Extracting Animated Meshes with Adaptive Motion Estimation
Nizam Anuar and Igor Guskov
University of Michigan, Ann Arbor, USA
Abstract pling from frame to frame. Reconstruction of ani-
mated mesh sequences from such data is an impor-
We present an approach for extracting coherently tant problem we aim to address in this paper.
sampled animated meshes from input sequences of
incoherently sampled meshes representing a con-
tinuously evolving shape. Our approach is based
on multiscale adaptive motion estimation proce-
dure followed by propagation of a template mesh
through time. An adaptive signed distance volumes
are used as the principal shape representation, and
a Bayesian optical ﬂow algorithm is adapted to the
surface setting with a modiﬁcation that diminishes
the interference between unrelated surface regions.
Additionally, a parametric smoothing step is em-
ployed to improve the sampling coherence of the
model. The result of the proposed procedure is a Figure 1: Comparison of trajectories for three meth-
single animated mesh. We apply our approach to ods of particle propagation for the jump sequence:
the human motion data. ﬁtting, temporal prediction followed by ﬁtting, our
estimated ﬂow followed by ﬁtting. The left two
propagations fail, the right one succeeds to preserve
1 Introduction the surface sampling.
Animated meshes are widely employed in char- We restrict our effort to the simpler scenario of
acter animation, visualization, and computational unchanging topology both in the input data and in
simulation applications. An animated mesh is a the desired output. A single mesh template of the
sequence of meshes with the same connectivity same connectivity is propagated through time and is
whose vertex positions change in time. This is a ﬁtted to the input surface data in every frame of the
convenient representation for dynamically chang- sequence. The main challenge is to establish corre-
ing shapes, with many processing, rendering, and spondence between consecutive shapes. Computing
compression tasks easily handled. For instance, such correspondence relation constitutes the main
modern compression methods can take advantage contribution of this paper. We assume that the in-
of the temporal coherence present in animated mesh put surfaces are closed and without boundaries; the
data, resulting in a very compact shape represen- ﬁrst step of the algorithm converts each input shape
tation [IR03][KG]. Unfortunately, several recently into an adaptive volumetric representation with a
introduced state of the art dynamic shape acqui- signed distance transform. Once a volumetric rep-
sition methods [SMP03][ZCS03] do not produce resentation is obtained, we run a Bayesian motion
their result in the animated mesh form; rather a se- estimation procedure similar to a differential optical
quence of meshes of varying connectivity is pro- ﬂow approach from image processing. The result-
duced. Volumetric morphing and isosurface extrac- ing vector ﬁeld deﬁned on the surface is used for the
tion are also examples of applications that produce initial propagation of the mesh template. The fur-
evolving surfaces that are not meshed consistently ther ﬁtting and parametric smoothing steps result in
in time, and have changing connectivity and sam- a temporally coherent animated mesh sequence.
VMV 2004 Stanford, USA, November 16–18, 2004
Our main contribution is to show that for contin- steps.
uously evolving surface data the propagation by an The estimation of the surface ﬂow has also been
adaptively estimated ﬂow ﬁeld can help to establish used as a part of non-rigid 3D scene reconstruction
temporally coherent meshing of the dynamic sur- in the shape and motion carving approach of Vedula
face. et al [VBSK00]. In their work, the ﬂow ﬁeld is ex-
tracted based on the visibility and color consistency
1.1 Related work of voxels, and the shape information is extracted si-
multaneously. In our approach, no color informa-
A similar problem of creating a single coherent ge- tion is currently used, and the shapes are known be-
ometry representation for a number of shapes has forehand.
been recently addressed by computer graphics re- Our approach can be considered as a remeshing
searchers. For instance, when several range scans of procedure for time-dependent shape data. Briceno
the same individual are given, Allen et al.[ACP02] et al. [BSM∗ 03] have implemented dynamic
solve the problem of ﬁtting a subdivision surface remeshing for the purpose of mesh compression,
template to create a posable 3D model. That work however in their work all the input meshes share the
was extended to handle the parameterization of same connectivity. In contrast, our input meshes do
whole-body data for multiple humans [ACP03]. In not have to be of the same connectivity.
