C ∞ (R, R) be such that a = 0 on (-∞, 1] 1 on [2, +∞) and a ≥ 0 on R. (2)
There exists > 0 such that for all α ∈]1 -, 1 + [, the function Vα defined as

Vα (e) = +∞ 0 1 t 3 (a • V 1 )(t r1(α) e 1 , . . . , t rn(α) e n )dt (3)
if e ∈ R n \{0} and Vα (0) = 0 is well defined, radially unbounded, of class C 1 (R n , R), and satisfies a) Vα (δ

r(α) λ e) = λ 2
Vα (e), for all e ∈ R n and λ > 0.

b) ∇ Vα (e), Ae-F (S -1 ∞ (1)C T , e) ≤ -γ( Vα (e)) 1+α
2 , for all e ∈ R n , where γ > 0. where F ,C and δ r(α) λ are defined in [START_REF] Ménard | A global high-gain finite-time observer[END_REF].

Proof of proposition 1. Let, α ∈]1 -1 n , +∞[, proceeding as in [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF], one directly shows that Vα is well defined, radially unbounded, C 1 on R n , and homogeneous of degree 2 with respect to the weights r(α). Then, only point b) remains to prove. Following the same lines as in [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF], there exists l, L > 0 such that for all e ∈ e ∈ R n | Vα (e) = 1 , one has n , +∞[. The function g is continuous, (e, t) belongs to a compact set and there exists γ 1 > 0 such that the image of g is included in ] -∞, -γ 1 [ for (e, t) ∈ {e ∈ R n , Vα (e) = 1} × {t ∈ [l, L]} and α = 1 (since it corresponds to the linear case). We can then apply Lemma 26.8 in [START_REF] Munkres | Topology[END_REF] (tube lemma) which gives the existence of > 0 such that for all (e, t, α)

∇ Vα (e), Ae-F (S -1 ∞ (1)C T , e) = L l 1 t α+2 a V 1 δ r(α) t e × ∇V 1 δ r(α) t e , Aδ
∈ {e ∈ R n , Vα (e) = 1} × {t ∈ [l, L]}×]1 -, 1 + [, g(e, t, α) ≤ -γ 1 . Then we have ∇ Vα (e), Ae -F (S -1 ∞ (1)C T , e) ≤ -γ 1 L l 1 t α+2 a V 1 δ r(α) t e dt ≤ -γ Vα (e) 2+α-1 2 (4)
where

γ > 0 is a lower bound of γ 1 L l 1 t α+2 a V 1 δ r(α) t
e dt for (e, α) ∈ {e ∈ R n , Vα (e) = 1}×]1 -, 1 + [. Since Vα is homogeneous of degree 2 with respect to the weights r(α), inequality (4) is valid for all e ∈ R n . Now that a new candidate Lyapunov function has been defined, we explain how it will be used to correct the proof of Theorem 2 in [START_REF] Ménard | A global high-gain finite-time observer[END_REF]. Please note that part 1 of the proof is correct, then it has already been proved that every trajectory starting from e 0 ∈ R n enter the ball B . S∞(θ) (1) after time T 1 (e 0 ) = log(1/V (e 0 ))/κ(θ) (see equation (18 * )). Denote ē = ∆ θ e, where ∆ θ = diag 1 1 θ . . . 1 θ n-1 , in the remaining, we will show that for every θ ≥ θ 2 = 2 γ (M 1 + 2), there exists > 0 such that the following inequality

Vα (ē) ≤ - γ 2 θ -1 Vα (ē) 2+α-1 2 + M 1 Vα (ē) (5)
holds for every ē ∈ B . S∞(1) (1), α ∈]1 -, 1[ , where M 1 > 0 is a constant independent of θ. This inequality replaces inequality (19 * ). Inequality (5) directly implies that the error system (11*) is finite time stable on B . S∞ (θ) [START_REF] Munkres | Topology[END_REF]. Thus, after time T 1 (e 0 ), the error enters B . S∞ (θ) (1) and after time T 1 (e 0 ) + T 2 (e 0 ) the error reaches the origin, where the settling time T 2 (e 0 ) is bounded as follows

T 2 (e 0 ) ≤ ln 1 -M1 γ 2 θ-1 Vα (e 0 ) 1-2+α-1 2 M 1 ( 2+α-1 2 -1) . ( 6
)
The remaining of the corrected proof is very similar to the original one. The dynamics of ē is given by ė = θ Aē -F S -1 ∞ (1)C T , ē -ρS -1 ∞ (1)C T C ē +∆ θ D(x, x, u).

One has

Vα (ē) = θ W1 + W2

with W1 = ∇ Vα (ē), Aē-F (S -1 ∞ (1)C T , ē)-ρS -1 ∞ (1)C T C ē and W2 = ∇ Vα (ē), ∆ θ D(x, x, u) . Following the same lines as in [START_REF] Ménard | A global high-gain finite-time observer[END_REF], one can show that there exists θ 2 ≥ 0 such that for every θ ≥ θ 2 , there exists > 0 such that for all α ∈]1 -, 1[ inequality (5) holds true.

  (e, t, α) ∈ {e ∈ R n , Vα (e) = 1} × {t ∈ [l, L]}×]1 -1