SIGGRAPH 2014論文紹介 - Sound & Light + Fabrication Session

SIGGRAPH 2014輪講会
Sound & Light Session
+
Fabrication Session
Yamo
@yamo_o
7/29

このスライドは2014/7/29に行われたSIGGRAPH 2014 Preview Seminar
で発表した資料に補足説明を加えたものです。
!
Sound & Light SessionとFabrication Sessionの論文を紹介しています。
!
!
Sound & Light Sessionではサウンドレンダリング関係の、
Fabrication Sessionでは3Dプリンタ関係の論文が集められています。

Parametric Wave Field Coding for Precomputed Sound Propagation
Nikunj Raghuvanshi, John Snyder
Parametric Wave Field Coding for Precomputed Sound Propagation
Nikunj Raghuvanshi John Snyder
Microsoft Research
Figure 1: Our wave coding transforms 7D pressure fields (dependent on source/listener location and time) generated by numerical wave simulation to time-
invariant 6D fields based on four perceptual parameters. Consistent with everyday experience, these parameters vary smoothly in space, aiding compression.
Scene geometry (‘Deck’) is shown on the left, followed by a 2D slice of the parameter fields for a single source (blue dot). Direct sound loudness (LDS)
exhibits strong shadowing while early reflection loudness (LER) captures numerous scattered/diffracted paths, and consequently shadows less. Low LDS
combined with high LER conveys a distant and/or occluded source. Early decay time (TER) and late reverberation time (TLR) together indicate scene size,
reflectivity and openness. TLR is spatially smoother than TER, being determined by many more weaker and higher-order paths in this complex space.
Abstract
The acoustic wave field in a complex scene is a chaotic 7D function
of time and the positions of source and listener, making it diffi-
1 Introduction
Numerical wave simulation generates environmental sound effects
of compelling realism that complement visual effects, reveal infor-
空間の全ての位置に音源と受音点を置いたときのインパルスレスポンス(残響)のコンパクトな表現を提案しています。
メリットはパラメトリックにIRを表現してるので全ての点での前計算をしなくても補間ができるということ。

ある音源からある受音点へのIRはDirect Sound, Early Reflection,
Late Reverberation, 減衰率という4つの支配的なパラメータで特徴付
けることができます。
Figure 2: Path-dependent propagation effects.
moves the the computation’s dependence on the number of sources,
fa
ra
T
s
s
re
in
te
(m
T
re
o
th
b

IRを全部保存しておくんじゃなくて、この4パラメータを時間ドメイン差分法で前計算し
たIRから閾値適切に設定して抽出してランタイム時に使ってる。
parametric
representation
direct sound
(DS)
early reflections
(ER)
late reverberation
(LR)
5ms 200ms
LDS LER
pressure𝑃(Pa)
Impulse Response (IR)
10log∫𝑃(dB)
Loudness parameters
TER =
TLR =
Decay time parameters
Figure 3: Parametric IR encoding schematic (time not to scale). The four parameters we extract are shown in green on the right for an IR shown on the left.
precomputation to higher frequencies. Geometric and numerical
wave techniques are compared in [Siltanen et al. 2010a].
Real-time wave acoustics Prior work proposes wave-based pre-
computation with real-time auralization [Raghuvanshi et al. 2010].
Our technique reduces memory by orders of magnitude (see Sec-
tion 7), and accelerates the run-time. It also allows a true 3D
(rather than 2D) sampling of source positions to support a ﬂying
source/listener. More recent work on large, outdoor spaces [Mehra
et al. 2013; Yeh et al. 2013] requires that either the listener or the
sources be static. These techniques also require manual partitioning
3 Precomputed Sound Simulation
The input to our system is the scene geometry represented as a
“triangle soup” with associated materials, supporting typical game
maps. Scene triangles are voxelized into a 3D occupancy grid for
simulation, along with their material codes. The maximum desired
simulation frequency, ⌫max, determines the cell size . The deci-
sion is based on memory and computational constraints. We use
the ARD solver [Raghuvanshi et al. 2009] which determines voxel
size via = 3/8 min, where min = c/⌫max is the minimum
wavelength and c is the speed of sound. This represents 2.7 sam-
Parametric Wave Field Coding for Precomputed Sound Propagation • 38:3

High-Order Diffraction and Diffuse Reflections
for Interactive Sound Propagation in Large Environments
High-Order Diffraction and Diffuse Reflections for Interactive Sound Propagation
in Large Environments
Carl Schissler⇤
Ravish Mehra†
University of North Carolina at Chapel Hill
Dinesh Manocha‡
Figure 1: Our high-order diffraction and diffuse reflection algorithms are used to generate plausible sound effects at interactive rates on
large static and dynamic scenes: (left) interior office (154K triangles); (center) oil refinery (245K triangles); (right) city (254K triangles).
Abstract
We present novel algorithms for modeling interactive diffuse re-
flections and higher-order diffraction in large-scale virtual environ-
entertainment. In order to improve realism and immersion, it is im-
portant to augment visual perceptions with matching sound stimuli
and auralize the sound fields. The resulting auditory information
can significantly help the user evaluate the environment in terms of
Carl Schissler, Ravish Mehra, Dinesh Manocha
前計算無しで広いシーンでの残響をインタラクティブに計算する研究。
レイベースドでやっていて、障害物が動く動的なシーンにも対応できるのが特徴です。

S
L
(r1, s1)
(r2, s2)
(r0, s0)
T1
T2
T0
111)))
S
Figure 2: An example set of 3rd-order diffuse ray paths. Rays
leave the sound source S, hit the sequence of surface patches
where vi(~p, t) is
[0, 1], which ind
~p. This formulat
sampling noise t
gree of surface s
ing equation. In
mainly limited t
factors at runtim
complexity grow
them unsuitable
tion in dynamic s
Our approach co
division to redu
tions. We reuse t
光でも音でもレイベースドの手法で必要になるのは観測点で有効となるレイの割合を増やす事
です。
前の時間でのレイのパスを覚えておいて、かにモデルが動いた場合にも使い回して計算効率
を上げています。
レイのパス使い回す手法はレイトレーシングでは珍しくないですが、音の場合に異なるのは
レイに対していつどこで発せられた音なのかという情報が載っていることで、使い回すときに
も注意しなければいけません。

E
e1
e2 e3 e4
e5
e6
e7
e8
ee5555
1
e2 e33 eeee44
ee6666
e
e9
e10
Figure 4: A top-down view of a portion of a diffraction edge visibil-
ity graph for a small village scene, shown here for edge E. Edges
e1..10 are visible to edge E and intersect the gray-shaded areas that
represent the shadow regions for E. Our approach only considers
Figure 5: A sec
i1 and i2 are o
last image posit
the shortest pat
tions lie on the
The problem o
difﬁcult due to
他には観測点から隠れてる面はカリングしたりして計算コストを抑えるのもまぁありがち

Input Mesh Surface Voxelization Marching Cubes Surface Decimation + Merge Edges Build Edge Visibility Graph
Figure 6: Stages of our simplification algorithm: surface-voxelization of input mesh; isosurface extractions; surface decimation based on
edge-collapses; merge collinear diffraction edges; visibility graph computation
Runtime Computation: At runtime, our algorithm uses the pri-
mary rays traced in the diffuse step to determine a set of triangles
visible to each source. For each visible triangle, we check to see
if it has any diffraction edges. If so, we search the corresponding
visibility graph, moving towards the listener, with that edge as the
starting point. The recursive graph search proceeds in a depth-first
manner until a maximum depth is reached, at which point the search
backtracks and checks other sequences of edges. At each step in the
the wavelength. There has been some work on simplifying geomet-
ric models or use of level-of-detail techniques for acoustic simula-
tion [Siltanen et al. 2008; Pelzer and Vorländer 2010; Tsingos et al.
2007]. However, a key challenge in the field is to automatically gen-
erate a simplification that preserves the basic acoustic principles,
including reflections, scattering and diffraction. For example, some
techniques based on geometric reduction applied to room models
can change the reverberation time of the simplified model [Siltanen
音の場合は周波数ごとに独立してシミュレーションしないといけないんですけど、周波数に
よって十分な3Dモデルの解像度が異なります。
具体的には高い周波数のシミュレーションを行うときほど境界は細かくしないといけないの
で、この研究では空間をレベルセットで離散化してマーチングキューブで表面生成してます。
マーチングキューブのグリッドの解像度を周波数ごとに変える事で効率を上げています。