both cases, a sparse set of 3D markers is used for Our work is also related to the volumetric mor-
model registration. Related work on ﬁnding consis- phing with one distinction: we assume that our
tent parameterizations of dissimilar shapes by Praun meshes are instances of a continuously evolving
et al. [PSS01] has also used user-deﬁned feature shape, while the main difﬁculty in morphing is
markers in a multiresolution remeshing procedure. the ability to handle drastically different shapes
Neither of the studies mentioned above explic- [WB98][BP98].
itly considers a continuously evolving surface. In
this paper, we concentrate our effort on extracting
a single animated mesh from a continuously evolv- 2 Overview
ing shape sequence. We do not assume the avail-
ability of markers or surface textures in the origi- The input data for our algorithm is a sequence of
nal data, with the hope that strong shape coherence triangular meshes M(t) that represent some non-
will give enough information for extracting surface rigidly moving closed surface. Meshes M(t) are
motion. We split our problem into the motion esti- not required to have the same connectivity; for in-
mation and the mesh propagation parts. The mo- stance, they could come from a marching cubes ex-
tion estimation problem has long been studied in traction, run independently at every time frame.
the computer vision community [BB81]. In partic- Our goal is to create an animated mesh sequence
ular, several efﬁcient optical ﬂow algorithms were A(t) that approximates M(t) and such that all the
introduced [SS96][Sim93]. We extend the multi- meshes A(t) have the same connectivity. Our ap-
scale optical ﬂow algorithm of Simoncelli [Sim93] proach will be to start with a template mesh A(0)
to work on adaptive signed distance volume repre- that approximates the initial mesh M(0), and prop-
sentations and modify it to remove unneeded inter- agate it through time frame by frame. We observe
actions between unrelated surface patches. that, without global motion estimation step, simple
Our motion estimation and surface ﬁtting proce- heuristics (like ﬁtting a previous frame to the next,
dures require a volumetric shape representation, and or using constant velocity assumption to predict the
an efﬁcient conversion of input polygonal data is location of the next frame) fail miserably on non-
necessary for the overall good performance of the trivial motions (see Figure 1).
algorithm. We employ an adaptive shape represen- Therefore, we employ a motion estimation pro-
tation similar to the ADF representation of Frisken cedure that operates on the sequence of adaptive
et al. [FPRJ00]; due to some differences and to signed distance volumes g(t) approximating input
avoid confusion we generically call it ASDV (adap- mesh data. An adaptation of the differential opti-
tive signed distance volume). We convert our input cal ﬂow procedure is applied to consecutive frames
meshes into a sequence of ASDV datasets and use of adaptive signed distance volumes to obtain the
them for both motion estimation and surface ﬁtting surface ﬂow at every frame of the sequence; the
extracted ﬂow is represented on the same adaptive termined on the coarse scale with ever smaller cor-
octree structure as the signed distance, and repre- rections applied on ﬁner levels. In our algorithm we
sents the correspondence relation between consec- adopted a version of the Bayesian Multi-Scale Dif-
utive surfaces in the sequence. Any motion estima- ferential Optical Flow (BMSDOF) algorithm of Si-
tion algorithm only gives an approximate mapping moncelli [Sim93]. The following section will give
between two surfaces; however, we ﬁnd that a com- a synopsis of BMSDOF, after which we shall de-
bination of propagation via extracted ﬂow and sur- scribe our implementation of it in the adaptive vol-
face ﬁtting produces coherently sampled animated umetric case.
Our combined propagation procedure is per- 3.1 Basics of the BMSDOF algorithm
formed as follows:
1. Convert input meshes into adaptive signed dis- BMSDOF algorithm [Sim93] belongs to the fam-
tance volumes g(t). ily of differential motion estimation algorithms that
2. Extract surface ﬂow v(t) for every frame of rely on solving the brightness constancy constraint
the sequence. equation:
3. Propagate every vertex of the template mesh gt + (v · )g = 0, (1)
A(t) with the computed ﬂow from its positions that is, ﬁnding v(x, t) for a given g(x, t). In our
at time t to the new positions at time t + 1. case, g(x, t) represents a volumetric signed dis-
4. Apply parametric smoothing and ﬁtting until tance function whose zero level surface approxi-
convergence similar to [WDSB00] (also see mates the shape of M(t).
