(Generation and) Recognition of Digital Planes using Multidimensional Continued Fractions

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(Generation and) Recognition of Digital Planes using Multidimensional Continued Fractions

Thomas Fernique

LIRMM – Univ. Montpellier 2

Lyon, April 16, 2008

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Introduction

Question: planarity of a given discrete object?

Many approaches already exist.

Here: generalization of Wu-Troesh algorithm for discrete lines.

Principe: local planarity checking and “backwards zooms”.

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Stepped planes Local properties Recoding A hybrid algorithm

1 Stepped planes

2 Local properties

3 Recoding (Backwards zooms)

4 A hybrid algorithm

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1 Stepped planes

2 Local properties

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Stepped planes Local properties Recoding A hybrid algorithm Face

Let (~e1, . . . , ~ed) be the canonical basis ofR^d. Faceof type i ∈ {1, . . . ,d}located at~x ∈Z^d:

e3

e1

e2 x

e3

e1

e2 x

e3

e1

e2 x

Discrete objectshere considered: unions of such faces.

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Stepped planes Local properties Recoding A hybrid algorithm Plane

Stepped planeP_~_α,ρ: boundary of the union of unit cubes ofZ^d intersecting the real half-space{~x | h~x|~αi ≤ρ}.

Vertices ofP_~_α,ρ: discrete standard planeof parameters (~α, ρ).

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Stepped planes Local properties Recoding A hybrid algorithm Patch

Patchof a stepped plane: union of faces included in this plane.

Let us stress that stepped planes can share patches.

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Stepped planes Local properties Recoding A hybrid algorithm Parameters

More generally, given a union of facesB,P(B) denotes the set of parameters of stepped planesB is a patch of:

P(B) ={(~α, ρ)∈R^d+\{~0} ×R | B ⊂ P_~_α,ρ}.

We speak aboutadmissibles parametersof B. It is a convex polytope ofR^d+1, non-empty iff Bis a patch of stepped plane.

Recognition of B: computation ofP(B).

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1 Stepped planes

2 Local properties

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Stepped planes Local properties Recoding A hybrid algorithm Principe

Stepped plane: constrained and regular object.

Looks planar.

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Stepped plane: constrained and regular object.

Does not look planar.

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Stepped planes Local properties Recoding A hybrid algorithm Run

Definition (run)

An (i,j)-run of a union of faces Bis a maximal sequence of faces of typei, aligned in the direction~e_j and included in B.

(1,3)-runs of length 2 and 3.

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Proposition (Berth´e-F. 2007)

(i,j)-runs of stepped planeP_~_α,ρ have length bα_i/α_jc or dα_i/α_je.

If it is a stepped plane with normal~α, then 2< α₁/α₃ <3.

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Proposition (Berth´e-F. 2007)

(i,j)-runs of stepped planeP_~_α,ρ have length bα_i/α_jc or dα_i/α_je.

It is not a stepped plane (incompatible lengths of runs).

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Stepped planes Local properties Recoding A hybrid algorithm Recognizability

Planes and patches do not always have identical runs:

?

Weak conditions info. on the (eventual) normal vector. We speak about a recognizable union of faces.

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Stepped planes Local properties Recoding A hybrid algorithm Recognizability

Planes and patches do not always have identical runs:

Weak conditions info. on the (eventual) normal vector.

We speak about a recognizable union of faces.

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1 Stepped planes

2 Local properties

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Recoding: map ˜T on union of faces which satisfies:

B patch ofP iff ˜T(B) patch of ˜T(P);

P(B) can be (easily) deduced fromP( ˜T(B));

T˜(B) contains (much) lesser faces thanB.

Aim: reducing the recognition ofBto the one of ˜T(B).

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More precisely, recoding bydual maps(see abstract):

T˜(B) =E₁^∗(σB)(B).

Choice ofσB relies on the runs of B (if recognizable).

Intuition of the “whole” principe:

Compute the common part of multi-dimensional continued fraction expansion (Brun’s algorithm) of the normal vectors of stepped planesB is a patch of.

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More precisely, recoding bydual maps(see abstract):

T˜(B) =E₁^∗(σB)(B).

Choice ofσB relies on the runs of B (if recognizable).

Intuition of the “whole” principe:

Compute the common part of multi-dimensional continued fraction expansion (Brun’s algorithm) of the normal vectors of stepped planesB is a patch of.

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Stepped planes Local properties Recoding A hybrid algorithm Problems with boundaries

Problem: T˜ can be applied only with particular boundaries.

This condition generally does not hold.

Solution: use the following equivalence notion: B ∼ B⁰ ⇔ P(B) =P(B⁰). The equivalence classes are generally big.

allows to choose an equivalent patch with suitable boundaries. Note: each equivalence class has a structure of semi-lattice

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Solution: use the following equivalence notion:

B ∼ B⁰ ⇔ P(B) =P(B⁰).

The equivalence classes are generally big.

allows to choose an equivalent patch with suitable boundaries.

Note: each equivalence class has a structure of semi-lattice

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Solution: use the following equivalence notion:

B ∼ B⁰ ⇔ P(B) =P(B⁰).

The equivalence classes are generally big.

allows to choose an equivalent patch with suitable boundaries.

Note: each equivalence class has a structure of semi-lattice

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A stepped plane containing this has (1,3)-runs of length 2 and 3.

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We thus can extend (1,3)-runs of length less than 2,

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and “close” those of length 3.

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We can proceed similarly for (1,2)-runs,

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We can proceed similarly for (1,2)-runs,

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and for (3,2)-runs.

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We can then repeat these operations.

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If the initial number is finite, then the process converges.

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We could also remove ”deducible” faces. Thus, large choice.

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1 Stepped planes

2 Local properties

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Stepped planes Local properties Recoding A hybrid algorithm Pseudo-code

Algorithm:

1. whileB recognizable do

2. B ←˜ suitable equivalent of B;

3. B ←recoding of ˜B;

4. endwhile;

5. computeP(B) thanks to another algorithm;

6. deduce the parameters of the initialB;

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Stepped planes Local properties Recoding A hybrid algorithm Complexity

Each loop:

reading runs + choosing an equivalent + recoding: O(|B|).

Number of loops:

IfBis a patch of stepped plane of normalα~ = (p₁/q, . . . ,p_d/q):

logd+2

d+1(p1+. . .+p_d+q).

Otherwise, letD be the side of a bounding box for B. Then, either Bis a patch of a stepped plane of normal ~α= (p1/q, . . . ,pd/q) withp_i,q ≤D, or it is not planar. Hence the general bound:

log^d+2

d+1((d+ 1)D).

Total complexity: quasi-linear ifD is polynomial in |B|.

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Each loop:

Number of loops:

logd+2

d+1(p1+. . .+p_d+q).

log^d+2

d+1((d+ 1)D).

Total complexity: quasi-linear ifD is polynomial in |B|.

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Each loop:

Number of loops:

logd+2

d+1(p1+. . .+p_d+q).

log^d+2

d+1((d+ 1)D).

Total complexity: quasi-linear ifD is polynomial in|B|.

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Stepped planes Local properties Recoding A hybrid algorithm Example

Union of 407 faces: is it a patch of stepped plane?

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Recognizable union computation of a suitable equivalent.

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Recoding union of 216 faces. Is it a patch of stepped plane?

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Unrecognizable union use another algorithm.

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Conclusion

Advantages of recognition by recoding:

new generalization of a result for discrete lines;

good theoretical complexity;

allows partial recognition (convergence of continued fractions);

“backwards” recoding generation (see abstract).

Open problems:

practical complexity?

robustness?