Eigenmode Compression for Modal Sound Models
Eigenmode Compression for Modal Sound Models
Timothy R. Langlois Steven S. An Kelvin K. Jin Doug L. James
Cornell University
Figure 1: Eigenmode Compression: (Left) This complex Heptoroid model’s displacement eigenmode matrix has 194 audible modes, 81884
vertices, and consumes 186 MB. By approximating each eigenmode with moving least squares (MLS), and nonlinearly optimizing the control
Timothy R. Langlois, Steven S. An, Kelvin K. Jin, Doug L. James
物体の衝突音なんかを生成する際に一般的な方法は、一定の周波数で振動する単純な変形モー
ドの足し合わせとして表現することです。
!
人間の可聴域は20∼20kHzなのでその範囲で実用的なモードの数は数千程度、これは一つのオ
ブジェクトで数GBほどになります。
実際にゲームなんかで一つのシーンに一種類のオブジェクトしか無いなんてことは無いわけで
これはオブジェクトの数だけメモリ上に置いておかなくてはいけません。なのでちょっと現実
的に使うのは難しいですね。
!
この研究ではこのモードのデータを1/100程度に圧縮する方法を提案しています

sociated displacement values w = (wi)i=1 from which an MLS
approximation can reconstruct the original mode accurately. Once
we have them, we can evaluate a scalar component of the mode
at x, by first constructing an m-degree polynomial, f(p x) =
cT
b(p x) 2 ⇧3
m, with d = (3+m)!
3!m!
coefficients c 2 Rd
, where
b(p x) 2 Rd
is a vector of monomial basis functions. Given
the control parameters p and w, the coefficients c are computed by
minimizing the MLS error,
c⇤
= arg min
c
nX
i=1
[wi f(pi x)]2
✓(pi x). (3)
Each xyz component of f is computed separately, by replacing wi
with wi,x, wi,y or wi,z in (3); we use a QR factorization to solve
the least-squares problem, which involves solving with three (xyz)
right-hand sides. Once f is fitted, the mode approximation at vertex
x is simply f(0). The weighting function ✓ controls the influence
of each control point; we use the adaptive ✓(v) = exp( ||v||2
/h2
)
defined in [Pauly et al. 2002], where h = r/3, with r the radius of
the enclosing sphere of the k nearest neighbors of x. This weight
function allows the approximation to adapt to varying control point
densities, and also improves performance since it is essentially zero
for control points with ||pj x|| > r.
Sin
w
squ
We
Le
ple
to
Mu
few
no
co
thi
to
thi
ve
Th
bia
tio
set
ent
de-
te,
ch
an
as-
LS
ce
de
=
ere
en
by
(3)
Number of points
Figure 5: MLS error convergence versus n: Adaptive MLS pro-
vides fair compression at the target error, "goal = 0.084, but our
optimized MLS fit requires even fewer control points (lower n).
4.2 Control Point Optimization
By further optimizing the n control points and weights, we can sig-
nificantly improve compression over adaptive MLS (see Figure 5).
Since the MLS approximation, ˜u, is a function of the controls p,
w 2 R3n
, we optimize their values using the nonlinear least-
squares optimization of eigenmode error at vertices V,
min
p,w
||˜u(p, w) u||2
2 = min
p,w
X
i2V
||˜ui ui||2
2. (4)
We perform this nonlinear least-squares optimization using the
Levenberg-Marquardt (LM) algorithm; we use the Ceres Solver im-
plementation [Agarwal et al. ] which uses automatic differentiation
to compute the Jacobian J = r˜u(p, w).
圧縮する方法は2つで、まず１つ目は各変形モードの全ての頂点で値を保持しないで
Moving Least Squareでパラメトリック表現します。
この論文ではこのMoving Least Square問題を非線形最小二乗問題に置き換えて
Levenberg-Marquardt法で解いています。Moving Least Squareをそのまま解かないのは
こっちのほうが制御点の数を少なくできるからだそうです。
u: mode vector
w: weight
p: control points
x: displacements

Mode 3, 1.9 kHz Mode 15, 9.3 kHz Mode 33, 16 kHz
Figure 6: Intra-mode symmetry examples: (Left) mirror symme-
try; (Middle) 4-way rotational symmetry, plus several mirror sym-
metries; (Right) cylindrical symmetry.
5.1 Intra-mode symmetry
The first symmetry we exploit is intra-mode or self symmetry (see
Figures 2 and 6). We slightly modify the geometric-symmetry
method of [Martinet et al. 2006] to detect object and eigenmode
symmetries simultaneously. Instead of a purely geometric gener-
alized moment function, we compute the generalized geometry-
(a)
Figure 7: Intr
mode (a), we de
mode. In (b), a
mirror symmetr
needed to match
use a least-squa
onal) displacem
u(x)=T u(R
Symmetry tole
eigenmode sym
ical eigenanalys
mode symmetry
ee
y
de
r-
y-
5)
n
ic
6)
7)
l.
i-
n-
or
e-
l-
i-
n
m-
e-
y
re
is
b-
r-
o
o
v-
use a least-squares solve on vertex data to estimate any (orthog-
onal) displacement transformation, T (with kT k2 ⇡ 1) such that
u(x)=T u(R x).
Symmetry tolerances: Given the approximate nature of discrete
eigenmode symmetry (due to meshing, MLS interpolation, numer-
ical eigenanalysis, etc.) we use a tolerance when confirming eigen-
mode symmetry; in our results, we use 0.02.
5.2 Inter-mode symmetry
Beyond symmetry within a single mode, an interesting character-
istic of cylindrically and n-way rotationally symmetric objects is
that they can have degenerate eigenmodes, i.e., modes with near-
equal eigen-frequencies, which form rotationally congruent pairs
(see Figures 2 and 8). If we can detect a congruent pair (j, j0
), we
only need to store one of them along with the relative rotation which
maps one to the other. We detect these pairs by summarizing the an-
gular structure of the modes in a low-dimensional Fourier basis to
find a candidate rotation, and then perform a rigorous verification
of the candidate. Furthermore, we observe that congruent pairs are
usually close to each other in frequency, so instead of doing this
for all pairs (j, j0
), we only do it for pairs such that j0
= j + 1
(assuming modes are numbered in order of increasing frequency).
Mode 10 Mode 11 Mode 19 Mode 20
7 kHz 7 kHz 11.78 kHz 11.78 kHz
Figure 8: Inter-mode symmetry: Pairs of rotationally congru-
ent eigenmodes (shown here for Lego and Wine Glass models) just
need to store one of the modes and a relative rotation.
For a given pair of modes (j, j0
), we first focus on the problem of
finding a best rotation angle j,j0 about a known symmetry axis.
For mode j, we compute Fourier-like moments,
aj
m =
R
S
kuj
(x)k eim (x)
dSx, m = ¯m . . . ¯m
that describe the mode’s amplitude variation about the rotation axis,
２つ目のアプローチはモデルの対称性に着目したものです。
これは人口の3Dモデルが大概シンメトリーな形状をしてるから有効、らしいです。
Intra-mode symmetry Inter-mode symmetry
これはさらに２通りに分かれて、単一の変形モードの中での対称性と
異なる2つのモードにおける対称性があります。

(a) (b) (c)
Figure 7: Intra-mode symmetry example: Starting with a full
mode (a), we detect symmetries and only save a small patch of the
mode. In (b), a 4-way rotational symmetry is used, and in (c), a
5.1 Intra-mode symmetry
The first symmetry we exploit is intra-mode or self symmetry (see
Figures 2 and 6). We slightly modify the geometric-symmetry
method of [Martinet et al. 2006] to detect object and eigenmode
symmetries simultaneously. Instead of a purely geometric gener-
alized moment function, we compute the generalized geometry-
eigenmode moment function of order 2p,
M2p
(ˆv) =
Z
s2S
||s ⇥ ˆv||2p
||u(s)||2p
ds, (5)
where s is a vector from the surface’s center of mass to a point on
the surface, S. It follows that their real-valued spherical harmonic
representation is given by
M2p
(ˆv) =
p
X
l=0
2lX
m= 2l
C2p
2l,m Y m
2l (ˆv), (6)
C2p
2l,m = Sl
p
Z
s2S
ksk2p
||u(s)||2p
D0,m
2l (Rs) ds, (7)
where formulae for Sl
p and D0,m
2l (Rs) are given in [Martinet et al.
2006]. By searching among roots of rM2p
(ˆv)=0, we find candi-
date symmetry axes, and then classify the symmetries of the eigen-
mode magnitudes as either cylindrical, n-way rotation, or mirror
symmetries as described in [Martinet et al. 2006]. In our imple-
mentation, we use order 2p = 8 moment functions.
u(x)=T u(
Symmetry t
eigenmode s
ical eigenana
mode symme
5.2 Inter-m
Beyond sym
istic of cylin
that they can
equal eigen-
(see Figures
only need to
maps one to t
gular structu
find a candid
of the candid
usually close
for all pairs
(assuming m
モデルの対称性をみつけるのは過去にたくさん研究があって、ここで使ってるのは簡単に
言うとモデル形状の対称性はモデルをシェルだとみなしたときのモーメントの対称性とし
てとらえると計算できる[Martinet et al. 2006]、っていう方法です。
Pizza cut
対称性がみつかるとこんなふうにピザみたいにモードを分割して対称軸と一緒に保存して
サイズを削っていきます。
さらに変形モードというのは物体の内部の振動なんですけど、物体から放射された後の音
の伝搬についても変形モードと同じ対称性を持っているのでここでもサイズ削減できます
よ、っていうことが書かれています。

