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Topologically Consistent Leafy Tree Morphing


Abstract and Figures

We present a novel morphing technique to generate pleasing visual effects between 2 topologically varying trees while preserving the topological consistency and botanical meanings of any in-between shapes as natural trees. Specifically, we first efficiently convert leafy trees into botanically inspired chain-lobe representations in an automatic way. With the aid of branching-pattern aware, one-to-many correspondences between branches and leaves, we hierarchically interpolate branches of in-between trees while maintaining their topological consistencies. Finally, we simultaneously interpolate foliage, specifically every single leaf, during the morphing process, avoiding the generation of unpleasant “floating” leaves. We demonstrate the effectiveness of our approach by creating visually compelling tree morphing animations, even between cross-species.
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Received: 12 March 2017 Accepted: 18 March 2017
DOI: 10.1002/cav.1761
Topologically consistent leafy tree morphing
Yutong Wang1Luyuan Wang1Zhigang Deng2Xiaogang Jin1
1State Key Lab of CAD&CG, Zhejiang
University, Hangzhou, China
2Computer Science Department, University
of Houston, Houston, TX, USA
Xiaogang Jin, State Key Lab of CAD&CG,
Zhejiang University, Hangzhou 310058,
Funding information
National Natural Science Foundation of
China, Grant/Award Number: 61472351
We present a novel morphing technique to generate pleasing visual effects between 2
topologically varying trees while preserving the topological consistency and botan-
ical meanings of any in-between shapes as natural trees. Specifically, we first
efficiently convert leafy trees into botanically inspired chain-lobe representations
in an automatic way. With the aid of branching-pattern aware, one-to-many cor-
respondences between branches and leaves, we hierarchically interpolate branches
of in-between trees while maintaining their topological consistencies. Finally, we
simultaneously interpolate foliage, specifically every single leaf, during the morph-
ing process, avoiding the generation of unpleasant “floating” leaves. We demonstrate
the effectiveness of our approach by creating visually compelling tree morphing
animations, even between cross-species.
shape morphing, special effects, topological consistency, tree modeling
During the past decades, morphing has consistently proven
its usefulness in generating breath-taking visual effects and
has become an indispensable tool in the special effects and
animation industry. Similarly, the task of morphing between
topologically varying trees opens up to efficiently creating
aesthetically special effects for many unconventional anima-
tion applications, such as nurturing the mysterious atmosphere
of the extraterrestrial environment in games and movies.
The majority of existing efforts have been put in gener-
ating morphing between man-made objects, animals, even
human portraits (please refer to Alexa1for a comprehensive
survey), and they have produced pleasing results. However,
none of them is capable of generating the morphological and
topological transformations between trees. The reasons are as
follows: (a) they do not especially establish topological-aware
correspondences between branches; (b) they do not explicitly
maintain the hierarchically consistent morphing trajectories,
resulting in disconnected “floating” branches during the mor-
phing process;(c) they do not handle the problem of gener-
ating smooth transitions between foliage. In addition to the
fluid topological and geometric transformations maintained
in the existing works, morphing between tree requires special
care on in-between trees’ botanical features, such as cohesive
branching patterns, reasonable foliage shapes, and biomass
distributions. Indeed, none of the existing methods is able to
ensure the preservation of the above high-level tree features
during the morphing process.
The main technical challenges to the tree morphing
problem are twofold: (a) establishing botanical-aware, topo-
logically consistent correspondences between different tree
species; (b) generating botanically plausible foliage morph-
ing trajectories that avoid producing floating leaves during the
morphing process.
In this paper, we propose a novel tree morphing tech-
nique, which takes two topologically varying leafy trees
as input and efficiently generates botanically sound mor-
phing results by preserving the topological consistency of
in-between trees; an example result is shown in Figure 1. To
the best of our knowledge, our work reports the first approach
that seamlessly integrates foliage morphing into the tree ani-
mation framework and demonstrates the ability of generating
smooth foliage transformations even between different tree
Comput Anim Virtual Worlds. 2017;28:e1761. Copyright © 2017 John Wiley & Sons, Ltd. 1of10
2of10 WAN G ET AL.
FIGURE 1 Morphing between a Eucalyptus and a Staphylea pinnata by our approach
species; see Section 7. Our work has three main contribu-
tions: (a) an efficient clustering algorithm to automatically
segment foliage into botanically plausible groups, that is,
lobes; (b) a branching pattern-aware correspondence estab-
lishment method to allow sound one-to-many matches among
branches; (c) a novel foliage morphing algorithm to not only
avoid producing unpleasant floating leaves but also produce
time-varying foliage details, including geometric and textural
transformations of every single leaf.
