GEncode: Geometry–driven compression in arbitrary dimension and co–dimension

THOMAS LEWINER1,2 , MARCOS CRAIZER1 , HELIO LOPES1 , SINESIO PESCO1, LUIZ VELHO3 ANDESDRAS MEDEIROS3

1 Department of Mathematics — Pontifıcia Universidade Catolica — Rio de Janeiro — Brazil2 Geometrica Project — INRIA – Sophia Antipolis — France

3 Visgraf Project — IMPA — Rio de Janeiro — Brazil{tomlew, craizer, lopes, sinesio}@mat.puc--rio.br. {lvelho, esdras}@visgraf.impa.br.

Abstract. Among the mesh compression algorithms, different schemes compress better specific categories ofmodel. In particular, geometry–driven approaches have shown outstanding performances on isosurfaces. It wouldbe expected these algorithm to also encode well meshes reconstructed from the geometry, or optimized by ageometric re–meshing. GEncode is a new single–rate compression scheme that compresses the connectivity ofthese meshes at almost zero–cost. It improves existing geometry–driven schemes for general meshes on bothgeometry and connectivity compression. This scheme extends naturally to meshes of arbitrary dimensions inarbitrary ambient space, and deals gracefully with non–manifold meshes. Compression results for surfaces arecompetitive with existing schemes.Keywords: Geometric compression. Mesh compression. Arbitrary dimensional meshes.

1(a): circumradius 1(b): quantized radius 1(c): # candidates 1(d): candidate position

Figure 1: Compression of a scanned mechanical piece: once the geometry is decoded, the decoder attaches a triangle e ] w to theactive border edge e. w is identified by the circumradius ρ(e ] w) of the original triangle which is divided by the length of e to becompact its histogram, which reduces the entropy. The position of w in the list of vertices with the same quantized radius is thenencoded.

1 IntroductionThe recent developments in Computer Graphics require

an increasing amount of processing time and generate big-ger and bigger meshes. This development has been fol-lowed by compression algorithms which improved on thecompression ratio, the range of models that can be encoded,the ease of use and the algorithm performances. However,they are still not fully adapted to the wide variety of mod-els and applications of Computer Graphics: scans in artistic

Preprint MAT. 14/05, communicated on May 15th, 2005 to the Depart-ment of Mathematics, Pontifıcia Universidade Catolica — Rio de Janeiro,Brazil. The corresponding work was published in the proceedings of theSibgrapi 2005, IEEE Press, 2005.

and archaeological modelling, isosurfaces for medical andmathematical visualization, re–meshed models for reverseengineering, finite–element meshes for simulation, high–dimensional meshes for solid representation, meshes withhigh co–dimension for non–linear optimization. . . The per-formances of each compression algorithm highly dependon the nature of the model. Thus, specific compressionschemes should correspond to specific applications, in par-ticular for the most time–consuming mesh generation algo-rithm: reconstruction and the re–meshing.

Related works. The first mesh-compression algorithmswere connectivity–driven, in the sense that the geometry en-

T. Lewiner, M. Craizer, H. Lopes, S. Pesco, L. Velho and E. Medeiros 2

Figure 2: 3 ways of attachinga simplex to a triangulation.

boundary

edges1–simplex

0–simplex

2–simplex

Figure 3: A 2–manifold simplicial com-plex.

non–manifold

vertex (pinch)non–pure edge

Figure 4: A non–pure, non–manifoldcomplex.

coding depends on the connectivity encoding rules.Amongthose, the Edgebreaker [24, 21, 18, 16] performs well ongeneric mesh, with guaranteed practical worst–case closeto the theoretical optimum [14]. On the other side, Va-lence Coding [26, 7, 13] performs very well on mesheswith a regular connectivity and has been widely extendedsince the original work. It has a theoretical asymptoticcompression rate close to the optimum [1], but performsvery well in practice. Some singularities of the mesh canfurther be handled by specific algorithm, in particularfor the non–manifold case [10, 22]. These connectivity–driven approaches can be extended to higher dimension, butthe complexity of the codes increases dramatically. Evenfor tetrahedral meshes, the extensions of the surface ap-proaches [11, 23, 12] are quite complex.