Inverse-Foley Animation: Synchronizing rigid-body motions to sound
Timothy R. Langlois, Doug L. James
録音された音からそれを表現できるような剛体アニメーションを逆に作る研究です。
条件をかなり絞っていて、剛体は無限平面上に自由落下、途中に障害物などは一切無いことを
仮定しています。ここがもっと自由度が高くなると実用的なものになると思います。

流れとしては、ある3Dモデルをいろんな角度と速度, 角速度で平面に落としてシ
ミュレーションして、モデルがバウンドして静止するまでの動きをデータベースと
して持っておいて、あとは音に合わせて違和感ないようにそれらを補間しつつ繋げ
てやる、ということをしています。
Figure 2: Overview
with the input sound. Our contact-event graph can be searched for
plausible motion paths, and supports constraints so the ﬁnal con-
tact event occurs at a contact-event node which is a terminal resting
state. Additional contact constraints can be also be introduced, such
as to make an object land in a particular orientation, or to match
contact locations observed in a video capture (see Figure 14). An
overview of our approach is shown in Figure 2.
Using our approach we were able to generate plausible rigid-body
animations with realistic synchronized sound. Our system has suc-
cessfully synthesized motions for dozens of objects (see Figure 13)
and hundreds of sounds, many of which would be hard to synthesize
sounds for digitally, e.g., a scruffy bulb of garlic.
Our technique also provides a new way for animators to use sound
to design physics-based animations. Unlike in space-time keyfram-
ing or other motion control techniques, our method only requires
guess to help nonlinear optimization methods converge. In con-
trast, Inverse-Foley Animation is essentially a time-based sketch,
which lacks spatial information to help nonlinear optimization.
Random sampling techniques have been used to explore the space
of initial conditions and other simulation parameters, that could
produce desired outcomes [Tang et al. 1995]. Barzel et al. [1996]
introduced the idea of plausibility for animations, arguing that there
can be many acceptable simulations. Markov chain Monte Carlo
(MCMC) has been used to sample animations satisfying speciﬁed
constraints [Chenney and Forsyth 2000]. Similar sampling methods
can be used to optimize contact-event times, however downsides
are that optimization times can be long, and that some methods
(such as MCMC) require extensive parameter tuning. In addition,
we found that forward sampling methods have a hard time hitting
all of the contact event times, necessitating frequent restarts, and
(a) (b) (c) (d)
Figure 9: Contact Registration (top down view): (a) To transition from state i to state j+1, we register state j to state i. (b) First a
planar translation is applied to align j with i. (c) Then a planar rotation is applied to minimize the orientation error between i and j.
(d) Then we can evaluate the transition from i to j+1.
boundary value problem, and we use a forward shooting method

録音された音からいつどのくらいの間衝突しているか、っていう情報はユーザに手作業でアノテー
ションさせています。アノテーションは２種類あって、単純に一回だけ床に当たってバウンドする
のと、短い間隔でスライドしながら連続的に床に衝突する場合を指定します。
Lee and
to mod-
usly. In
nstrained
ng tech-
imulated
motions
ories due
s can im-
Bhat et
eo using
ree-ﬂight
Figure 4: Input sound with user-annotated contact-event times.
Contact event amplitudes: Given the user-annotated sound sig-
nal, we can automatically estimate contact event amplitudes, ¯a1,
. . ., ¯an, from the sound signal. For a discrete event at time ¯tk,
we set the amplitude ¯ak to be that of the nearest peak in a small

Contact Event Graph Approach
(d) (e)
Figure 7: Motion Sampling: To sample initial simulation param-
eters, we (a) sample a random orientation, (b) push the object into
contact, (c) sample linear (red) and angular (blue) velocities, (d)
simulate forwards until the object comes to rest, and (e) simulate
backwards to obtain pre-contact ballistic motion.
5.2 Contact-Event Graph Construction
The motion database is turned into a contact-event graph where
each node represents a contact event, and edges represent inter-
contact motions that transition between these contact states (see
Figure 8). The weight on each edge represents how expensive each
transition is, which measures both rigid-body contact state errors,
as well as synchronization errors when used for a specific contact-
event time.
Figure 8: Contact nodes and edges: Nodes represent contact
states, and edges represent transitions between these states. Solid
Figure 10: Transitions: G
i ! i+1 and j ! j+
registering j to i and co
5.3.1 Registering contac
To evaluate contact state sim
or to evaluate a motion tra
tational invariance on the
thereby increasing the fit qu
j, we can transform state j
registration of the two cont
and tj+ (see Figure 9). Th
translation that best aligns
and (2) a planar rotation th
qj+. There is an analytica
2001].
5.3.2 Motion Connection
Given two similar contact s
istered, we can smoothly t
computing a modified rigid
However, unless these two
terpolating the rigid-body t
tational slerp or other meth
Kumar 2002; Hofer and Po
cal distortion for motions w
velocities. Instead, we pro
computing a perturbation to
the resulting motion match
tion at j + 1. In our implem
turbations separately for th
PlasticSpoon TapeDispens
Figure 13: Virtual models (left) and r
over the n-contact motion sequence,
Score = (
Y
i
fvfqftfaflfc)
1
6n .
To shed light on motion quality, we also report the Score i
dent of the sound amplitude factors,
Scorew/o sound = (
Y
i
fvfqft)
1
3n .
Optional speedups: Since searching large contact-event
can be slow, we use several optional speedups. To avoid
ing time exploring obviously poor transitions, we only
edges where log(fv) > 10 and log(fq) > 10. Furth
we use the k-d tree to quickly find and search only the
Figure 13: Virtual models (left) and real-world objects (right) used in over
e n-contact motion sequence,
Score = (
Y
i
fvfqftfaflfc)
1
6n . (12)
d light on motion quality, we also report the Score indepen-
the sound amplitude factors,
Scorew/o sound = (
Y
i
fvfqft)
1
3n . (13)
nal speedups: Since searching large contact-event graphs
slow, we use several optional speedups. To avoid spend-
me exploring obviously poor transitions, we only explore
where log(fv) > 10 and log(fq) > 10. Furthermore,
e the k-d tree to quickly find and search only the best 5
n/transitions of each node, thereby reducing the number of
ons that must be considered during the branch-and-bound
For each recorded sound, we also use an “early exit” condi-
at terminates the search (and returns the found motion) if the
Score is sufficiently high
0.3. We also enforced
out on potentially infeasi
Lazy evaluation of blen
for edges during the grap
blends for non-simulatio
very high, e.g., in graphs
require many hours of N
blends for edges used in
introduce ground interpe
similar motions. Since i
ersome it is not allowed
feasible, and then once
its blends, then if any e
recompute the optimal su
duced enormously, and t
are infeasible. We note
and not the input sound.
一番違和感無い動きを計算するような最適化問題なので当然最小化すべき目的関数があって
この研究では、音とのシンクロ率と動きのエラー関数の値の積を使っています。
動きのエラー値っていうのはある状態から他の状態への遷移しにくさ、みたいなものです。