Morphing. To date, a large number of morphing tech-
niques (both 2D and 3D) have been proposed to continuously
transform one shape into another. Two-dimensional morph-
ing techniques, including polygonal morphing2,3 and image
morphing,4suffer from the lack of the three-dimensional
information of shapes and may produce unpleasant ghost
artifacts when applied to 3D shapes. Therefore, many mor-
phing methods specialized for 3D shapes have been proposed
(please refer to Alexa1for a comprehensive survey). How-
ever, none of the existing methods has explicitly handled the
tree morphing problem. The main reason is that morphing
between trees requires the preservation of high-level botanical
information (such as branching patterns, topological consis-
tency, and botanical plausibility) of the in-between trees,
which has not been addressed in all the previous morphing
Tree animations. Despite its long history, the study of tree
animations remains to be active. Existing literature can be
roughly classified into two categories: tree growth animations
and interactions between trees and the environment. The early
X-frog system5allows users to define several keyframes that
describe the developmental stages of trees, and then inter-
polate them to generate tree growth animations. Later, Pirk
et al6automatically infer the developmental parameters from
one single static tree and then interpolates them to generate
the tree growth animations. Environmental factors, includ-
ing obstacles,7,8 lighting,8winds,9,10 and competition with
other trees11 have been studied to date to realistically ani-
mate interactions between trees and the environment. How-
ever, none of them is capable of generating fluid morphing
between topologically varying trees. The reason is three-
fold: the developmental parameters of the growth model are
species-dependent; the topologically consistent correspon-
dences between different tree species are hardly addressed;
and the smooth transformations between foliage are not con-
sidered during the interpolation.
Creative tree modeling. Several exploratory approaches,
such as previous studies,12–15 have been proposed to assist
users to efficiently create high-quality 3D shapes. The recent
approach by Wang et al.16 is the most relevant work to ours.
It utilizes blending techniques as a novel shape creation tool,
targeting on generating as many morphologically diverse trees
as possible. However, it only establishes either one-to-one
or one-to-none correspondences between branches because
of its bipartite matching algorithm. Without considering the
branching patterns, visually unpleasant branches with abnor-
mally large substructures violating the biomass distributions
might be generated during the blending process, see Figure 2.
Furthermore, they do not tackle the problem of producing a
fluid transformation between foliage.
FIGURE 2 Morphing results generated by Wang et al.16 (top row) and our method (bottom row)
WAN G ET AL.3of10
FIGURE 3 Approach overview
The goal of tree morphing is to generate a sequence of
in-between trees that not only fluidly transform from the
source to the target but also maintain their topological con-
sistency and botanical plausibility. Figure 3 illustrates the
main steps of our approach. Given two leafy trees (Figure 3a),
line skeletons are extracted using Oscar et al.,17 which are
further segmented into branching-pattern-aware chain groups
based on a hybrid ordering strategy. After constructing the
branching-pattern aware topology tree (the BPTT) to pre-
serve the branching patterns and hierarchical topology of
chain groups, foliage are automatically clustered into lobes,
associating leaves with chains (Figure 3b). Based on the
BPTTs, branching-pattern aware, one-to-many correspon-
dences are hierarchically established in two steps: set up
either one-to-one or one-to-none correspondences between
chain groups by solving a minimum weight bipartite matching
problem, and then build one-to-many matches among chains
within the matched chain groups using a greedy algorithm.
Meanwhile, one-to-many correspondences between leaves
within the matched chains are also established by the greedy
algorithm (Figure 3c). The in-between leafy trees are hier-
archically interpolated by maintaining their topological con-
sistency, resulting in visually pleasing morphing results; see
Figure 3(d).
4.1 BPTT
Hybrid branch orderings. Line skeletons of the input trees
are first extracted using the Laplacian contraction method.17
On the basis of the classic strand model,18 we describe their
branching hierarchies using a hybrid of the gravelius and
the weibull orderings. It labels branches sharing the same
branching patterns with the same orders, contributing to an
efficient identification of branching patterns.
Branch orders are hierarchically assigned from the root
branches (with the order 1) towards tip branches. Given a
branch pwith ordering D(p), the branching angles and radii of
its child branches {ci}are computed, and the ordering D(ci)
is assigned using Equation 1:
D(ci)=Dw(ci),if 𝜎𝛼𝜃and 𝜎rrt,
where 𝜎𝛼and 𝜎rdenote the standard variance of the branching
angles and radii, respectively. 𝜃and rtare the user-defined
thresholds, and Dg(·) and Dw(·) are the gravelius and weibull
ordering functions; see Equation 2.
Dg(ci)=D(p),if 𝛼ci=𝛼min,
D(p)+1,if 𝛼ci𝛼min,
where 𝛼ciis the branching angle and 𝛼min is the minimum
branching angle. The algorithm repeats until it reaches tip
BPTT. In real world, the growth of branches generally
exhibits one of three patterns19 (see inset). With alternative
branching, exact one branch
grows at a branching point.
In contrast, with opposite
and whorled branching, mul-
tiple branches grow at the
same branching point and are
arranged into special forms. Therefore, they should be con-
sidered as a group rather than individuals.
We use BPTT, to describe the topological hierarchies
among branches. Similar to Pirk et al.6and Wang et al.,16
consecutive branches sharing the same ordering are defined
as chains, which are then described by seven parameters listed
in Table 1. Branching patterns are identified using Equation3,
which cluster chains into groups, denoted as a set of GCi:
alternative,if #=1,
opposite,if #=2and𝛼dvg 180<𝜃,
where #denotes the number of chains sharing the same order-
ing while growing at the same location, 𝛼dvg is the average of
divergence angles 𝛼dvg between chains (Figure 4a), and 𝜃is a
user-defined threshold, which is experimentally set to 30in
this paper.