Section 4). Typically, one solves the equation (1) in a local
The following two sections describe the details neighborhood of a point q by adding regularization
of our surface motion estimation algorithm and our term on the magnitude of the estimated v and ob-
mesh propagation procedure. taining a least-square solution to the system of con-
straints posed by (1) on a discrete stencil around q,
with the resulting estimator expressed as:
3 Motion estimation
v (q, t) = −Mreg
˜ wk b(k) (2)
The purpose of the motion estimation step is to pro-
duce a vector ﬁeld v(t) that approximates the mo-
tion of the surface at time t; that is, v(t) should be with Mreg = σI + wk M (k),
such that propagating a particle positioned on sur- k∈K(q)
face M(t) at time t along the ﬂow vectors v(t)
where σ is a small regularization coef-
should place it on the surface M(t + 1) at time
ﬁcient, the ﬁlter wk represents Gaussian
t + 1. For more precision, we employ a multi-step
averaging, and the summation is over the
Runge-Kutta like procedure that uses both v(t) and
stencil K(q); an example of K(q) is pic-
v(t + 1) when propagating between frames t and
tured as blue dots on the left, q being the
t + 1. Therefore, we need the vector ﬁeld v(t) to
yellow dot in the center. The matrices M (x) and
be deﬁned not only on the surface but also in some
vectors b(x) are deﬁned as follows:
neighborhood around it. An adaptive representa-
tion of v(t) that has most precision on the surface 2
gx g x gy g x gz
M(t) and gradually coarsens away from the surface def
M = g y gx gy2 def
gy gz , b = g y gt .
is then appropriate. gz gx gz gy gz2
gz g t
It is easy to see that the problem of ﬁnding v(t)
is not well deﬁned without some regularization, as Given the values of g on a regular grid, we can
there are many vector ﬁelds that propagate the sur- approximate M (x) and b(x) at the centers of grid
face of one frame onto the surface of the next one. cells using central differences.
One way to regularize the problem is to require the The simple local algorithm described above
estimated ﬂow to be smooth, and many practical al- would typically fail to discover motions on the scale
gorithms for ensuring such smoothness are built in a larger than the grid step size. Typically, a coarse-
coarse to ﬁne fashion, so that the overall ﬂow is de- to-ﬁne approach would give better results, by ﬁrst
estimating the ﬂow on the coarse level and then to the appropriate “previous” and “next”
using this solution as a prior for the ﬁner level. ﬁelds of the frame t ASDV data structure.
BMSDOF handles such prior information by pre- Thus, the values of three consecutive
warping. This can be described as follows: sup- frames g(·, t − 1), g(·, t), g(·, t + 1) are
pose a prior ﬂow estimate u is known, and one sampled at the same spatial locations and
would like to improve that estimate given the ac- stored within a single adaptive octree.
tual data g. We express the updated ﬂow v as (b) Compute the ﬂow v L0 (x, t) at all the ver-
the sum of u and the unknown residual w so that tices of the cells at level L0 and below.
v(x, t) = u(x, t) + w(x, t). It can be shown that See Section 3.4.
the ﬁne scale residual w satisﬁes the following con- (c) Linearly interpolate the ﬂow v l−1 (x, t)
straint: onto vertices of the next ﬁner level l. This
produces the prior ﬂow ul (x, t).
W[u, g, t](x, s) + (w · )W[u, g, t](x, s) = 0, (d) Warp the distance ﬁelds from frames t−1
(3) and t + 1 using (4) and assign them to the
where W[u, g, t](x, s) represents the ﬁeld g warped appropriate “previous” and “next” ﬁelds
from time t to time s using ﬂow u: of the frame t ASDV data structure. Use
the prior ﬂow ul (x, t) for warping.
W[u, g, t](x, s) = g(x + (t − s)u, t) (4) (e) Repeat the estimation step for residual
ﬂow wl (x, t) at all the cells of level l and
Equation (3) has the same form as (1), except that below using (3) (see also Section 3.4).
a pre-warped version of the distance ﬁeld g is used. Update the current ﬂow with the result
Hence, equation similar to (2) can be used to solve v l = ul + wl . If not done go to the step
for the residual correction w. 3c.
Applying this warp-update procedure in a coarse-
to-ﬁne fashion results in the BMSDOF for image
sequences described in [Sim93]. The following sec- 3.3 Adaptive Signed Distance Volume
tion will describe its realization in the adaptive esti- Construction
mation of surface ﬂow.