Contact Event Graph Approach
(d) (e)
Figure 7: Motion Sampling: To sample initial simulation param-
eters, we (a) sample a random orientation, (b) push the object into
contact, (c) sample linear (red) and angular (blue) velocities, (d)
simulate forwards until the object comes to rest, and (e) simulate
backwards to obtain pre-contact ballistic motion.
5.2 Contact-Event Graph Construction
The motion database is turned into a contact-event graph where
each node represents a contact event, and edges represent inter-
contact motions that transition between these contact states (see
Figure 8). The weight on each edge represents how expensive each
transition is, which measures both rigid-body contact state errors,
as well as synchronization errors when used for a specific contact-
event time.
Figure 8: Contact nodes and edges: Nodes represent contact
Figure 10: Transitions: G
i ! i+1 and j ! j+
registering j to i and co
5.3.1 Registering contac
To evaluate contact state sim
or to evaluate a motion tra
tational invariance on the
thereby increasing the fit qu
j, we can transform state j
registration of the two cont
and tj+ (see Figure 9). Th
translation that best aligns
and (2) a planar rotation th
qj+. There is an analytica
2001].
5.3.2 Motion Connection
Given two similar contact s
istered, we can smoothly t
computing a modified rigid
However, unless these two
terpolating the rigid-body t
tational slerp or other meth
Kumar 2002; Hofer and Po
cal distortion for motions w
velocities. Instead, we pro
computing a perturbation to
the resulting motion match
tion at j + 1. In our implem
turbations separately for th
PlasticSpoon TapeDispens
Figure 13: Virtual models (left) and r
over the n-contact motion sequence,
Score = (
Y
i
fvfqftfaflfc)
1
6n .
To shed light on motion quality, we also report the Score i
dent of the sound amplitude factors,
Scorew/o sound = (
Y
i
fvfqft)
1
3n .
Optional speedups: Since searching large contact-event
can be slow, we use several optional speedups. To avoid
ing time exploring obviously poor transitions, we only
edges where log(fv) > 10 and log(fq) > 10. Furth
we use the k-d tree to quickly find and search only the
Figure 13: Virtual models (left) and real-world objects (right) used in over
e n-contact motion sequence,
Score = (
Y
i
fvfqftfaflfc)
1
6n . (12)
d light on motion quality, we also report the Score indepen-
the sound amplitude factors,
Scorew/o sound = (
Y
i
fvfqft)
1
3n . (13)
nal speedups: Since searching large contact-event graphs
slow, we use several optional speedups. To avoid spend-
me exploring obviously poor transitions, we only explore
where log(fv) > 10 and log(fq) > 10. Furthermore,
e the k-d tree to quickly find and search only the best 5
n/transitions of each node, thereby reducing the number of
ons that must be considered during the branch-and-bound
For each recorded sound, we also use an “early exit” condi-
at terminates the search (and returns the found motion) if the
Score is sufficiently high
0.3. We also enforced
out on potentially infeasi
Lazy evaluation of blen
for edges during the grap
blends for non-simulatio
very high, e.g., in graphs
require many hours of N
blends for edges used in
introduce ground interpe
similar motions. Since i
ersome it is not allowed
feasible, and then once
its blends, then if any e
recompute the optimal su
duced enormously, and t
are infeasible. We note
and not the input sound.
これをいわゆるHMMと同じ感じで最小コストの経路を探索してやります。
グラフつくるためには近い状態のノードを探して枝で繋いでやる必要がありますが、この研
究ではこれを剛体のパラメータ空間における、２つの12次元k-dツリーを使って探索してい
ます。なんで2つかというと、物体が静止する、っていう状態と運動途中を分けて考えてい
るからです。

面白いのは、音楽なんかを与えて物理的にそれらしく動く、音楽を演奏する
剛体オブジェクトみたいなのが作れたりします。

Bridging the Gap: Automated Steady Scaffoldings for 3D Printing
Left: Scaffolding for the DNA model. Middle: After
ight: After cleanup. Print time: 3h36, 8.7m of filament.
oints through thin beams while our approach builds a full
r this object, with both the Makerware and MeshMixer
had to use a raft for the part to remain stable on the
In all other cases we use significantly less plastic than
. Our print times are comparable to Makerware, which
part due to the printing of the many small connectors.
print times could be significantly reduced by grouping
supported by a same bridge into continuous connectors.
itional Results
Figure 16: Models printed with our technique, scaffoldings on the
left and cleanup model on the right. Top: The Gymnast model.
Middle: The curved Hilbert cube model. Bottom: The 5cm Bunny
Peel model. Hilbert cube model: thingiverse.com/thing:16343 by tbuser.
Bunny peel model: thingiverse.com/thing:131054 by user meshmixer.
in approximatively 12 % of cases, leaving hanging filament in the
print. This is visible in figures showing the print before cleanup.
This has little impact on surface quality as falling filament cools
quickly and does not bond with the surface below.
7 Conclusion
We have shown how to exploit a specific property of FFF printers
— their ability to print bridges across gaps — to construct reliable
scaffoldings. Their geometry gives to our scaffolding interesting me-
chanical properties that makes them sturdier and more stable, even
at the smallest thickness ensuring that they print correctly. Our struc-
tures could probably benefit other processes such as stereolithogra-
phy — but the set of requirements are different.
Further reducing the quantity of material usage while preserving
reliability will require a precise modeling of the mechanical prop-
erties of the structure and object throughout the print process. This
is a challenging task since the plastic deposited in layers has an
anisotropic behavior which we expect to become highly nonlinear
on thin slanted structures. This is nevertheless an exciting venue
of future work. In the meantime our technique provides a simple
Jérémie Dumas, Jean Hergel, Sylvain Lefebvre
積層型の3Dプリントするときのサポート材の構造を最適化をする研究。
ブリッジとそれを支える垂直な柱からなるサポート材を自動で生成します。

ALLAIRE, G. 2006. Conception optimale d
ISBN 3-540-36710-1.
ALLEN, S., AND DUTTA, D. 1995. Determ
of support structures in layered manufactu
ALLISON, J. W., CHEN, T. P., COHEN, A.
SNEAD, D. E., AND VORGITCH, T. J.,
comparison slice. US Patent 5854748, 3D
CHALASANI, K., JONES, L., AND ROSCO
generation for fused deposition modelin
Fabrication Symposium, 229–241.
CHENG, W., FUH, J., NEE, A., WONG
MIYAZAWA, T. 1995. Multi-objective opti
ing orientation in stereolithography. Rapi
1, 12–23.
EGGERS, G., AND RENAP, K., 2007. Met
automatic support generation for an obje
a rapid prototype production method. US
Materialize.
FRANK, D., AND FADEL, G. 1995. Expert
of the preferred direction of build for rapid
Journal of Intelligent Manufacturing 6, 5,
HEIDE, E., 2011. Method for generating and
tures with deposition-based digital manu
US Patent 20110178621 A1.
HUANG, X., YE, C., MO, J., AND LIU, H.
support generation algorithm for fused de
inghua Science and Technology 14, S1, 22
HUANG, X., YE, C., WU, S., GUO, K., AND
wall structure support generation for fuse
The International Journal of Advanced M
ogy 42, 11-12, 1074–1081.
左が提案手法で右がMeshMixerで出力した従来の木構造サポート。木構造だと見た目から
して強度に不安がありそうな感じします。

ional bias — such as axis aligned bridges only — would generate
a larger number of pillars when supporting features at an angle. We
also note that while thin slanted pillars become less reliable as their
ength increases, they can print reliably on short distances. This
s particularly useful when trying to support several points with a
rectilinear bridge: perfect alignments are unlikely. Our algorithm
herefore has the ability to connect vertical pillars to other elements
by adding a small slanted connector at their top.
5.1 Bridge Gain and Score
Our algorithm enumerates and selects new bridges that improve
he current solution. It therefore requires a function to estimate
he benefit of a new bridge. We approximate the bridge benefit
by counting the gain and loss in terms of pillar and bridge length.
Z
hb
wb
lmin
lmax
Following notations in the inset, a bridge of
ength wb at height hb supporting k elements
provides a gain of Gain(b) = (k 2)hb wb.
Clearly, only bridges supporting more than two
points can be beneficial. Our algorithm only
nserts bridges where Gain(b) > 0.
When deciding which bridge to insert we com-
pute a score for each bridge. The score is:
Score(b) = Gain(b) k · lmax(b), where
max(b) is computed as the maximum length
of the structure connecting an element above to the bridge. It takes
nto account non vertical parts that may occur when an element
above is not in the vertical plane of the bridge. The score penalizes
uneven distributions of connection lengths above the bridge. The
CoM disk is
nlarging the
the arbitrary
ons may no
and tag the
may not be
mall bridge
ure 6, right,
scaffolding
m the stabil-
en enlarged
omated raft
ngs for this
of the structure connecting an element above to the
into account non vertical parts that may occur wh
above is not in the vertical plane of the bridge. The
uneven distributions of connection lengths above th
bridge giving the best (possibly negative) score will
In cases where the bridge extremities are above the
the free length of each vertical pillar instead of the
Gain(b) = k · hb h1 h2 wb with h1 and h2 th
pillars before reaching the object.
lmin (see the inset) is a parameter fixing the minim
tween a bridge and a supported point (1.6 mm in
tation). Note that lowering the bridge would only
Thus bridges have maximal gain at a distance lmin b
of the elements they support. This provides a wa
enumerate possible bridge heights.
5.2 Construction Algorithm
ブリッジの強度はサポートする点の数kと幅wと高さhの関数になっていて、こ
れを最大化するようにブリッジを配置していきます。
サポートが必要な点は、ノズルの直径の半分以上が下のレイヤーからはみ出
した点として定義されています。

X
Y
sweep
bridges?
Figure 7: Two bridges and an isolated point as well as their cor-
responding anchoring segments for a sweep along the X axis. The
green squares are events considered during the sweep. The pur-
ple line illustrates the YZ sweep plane when examining one event.
Algorithm
Input: A
br
Output:
1 Initialize
2 while tru
3 best
4 for i
5 S
6 P
c
7 Q
8 w
平面をスイープさせてサポート点群をスキャンしていって、繋げられる２点が
見つかるとブリッジを生成、それで強度が上がれば採用っていうGreedyなア
プローチでやってるようです。