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TABLE 1 Parameters of chains
Name Description
chn piecewise linear representation of its geometry, including
radii information
OL the shape of the outer lobe
OR outer lobe ratio, denoting the lowest location where the
substructures grow
IL the shape of the inner lobe
IR inner lobe ratio, denoting the location where the leaves grow
vbranching vector
pgrowth location
FIGURE 4 Lobes (a) and chain parameters (b)
In the spirit of the multiscale topology tree,16 whose nodes
represent chains and edges encode their hierarchies, we take
one step further to preserve the hierarchy of branching pat-
terns and define the nodes of the BPTT as a quintuple:
where GCiis the group of chains exhibiting a certain branch-
ing pattern, GOLidenotes the group outer lobe of the group,
GRiis group lobe ratio, Gviis the average branching vector of
the chains, and Gpiis the growth location of the group. Niand
Njare connected by a BPTT edge if all the chains encoded by
Njcould find their parents in Ni.
4.2 Lobe geometry
We observe that people subconsciously tend to perceive trees’
foliage by dividing it into smaller parts according to their
branches, namely, the lobes. The existing lobe extraction
method20 usually takes nontrivial efforts to obtain an ideal
set of lobes because the thresholds differentiating the foliage
from the branches are needed to be manually set and tuned
according to the tree species. Instead, we propose an auto-
matic outer-to-inner lobe extraction method, which hierarchi-
cally groups leaves into lobes using K-means clustering.
Because leaves grow on the branches, we assume that the
foliage of a chain generally follows a crownlike shape formed
by its substructures. Three types of lobes are defined in our
approach; see Figure 4(b). The outer lobe of a chain (OL)
is defined as the leaves covering all of its substructure. The
group outer lobe (GOL), describing the general appearance
of a chain group’s foliage, equals to the union of OLsofthe
chains within the group. The inner lobe (or simply the lobe),
denoted as IL, defines the leaves growing at the tip of the
chain, that is, the actual leaves belonging to a chain.
Our goal is to extract ILs for chains. In order to main-
tain the botanical consistency, ILs are extracted hierarchically
downwards the BPTT in the following three steps.
Step1: initialization. Starting from the root node Ni
of the BPTT, the algorithm initializes its GOLlas all
the leaves in the foliage, where l
denotes the level of the node in the
BPTT, and l=0 for the root node.
Then, the algorithm collects its chil-
dren and computes the center umand
radius r
mof the crownlike shape formed by the substructures
of chains for the mth child chain group, which are the candi-
date centers and radii of their potential foliage. Besides, for
any chain chnnGCi, the algorithm assigns its potential
foliage center vnas the tip of the chain, and the radius r′′
10%of the chain’s length (refer to inset).
Step 2: clustering. Leaves within the GOLlare itera-
tively divided into (m+n)groups by K-means clustering
method, where mis the number of child chain groups and n
is the number of chains contained in GCi. Initially, cluster
centers are set to be UV,whereU={u0,u1,,um}and
V={v0,v1,,vn}. Unlike the standard K-means clustering
algorithm, we associate each cluster with a distance threshold
to reduce the influence of outliers, which is experimentally
initialized as 35%of the radius r
iand increased by 10.5%
during every iteration. For each unassigned leaf, the algorithm
measures its Euclidean distances to all the clusters and assigns
it to the nearest cluster when the distance is smaller than
the distance threshold. Then, the algorithm updates the clus-
ter centers and relaxes the distance threshold by 10.5%.The
clustering repeats until all of the leaves within the GOLlare
Step 3: iteration. As a result, leaves assigned to clusters
that are initialized with uiUdefine the global outer lobe
jof the child group and serve as the input of the next
iteration, whereas the others are the inner lobes of the chains
contained in the group GCi. Then, the algorithm moves down-
wards the BPTT and repeats the procedure for the child groups
at level l+1. The algorithm terminates when all of the leaves
are grouped into ILs.
4.3 Leaf geometry
The foliage usually consists of thousands of leaves. Due
to the large amount, we adopt the notion of billboard
clouds21 and generalize leaves into
textured quads. Specifically, a leaf is
represented as a septuple as li∶=<
chnidirepresents the chain it grows on,
qiis the leaf position relative to its
corresponding chain, wiand hiare the
WAN G ET AL.5of10
width and height of the leaf, respectively, r
denotes the leaf’s 3D orientation, and Tidenotes its texture.
Considering the variety of branching patterns in trees, we
establish one-to-many (including one-to-one matching) and
one-to-none matchings between chains, see Figure 3(c),
where corresponding chains are rendered in the same color.
Other than constraining the matching candidates to be
nodes at the same level (Rule 1) and have matched parents
(Rule 2),16 we also require that only those chains belonging
to the matched groups are supposed to be matched in order to
maintain their topological consistency (Rule 3).
The cost of matching two BPTT nodes is measured by their
similarities (see Equation (4)):
where · denotes a distance function, nk
iand nk
jare the
kth element in node Niand Nj, respectively, and wkis the
weight term. Specifically, we use the Hausdorff distance, the
classic shape similarity measurement, to evaluate the simi-
larities between two GOLs. Similarities between branching
vectors, GRs, number of chains within the groups, and average
divergence angles are computed by the Euclidean distances.