On the contrary, geometry–driven approaches [9] handlegracefully complex connectivity. Still, the compression ra-tios of the geometry are not yet optimal, as these schemesare quite new to the community. However, for the case ofisosurfaces, specific compression schemes [25, 15, 17] out-perform any connectivity–driven approach.

Contribution. In the case of scanned models, the recon-struction algorithm computes the connectivity of the sur-face only from the geometry. Therefore, a geometry–drivenapproach for these cases should not waste any byte in con-nectivity. This is the goal of this work, which proposesa new geometry–driven approach of meshes named GEn-code. GEncode actually encodes a mesh based on a localgeometric criterion (Figure 1), and if the encoded modelcan be reconstructed with this criterion, GEncode willcompress its connectivity without wasting any byte in thecompressed stream. The geometry is encoded with an im-proved method on binary space partitions, which takes thebest parts of [9] and [6]. This scheme is very general, anddeals with simplicial complexes of arbitrary dimension inany dimensional space, even if the complex is non–pure,non–manifold or non–orientable. This information is onlyused by the coder and the decoder to optimize the algo-rithm and the compression ratio. For surface, the resulting

compression ratios are competitive with the Edgebreakerwith the parallelogram prediction.

Overview. This work is organized as follow. Section 2 Ba-sic concepts recalls the basic notions of triangulations, ex-pressed in arbitrary dimension as our scheme works at thatlevel of generality. Then section 3 GEncode describes theproposed scheme and its extensions. Finally, section 4 Re-sults and comparisons shows compression results on commonsurface models, and compares them with the Edgebreaker.

2 Basic conceptsThis section introduces the basic definitions that are

used in this work, especially the notion of triangulations(or simplicial complexes) [3]. Among these, the class ofsimplicial manifolds is the most widely used. They canbe constructed in an incremental manner (usually calledadvancing front triangulation), which entails most of themesh decoding algorithms.

Triangulations. A triangle, or 2–simplex, is the convexhull of 3 affine independent points. More generally, an n–simplex σ is the convex hull of (n + 1) affine independentpoints in the space, called the vertices of σ. A sub–simplexτ of σ is the convex hull of a proper subset of its vertices.We will say that σ is incident to τ and τ is bounding σ.

A complex is a coherent collection of simplices, wherecoherence means that the collection contains the sub–simplices of each simplex and the intersection of any twosimplices. A complex K is pure of dimension d if everysimplex in K is of dimension d or is a sub–simplex of asimplex of dimension d belonging to K. The geometry ofa complex usually refers to the coordinates of its vertices(0–simplex), while its connectivity refers to the incidenceof higher simplices on these vertices. A simplex σ can beattached to a complex K by identifying a collection of sub–simplices of σ with some of the simplices of K (Figure 2).Such operation can alter the topology of the complex, andits manifoldness.

Manifolds. A simplicial d–manifold M is a pure simpli-cial complex of dimension d where for each vertex v, the

The corresponding work was published in the proceedings of the Sibgrapi 2005, IEEE Press, 2005.

3 GEncode: Geometry–driven compression in arbitrary dimension and co–dimension

Figure 5: Encoding of a tetrahedral mesh in R3 (left) and R3 (left). The 1986 vertices of the solid sphere (left, courtesy of PierreAlliez) were compressed by [9] with 3433 bytes, by [6] with 3877 bytes and by our method with 3429 bytes. The 192 vertices of theCartesian product of a sphere and a circle , (right, courtesy of Helio Lopes) were compressed by [9] with 649 bytes, by [6] with 980bytes and by our method with 646 bytes. The coloured lines represent part of the traversal of the mesh by the connectivity encoder.

union of each open simplex containing v is homeomorphicto an open d–ball B or the intersection of B with a closedhalf–space. This implies that each (d−1)–simplex boundseither one or two d–simplices. The set of (d−1)–simplicesbounding only one d–simplex is called the border of M .

For example, a 2–complex is a surface (i.e. 2–manifold)if it has only vertices, edges and faces (0– 1– and 2–simplices respectively) ; if each edge bounds either one ortwo triangles ; and if the border does not pinch. Figure 3shows an example of 2–manifold and Figure 4 illustrates a2–complex that is neither pure nor a manifold.

