Vol. 3, No. 4, 2014
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Journal of Computer Graphics Techniques

A Simple Method for Correcting
Facet Orientations in Polygon Meshes
Based on Ray Casting
Kenshi Takayama

Alec Jacobson

Department of Computer Science
ETH Zurich

Department of Computer Science
Columbia University

Ladislav Kavan

Olga Sorkine-Hornung

Department of Computer and Information Science
University of Pennsylvania

Department of Computer Science
ETH Zurich

Abstract
We present a method for fixing incorrect orientations of facets in an input polygon mesh, aproblem often seen in popular 3D model repositories, such that the front side of facets is visible from viewpoints outside of a solid shape represented or implied by the mesh. As opposed
to previously proposed methods which are rather complex and hard to reproduce, our method
is very simple, only requiring sampling visibilities by shooting many rays. We also propose
a simple heuristic to handle interior facets that are invisible from exterior viewpoints. Our
method is evaluated extensively with the SHREC’10 Generic 3D Warehouse dataset containing 3168 manually designed meshes, and is demonstrated to be very effective.

1.

Introduction

Polygon meshes are fundamental to computer graphics and other areas. A polygon
mesh generally consists of a set of 3D vertex positions and a set of polygonal facets
each defined by an ordered list of vertex indices. The ordering of vertices of a facet determines its orientation; i.e., by convention, if we see the ordering of vertices of a facet
as counterclockwise from a particular viewpoint, we consider the facet to be showing
its front side. Having “correct” facet orientations is essential for various applications
such as back face culling, rendering with two-sided colors, inside-outside segmentation [Jacobson et al. 2013], and collision response [Harmon et al. 2008]. Here, the“correctness” of facet orientations is defined as follows: a facet’s orientation is correct
if its front side corresponds to the “outside” of a solid object represented or impliedby the mesh. This definition is not applicable to a mesh which does not represent1Vol. 3, No. 4, 2014
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Figure 1.

A few examples of meshes from the SHREC’10 Generic 3D Warehousedataset [Vanamali et al. 2010] with incorrect facet orientations (top row), and the results of
our automatic correction method (bottom row).

or imply a solid object, a common example of which is a “single-faceted” geometry
representing an infinitely thin part with only one sheet of facets, see Figure 7. Suchcases are excluded from the scope of this paper.
Unfortunately, it is still not uncommon today to find polygon meshes with incorrect facet orientations on the Internet. One of the main reasons is that many popularpolygon-based 3D modeling tools such as SketchUp (as opposed to other tools such
as ZBrush and CATIA which are primarily designed for creating solid shapes) use
double-sided lighting which ignores facet orientations, thus allowing the user to create polygon meshes with arbitrary facet orientations. Incorrect facet orientations may
also arise when compositing multiple meshes from different sources. The top row of
Figure 1 shows a few examples of such incorrect facet orientations found in the popular SHREC’10 Generic 3D Warehouse dataset [Vanamali et al. 2010]. In this paper,
we propose an automatic method for correcting such facet orientations as shown inthe bottom row of Figure 1.
1.1.

Previous methods

There exist three methods in the literature for correcting facet orientations in polygon
meshes. Murali and Funkhouser’s method [1997] computes a discrete harmonic
function over a tessellation of the 3D space (either via Binary Space Partitioning or
tetrahedralization) whose sign encodes whether a given 3D position is inside (positive) or outside (negative) of a solid shape implied by the input polygon mesh. Eachfacet can be reoriented by integrating the function values over its area on its front andback sides separately and comparing the results. Because the function values on thetwo sides of every facet are constrained to have the same absolute value but opposite

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+1

−1

(a)

(b)

(c)

(d)

Figure 2. An example in 2D (representing a pants with an accessory) where Murali andFunkhouser’s method [1997] fails because of overlapping facets: (a) input facets, (b) visualization of the harmonic function, (c) binary coloring of the function, (d) result of orienting
each facet based on the sum of the function values on its two sides.