Without loss of generality, matching nodes that violate any
of the rules is defined to be a positive infinity. Likewise,
the similarity between chains is defined by the weighted
sum of similarities between their feature parameters in
Table 1.
Similar to Wang et al.,16 we employ a top-down matching
strategy, which hierarchically establishes correspondences
based on the BPTTs. However, the correspondences at each
level are established via a two-phase protocol in order to pre-
serve their branching patterns: (1) constrained by the Rules
1 and 2, establish either one-to-one or one-to-none corre-
spondences between the BPTT nodes (i.e., the chain groups)
by solving a minimum weight bipartite matching problem as
Wan g et al .16; (2) according to the Rule 3, greedily match
chains with the most similar ones for the matched groups,
resulting in one-to-many correspondences.
Correspondences between chains also indicate matches
between their ILs. Leaves within the matched ILs are greed-
ily matched by finding the most similar ones. The similarity
between two leaves is equivalent to the weighted sum of
similarities between their leaf positions, widths, heights, and
6.1 Morphing trunks
Correspondences induce morphological transformations.
Five transformation operations are defined in this work; see
Figure 5. Inspired by Wang et al.,16 we apply morph oper-
ations to the one-to-one matched chains and grow or wilt
operations to chains, which have no corresponding sources or
targets. The transformation of one-to-many matched chains is
achieved by split or merge operations. The in-between trunks
are interpolated and then reconstructed using the method of
Alsweis et al.11
Specifically, a psplit of a chain is performed in four
steps: (a) make pcopies of the chain, denoted as SP =
{chn0,chn1,,chnp}; (b) at time t∈[0,1],segment
chniSP at the arc-length parametric location (1t)
into two parts, the chn(1t)
iand the chnt
i; (c) compute the
interpolation weights for every points n
jon each part as
j=b·t,if n
t,if n
where b∈(0,1]is the slow-down coefficient, which is pro-
portional to the arc-length parametric value of the point n
(d) perform morph operation to each copy using the point
weight w
js. Being the inverse operation of the split task, the
merge of chains is performed similarly except that we pro-
gressively speed up the morphing of the nodes nearby the
chains’ roots by calculating the interpolation weight wjusing
min(1.0,(2.0bt)when n
6.2 Morphing foliage
The in-between leaves are computed by interpolating each
term of their feature vectors. The leaf position is equivalent
to the linearly interpolated parametric location on the newly
morphed chain. The width and height of the leaf are also lin-
early interpolated, and the orientations are computed through
quaternion interpolation. In order to avoid ghosting artifacts,
we separately interpolate contours and textons of the corre-
sponding leaves, and then superimpose the morphed texton on
the morphed contour and create the in-between leaf. Specif-
ically, leaf contours are extracted by the Canny boundary
FIGURE 5 Transformation operations
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FIGURE 6 Morphing between two leaves
FIGURE 7 Morphing between two Salix
FIGURE 8 Morphing between a Whitethorn acacia and an Acer japonicum
detection method.22 With the piecewise linear polygon repre-
sentations, the in-between leaf contours are interpolated using
2D polygon morphing method.3The in-between textons are
simply computed by pixel-wise linear interpolation. Although
more advanced leaf modeling method23 couldalsobeusedfor
our purpose, we experimentally found that our simple inter-
polation is sufficient to generate visually pleasing results; see
Figure 6.
We collected 60 real-world trees, including arbors*and
shrubs, from the internet (i.e.,
In addition, 10 designers’ hand-modeled virtual trees are
also incorporated into the dataset to evaluate the effec-
tiveness and flexibility of our approach. With randomly
paired inputs, our algorithm is capable of generating visually
pleasing and botanically plausible animations for both inner
species (Figure 7) and cross-species morphings (Figure 1
and Figure 8). Specifically, branching patterns are maintained
*An arbor is a woody plant that have an elongated, dominate stem, that is,
the trunk.
A shrub is a small woody plant that usually has multiple stems forming a
decurrent architecture.
during the morphing process, contributing to natural transi-
tions between matched branches; see Figure 1 and Figure 2.
Furthermore, the interpolation of leaves, including shapes
and textures, is performed in parallel to the trunk morph-
ing process based on the leaf–branch mappings, generating
smoothly transformed foliage sequences. Although the mor-
phing of flowers, fruits, and other factors are not explicitly
tackled currently, they can be interpolated in a similar way
as that of leaves, see Figure 9, where the source leafy tree
perched with butterflies smoothly transforms into the target
tree blooming purple flowers. One potential application of the
cross-species tree morphing is to enhance the supernatural
atmosphere when modeling the extraterrestrial environment
in games and movies. Please refer to our animation results in
the supplementary video.