Computational Light Routing: 3D Printed Optical Fibers For Sensing and Display
Boston
still few algorithms
olid object. Existing
mselves to light dif-
to fabricate objects
reflection to guide
sign algorithms to-
t between two arbi-
ansmission by min-
on while respecting
the influence of dif-
opagation in the vol-
fiber. Our methods
itrary shape, touch-
light distribution in
Graphics]: Compu-
ased modeling
Fig. 1. Total internal reflection happens because of the higher refractive
index in the core. This allows good propagation of light inside an optical
fiber.
1. INTRODUCTION
Despite recent advances there are still few fabrication techniques
and algorithms that let us control how light propagates inside a
solid object. Existing methods design surfaces that reflect [Weyrich
et al. 2009; Matusik et al. 2009] and refract light [Papas et al. 2011;
Finckh et al. 2010] or restrict themselves to reproducing light dif-
fusion in solid objects [Dong et al. 2010; Haˇsan et al. 2010]. We
2 • T. Pereira et al.
Fig. 2. We use 3D printing to fabricate objects with embedded optical fibers that route light between two interfaces. We use this pipeline it to create displays
of arbitrary shape, such as this animated face. Given a parameterized output surface (left), our algorithm automatically designs the fibers (middle-left) to
maximize light transmission. We use a micro-projector to input an image (inset) on the printed object’s (middle) flat interface, and it is routed to the surface
(middle-right). We also present a painting application in which fibers are used for sensing and display. The light from a touch-sensitive infrared pen (right) is
routed through the object to a camera.
THIAGO PEREIRA, SZYMON RUSINKIEWICZ, WOJCIECH MATUSIK
3Dプリンタで好きな形の立体形状ディスプレイを作る研究。

Boston
sign algorithms to-
t between two arbi-
ansmission by min-
on while respecting
ﬁber. Our methods
Graphics]: Compu-
ased modeling
ﬁber.
1. INTRODUCTION
一面だけ平面になっててそこに平面ディスプレイをくっつけると光ファイバーの束に光
が入ってきて, モデルの中を通って表面に表示されます。
光ファイバーなのでもちろんタッチパネルにすることもできます。

Boston
sign algorithms to-
t between two arbi-
ansmission by min-
on while respecting
ﬁber. Our methods
Graphics]: Compu-
ased modeling
ﬁber.
1. INTRODUCTION
2種類の材質を使って光ファイバーごとモデルを3Dプリントしてやります。
ユーザが与えるのはモデル表面のうちディスプレイ当てる平面と表示面のuvコーディネート。

Curvature
Computational Light Routing: 3D Printed Optical Fibers For Sensing and Display • 5
Fig. 8. Light propagation inside a poorly designed object cross-section.
Since we are imaging from the side what we actually observe is scattering
along the volume. While some light arrived at its destination even for com-
plex routes, much light is leaking and scattering through the volume. The
nal Light Routing: 3D Printed Optical Fibers For Sensing and Display • 5
ct cross-section.
erve is scattering
on even for com-
the volume. The
a high curvature
ent direction.
a measurement
ds on the bend-
the length of a
ent is not only
ding radius, but
o fittings to this
his exponential
d both in theory
fibers with di-
h [Gloge 1972;
1R), where at-
s to the straight
of 15% and 68
3%. While this
he exponential
decrease of ↵.
f our measured
ion. These are
raight segment
curved part of
measurement,
g error.
15 bending at-
0% and now 68
e 7 shows both
on coefficients
n coefficient as
nite radius.
n is greatly re-
vates our algo-
otal internal re-
tion introduces
d printer voxel
use light to leak
fibers to route
Fig. 9. Sample results show curvature optimized routes while respecting
user provided parametrization constraints. The input and output surfaces
can be arbitrary as shown in these cylinder and sphere routings.
Fig. 10. On the right, fibers generated by minimizing the thin-plate energy
resulting in higher curvature in concentrated regions. On the left, minimiz-
ing the third derivative energy which results in more uniform curvature.
Plots display color coded curvature at different scales.
parameterization. We also show how to incorporate additional de-
grees of freedom into our optimization by automatically selecting
a parameterization of the volume’s flat interface (Subsection 4.3).
We propose an implicit formulation for the routing problem. Our
algorithm receives as input both an input and an output surface,
together with their u, v parameterizations. It then calculates u, v
coordinates for every point in space by solving a variational prob-
lem. Each fiber can be seen as the set of points in space that have a
given u0, v0 coordinate — in other words, a level set. In our current
formulation, we solve for both u and v as separate optimization
problems, so from now on we will only discuss u.
Figure 9 shows sample results of our algorithm for motivation.
The green and blue points represent the input and output surfaces
n.
g
m-
he
re
nt
d-
a
y
ut
is
al
y
i-
2;
t-
ht
8
is
al
↵.
d
e
nt
of
t,
t-
8
h
ts
as
e-
o-
e-
es
el
Fig. 9. Sample results show curvature optimized routes while respecting
user provided parametrization constraints. The input and output surfaces
can be arbitrary as shown in these cylinder and sphere routings.
Fig. 10. On the right, fibers generated by minimizing the thin-plate energy
resulting in higher curvature in concentrated regions. On the left, minimiz-
ing the third derivative energy which results in more uniform curvature.
Plots display color coded curvature at different scales.
parameterization. We also show how to incorporate additional de-
grees of freedom into our optimization by automatically selecting
a parameterization of the volume’s flat interface (Subsection 4.3).
We propose an implicit formulation for the routing problem. Our
algorithm receives as input both an input and an output surface,
together with their u, v parameterizations. It then calculates u, v
coordinates for every point in space by solving a variational prob-
lem. Each fiber can be seen as the set of points in space that have a
given u0, v0 coordinate — in other words, a level set. In our current
formulation, we solve for both u and v as separate optimization
erization jointly with fiber routing
uch less curvature (last row) and
w). On the top right, we show the
timized fiber placement.
o keep the base from growing too
u, v and solve for the corresponding x, y position to start the fi
When there are multiple answers we found it adequate simply
choose the one nearest to the projected center of the mesh.
Base layout introduced an undesirable side effect when desi
ing an inward looking hemisphere that routes light from a pl
(Figure 14, right). Both our energies force the base to grow v
large, since that reduces both curvature and compression (Fig
13, left). Since it is impractical to make these very large obje
we added an extra objective term to keep stretch low. This te
works as a weak quadratic prior that pulls ux, uy, vx, vy towa
their mean values on the target surface. Figure 13 shows how
term provides control over stretch. Both curvature and compress
are volumetric terms, while stretch is an area term. We norma
all energies by volume and area respectively before adding them
After the addition of the base layout constraints and the stre
energy term, our optimization problem is written below.
minimize
u
C(u) + wkK(u) + wsS(u)
subject to u(x) =
X
i
↵ihi(x), x 2 B,
u(x) = g(x), x 2 Q,
ru(x) · n(x) = 0, x 2 Q [ B.,
Using this algorithm, we routed and printed a few different s
faces (Figure 14). The parameters used and some summary sta
tics of routing quality including curvature and compression
Compression Stretching
u: displacements
綺麗に表示させるためにはいくつか条件があって、例えば
モデルの中を通る光ファイバーはできる限りまっすぐ伸び
ていたほうがロスが少ないので曲率は最小化する必要があ
ります。
また、光ファイバー同士の距離が近過ぎると干渉するので
できる限り距離は最大化(逆数を最小化)します。
あとは表示面での歪みも最小化してやる必要があるので3
つ目の項が入ってきます。
これを内点法で最小化してます。

An Asymptotic Numerical Method for Inverse Elastic Shape Design
otic Numerical Method for Inverse Elastic Shape Design
Xiang Chen⇤
Changxi Zheng†
Weiwei Xu‡
Kun Zhou⇤§
&CG, Zhejiang University †
Columbia University ‡
Hangzhou Normal University
objects greatly eases the design ef-
ired target shapes without thinking
ving this problem using classic it-
aphson methods), however, often
oward a desired solution. In this
c numerical method that exploits
cture of speciﬁc nonlinear material
magnitude faster than traditional
this method to compute rest shapes
e rest shape of an elastic object is
fabrication the real object deforms
e the performance and robustness
elastic fabrication experiments.
r Graphics]: Computational Geom-
ically based modeling;
D printing, ﬁnite element methods,
(a) (b) (c)
3754 seconds 7 seconds
(d) (e)
Figure 1: Plant: Top: Our method computes the rest shape of a
Xiang Chen, Changxi Zheng, Weiwei Xu, Kun Zhou
やわからめの材質でモデルを3Dプリントした場合に例えばこんな木だと垂れ下がってしまうの
で、力が加わったときの変形後の形状をユーザが指定して、そこから変形前の3Dプリントすべ
き元形状を求める研究。