We use an adaptive signed distance volume repre-
sentation to store our distance and ﬂow ﬁeld data.
3.2 Adaptive motion estimation algorithm Given a mesh for frame t we extract an adaptive oc-
Our adaptive motion estimation algorithm has the tree whose leaves at the ﬁnest level contain the sur-
following three steps: face, and whose structure satisﬁes the 26-restriction
1. Compute ASDV for three consecutive geome- criteria that ensures that none of the 26 possible
try frames M(t − 1), M(t), and M(t + 1). neighbors of a cell is at a level that differs by more
The result of this procedure is the adaptive vol- than one from the cell’s own level. This restriction
umetric approximation of the signed distance is important for more gradual transition from the re-
function for these three frames g(·, t − 1), ﬁned surface region to the far ﬁeld. Our adaptive
g(·, t), and g(·, t + 1). For details, see Sec- structure will typically have more reﬁnement than
tion 3.3 the ADF from [FPRJ00] because we need a ﬁne grid
2. Run an approximate medial axis detection pro- near all the surface locations for accurate approxi-
cedure and tag the corresponding cells of the mation of the distance ﬁeld derivatives that are used
ASDV of the frame t. For details, see Sec- in the ﬂow estimation procedure.
tion 3.4. Our ASDV extraction proceeds as follows: ﬁrst,
3. Run the adaptive version of the ﬂow extrac- a scan conversion procedure is performed at a pre-
tion procedure starting from some level L0 . deﬁned ﬁnest level of the hierarchy, each intersected
(All the computations are performed within cell is inserted into the octree causing it to reﬁne ap-
the ASDV structure of the frame t.) This re- propriately. Then the ﬁne-to-coarse restriction en-
quires the following ﬁve substeps: forcing procedure is performed resulting in the ﬁ-
(a) Interpolate the distance ﬁelds from nal octree structure of the ASDV (see the pictures
frames t − 1 and t + 1 and assign them on the left). The inside/outside testing is performed
that assigns inside/outside ﬂag for all the vertices dial axis. In our experiments we deﬁne med = 0.9,
of ASDV, additionally the vertices of the leaf cells the resulting leaf medial axis cells of the octree are
intersected by the surface are assigned signed dis- shown in the ﬁgure on the left: only the leaves
tance values. At this point a fast marching method starting with level seven and up are shown. One
[Set99] is applied to assign signed distance values can clearly see the three main components of the
to all the remaining vertices of the ASDV structure. approximate medial axis separating two legs from
each other and separating arms from the body. We
3.4 Restricted Flow Estimation only use the outside portion of the medial axis, so
that the motion coherence is preserved inside the
The ﬂow ﬁeld at time t is used to prop- body, while the unneeded interference through the
agate the mesh vertices that lie on the “thin air” is diminished.
surface, hence we need the full preci- We use this information in our ﬂow
sion ﬂow to be computed near the sur- estimation routine by excluding the gra-
face, while away from the surface we dient values in cells that are “behind
can use a coarser less accurate approx- the medial axis” from the averaging in
imations. Therefore, we use the struc- equation (2). For an illustration, con-
ture of the surface ASDV at time t to sider the two-dimensional case for the four-by-four
compute and store the ﬂow at time t. averaging mask. The following simple criteria are
BMSDOF algorithm can be easily used to determine whether a cell’s gradient value
adapted to work on the adaptive vol- is to be included in the averaging: a red cell is in-
ume representation. The only differ- cluded only if it is not on the medial axis. A blue
ence is that the local evaluation of dis- cell is included only if it is not on the medial axis
tance ﬁeld may require traversing the and the red cell separating it from the center of the
octrees in order to compute a needed distance value. stencil is not on the medial axis. In the special case
Within a single level, the procedure starts by com- when all the red cells are on the medial axis, we
puting the gradients of all the cells in the four-by- take the averaging only on the central two-by-two
four cell stencil around a given vertex. Some of stencil. Once the inclusion is determined we renor-
the cells covered by the stencil may not be subdi- malize the mask coefﬁcients to sum up to one.