Runtime statistics. We have implemented our morph-
ing algorithm in C++ on a desktop equipped with Intel©
Core i7 clocked at 3.50 GHz, 8 GB of RAM, and NVIDIA©
Geforce GTX 660 GPU. The runtime statistics are presented
in Table 2. In sum, our method demonstrates its efficiency
of generating smooth morphing results on an off-the-shelf
Comparison with topological-aware tree-blending
method. Figure 2 compares our approach with the most rele-
vant work by Wang et al.,16 which generates diverse inspiring
novel trees by blending among topologically varying trees.
WAN G ET AL.7of10
FIGURE 9 A morphing scene, where the source leafy tree perched with butterflies smoothly transforms into the target tree blooming purple flowers
TABLE 2 Runtime statistics
Input statistics Preprocessing Set up Morphing
Examples # of chains # of leaves Ordering (s) Lobe extraction (s) matches (s) (per frame, s)
Figure 1 5,220/20,195 4,690/5,919 1.82 3.12 0.49 11.35
Figure 3 4,507/4,590 4,436/3,765 0.61 0.99 0.19 1.52
Figure 8 5,660/4,370 2,937/2,844 1.34 0.97 0.55 7.31
Figure 7 7,782/21,138 6,459/2,874 4.39 1.81 0.85 7.92
Figure 9 5,224/8,272 2,351/5,564 1.64 1.61 0.81 10.08
In contrast to massively growing branches from the main
trunk using the method in Wang et al.16 (top row), our
algorithm gradually splits the source main trunk into mul-
tiple stems, producing a more natural morphing sequences
(bottom row).
In addition to visual comparison, we quantify the “natu-
ralness” of the morphing sequences using the trunk support
weight in the well-known pipe model theory.24 Denoted as
T(z), it defines the weight sustained by a trunk as all the
weights of leaves and branches from the top of its substruc-
tures to a given location zalong the trunk. Because people
are sensitive to the dominant trunks of trees, we compute
the T(OR)s for the main trunks of every in-between tree and
take the heaviest weight as the maximum main trunk support
weight (MTSW). Without loss of generality, we normalize
the MTSW by the total weight of the tree. Figure 10 plots
the MTSWs of the in-between trees generated by both meth-
ods against the morphing process. Evidently, our method (red
FIGURE 10 Comparison of the MTSWs of the morphing sequences
generated by our method and11
plot) smoothly “shifts” the weight of leaves and branches sus-
tained by the source main trunk to multiple target stems under
the splitting operations. By contrast, the results by Wang
et al.16 (blue plot) shows steep weight drops at the interval of
[30%,40%]and a slow raise at the interval of [70%,100%].
8of10 WA N G ET AL.
FIGURE 11 Comparison between our method (top) and the baseline method (bottom)
The above can be explained by our branching pattern pre-
serving one-to-many correspondences (see the red circled
trees in Figure 2), where the multiple stems exhibiting the
whorled branching patterns are grouped and interpolated at
the same time during the whole morphing process, leading
to naturally sharing of the trees’ weights. In contrast, the
correspondence establishment scheme in Wang et al.16 only
allows either one-to-one or one-to-none matches between
branches; see the blue circled tree in Figure 2. Therefore,
two other unmatched main stems should be interpolated by
the grow operations during the morphing process. Because of
their young age and small radii, the gradually grown chains
are supported by the main trunk during early morphing time
(e.g., [0%,30%]).Inthemiddletime(e.g.,[30%,40%]), when
their radii reach a certain threshold, the ordering strategy
marks them as main stems. As a result, they share the weight
originally supported by the main trunk, leading to the sudden
drop of the MTSW. Because the tree weight are nonuni-
formly shifted to growing stems, the MTSWs might be lower
than the target ones, see the MTSW values at the later time
(e.g.,[70%,100%]). In a nutshell, our method not only pre-
serves branching patterns during the morphing process but
also maintain a smooth MTSW change, contributing to more
natural and pleasant animations compared to Wang et al.16
Comparison with baseline foliage morphing method.
Because few, if any, existing works have put explicit efforts
on the smooth morphing between foliage, we implemented
a naive foliage morphing method as the baseline by linearly
interpolating randomly matched leaves and compared it with
our method, see Figure 11. Apparently, the baseline method
produces floating leaves (bottom row, circled leaves), leading
to noticeable visual artifacts.
In this paper, we present the first approach to efficiently gen-
erate smooth morphing sequences between two leafy trees.
Unlike the conventional tree modeling techniques,6,8,16 which
usually emphasize trunk modeling but treat foliage model-
ing as postprocessing operations, we seamlessly integrate
trunk morphing and foliage morphing into a unified animation
framework and generate a novel type of visual special effect.
We also develop an automatic foliagesegmentation method to
efficiently associate leaves with branches using the structure
lobe. Our method is advantageous in preserving both geomet-
ric and topological consistency of the in-between trees during
the morphing process. Therefore, it is highly flexible to create
visually compelling morphing effects between topologically
varying leafy trees.
Limitations and future work. Our current method still
leaves room for improvement. Because our goal is to gener-
ate visually pleasing morphing effects, issues such as smooth
ramifications25 and collision-free branches26 are not explic-
itly addressed. In terms of the lobe extraction, environmental
factors, such as lighting conditions, wind effects, and so forth,
should be considered in the future. In addition, we also plan to
incorporate user interaction to our morphing pipeline, where
users may control the general crown shapes of the in-between
shapes to meet their artistic needs.