An Asymptotic Numerical Method for Inverse Elastic Shape Design
otic Numerical Method for Inverse Elastic Shape Design
Xiang Chen⇤
Changxi Zheng†
Weiwei Xu‡
Kun Zhou⇤§
&CG, Zhejiang University †
Columbia University ‡
Hangzhou Normal University
objects greatly eases the design ef-
ired target shapes without thinking
ving this problem using classic it-
aphson methods), however, often
oward a desired solution. In this
c numerical method that exploits
cture of speciﬁc nonlinear material
magnitude faster than traditional
this method to compute rest shapes
e rest shape of an elastic object is
fabrication the real object deforms
e the performance and robustness
elastic fabrication experiments.
r Graphics]: Computational Geom-
ically based modeling;
D printing, ﬁnite element methods,
(a) (b) (c)
3754 seconds 7 seconds
(d) (e)
Figure 1: Plant: Top: Our method computes the rest shape of a
Xiang Chen, Changxi Zheng, Weiwei Xu, Kun Zhou
ANM自体は以前からあるんですけど、それを物理シミュレーションに利用したのがこの論文の
新しいところです。
特にNeo-Hookeanっていう超弾性体のモデルでANMを使えるように定式化してます。

rse shape design tool automatically computing a rest shape
orms into a desired target shape under given external forces
ure 1(a-c)).
eneral, the inverse shape design problem amounts to solvi
c equilibrium equation,
f(x, X) + g = 0,
xchen.cs@gmail.com, kunzhou@acm.org
cxz@cs.columbia.edu
weiwei.xu.g@gmail.com
Corresponding author
解くべき問題はこんな式で表されます。gは重力だとか外力でfがモデル内部
の力です。
ある外力を与えたときにユーザが変形してほしいと思うときの変位xから変
形前の変位Xを求めたいわけです。
!
f: internal forces
x: displacements after deformed
X: displacements at rest pose
g: external forces (ex gravity)

1 Asymptotic Numerical Method
r goal is to solve Eq. (2), in which x is provided by the u
d X is the unknown rest shape. First, consider a parameteri
sion (a so-called homotopy) of Eq. (2),
f(x, X) + g = 0,
ere is a loading parameter in the range [0, 1]. When =
solution of Eq. (5) is clearly X = x up to a rigid transfor
n, since only an undeformed shape produces a vanishing inte
ce. When = 1, its solution is what we desired, i.e., the solu
Eq. (2). The basic idea of ANM is derived from numerical con
ion methods [Allgower and Georg 1990]: in a step-wise man
hanges the parameter by following an implicitly deﬁned cu
a) starting from (0) = 0. At each step, a new a is selected
これをこんなふうに新しい変数をつけて表します。解きたいのは当然λが1の場
合です。あとλ=0のときは変形しないので既知です。
提案手法は、これを最初からいきなりλ=1で解くよりもλを0から少しずつ1に
向かって近づけながら解いていったほうがずっと速く解ける、というものです。
!
f: internal forces
x: displacements after deformed
g: external forces (ex gravity)
λ: [0,1]

Figure 3: Visualization of ANM Steps: The x-axis indicates the norm of kX(a) xk2, while the y-axis indicates the corresponding (a)
such that X(a) solves f(x, X(a)) + (a)g = 0. Given a target cactus shape named as bifur3 (in the inset), the ANM first computes an
asymptotic expansion of (X(a), (a)) (green curve in the first figure) at a = 0 to track the solution branch locally. It then changes the value
of a along the expansion branch as far as possible, until the convergence radius is reached. From there, it refines the solution and creates a
new expansion (green curve in the second figure). This process is repeated, until (a) = 1 is reached.
solution X, causing slow convergence or even instability in the
iterative solver. We refer the reader to [Allgower and Georg 1990]
for a detailed explanation of the motivation of introducing a.
Tracking Solution using Asymptotic Expansions. Consider a sin-
gle step of ANM. Let a0 denote the current parameter value of a
step. We have 0 = (a0) and the corresponding solution X0 that
satisfies f(x, X0) + 0g = 0. Without an explicit definition of
(a), ANM expresses (a) and its corresponding solution using a
power series expansion around a0
  nX 
4.2.1. Mathematical Insights Before diving into our derivation
details of computing the coefficients {Xk, k} for Eq. (6), we first
present the critical insights that lead to fast solves for the coefficients.
Suppose for a moment the force function f has a quadratic form of
X. Namely,
f(x, X) = L0 + L[X] + Q[X, X], (7)
where L[?] and Q[?, ?] are respectively a linear and bilinear vector
valued operators of vector inputs. Substituting the expansion (6) of
X(a) into this expression yields a quadratic series,
Algorithm 1 ANM Tracing
Set X0 = x, 0 = 0, a0 = 0; {initial starting point}
while < 1 do
Solve the polynomial coefficients {Xk, k}, k = 1...n;
Calculate reliable change of a based on residual estimation;
Refine X(a) by Newton-Raphson method;
Set X0 = X(a), 0 = (a), a0 = a;
end while
However, both Xk and
constrained linear system w
in (10). To get a full-rank sy
as suggested by Cochelin e
(X(a) X0)T
Essentially, this constraint
the projection of state incr
tangent vector (X1, 1). Af
and (a) into Eq. (11), and equating the coefficients of p
(a a0), we have one more equation for every (Xk, k),
XT
k X1 + k 1 = k1, k = 1...n,
where k1 is the Kronecker delta: k1 = 1 if k = 1,
otherwise. When k = 1, Eq. (12) requires [X1, 1] t
unit 2-norm. We therefore simply normalize one solutio
under-constrained linear system (9). When k > 1, pu
constraint (12) together with Eq. (10) yields a full-rank linea
of (Xk, k) (see details in Appendix A).
We note that all the linear systems for (Xk, k) are ver
solve. Indeed, as detailed in Appendix A, all the linear sys
A

Xk
k
= bk,
share the same matrix A, which is exactly the Jacobian of
linear function f(x, X) + g at (X0, 0). Namely,
A =
@
@(X, )
(f(x, X) + g)
(X , )
.
e 3: Visualization of ANM Steps: The x-axis indicates the norm of kX(a) xk2, while the y-axis i
hat X(a) solves f(x, X(a)) + (a)g = 0. Given a target cactus shape named as bifur3 (in the in
totic expansion of (X(a), (a)) (green curve in the first figure) at a = 0 to track the solution branch
long the expansion branch as far as possible, until the convergence radius is reached. From there, it r
xpansion (green curve in the second figure). This process is repeated, until (a) = 1 is reached.
on X, causing slow convergence or even instability in the
ve solver. We refer the reader to [Allgower and Georg 1990]
detailed explanation of the motivation of introducing a.
ing Solution using Asymptotic Expansions. Consider a sin-
ep of ANM. Let a0 denote the current parameter value of a
We have 0 = (a0) and the corresponding solution X0 that
es f(x, X0) + 0g = 0. Without an explicit definition of
ANM expresses (a) and its corresponding solution using a
series expansion around a0

X(a)
(a)
⇡

X0
0
+
nX
k=1
(a a0)k

Xk
k
, (6)
re n is the truncation order; the set of coefficients,
k}, k = 1...n, are what we need to compute at the current
After establishing this local power series, we start to change a
d the value satisfying (a) = 1. Inevitably, as we move a away
4.2.1. Mathematical Insights B
details of computing the coefficien
present the critical insights that lead
Suppose for a moment the force fu
X. Namely,
f(x, X) = L0 + L
where L[?] and Q[?, ?] are respec
valued operators of vector inputs.
X(a) into this expression yields a
f(x, X(a)) = L0 + L[X0] + Q
+ (a a0) (L[X1
+
nX
k=2
(a a0)k
L[Xk] + 2Q[X
a: implicit parameter
(a a0), we have one more equation for every (Xk, k)
XT
k X1 + k 1 = k1, k = 1...n,
otherwise. When k = 1, Eq. (12) requires [X1, 1]
unit 2-norm. We therefore simply normalize one soluti
under-constrained linear system (9). When k > 1, p
constraint (12) together with Eq. (10) yields a full-rank line
We note that all the linear systems for (Xk, k) are ve
solve. Indeed, as detailed in Appendix A, all the linear sy
A

Xk
k
= bk,
share the same matrix A, which is exactly the Jacobian o
A =
@
@(X, )
(f(x, X) + g)
(X0, 0)
.
Recall that as described in §4.1, after we change a to a n
このλと求めるレストポーズXはaっていうパラメータの多項式で表現されていてa=0
のときλは0になります。

  nX 
X. Namely,
f(x, X) = L0 + L[X] + Q[X, X], (7)
while < 1 do
Set X0 = X(a), 0 = (a), a0 = a;
end while
(X(a) X0)T
XT
k X1 + k 1 = k1, k = 1...n,
A

Xk
k
= bk,
A =
@
@(X, )
(f(x, X) + g)
(X , )
.