vided. In such cases the value of the gradient from
the coarser level is used. Because of the restriction
criteria used to build our ASDV structure, we can 4 Mesh propagation
only have a single level difference between the cells
of the stencil, which is beneﬁcial for the quality of Our ﬂow-based surface propagation extracts the
derivative estimation. shape evolution, however in ﬂat surface regions
In order to diminish the ef- there exists an inherent uncertainty in estimating
fects two close but differently the tangential component of surface motion. In or-
moving surfaces can have on der to improve the parametric coherence of the ex-
each other, we compute an ap- tracted mesh sequence we employ a variant of the
proximate medial axis by look- mesh smoothing technique of [DMSB99], modiﬁed
ing at the gradient of the signed so that only the tangential smoothing component is
distance function within each used as described in [WDSB00]. We compute the
cell (for an overview of approxi- coefﬁcients of the smoothing operator based on the
mate medial axis algorithms see geometry of the mesh in the ﬁrst frame. The appli-
[MFV98]). In the cells away cation of the parametric smoothing improves mesh
from the medial axis the abso- quality in ﬂat regions of the propagated mesh and
lute value of the gradient vec- tries to preserve the sampling pattern of the mesh
tor should be close to one (up from the ﬁrst frame.
to approximation error), thus we Special care should be taken when two surfaces
tag all the cells with the absolute are close to each other. In order to eliminate spu-
value of gradient below some rious jumping of vertices onto the wrong side, we
predeﬁned threshold med as intersecting the me- compare the normal vectors computed from the
mesh with the normal derived from ASDV. If those the ﬁrst frame and propagate it through long se-
vectors point in the opposite directions, we reposi- quences of motions given in the original data (the
tion the offending vertex to the point predicted by its human motion sequence we processed contained
immediate mesh neighbors. This solves the prob- 685 frames). A typical result of the resulting ani-
lem for the underarm regions of human shape data. mated mesh is given in the supplementary archive.
Figure 5 shows the mesh after 200 time steps of
propagation. Note that no texture or marker infor-
5 Results and discussion mation is made available to the algorithm, thus one
cannot expect to completely preserve sampling pat-
We have implemented a library of adaptive motion terns of the original mesh template.
estimation for evolving meshes. The typical tim- We used two versions of the human template:
ings for a 2GHz Pentium PC processing the hu- the original ﬁrst frame of the human skin data with
man motion data (mesh sizes around 16K vertices 16K vertices and a reﬁned version with 60K ver-
each frame): level 9 ASDV construction for a single tices. For the coarser template, we had problems
frame – approximately 20 seconds, the ﬂow compu- with the smoothing of the extremities of the shape
tation on all the levels – approximately 2 seconds. (hands and feet). We improved the performance of
The surface ﬁtting step typically takes less than a the algorithm on those regions by selecting them in
second. We computed the ﬂow starting with level the ﬁrst frame and decreasing the amount of para-
six. The typical surface reconstruction L2 errors as metric smoothing in those regions by the factor of
reported by Metro [CRS98] were about 3 × 10−4 ten. For the reﬁned version of the template, there
relative to the diagonal of the bounding box of the were no problems on the extremities. Examples of
mesh (that is, the reported L2 error is divided by both propagated templates are provided with sup-
the length of the bounding box diagonal of the ﬁrst plementary archive.
frame). We have checked the performance of our ani-
mated mesh extraction algorithm with the test ex-
ample of a jumping cactus. Figure 3 shows the
trajectories of the original animated model (which
had the same connectivity) and the model extracted
from the corresponding signed distance data using
our motion estimation algorithm. We see that while
each individual trajectory may not be close, the
overall character of motion and the spatial distribu-
tion of particles are similar. Figure 4 shows that the
relative L2 error stays within 0.04%.
Figure 2: Shaded version of the propagated tem-
plate for the human sequence.
We applied our animated mesh extraction algo-
rithm on several datasets, ranging from simple test
cases like jumping cactus sequence to the captured
deforming human skin data from [SMP03]. We Figure 3: Comparison of the trajectories of the ver-
analyzed the performance of the algorithms using tices of the original animated cactus (left) and the
several evaluation criteria including visual quality, cactus extracted with our algorithm (right). Both
mesh distortion, and error with respect to the orig- models are rendered at the end of the jump se-
inal data. Our experiments take the mesh from quence.