We thank the anonymous reviewers for their constructive
comments that helped improve this paper. Xiaogang Jin was
supported by the National Natural Science Foundation of
China (Grant 61472351).
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Yutong Wang received her BSc
degree in software engineer-
ing from Chongqing University,
China, in 2012. She is currently
a Ph.D candidate at the State Key
Lab of CAD&CG, Zhejiang Uni-
versity. Her main research inter-
ests include creative modeling,
computer animation, and sketch-based modeling.
Luyuan Wang received her
BSc degree in digital media
from Hunan University, China,
in 2016. She is currently a Ph.D
candidate at the State Key Lab
of CAD&CG, Zhejiang Univer-
sity. Her main research interests
include creative modeling and
virtual try-on.
Zhigang Deng is currently a Full
Professor of Computer Science
at the University of Houston
(UH) and the Founding Director
of the UH Computer Graphics
and Interactive Media (CGIM)
Lab. His research interests
include computer graphics,
computer animation, virtual human modeling and ani-
mation, and human computer interaction. He earned
his Ph.D. in Computer Science at the Department
of Computer Science at the University of Southern
California in 2006. Prior to that, he also completed
B.S. degree in Mathematics from Xiamen Univer-
sity (China) and M.S. in Computer Science from
Peking University (China). He is the recipient of a
number of awards including ACM ICMI Ten Year
Technical Impact Award, UH Teaching Excellence
Award, Google Faculty Research Award, UHCS
Faculty Academic Excellence Award, and NSFC
Overseas and Hong Kong/Macau Young Scholars
Collaborative Research Award. Besides being the
CASA 2014 Conference General Co-chair and SCA
2015 Conference General Co-chair, he currently
serves as an Associate Editor of several journals
including Computer Graphics Forum and Computer
Animation and Virtual Worlds Journal. He is a senior
member of ACM and a senior member of IEEE.
10 of 10 WAN G ET AL.
Xiaogang Jin is a professor of
the State Key Lab of CAD&CG,
Zhejiang University, China.
He received his BSc degree in
computer science in 1989, MSc
and PhD degrees in applied
mathematics in 1992 and 1995,
respectively, all from Zhejiang
University. His current research interests include traffic
simulation, insect swarm simulation, physically based
animation, cloth animation, special effects simula-
tion, implicit surface computing, computer-generated
marbling, nonphotorealistic rendering, and digital
geometry processing. He received an ACM Recogni-
tion of Service Award in 2015. He is a member of the
How to cite this article: Wang Y, Wang L, Deng
Z, Jin X. Topologically Consistent Leafy Tree Mor-
phing. Comput Anim Virtual Worlds. 2017;28:e1762.

Supplementary resource (1)

... 5 Consequently, topology-aware approaches have been proposed to generate smooth transformations between trees, achieving visually compelling special effects. [6][7][8] However, they do not put explicit efforts into preserving a sequence of morphologically continuous crowns when transforming between trees with specialized crown shapes, leading to less meaningful in-between trees (see Figure 9, top row). ...
... To tackle the challenge, we propose a self-adaptive multiscale cage-based deformation strategy to hierarchically propagate the transformations of crowns to tree branches. Specifically, the input leafy tree is first abstracted into a skeleton-lobe representation, 7 where the foliage is clustered into canonical geometries (i.e., lobes) in order to describe the tree in a hierarchical multiscale manner. The morphologically meaningful and continuous in-between trees are achieved by first gradually deforming the tree's crown according to the designer's sketch and then hierarchically propagating the deformation to tree branches based on the self-adaptive multiscale cage tree (MCT), in which the nodes are the view-dependent silhouettes serving as cages and the edges maintain their topological hierarchies. ...
... Topology-aware tree animations. Recent works 6,7 have created topologically consistent animations between two topologically varying trees. However, the work of Wang et al. 6 does not handle the animations of foliage. ...
We present a novel and intuitive sketch‐based tree animation technique, targeting on generating a new type of special effect of smoothly transforming leafy trees into morphologically different new shapes. Both topological consistencies of branches and meaningful in‐between crown shapes are preserved during the transformation. Specifically, it takes a leafy tree and a user's sketch describing the silhouette of the desired crown shape under a certain viewpoint as the input. Based on a self‐adaptive multiscale cage tree representation, branches are locally transformed through a series of topology‐aware deformations, and the resulting tree conforms to the user‐designed shape, demonstrating better aesthetics compared to global single‐cage‐based methods. By interpolating the transformations, we are able to create visually pleasing shape‐preserving animations of trees transforming between two crown shapes. Our proposed framework also provides an efficient way to interactively edit leafy trees toward desired shapes, demonstrating its potential to leverage existing tree modeling frameworks by providing flexible and intuitive tree editing operations.
... The graphical objects are often described by triangular meshes. Therefore, a 'smooth' transition indicates that during the process, no geometrical failures are generated, e.g., large distortion, non-manifold, self-intersections, etc. [19][20][21]. In geometrical point of view, this process forms a smooth deformation both on the exterior boundaries (the mesh's boundary) and interior boundaries (triangle edges inside the mesh) of a geometry, which is realized by moving a selection of vertices (also known as hot points) of the source geometry towards their corresponding vertices of the target geometry. ...