Advancing front triangulations. Any triangulation canbe constructed by attaching a sequence of simplices toan initial vertex. This property is intensively used in sur-face reconstruction with advancing front triangulation al-gorithms [19]. The problem here is close to the geometry–driven approach to mesh compression: once the vertices aretransmitted, the algorithm must create a complex with thegiven vertices. For reconstruction, the final complex mustapproximate the original model, without additional infor-mation. For compression, the decoder uses the coded infor-mation to exactly match original model.

An example of reconstruction algorithm that we willuse to illustrate our algorithm is the Ball Pivoting Algo-rithm [4]. Having the position of each vertex and a givenradius α, the algorithm reconstructs the surface by succes-sively attaching one triangle t = e]w to a border edge e ofthe complex being constructed. The vertex w is chosen tominimize the circumradius G(e, w) = ρ(t) and preserve themanifoldness of the complex. If t has circumradius greaterthan α, e is left as a border edge. Actually any local geom-etry criterion G(e, w) can be used in order to choose w.

3 GEncodeThe proposed encoding scheme works similarly to re-

construction algorithms, although it encodes continuously

the differences of the original and reconstructed mesh. Itfirst encodes the geometry of each vertex, in any ambientspace, using a simple k–d tree coding that is an improve-ment of both [9, 6] (Figure 5). The algorithm encodes aninitial simplex and then works as an advancing front trian-gulation, attaching at each step a simplex e ] w to a bordersimplex e. The difference is that w is not always the one thatminimizes the geometric criterion G(e, w), but is encodedby its position in a list of candidates. These candidates arechosen from an encoded bound on G(e, w). To minimizethe entropy, that list is ordered by the geometric criterion.

(a) Geometry encoding.

For the geometric encoding, we use a simple tree cod-ing technique. This techniques works for vertices with anarbitrary number of coordinates, allowing encoding meshesof arbitrary co–dimension. As in [9] and [6], the space isdivided with a particular binary space partition where eachseparator is perpendicular to the axis (i.e. a k–d tree): theaxis alternates from one level to the other (X,Y,Z,X,Y. . . inR3), and the separator is always positioned at the middle ofthe node (Figure 6, Figure 6 and Figure 6). The subdivisionis performed until each node contains only one vertex.

5

5

⇒ 0

4⇒ 1

2

⇒ 2

1 1

⇒ 1⇒ 1

Figure 6: Geometry encoding of [9]: the encoder encodes 5 on32 bits, then 5 on dlog2(6)e bits, then 4 on dlog2(6)e bits. Theright vertex position is then encoded. Then 2 is encoded ondlog2(5)e bits, 1 on 1 bit and 1 bit again. Then each remainingvertex is encoded.

In [9], each node of the binary space partition is encodedby the number of vertices #vl its left children contains. Thenumber of vertices of the other node #vr is simply deduced

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T. Lewiner, M. Craizer, H. Lopes, S. Pesco, L. Velho and E. Medeiros 4

by difference from the number of the known vertices of theparent’s node #vf . This technique wastes many bits at thebeginning of the encoding, as the number of nodes must beencoded on dlog2(#vf + 1)e bits (Figure 6). When there isonly one vertex per node, is is encoded with 1 bit per level,which is optimal.

+

0

+ ++

+

+ +

++ ++

0

0

Figure 7: Geometry encoding of [6]: the encoder encodes thefollowing sequence: +0, ++, ++, 0+, ++, ++, 0+. Thenfollow 0+ and +0 to reach the the desired number of bits.

[6] encodes each node by one of 3 symbols: ++ ifboth children contain at least one vertex, +0 if only theleft child contains a vertex, and 0+ if only the right onecontains a vertex (Figure 7). Note that at least one childmust contain a vertex, as the parent did. The encoding stopsat a predefined level. This method spends more bits at theend of the encoding, since the decoder doesn’t know whenthere is only one vertex in a node. Therefore, the encodersends log2(3) bit for each level, which is greater than the 1bit of [9] for the last part.

+

0

+ 1+

+

1 1

11

Figure 8: Our geometry encoding: the encoder encodes the fol-lowing sequence: +0, +1 ant the right vertex is fully encoded.Then ++, 11, 11. The position of each remaining vertex is thenencoded.