X(a)
(a)
⇡

X0
0
+
nX
k=1
(a a0)k

Xk
k
, (6)
X. Namely,
f(x, X) = L0 + L
f(x, X(a)) = L0 + L[X0] + Q
+ (a a0) (L[X1
+
nX
k=2
(a a0)k
L[Xk] + 2Q[X
XT
k X1 + k 1 = k1, k = 1...n,
A

Xk
k
= bk,
A =
@
@(X, )
(f(x, X) + g)
(X0, 0)
.
で、aを少しずつ増加させていくとλとXも変化していって、ある程度までいくと誤差
が増大します。

  nX 
X. Namely,
f(x, X) = L0 + L[X] + Q[X, X], (7)
while < 1 do
Set X0 = X(a), 0 = (a), a0 = a;
end while
(X(a) X0)T
XT
k X1 + k 1 = k1, k = 1...n,
A

Xk
k
= bk,
A =
@
@(X, )
(f(x, X) + g)
(X , )
.

X(a)
(a)
⇡

X0
0
+
nX
k=1
(a a0)k

Xk
k
, (6)
X. Namely,
f(x, X) = L0 + L
f(x, X(a)) = L0 + L[X0] + Q
+ (a a0) (L[X1
+
nX
k=2
(a a0)k
L[Xk] + 2Q[X
XT
k X1 + k 1 = k1, k = 1...n,
A

Xk
k
= bk,
A =
@
@(X, )
(f(x, X) + g)
(X0, 0)
.
この誤差が閾値を超えたところでNewton-Raphson法を使ってλとXを修正してやっ
て、またそこを初期値としてaを変化させます。

  nX 
X. Namely,
f(x, X) = L0 + L[X] + Q[X, X], (7)
while < 1 do
Set X0 = X(a), 0 = (a), a0 = a;
end while
(X(a) X0)T
XT
k X1 + k 1 = k1, k = 1...n,
A

Xk
k
= bk,
A =
@
@(X, )
(f(x, X) + g)
(X , )
.

X(a)
(a)
⇡

X0
0
+
nX
k=1
(a a0)k

Xk
k
, (6)
X. Namely,
f(x, X) = L0 + L
f(x, X(a)) = L0 + L[X0] + Q
+ (a a0) (L[X1
+
nX
k=2
(a a0)k
L[Xk] + 2Q[X
XT
k X1 + k 1 = k1, k = 1...n,
A

Xk
k
= bk,
A =
@
@(X, )
(f(x, X) + g)
(X0, 0)
.
これがλ=1になるまで繰り返せばめでたく正しいレストポーズXが求まっている、と
いうのが大枠です。

この手法のいいところはxと Xをひっくり返せばそのまま逆問題だけじゃなく普通
の静的解析にも使えるという点で、いろんな3次元のインタラクティブな物理シ
ミュレーションを高速化できるよと言っています。
Figure 7: Phone Holder: We compute a rest shape of a phone holder (a) based on its target shape under working forces (b) for clamping a
cell phone. We then fabricate the computed rest shape (c). As shown in (d), its mouth clamps a cell phone tightly as predicted.
Figure 8: Hanger: We compute a rest shape of hanger model (a) given its target shape under gravity (b) and target shape under working
forces (c). The fabrication of the rest shape has a horizontal bottom bar under gravity (d). The shape under the target work load (e) is visually
similar to the designed target shape. The weight of cloth is shown in (f).
Figure 9: Multi-target Dinosaur: We compute a rest shape for the dinosaur model given multiple designed target shapes (top row). When

Build-to-Last: Strength to Weight 3D Printed Objects
Lin Lu, Andrei Sharf, Haisen Zhao, Yuan Wei, Qingnan Fan, Xuelin Chen, Yann
Savoye, Changhe Tu, Daniel Cohen-Or, Baoquan Chen
Lin Lu1⇤
Andrei Sharf2
Haisen Zhao1
Yuan Wei1
Qingnan Fan1
Xuelin Chen1
Yann Savoye2
Changhe Tu1
Daniel Cohen-Or3
Baoquan Chen1†
1
Shandong University 2
Ben-Gurion University 3
Tel Aviv University
Figure 1: We reduce the material of a 3D kitten (left), by carving porous in the solid (mid-left), to yield a honeycomb-like interior structure
which provides an optimal strength-to-weight ratio, and relieves the overall stress illustrated on a cross-section (mid-right). The 3D printed
hollowed solid is built-to-last using our interior structure (right).
Abstract
The emergence of low-cost 3D printers steers the investigation of
new geometric problems that control the quality of the fabricated
object. In this paper, we present a method to reduce the material
cost and weight of a given object while providing a durable printed
model that is resistant to impact and external forces.
We introduce a hollowing optimization algorithm based on the
concept of honeycomb-cells structure. Honeycombs structures are
known to be of minimal material cost while providing strength
Links: DL PDF
1 Introduction
Recent years have seen a growing interest in 3D printing technolo-
gies, capable of generating tangible solid objects from their digital
representation. Typically, physically printed objects are built by
successively stacking cross-section layers of powder-based mate-
rial. Layers are generated through fused-deposition modeling and
liquid polymer jetting. Hence, the production cost of the result-
ing model is directly related to the volume of material effectively
“Honeycombs structures are known to be of minimal material
cost while providing strength in tension”
材料のコストを抑えつつ、できるだけ軽く、かつ頑丈なモデルを3Dプリンタで出
力できるように最適化する研究です。
honeycomb-cells structure、つまり蜂の巣状の内部構造をモデルに持たせます。

Lin Lu, Andrei Sharf, Haisen Zhao, Yuan Wei, Qingnan Fan, Xuelin Chen, Yann
Savoye, Changhe Tu, Daniel Cohen-Or, Baoquan Chen
Lin Lu1⇤
Andrei Sharf2
Haisen Zhao1
Yuan Wei1
Qingnan Fan1
Xuelin Chen1
Yann Savoye2
Changhe Tu1
Daniel Cohen-Or3
Baoquan Chen1†
1
Shandong University 2
Ben-Gurion University 3
Tel Aviv University
Figure 1: We reduce the material of a 3D kitten (left), by carving porous in the solid (mid-left), to yield a honeycomb-like interior structure
which provides an optimal strength-to-weight ratio, and relieves the overall stress illustrated on a cross-section (mid-right). The 3D printed
hollowed solid is built-to-last using our interior structure (right).
Abstract
The emergence of low-cost 3D printers steers the investigation of
new geometric problems that control the quality of the fabricated
object. In this paper, we present a method to reduce the material
cost and weight of a given object while providing a durable printed
model that is resistant to impact and external forces.
We introduce a hollowing optimization algorithm based on the
concept of honeycomb-cells structure. Honeycombs structures are
known to be of minimal material cost while providing strength
Links: DL PDF
1 Introduction
Recent years have seen a growing interest in 3D printing technolo-
gies, capable of generating tangible solid objects from their digital
representation. Typically, physically printed objects are built by
successively stacking cross-section layers of powder-based mate-
rial. Layers are generated through fused-deposition modeling and
liquid polymer jetting. Hence, the production cost of the result-
ing model is directly related to the volume of material effectively
“Honeycombs structures are known to be of minimal material
cost while providing strength in tension”
まぁいわゆるボロノイ的なあれですね。
材料の量と強度のトレードオフをユーザがコントロールできるところが売りのようです。

(a) (b) (c) (d) (e)
Figure 2: Given a 3D shape of a shark and external forces we compute an initial stress map (a) and generate a corresponding interior point
distribution (b). We compute the lightest interior that sustains the given stress through an optimization process. We show here two steps (c-d)
of the optimization and an optimal strength-to-weight ratio in (e).
set of sites, the Voronoi diagram deﬁnes a space partitioning in-
to closed-cells of nearest regions with respect to the sites [Voronoi
1908]. As the number of sites increases, Voronoi cells converge
to hexagonal honeycomb-like shapes [Bronstein et al. 2008], pro-
ducing a structure of high strength-to-weight ratio for any mate-
the object strength and interior amount of material.
2 Related Work
Recent years have shown a growing interest in 3D printing tech-
最初にストレスマップというのを作ります。これはモデル内部にかかる内力
の分布を表しています。
この分布からボロノイテッセレーションしてまず初期状態とします。
ここからセルの総数とセル内部の空洞部の割合をアダプティブなモンテカル
ロ法を使って最適化してやります.
stress map

Results
Figure 10: We build-to-last and 3D print our models as well as their hollowed honeycomb-like interiors. A standard key is as the size
reference.
Model
Solid Vol.
(cm3
)
Result Vol.
(cm3
)
Ratio
(%)
Stress
(N/m2
)
Chair 719.24 472.03 65.6 4.00e7
Cup 214.4 89.33 41.7 4.01e7
Fertility 54.24 20.02 36.9 4.01e7
Hangingball 226.66 58.5 25.8 2.65e7

Spin-It: Optimizing Moment of Inertia for Spinnable Objects
Moritz Bacher, Emily Whiting, Bernd Bickel, Olga Sorkine-Hornung
3Dプリンタで好きな形のコマとかヨーヨーみたいな回転体を作ることができるってやつです。
Moritz B¨acher
Disney Research Zurich
Emily Whiting
ETH Zurich
Bernd Bickel
Olga Sorkine-Hornung
ETH Zurich
(a) unstable input (b) hollowed, optimized model (c) our spinning top design (d) elephant in motion
Figure 1: We introduce an algorithm for the design of spinning tops and yo-yos. Our method optimizes the inertia tensor of an input model
by changing its mass distribution, allowing long and stable spins even for complex, asymmetric shapes.

