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Received: 12 March 2017 Accepted: 18 March 2017

DOI: 10.1002/cav.1761

SPECIAL ISSUE PAPER

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

Correspondence

Xiaogang Jin, State Key Lab of CAD&CG,

Zhejiang University, Hangzhou 310058,

China.

Email: jin@cad.zju.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 61472351

Abstract

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.

KEYWORDS

shape morphing, special effects, topological consistency, tree modeling

1INTRODUCTION

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. wileyonlinelibrary.com/journal/cav Copyright © 2017 John Wiley & Sons, Ltd. 1of10

https://doi.org/10.1002/cav.1761

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.

2RELATED WORK

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

methods.

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

3APPROACH 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).

4PROCESSING TREE MODELS

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 𝜎𝛼≤𝜃and 𝜎r≤rt,

Dg(ci),otherwise,(1)

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 𝛼ci≠𝛼min,

Dw(ci)=D(p),

(2)

where 𝛼ciis the branching angle and 𝛼min is the minimum

branching angle. The algorithm repeats until it reaches tip

branches.

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,

whorled,otherwise

(3)

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 30◦in

this paper.

4of10 WA N G ET AL.

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:

Ni∶=<GCi,GOLi,GRi,Gvi,Gpi>,

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 chnn∈GCi, the algorithm assigns its potential

foliage center vnas the tip of the chain, and the radius r′′

nas

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 U∪V,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′

i∕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

assigned.

Step 3: iteration. As a result, leaves assigned to clusters

that are initialized with ui∈Udefine the global outer lobe

GOLl+1

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∶=<

chnidi,qi,wi,hi,r′

i,s′

i,t′

i,Ti>,where

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′

i,s′

i,t′

itogether

denotes the leaf’s 3D orientation, and Tidenotes its texture.

5CORRESPONDENCES

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)):

CNi,Nj=kwk·nk

i−nk

j,(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

orientations.

6MORPHING

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

∀chni∈SP at the arc-length parametric location (1−t)

into two parts, the chn(1−t)

iand the chnt

i; (c) compute the

interpolation weights for every points n′

jon each part as

w′

j=b·t,if n′

j∈chn(1−t)

i,

t,if n′

j∈chnt

i,

where b∈(0,1]is the slow-down coefficient, which is pro-

portional to the arc-length parametric value of the point n′

j;

(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,(2.0−b)·t)when n′

j∈chn(1−t)

i.

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

6of10 WA N G ET AL.

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.

7RESULTS AND EVALUATIONS

We collected 60 real-world trees, including arbors*and

shrubs†, from the internet (i.e., http://www.evermotion.org/).

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

computer.

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

Time

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.

8CONCLUSION

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.

ACKNOWLEDGEMENTS

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

IEEE and ACM.

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.

https://doi.org/10.1002/cav.1761