Our technique takes the best part of both. First, it en-codes each node by one of 6 symbols: ++ if both childrencontains more than one vertex, +1 and 1+ if one child con-tains more than one vertex, and the other only one, 11 ifthey both contain only one vertex, and +0 and 0+ if onechild contains more than one vertex, and the other child isempty (Figure 8). With this encoding, the encoder detectswhen there is only one vertex in a node, and then uses thetechnique of [9]. The symbols do not have the same proba-bility and the coder takes that into account. Moreover, theseprobabilities are used differently depending on the level ofthe node to encode: nodes closer to the root are more fre-quently of type ++, whereas those are rare when goingcloser to the leaves.

(b) Connectivity encoding for surfaces.

Our scheme decodes the connectivity in an advancingfront manner, attaching at each step a triangle e ] w toan edge e of the active border, or by removing e from

the active border. More precisely, once the first triangle ofthe connected component is decoded, its three boundingedges are added to the active border. Then, one edge ofthe active border is selected, for example the longest one(as described below). The decoder then receives from theencoder two numbers to choose w: the quantized radius andthe candidate position. The quantized radius will serve tobuild up a list of candidates, or to indicate that e was onthe border of the encoded mesh in which case e is removedfrom the active border. Then the candidate w is identifiedby its position inside the candidates list. The decoder thenattaches the triangle e ] w and adds its edges to the activeborder (Algorithms.1 and 2).

Algorithm 1 GEncode1: priority queue q // q ordered by edge length2: q.enqueue( edges of the first triangle )3: while NOT q.empty() do // main loop4: e := q.top() // the longest edge of the border5: w such that e]w is a triangle and NOT w.encoded()6: r := G(e, w) // ρ(e ] w)/length(e)7: [Gmin,Gmax] := quantize(r) // quantization interval8: encode Gmax // encodes the quantized radius9: candidates := select( bsp, e, [Gmin,Gmax] ) // looks

in the binary space partition for candidates10: sort (candidates, G ) // optimal G(e ] w) first11: encode( position of w in candidates ) // candidate #12: q.enqueue( edges of e ] w 6= e ) // front update13: end while

Algorithm 2 GDecode1: priority queue q // q ordered by edge length2: q.enqueue( edges of the first triangle )3: while NOT q.empty() do // main loop4: e := q.top() // the longest edge of the border5: decode Gmax // decodes the quantized radius6: candidates := select( bsp, e, [Gmin,Gmax] ) // looks

in the binary space partition for candidates7: sort (candidates, G ) // optimal G(e ] w) first8: w = decode( position in candidates ) // candidate #9: add triangle e ] w // advancing front

10: q.enqueue( edges of e ] w 6= e ) // front update11: end while

Quantized radius. The decoder has to choose which ver-tex w of the encoded mesh was adjacent to e. This informa-tion is encoded by the index of w inside a list of candidates.This list could be the list of all vertices, but that would betoo expensive. In order to minimize the number of elementsof the candidate list and the time used to create that list, thegeometric criterion G(e, w) of w is quantized and encoded.

The corresponding work was published in the proceedings of the Sibgrapi 2005, IEEE Press, 2005.

5 GEncode: Geometry–driven compression in arbitrary dimension and co–dimension

9(a): # candidates 9(b): candidate position

Figure 9: The quantization of the geometrical criterion affectsthe entropy of the candidate position.

If the edge e was on the border of the original mesh, a spe-cific symbol is encoded. In the implementation, the binaryspace partition of the geometric encoding is used to accel-erate the search for candidates.

candidate #3

would create a non–

manifold

vertices out of range

candidate #1

candidate #2

G(e, w)e

w

Figure 10: Candidate selection from a given geometric crite-rion (here the circumradius).

Candidate position. The quantized radius is just an inter-val [Gmin,Gmax] that contains G(e, w). The decoder thenenumerates all the candidates vi /∈ e that could fit in this in-terval : G(e, vi) ∈ [Gmin,Gmax], and order this list of can-didates by G: G(e, vi) < G(e, vi+1) (Figure 10). For typicalmeshes, this ordering lowers drastically the entropy.

Manifoldness restriction. If we know the original meshwas a manifold, we can remove from the list the candidatesthat would create a non–manifold triangulation, i.e. thevertices that are not in the active border. Moreover, oncean edge of the active border has been processed, it cansafely be removed from the active border, as it has eitherbeen marked as a border edge or is already bounding twotriangles.