signs, the function value vanishes where the input mesh has open boundaries or intersections, leading to unconfident indication of insideness or outsideness. In addition,
this method assumes that every facet lies on the boundary between inside and outsideof the implied solid shape, but such an assumption does not hold for meshes that are
not meant to imply a solid shape. Even if the mesh does represent a solid shape, theassumption is wrong if there are some overlapping facets, often leading to unintended
facet orientations (Fig. 2). Additionally, robustly tessellating the 3D space while conforming to an arbitrary polygon mesh is far from trivial, which is another drawbackof this method.
Borodin et al.’s method [2004] and Zhou et al.’s method [2008] both sample
visibilities of facets by shooting many rays from outside the bounding box of the input
mesh to get an “initial guess” of facet orientations, followed by subsequent steps of
greedy aggregation (Borodin et al.) and graph-cut (Zhou et al.) to obtain final facet
orientations. However, it is unclear to what degree these subsequent steps improve
facet orientations computed in the visibility sampling step. In fact, Borodin et al.
claim in their Figure 9 that visibility sampling alone is insufficient and problematic,
which we question because visibility sampling should orient facets such that as few
as possible facets show their back sides to exterior viewpoints. Reproducing andanalyzing their results seem difficult, since quite a few hand-tuned parameters are
involved in these methods, and their reference implementations and datasets are notreadily available.
In this paper, we present a very simple method based solely on visibility samplingfor correcting facets in polygon meshes. We evaluate our method with the SHREC’10
Generic 3D Warehouse dataset and show that our method works sufficiently well for
all the cases where the problem is well defined.

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2.
2.1.

MethodVisibility sampling

We follow Borodin et al.’s strategy for visibility sampling (described in Section 3.4.1of their paper). The basic idea is to orient each facet such that its front side is more visible from outside, which is determined by shooting many rays from the facet. Specifically, for each facet, a large number of points proportional to the facet’s area are
randomly sampled on the facet as the ray origins, and for each ray origin, a ray direction is randomly sampled. Our modification to Borodin et al.’s method is that foreach ray origin and direction, we shoot two rays in opposite directions to ensure thatthe same number of rays are shot for both sides of the facet. If a ray shot on thefacet’s front (resp. back) side does not intersect with any other facets in the mesh, thecorresponding counter cfront (resp. cback ) is incremented. After shooting all the rays,
the facet is flipped if cfront < cback .
The above method cannot make any decisions for facets thatare completely occluded and invisible from outside (often representing some internal
structures, see Figure 8) because cfront and cback are both 0. We propose a simple
heuristic to handle such cases, where the basic idea is to orient an interior facet suchthat the front side has more “free” space than the back side. Specifically, in the rayshooting process as explained above, if a ray shot on the facet’s front (resp. back)side intersects with some other facet, the distance between the ray origin and theintersecting point is accumulated to the corresponding counter dfront (resp. dback ).
After shooting all the rays, the facet is flipped if
Handling interior facets.

(cfront < cback ) or ((cfront = cback ) and (dfront < dback )).
2.2.

(1)

Ray intersection parity sampling

If the input mesh implies a solid shape and it has a narrow cavity invisible from theoutside, the above heuristic method may orient the facet at the cavity in a way that
contradicts with the implied solid shape (Fig. 3b). For such cases, although they
seldom occur and are mostly of theoretical interest (see Section 3), we propose analternative method based on ray intersection parity (Fig. 3c), which is achieved witha slight modification to the above method. Specifically, the system employs the sameray shooting process as the above method, but this time, for each ray, it detects all
the intersections with all the facets (i.e., without stopping at the first intersection),and accumulates the number of intersections modulo 2 to counters pfront or pback corresponding to the ray direction. After shooting all the rays, the facet is flipped if
pfront > pback .
Because this method assumes that the input mesh implies a solid shape, it may
produce some unintuitive facet orientations if the assumption is violated, similar to4Vol. 3, No. 4, 2014
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+1

(a)

(b)

(c)

0

(d)

(e)

(f)

Figure 3. 2D examples demonstrating the effect of the alternative method based on ray in-tersection parity. (a, d) Input facets. (b, e) Result of the visibility sampling method with theheuristic to handle the interior facet. (c, f) Result of the alternative method based on ray intersection parity. The color map visualizes the generalized winding number function [Jacobson
et al. 2013] defined by the given facet orientations.

Murali and Funkhouser’s method [1997]. If the input mesh contains both overlapping facets and facets representing internal cavities invisible from outside, neither of
the above two methods produce facet orientations that we would expect (Fig. 3e-f).
However, such cases were never observed in the dataset we used in the evaluation.
3.

Evaluation

We used the SHREC’10 Generic 3D Warehouse dataset [Vanamali et al.
2010] to evaluate the performance of our method. This dataset was created by firstcollecting SketchUp (*.skp) files from the Google 3D Warehouse and then manuallyprocessing them (e.g., decomposing a scene with many related objects into individual
objects) so that the dataset is suitable for benchmarking shape retrieval algorithms. It
contains 3168 meshes, most of which are far from being clean and watertight: they
very often have various types of defects such as disconnected facets with randomorientations and self-intersecting/coplanar facets. We believe this dataset to be a fairly
good representation of meshes manually created by general users.
Note that meshes in the dataset often contain single-faceted facets representing
infinitely thin objects (e.g., fish fins) or facets invisible from outside representing interior objects (e.g., car seats). In such cases, the “correct” facet orientations can only
Dataset.