Figure 5 shows the consecutive four frames of the processed surface data corresponds to the “skin”
human data to give an idea of the amount of motion of a real deforming object. For such closed sur-
between frames the algorithm is able to handle sat- face data it is natural to use signed distance vol-
isfactorily. Figure 4 shows the plot of the relative umes. An easy extension of the algorithm can em-
L2 error (w.r.t. the bounding box diagonal of the ploy the non-signed (positive) distance volumes to
ﬁrst frame) for the human and cactus sequences. handle open surfaces with smooth and temporally
The supplementary video shows the animation of coherent boundaries. We have experimented with
the extracted mesh textured in the ﬁrst frame and simple animated meshes with boundaries, and used
animated using a publicly available rendering pack- additional distance ﬁelds corresponding to bound-
age Jot [Jot]. Some frames of the animation are ary curves in order to guide the ﬂow estimation pro-
shown in Figure 6. While there is a certain amount cedure so that it prefers to map boundaries onto
of texture sliding within the surface, the overall re- boundaries in consecutive frames. From the prac-
sult is reasonably stable. Note that the area near the tical perspective, however, it is much more impor-
neck is noisy due to the chaotic neck region mo- tant to handle surfaces with noisy boundaries such
tion in the original data. The shaded version of the as the ones coming from real-time structured light
mesh without texture is also provided with supple- acquisition [ZCS03]. Noisy boundaries should not
mentary material, illustrating good approximation be mapped onto each other from frame to frame, in
of the original surface data, several frames of that fact, their effect on the ﬂow should be diminished
sequence are shown in Figure 2. as much as possible. This would only be possible
if the additional color information is provided, or
L2 error of the human sequence
if the processed surfaces have signiﬁcant geometric
detail. For the acquired data, the ﬁrst option seems
L2 error rel. to bbox diag
0.0005 6 Conclusions and future work
430 530 630
730 830 930
We introduced an animated mesh extraction algo-
rithm that can handle long sequences of evolving
L2 error of the cactus sequence
closed shapes and produce sequence of meshes of
the same connectivity approximating the input sur-
L2 error rel. to bbox diag
0.0003 face motion data. The resulting meshes can serve as
approximations to the original shape motion, with
advantage of having the same connectivity and tem-
poral coherence. Such meshes can be represented
0 very compactly with the recent animated mesh com-
0 50 100 150 200
Frame number pression approaches.
There is much work remaining for handling
Figure 4: Left: plot of L2 error for the ﬁrst 500 shapes of changing topology, which would lead to
frames of the human sequence. We observe that dynamically changing topology of the mesh being
most of the error comes from the neck area. Right: ﬁtted. Another important extension would involve
L2 error for the cactus jump. The error is mea- the handling of meshes with boundaries (often pro-
sured with the Windows version of the Metro pack- duced by the shape acquisition algorithms).
age [CRS98]; the reported relative L2 error is nor- Currently we propagate a single resolution ver-
malized with respect to the length of the bounding sion of the template through the whole sequence
box of the ﬁrst frame. of frames. This assumes that all the signiﬁcant
features of the shape are present in its ﬁrst frame.
While this is true for the human motion data we
Limitations of the algorithm Our approach is considered, different data may require the adapta-
currently limited to closed surfaces without bound- tion of the mesh template to the geometry of all the
aries, which is a reasonable assumption when the shapes appearing in a sequence. In particular, ani-
mated mesh sequence simpliﬁcation is a very rele- [KG] K ARNI Z., G OTSMAN C.: Compression of
vant problem to be addressed in the future. soft-body animation sequences. To appear in
Computers and Graphics, 2003.
[MFV98] M ALANDAIN G., F ERNANDEZ -V IDAL S.:
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[Jot] Jot home page: http://jot.cs.princeton.edu.
Figure 6: Frames from the accompanying video
showing a mesh textured in the ﬁrst frame, and ren-
dered in the consecutive frames of the extracted
animated mesh with the same texture assignment.
The overall positioning of the texture is similar to
Figure 5: Overview of the process. Top: original the ﬁrst frame. The neck area undergoes chaotic
meshes M(t−1), M(t), M(t+1), M(t+2); 2nd undulations in the original data causing excessive
row: one plane from ASDVs at level eight g(t − 1), stretching on the top.
g(t), g(t + 1), g(t + 2); 3rd row: the ﬂows v(t) and
v(t + 1), interpolated onto the corresponding sur-
faces; bottom: propagated template mesh at frames
t and t + 1.