Quantization for the Ball Pivoting criterion. We wouldlike our encoding algorithm to compress better the outputof reconstruction algorithms. For example, each triangleof a mesh reconstructed by the Ball Pivoting Algorithmminimizes the geometric criterion G(e, w) = ρ(e]w). Theradius can then be quantized in a more efficient manner byconsidering G(e, w) = ρ(e]w)/length(e) (Figure 11). Asρ is the circumradius, ρ(e ] w) > 1

2 length(e). Moreover,with this quantization, the candidates can be searched onlyin a ball centred of radius

√3 · Gmax at the midpoint of e.

The radius is then quantized in an exponential manner, witha parameter k that, experimentally, was optimum around 7:

q(e, w) =⌈log2

(k ρ(e ] w)length(e)

− k

2)⌉

, k = 7.

Figure 12: The circumradius encodgin can be expensive formeshes generated with a criterion different from G(e, w).

This quantization is actually a trade–off between an ex-pensive coding of the radius, which could identify com-pletely the vertex and reduce the candidate position (Fig-ure 9), and no quantization of the radius which will slowthe coders and increase the entropy of the candidate posi-tion. Note that if the mesh was reconstructed by the BallPivoting Algorithm, the quantized radius takes a uniquevalue, and the right candidate is always the first one. Theconnectivity is thus compressed with 0 byte.

Figure 13: Traversal of a sphere and of a Klein bottle models,from cold to hot colours: good orders can improve the com-pression.

Traversal strategy. These kinds of optimizations actuallydepend of which edge of the active border is chosen at

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T. Lewiner, M. Craizer, H. Lopes, S. Pesco, L. Velho and E. Medeiros 6

11(a): edge length 11(b): circumradius 11(c): quantized radius

Figure 11: Whereas the circumradius of a simple mesh has a high entropy (a), its circumradius divided by the edge length (b)has a very low entropy and can be efficiently quantized (c). However, when some of the triangles an edge to a distant vertex, as inFigure 12, their quantization become expensive.

each step. In particular for the Ball Pivoting case, the radiuswill be better quantized if the length of the active edge isbigger, as it optimizes the entropy of G(e, w) = ρ(e ]w)/length(e) (Figure 13).

(c) GEncoding in higher dimension

The algorithm we described here actually works for sim-plicial manifolds of any dimension d, and the generality ofthe geometry encoder allows working in any co–dimension.The algorithm is exactly the same: a first s–simplex is en-coded, and all its bounding (d−1)–simplices are enqueuedin the active border priority queue.

For the Restricted Ball Pivoting case [20], this queue issorted by the volume of the (d−1)–simplex. For each ac-tive (d−1)–simplex τ , the radius of the uncoded incidentd–simplex σ = τ ] w divided by the volume of τ is quan-tized and coded. In dimension greater than 3, the volumeof a simplex can be computed using the Cayley–Mengerdeterminant [5]. This determinant can be used to computethe circumradius of the simplex [5], but the computation inhigh dimensions is expensive and our implementation sub-stitutes it for dimension greater than 4 by the volume.

Based on that radius, the coder builds up a list of can-didates, using the binary space partition to optimize thesearch. The position of w in that list is then coded. When themesh is known to be a manifold, vertices that are not partof the active border are not considered as valid candidates,and the (d−1)–simplices are considered only once.

(d) GEncoding non–manifold objects

The encoding of non–manifold triangulations (as on Fig-ure 1) which are pure complex functions exactly as de-scribed in the previous section, without the manifoldnessrestrictions. The special symbol for border (d−1)–simplicesat the radius quantization step is used to tell the decoder thatthe active (d−1)–simplex has no more incident d–simplex.

For non–pure triangulations, i.e. simplicial complexes ofdimension d ≥ 2 having non–pure simplices of dimensionk < d not bounding any simplex, the situation is quite moredelicate. We encode first the pure part Kd of the complex,i.e. all the simplices of dimension d, with the above tech-nique. Then, we consider all the non–pure simplices of di-mension d−1 as a new non–manifold triangulation to en-code, considering the border of Kd as the active border.This procedure then repeats for dimension d−1, until reach-ing dimension 1.