... The X-FEM scheme is shown as Fig. 3, where the intersected finite elements are divided into two parts: solid material and weak material. The stiffness matrix can be calculated as: (19) where the contributions of the solid material and weak material to the whole element stiffness matrix, K S and K w , can be expressed as: ...
... The morphing technique in computer graphics community aims to create a smooth transition from one initial graphical object to a target graphical object with different geometric shapes (Beier and Neely 1992;Jacobson et al. 2011;Zhou et al. 2014), where triangular meshes are used to describe the graphical object. The "smooth" means that triangular mesh faults such as non-manifold edges and self-intersections (Li and Iyengar 2015;Baek et al. 2015;Wang et al. 2017) do not occur during morphing. The morphing process first selects vertices (also known as hot vertices) from the initial geometry boundary and then moves these hot vertices to the corresponding vertices on the boundary of the target geometry to create smooth morphing of the initial geometry into the target geometry. ...
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This article presents a way of optimizing the conduction topology for heat-generating structures by means of transformable triangular mesh (TTM) algorithm which is implemented in an explicit and geometrical way. Unlike the traditional optimization approaches, the proposed method capitalizes on the use of a special morphing algorithm to generate optimal topologies from a genus zero surface. In this method, the initial geometry is firstly converted into triangular mesh and stored as a half-edge data structure. Then, the mesh operations (i.e., subdivision, split, and refinement) are employed to activate the geometry to move, split, and deform upon the underlying finite element mesh so that the conduction topology can be achieved by optimizing the positions and orientations of the triangular grids. The unique feature of the mesh operation is the split, which makes the geometries have different number of faces, edges, vertices as the initial one, and therefore different genus number between these geometries. This method renders the optimization process more flexibility. Finally, some examples with verification results are presented to demonstrate that TTM algorithm is capable of proposing solutions having almost the same cooling effectiveness with less computing resources compared with the commonly used density approaches.
... One such study includes a modeling method that applies the CGA Shape grammar to the growth of digital plants based on the convolution sums of divisor functions [13]. Other examples include a study using hierarchical topology-preserving, blending, and morphing techniques for tree modeling [14,15], another study involving a method for modeling trees using local convolution surface approximation, and yet another study based on simulating the growth of plant leaves that are composed of Voronoi diagrams [17]. However, these studies have shown limitations in the context of effectively generating various digital plants, since they lack in the aspect of intuitiveness from the user's perspective. ...
This study proposes a method to develop an authoring system(GeoMaTree) for diverse trees that constitute a virtual landscape. The GeoMaTree system enables the simple, intuitive production of an efficient structure, and supports real-time processing. The core of the proposed system is a procedural modeling based on a mathematical model and an application that supports digital content creation on diverse platforms. The procedural modeling allows users to control the complex pattern of branch propagation through an intuitive process. The application is a multi-resolution 3D model that supports appropriate optimization for a tree structure. The application and a compatible function, with commercial tools for supporting the creation of realistic synthetic images and virtual landscapes, are implemented, and the proposed system is applied to a variety of 3D image content.
Enabling ordinary people to create high-quality 3D models is a long-standing problem in computer graphics. In this work, we draw from the literature on design and human cognition to better understand the design processes of novice and casual modelers, whose goals and motivations are often distinct from those of professional artists. The result is a method for creating exploratory modeling tools, which are appropriate for casual users who may lack rigidlyspecified goals or operational knowledge of modeling techniques. Our method is based on parametric design spaces, which are often high dimensional and contain wide quality variations. Our system estimates the distribution of good models in a space by tracking the modeling activity of a distributed community of users. These estimates drive intuitive modeling tools, creating a self-reinforcing system that becomes easier to use as more people participate. We present empirical evidence that the tools developed with our method allow rapid creation of complex, high-quality 3D models by users with no specialized modeling skills or experience. We report analyses of usage patterns garnered throughout the year-long deployment of one such tool, and demonstrate the generality of the method by applying it to several design spaces.
We present a new method to efficiently generate a set of morphologically diverse and inspiring virtual trees through hierarchical topology-preserving blending, aiming to facilitate designers’ creativity production. By maintaining the topological consistency of the tree branches, sequences of similar yet different trees and novel intermediate trees with encouragingly interesting structures are generated by performing inner-species and cross-species blending, respectively. Hierarchical fuzzy correspondences are automatically established between two or multiple trees based on the multi-scale topology tree representations. Fundamental blending tasks including morph, grow and wilt are introduced and organized into a tree-structured blending scheduler, which not only introduces the randomness into the blending procedure but also wisely schedules the tasks to generate topology-aware blending sequences, contributing to a variety of resulting trees that exhibit diversities in both geometry and topology. Most significantly, multiple batches of blending can be executed in parallel, resulting in a rapid creation of a large repository of diverse trees.