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Figure 4. A motorcycle model (left, named 2292.off in the dataset) whose backfacingness

is measured to be 0.2756 according to rendered images with two-sided coloring (right).

be decided by the authors of the meshes according to their intentions, which makes
it impossible to define the ground-truth facet orientations for the dataset. Therefore,
in the following, we do not discuss the performance of our method in terms of “accuracy”.
Nevertheless, we use a measure called backfacingness
which tells us how much of a given mesh’s facets are showing their back sides to
distant viewpoints, to measure the plausibility of the facet orientations at least whenviewed from outside. To compute this measure, we first render the mesh into a framebuffer with two-sided coloring and orthographic projection from a few view directions(Fig. 4). The measure is then obtained as the ratio of the number of pixels comingfrom the back side to the total number of drawn pixels. We found that the framebuffer
resolution of 10242 and 6 canonical view directions provide sufficiently good reliability in practice. Note again that this measure is irrelevant for single-faceted parts andinterior objects/cavities where neither of the two orientations is more plausible thanthe other in general.
Backfacingness measure.

We implemented a preprocessing step of extracting manifold patches
similar to Borodin et al.’s method (in Section 3.2 of their paper) where the basic idea
is to group facets connected by manifold edges (i.e., edges shared by exactly two
facets) into a manifold patch. However, we found that there are quite a few cases
where the extracted patch structures are rather unexpected because of the complicatedmesh connectivities designed by hand. Figure 5a shows an example where the facets
for the top cover and the bottom support of the piano model are grouped as patches
in an undesirable way. To circumvent this problem, we omit the patch extraction stepin a default setting and process individual facets independently (Fig. 5b). Note, however, that this setting can be inappropriate for meshes that represent thin parts withsingle-faceted facets; for such cases, omitting the patch extraction step may result insome neighboring facets oriented in opposite directions (Fig. 5d). Another prepro-

Preprocessing.

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(a)

(b)

(c)

(d)

Figure 5. Pros and cons of patch-wise and facet-wise processing. Patch IDs are visual-ized with different colors, and backfacing facets are rendered darker. In some cases, manifold
patches are extracted in an unexpected manner (a) because of the complicated mesh connectivities in the input meshes; in this case of the piano model (2371.off), there are non-manifold
edges in the base of the piano, and decomposing them into oriented manifold patches led to an
unexpected result with the top cover and the bottom support being grouped together, whereas
processing each facet independently is more appropriate (b). However, for single-faceted geometries whose front-side and back-side visibilities are mostly the same, such as the backseat of the chair model (938.off), extracting manifold patches and processing patch-wise ismore appropriate (c), otherwise the visibility-based decision for each facet can fluctuate (d).

cessing step we employ as a default setting is to eliminate facets with identical vertex
sets. Quite a few meshes in the dataset contain multiple facets that have the exact
same set of vertices (in the same or reversed orientation). Such facets can lead to thegeneration of numerous spurious patches, and may also confuse the ray intersectionengine.
We applied our method with the default settings (i.e., patch extraction being skipped, all facets having unique vertex sets) to all the 3168 meshes in the dataset.
The average and the standard deviation of the backfacingness measure are 0.19 and±0.26 for the original meshes, and 0.0026 and ±0.0063 for the output meshes, respectively. Figures 1 and 6 show some results where the output meshes are oriented
in a natural way. Figure 7 shows some output meshes with higher backfacingness
remaining. Note that all the facets showing their back sides belong to single-facetedgeometry representing very thin parts, and therefore there is no way to reduce thebackfacingness any further. Figure 8 shows some results containing interior objects
invisible from the outside, such as car seats. Note how our heuristic described in Section 2.1 works successfully for orienting interior objects. By manually and carefully
inspecting each mesh in the database, we confirmed that there are no cases where ouralternative method described in Section 2.2 is more suitable. Table 1 shows how
much backfacingness was reduced for each of these results.
Results.

7

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Figure 6. Results of our method. In each pair, the original facet orientations and the facet

orientations corrected by our method are shown on the left and on the right, respectively.

Figure 7. Examples where the backfacingness measures remain relatively high after process-ing, due to the single-faceted geometry.