4 Results and comparisons

Geometry Connectivityanimal 19.343 1.980art 19.561 1.491cad 18.566 1.682math 21.499 1.996medical 21.220 2.411scans 18.639 1.372original 19.334 2.246re–meshed 18.882 1.269all 19.089 1.717

Table 1: GEncode compression ratio, in bits per vertex. Theseresults are an average over 200 models, grouped by category.

GEncode intends to compress better meshes that have anice geometry. In the case of the Ball Pivoting criterion, thismeans that the simplices should be as equilateral as pos-sible, or at least a sub–triangulation of a Delaunay poly-hedron. This is the case of reconstruction algorithm [4]or some techniques of re–meshing [2]. The results of Ta-ble 1 shows this behaviour stands in practice, as the re–meshed models and the scans sculptures are better com-pressed than the other models (Figure 14). Although most

The corresponding work was published in the proceedings of the Sibgrapi 2005, IEEE Press, 2005.

7 GEncode: Geometry–driven compression in arbitrary dimension and co–dimension

of the results presented here are surfaces, the algorithm hasbeen implemented for any dimension and co–dimension.The only two features that were not implemented in highdimension are the non–pure compression and the manifold-ness recognition (which is a NP–hard problem). We did nothave enough geometrical models in high dimension to makea valid benchmark.

The compressor compares nicely to existing compres-sion scheme, although it is able to compress a wider rangeof models. Table 2 details some comparisons with the Edge-breaker algorithm [21, 18, 16] for the meshes illustratingthis work. These comparisons were made using the par-allelogram prediction for the Edgebreaker, with the samequantization for the vertices (12 bits per coordinate). Weare aware that many techniques improved on this parallel-ogram prediction [8]. As the difference is tight, the per-formance of connectivity–driven algorithm could best thiscurrent version of GEncode for surfaces. However, we be-lieve that there are many open directions to improve thisnew scheme that will maintain it competitive.

5 Future worksThis work provides a geometry–driven compression

scheme that compresses efficiently meshes resulting fromexpensive processing, in particular scanned and re–meshedmodels. This scheme actually encodes also nicely moregeneral models, and compares nicely to the existing com-pression schemes for the case of surfaces. Moreover, it gen-eralizes easily to arbitrary dimension and arbitrary ambientspace dimension, and to singular topological models.

The geometrical criterion of the algorithm is fundamen-tal for the compression and we can certainly improve it orcombine it with other criteria. For example for the CADmodels (Figure 15), the edge length could serve as a switchbetween criteria (Figure 16), which allows using differentones jointly. This work extends in a straightforward man-ner for non–simplicial meshes with convex cells. But weplan to extend it first to progressive and multi–resolution

Figure 14: Candidate position for scanned models: the connec-tivity is encoded almost at zero rate: # is not transmitted, andthus almost only 0 codes are encoded.

Figure 15: Number of candidates: with the Ball Pivoting crite-rion, CAD models are compressed faster if re–meshed.

meshes, not directly based on the k–d tree hierarchy, as ex-isting methods gave poor visual results [9, 15].

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Preprint MAT. 14/05, communicated on May 15th, 2005 to the Department of Mathematics, Pontifıcia Universidade Catolica — Rio de Janeiro, Brazil.

T. Lewiner, M. Craizer, H. Lopes, S. Pesco, L. Velho and E. Medeiros 8

#vertices EB Geometry EB Connectivity GEncode Geometry GEncode Connectivityterrain 16641 17.093 0.282 13.505 3.557mechanical 71150 ∗ ∗ 15.860 0.822david 24988 25.800 2.707 16.985 1.631horse 19851 24.856 3.012 18.216 1.193gargoyle 30059 20.992 2.316 18.379 0.894bunny 34834 17.050 2.178 18.767 1.182blech 4102 21.552 2.102 20.721 4.550fandisk 6475 19.555 2.254 21.519 2.630klein 4120 ∗ ∗ 22.110 3.436sphere 642 27.975 2.268 24.685 0.051rotor 600 31.667 3.693 24.078 6.438

Table 2: GEncode compression ratio compared with Edgebreaker. The mechanical piece is not a manifold object, and the Kleinbottle is not orientable, which prevented the Edgebreaker to work.

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