Several applications in shape modeling and exploration require identification and extraction of a 3D shape part matching a 2D sketch. We present CustomCut, an on-demand part extraction algorithm. Given a sketched query, CustomCut automatically retrieves partially matching shapes from a database, identifies the region optimally matching the query in each shape, and extracts this region to produce a customized part that can be used in various modeling applications. In contrast to earlier work on sketch-based retrieval of predefined parts, our approach can extract arbitrary parts from input shapes and does not rely on a prior segmentation into semantic components. The method is based on a novel data structure for fast retrieval of partial matches: the randomized compound k-NN graph built on multi-view shape projections. We also employ a coarse-to-fine strategy to progressively refine part boundaries down to the level of individual faces. Experimental results indicate that our approach provides an intuitive and easy means to extract customized parts from a shape database, and significantly expands the design space for the user. We demonstrate several applications of our method to shape design and exploration.
Conference Paper
This paper presents biologically-motivated a procedural method for the simulation of leaf contour growth and venation development. We use a mathematical model for simulating the growth of a plant leaf. Leaf tissue is regarded as a viscous, incompressible fluid whose 2D expansion is determined by a spatially varying growth rate. Visually realistic development is described by a growth function RERG that reacts to hormone (auxin) sources embedded in the leaf blade. The shape of the leaf is determined by a set of feature points within the leaf contour. The contour is extracted from photos by utilizing a Curvature Scale Space (CSS) Corner Detection Algorithm. Auxin transport is described by an initial auxin flux from an auxin source to an auxin sink that is gradually channelized into cells with high levels of highly polarized transporters. The leaf is presented as a triangulated double layer structure that consists of a Voronoi-Diagram that is discretised along the vein structures.
We present an interactive tree modeling and deformation system that supports an efficient collision detection and avoidance using a bounding volume hierarchy of sweep surfaces. Starting with conventional tree models (given as meshes), we convert them into sweep surfaces and deform their branches interactively while detecting and avoiding collisions with many other branches. Multiple tree models (sharing the same topology) can be generated with great ease using this sweep-based approach, and they can serve as a basis for the generation of a multiparameter family of trees. We demonstrate the effectiveness of our approach in an automatic generation of similar trees, the colonization of trees to form a forest, and the tree growth, aging, and withering simulations. Copyright © 2015 John Wiley & Sons, Ltd.
Physically based simulation can produce quality motion of plants, but requires an authoring stage to convert plant "polygon soup" triangle meshes to a format suitable for physically based simulation. We give a system that can author complex simulation-ready plants in a manner of minutes. Our system decomposes the plant geometry, establishes a hierarchy, builds and connects simulation meshes, and detects instances. It scales to anatomically realistic geometry of adult plants, is robust to non-manifold input geometry, gaps between branches or leaves, free-flying leaves not connected to any branch, spurious geometry, and plant self-collisions in the input configuration. We demonstrate the results using a FEM model reduction simulator that can compute large-deformation dynamics of complex plants at interactive rates, subject to user forces, gravity or randomized wind. We also provide plant fracture (with prespecified patterns), inverse kinematics to easily pose plants, as well as interactive design of plant material properties. We authored and simulated over 100 plants from diverse climates and geographic regions, including broadleaf (deciduous) trees and conifers, bushes and flowers. Our largest simulations involve anatomically realistic adult trees with hundreds of branches and over 100,000 leaves.
In this paper, we propose a local convolution surface approximation approach for quickly modelling tree structures with pleasing visual effect. Using our proposed local convolution surface approximation, we present a tree modelling scheme to create the structure of a tree with a single high-quality quad-only mesh. Through combining the strengths of the convolution surfaces, subdivision surfaces and GPU, our tree modelling approach achieves high efficiency and good mesh quality. With our method, we first extract the line skeletons of given tree models by contracting the meshes with the Laplace operator. Then we approximate the original tree mesh with a convolution surface based on the extracted skeletons. Next, we tessellate the tree trunks represented by convolution surfaces into quad-only subdivision surfaces with good edge flow along the skeletal directions. We implement the most time-consuming subdivision and convolution approximation on the GPU with CUDA, and demonstrate applications of our proposed approach in branch editing and tree composition.
Given a static tree model we present a method to compute developmental stages that approximate the tree's natural growth. The tree model is analyzed and a graph-based description its skeleton is determined. Based on structural similarity, branches are added where pruning has been applied or branches have died off over time. Botanic growth models and allometric rules enable us to produce convincing animations from a young tree that converge to the given model. Furthermore, the user can explore all intermediate stages. By selectively applying the process to parts of the tree even complex models can be edited easily. This form of reverse engineering enables users to create rich natural scenes from a small number of static tree models.
(Figure Presented) We present a novel method for combining developmental tree models with turbulent wind fields. The tree geometry is created from internal growth functions of the developmental model and its response to external stress is induced by a physically-plausible wind field that is simulated by Smoothed Particle Hydrodynamics (SPH). Our tree models are dynamically evolving complex systems that (1) react in real-time to high-frequent changes of the wind simulation; and (2) adapt to long-term wind stress. We extend this process by wind-related effects such as branch breaking as well as bud abrasion and drying. In our interactive system the user can adjust the parameters of the growth model, modify wind properties and resulting forces, and define the tree's long-term response to wind. By using graphics hardware, our implementation runs at interactive rates for moderately large scenes composed of up to 20 tree models.