Moritz Bacher, Emily Whiting, Bernd Bickel, Olga Sorkine-Hornung
去年make it standっていうモデルをバランスよくちゃんと立つようにする研究があり
ましたがそれの続き的な感じで回してみたらしい。
ちなみに需要があるのかどうかは。
ヨーヨーってミニ四駆と並んで10年周期ぐらいで流行ってるから次のブームのときに
は自分でオリジナルのデザインができる要素も入ってくると面白いのかも。
Moritz B¨acher
Emily Whiting
ETH Zurich
Bernd Bickel
Olga Sorkine-Hornung
ETH Zurich
(a) unstable input (b) hollowed, optimized model (c) our spinning top design (d) elephant in motion
Figure 1: We introduce an algorithm for the design of spinning tops and yo-yos. Our method optimizes the inertia tensor of an input model
by changing its mass distribution, allowing long and stable spins even for complex, asymmetric shapes.

1. 重心が回転軸上にあること
2. 重心位置が接地点になるべく近く、さらに軽いこと
3. 回転軸と最大となる主要慣性軸が平行であること
4. 最大となる主要慣性軸の大きさが他の主要慣性軸に比べて支配的であること
For yo-yos, the gravitational torq
spin if we neglect the effect of an
Moment of inertia is the analo
and measures the resistance to ro
ler’s equations from classical mec
2001]) conveniently describe the r
its body frame, whose axes are t
and the origin is c. Since there is
body (for c on the spinning axis)
with constant angular velocity if i
For an
an eq
inertia
the pr
E with
khbk
E’s pr
to its
mome
sum o
axes’ lengths (omitting a commo
the inset. Hence, the maximal pri
to the shortest axis hc, and we h
(a) (b) (c)
Figure 2: Spinning Yo-yos and Tops stably: For spinning tops,
the center of mass must lie on the user-speciﬁed spinning axis a,
otherwise it will cause an unbalanced external torque |⌧| = Mgd
relative to p (a). For slower angular velocities, the precession an-
モデルが回転するための条件

この条件を満たすようにモデル内部をオクツリーボクセルで埋めていきます
Figure 9: “Teapot”: (Left) Hollowed result showing voxelized inte-
rior mass and aligned axes using ftop = fyo-yo. (Middle) Lowering
of the center of mass shifts the mass distribution closer to the con-
tact point. If we include mass reduction (right), mass is reduced
inward out, resulting in the design with highest rotational stability.
Figure 11: “Dancing Couple”. (Top: left to right) I
with principal axes rotated away from spin frame; afte
the dominant primary axis is still not aligned; optimi
96:8 • M. Bächer et al.
employ a multi-resolution voxelization based on an adaptive
e, thereby significantly reducing the number of fill variables.
discretized volume integrals then become
s⌦ ⌦0 = s⌦
X
k
k s⌦k
e ⌦i =
S
k ⌦k is a partitioning of the interior into octree cells
The void space ⌦0
consists of all cells ⌦k for which k = 1.
Optimization approach
n our adaptive voxel discretization, since the fill values are bi-
the resulting minimization problem would be combinatorial.
der to take advantage of continuous optimization techniques,
ropose a relaxation approach that allows k to take on a con-
us value in the interval [0, 1].
goal of the optimization eventually is to assign binary fill values
ch voxel. In practice, we observed that fill variables k with
ctional value only occur on the boundary between voids and
regions. Hence, we sample these regions at a high resolution,
cells ⌦k
optimiza
Our functiona
To prevent an
uniform symm
neighboring ce
where is a
ftop( ) or fyo-
5.4 Implem
Cells overlapp
resent the co
the cell’s leve
ity to the bou
boundary
shell
interior
initialization
iterations
merge
split
boundary
Figure 3: Hollowing: (Left) Our input encloses a volume ⌦. By introducing voids ⌦0
, we can compensate for an unfavorable mass distribu-
tion. (Right) To reduce the number of variables and overall time complexity for our voids optimization, we summarize contributions of octree
Spin-It: Optimizing Moment of Inertia for Spinnable Objects • 96:5
s: objective function
β: [0,1]

ボクセルごとに目的関数を定義してこれが最小化されるように埋めていきます。βは0のとき空洞、1のとき完全
に埋まるようになってて、βが0と1の間のときにどんどん再帰的にオクツリーを細分化させていきます。
Figure 11: “Dancing Couple”. (Top: left to right) I
with principal axes rotated away from spin frame; afte
the dominant primary axis is still not aligned; optimi
employ a multi-resolution voxelization based on an adaptive
e, thereby significantly reducing the number of fill variables.
discretized volume integrals then become
s⌦ ⌦0 = s⌦
X
k
k s⌦k
e ⌦i =
S
k ⌦k is a partitioning of the interior into octree cells
The void space ⌦0
consists of all cells ⌦k for which k = 1.
Optimization approach
n our adaptive voxel discretization, since the fill values are bi-
the resulting minimization problem would be combinatorial.
der to take advantage of continuous optimization techniques,
ropose a relaxation approach that allows k to take on a con-
us value in the interval [0, 1].
goal of the optimization eventually is to assign binary fill values
ch voxel. In practice, we observed that fill variables k with
ctional value only occur on the boundary between voids and
regions. Hence, we sample these regions at a high resolution,
cells ⌦k
optimiza
Our functiona
To prevent an
uniform symm
neighboring ce
where is a
ftop( ) or fyo-
5.4 Implem
Cells overlapp
resent the co
the cell’s leve
ity to the bou
boundary
shell
interior
initialization
iterations
merge
split
boundary
Figure 3: Hollowing: (Left) Our input encloses a volume ⌦. By introducing voids ⌦0
, we can compensate for an unfavorable mass distribu-
tion. (Right) To reduce the number of variables and overall time complexity for our voids optimization, we summarize contributions of octree
s: objective function
β: [0,1]

Connex
Objet’s
olution
xes, re-
builds
upport-
cannot
oles in
hem af-
nd fab-
racters
esented
ximum
andard
rocess-
oading
g opti-
ization
frames
nd blue
inertia
Finally, two break-dancing Armadillos are shown in Fig. 8, one
spinning on his back shell, one on the tip of his ﬁnger. Our hollow-
ing successfully aligns the maximal principal axis of inertia with the
user-speciﬁed one, even if it is far off as for the Armadillo spinning
on his shell (compare left and right visualizations). Both Armadil-
los “dance” very stably around a, as we demonstrate in our video.
Figure 8: “Break-dancing Armadillos”: Through our hollowing
optimization, the Armadillos can perform spinning dance moves.
For each design, the unstable input (left), and the optimized stable
Figure 10: Yo-yo designs: (Left to right) 3D print; input model;
optimized output model after hollowing. (Top) “Cuboid”: Our op-
timization rotates the original principal axes frame about the mid-
magnitude axis. (Bottom) “Woven Ring”: The axis of dominant
Figure
with pr
the dom
ter defo
(green)
formati
ollowed result showing voxelized inte-
sing ftop = fyo-yo. (Middle) Lowering
e mass distribution closer to the con-
ss reduction (right), mass is reduced
esign with highest rotational stability.
Left to right) 3D print; input model;
hollowing. (Top) “Cuboid”: Our op-
l principal axes frame about the mid-
Woven Ring”: The axis of dominant
aligned to the spin direction.
e multiple densities. The interior of
onsists of tin-solder material with sig-
= 8.1 g/cm3
) compared to our printer
Figure 11: “Dancing Couple”. (Top: left to right) Initial design
with principal axes rotated away from spin frame; after hollowing,
the dominant primary axis is still not aligned; optimized result af-
ter deformation. (Middle: left to right) Initial (red) and deformed
(green) models; voxelization after hollowing; voxelization with de-
formation optimization. (Bottom) The 3D printed result.
中身埋めるだけじゃうまくいかないときは仕方ないのでサーフェースのほうを少し変形させます。
あと単独の材料で最適化がうまくいかない場合は密度の異なる複数の材料で埋めるのもサポートし
てます。

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