Figure 8. Results with meshes containing interior objects. In each set, the original andthe processed facet orientations are shown on the left and right columns, respectively. The
top and bottom rows show the exterior views and the cut-away views exposing the interior,respectively.

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Fig

Model (ID)

Backfacingness

Fig

Model (ID)

Backfacingness

1

Bicycle (103)
Cell phone (835)Chair 1 (994)
Truck 1 (2983)

0.3743 → 0.0085
0.3000 → 0.0287
0.6344 → 0.0880
0.2458 → 0.0344

6

Woman (3120)

0.5343 → 0.0003

7

Bird (164)
Chair 2 (994)
Guitar (1437)
Lamp 1 (1990)
Truck 2 (2867)

0.5535 → 0.0073
0.6344 → 0.0880
0.5556 → 0.0009
0.5868 → 0.0273
0.5693 → 0.0050

Chair 3 (914)Fish (1335)
Keyboard (1635)
Lamp (1953)

0.5755 → 0.3743
0.1499 → 0.1499
0.4819 → 0.4811
0.4803 → 0.2477

8

Bus (619)
Parts (1065)
Train (2800)
Truck 2 (2867)

0.0544 → 0.0001
0.0000 → 0.0000
0.3002 → 0.0020
0.5693 → 0.0050

6

Table 1. The reduction of backfacingness for each of the results in the paper. The ID of amodel denotes its file name in the dataset (e.g., a model with ID 103 is named as 103.off).

Our method is implemented in C++ using libigl1 and Embree2 libraries. The source code is included in the supplementary material. Processing thelargest mesh in the dataset, 1938.off with 16K vertices and 92K triangles, using 3
million rays takes 10 seconds with Intel Core i7 2.6GHz CPU and 8GB of RAM.

Performance.

4.

Conclusion

We presented a very simple method for correcting facet orientations in polygon meshes
and demonstrated its effectiveness through an evaluation with a large public meshdataset. One possible direction for future work would be to design an efficient userinterface for expressing semantic information about facet orientations where automatic algorithms have little confidence because of ambiguities.
Acknowledgements

We are grateful to Peter Kaufmann, Wenzel Jakob, Nobuyuki Umetani, Melina Skouras, Ilya Baran and Ryan Schmidt for their advice and feedback. This work was
supported in part by the ERC grant iModel (StG-2012-306877), by an SNF award
200021_137879, by an NSF grant IIS-1350330 and by a gift from Adobe Research.
Kenshi Takayama’s work was funded in part by JSPS Postdoctoral Fellowships for
Research Abroad. Alec Jacobson’s work was supported in part by an Intel Doctoral
Fellowship.
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176. 3

Index of Supplemental Materials• code/: C++ implementation of our algorithm configured with CMake.
• data/: Resulting meshes whose facet orientations are corrected by our algorithm,
plus reports on backfacingness for all the input and output meshes.
• py/: A couple of small Python scripts to batch process meshes.

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Author Contact Information
Kenshi Takayama
Department of Computer Science
ETH Zurich
CAB G 88, Universitaetstrasse 68092 Zurich, Switzerland
kenshi84@gmail.com

Alec Jacobson
Department of Computer Science
Columbia University
Mail Code 0401
500 West 120 St, Room 450
New York, NY 10027
jacobson@cs.columbia.edu

Ladislav Kavan
Department of Computer andInformation Science
University of Pennsylvania
Levine Hall 303, 3330 Walnut Street
Philadelphia, PA 19104-3409
ladislav.kavan@gmail.com

Olga Sorkine-Hornung
Department of Computer Science
ETH Zurich
CNB G 106.2, Universitaetstrasse 68092 Zurich, Switzerlandsorkine@inf.ethz.ch

Kenshi Takayama and Alec Jacobson and Ladislav Kavan and Olga Sorkine-Hornung, , Journal of Computer Graphics Techniques (JCGT), vol. 3, no. 4, 1–1, 2014
http://jcgt.org/published/0003/04/02/
Received:
Recommended:
Published:

July 4, 2013
July 4, 2013
July 4, 2013

Corresponding Editor: Bernd Bickel
Editor-in-Chief:Morgan McGuire

c 2014 Kenshi Takayama and Alec Jacobson and Ladislav Kavan and Olga Sorkine-Hornung (theAuthors).
The Authors provide this document (the Work) under the Creative Commons CC BY-ND 3.0
license available online at http://creativecommons.org/licenses/by-nd/3.0/. The Authors further grant permission reuse of images and text from the first page of the Work, provided thatthe reuse is for the purpose of promoting and/or summarizing the Work in scholarly venues
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