Cubic Stylization
HSUEH-TI DEREK LIU, University of Toronto, Canada
ALEC JACOBSON, University of Toronto, Canada

Fig. 1. Cubic stylization deforms a given 3D shape into the style of a cube while maintaining textures and geometric features. This can be used as a non-realistic
modeling tool for creating stylized 3D virtual world. We obtain 3D assets from sketchfab.com by smeerws and Jesús Orgaz under CC BY 4.0.
We present a 3D stylization algorithm that can turn an input shape into
the style of a cube while maintaining the content of the original shape. The
key insight is that cubic style sculptures can be captured by the as-rigidas-possible energy with an ℓ 1 -regularization on rotated surface normals.
Minimizing this energy naturally leads to a detail-preserving, cubic geometry.
Our optimization can be solved efficiently without any mesh surgery. Our
method serves as a non-realistic modeling tool where one can incorporate
many artistic controls to create stylized geometries.
CCS Concepts: • Computing methodologies → Mesh models; Meshgeometry models.
Additional Key Words and Phrases: geometry processing, geometric stylization, shape modeling
ACM Reference Format:
Hsueh-Ti Derek Liu and Alec Jacobson. 2019. Cubic Stylization. ACM Trans.
Graph. 38, 6, Article 197 (November 2019), 10 pages. https://doi.org/10.1145/
3355089.3356495

1

INTRODUCTION

The availability of image stylization filters and non-photorealistic
rendering techniques has dramatically lowered the barrier of creating artistic imagery to the point that even a non-professional user
can easily create stylized images. In stark contrast, direct stylization of 3D shapes or non-realistic modeling has received far lessattention. In professional industries such as visual effects and video
games, trained modelers are still required to meticulously create
non-realistic geometric assets. This is because investigating geometric styles is more challenging due to arbitrary topologies, curved
metrics, and non-uniform discretization. The scarcity of tools togenerate artistic geometry remains a major roadblock to the development of geometric stylization.

Authors’ addresses: Hsueh-Ti Derek Liu, University of Toronto, 40 St. George Street,Toronto, ON, M5S 2E4, Canada, hsuehtil@cs.toronto.edu; Alec Jacobson, University ofToronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada, jacobson@cs.toronto.edu.
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https://doi.org/10.1145/3355089.3356495

Fig. 2. The cubic style have been attracting artists’ attention over centuries,
such as the Serpend à’ Plumes found in Chichén Itzá (left), The Kiss by
Constantin Brâncus, i (middle), and the Taichi by Ju Ming (right). We obtain
images from wikimedia.com photographed by Jebulon under CC0 1.0, fromflickr.com by Art Poskanzer under CC BY 2.0, and from wikimedia.com
by Jeangagnon under CC BY-SA 3.0.
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

197:2• Hsueh-Ti Derek Liu and Alec Jacobson

Fig. 3. A digital art – Anicube – by Aditya Aryanto produces cubic style
images (right). Our method takes an input tiger (left) and outputs a “3D anicube” tiger while maintaining geometric details (middle). ©Aditya Aryanto(right). Used under permission.

We present cubic stylization which formulates the task as an energy optimization that naturally preserves geometric details while
cubifying a shape. Our proposed energy combines an as-rigid-aspossible (arap) energy with an ℓ 1 regularization. This energy canbe minimized efficiently using the local-global approach with alternating direction method of multipliers (ADMM). This variational
approach affords the flexibility of incorporating many artistic controls, such as applying constraints, non-uniform cubeness, and different global/local cube orientations (Sec. 4). Moreover, our method
requires no remeshing (Fig. 5) and generalizes to polyhedral stylization (Fig. 24). Our proposed tool for non-realistic modeling goesbeyond the 2D stylization and opens up the possibility of, for instance, creating non-realistic 3D worlds in virtual reality (Fig. 1).

2

RELATED WORK

Our work shares similar motivations to a large body of work on
image stylization [Kyprianidis et al. 2013], non-photorealistic rendering [Gooch and Gooch 2001], and motion stylization [Hertzmann
et al. 2009]. While their outputs are images or stylized animations,
we take a 3D shape as input and output a stylized shape. Thus we
focus our discussion on methods for processing geometry, including
the study of geometric styles and deformation methods that sharetechnical similarities.

Fig. 4. One can control the cubic stylization by incorporating constraints.
For instance, we can fix some parts of a shape to mimic the style of a Jaguar
metate from ancient Costa Rica (top) or add point constraints to mimic the
Assyrian Lamassu wall sculpture (bottom). ©Antiques & Artifacts LLC (top).
Used under permission.

Fig. 5. Our cubic stylization requires no remeshing, thus vertex attributes
such as textures are preserved during the optimization. Our arap term
encourges locally isometric deformations to help maintain nice textures.

In this paper, we focus on the specific style of cubic sculptures. The
cubic style is prevalent across art history, for instance the ancient
sculptures from the post-classic era (900-1250 CE), Maya sculptures,block statues in Egypt, and modern abstract sculptures such as theones from Constantin Brâncus, i and Ju Ming (Fig. 2). In addition,
the cubic style is a popular digital art, such as the award-winning
Anicube by Aditya Aryanto (Fig. 3). Complementing their presence
in art, cubic shapes also present themselves in fabrication and furniture purposes (Fig. 4). We contribute to the rich history of cubic
sculpting by providing a stylization tool that takes a 3D shape as
input and outputs a deformed shape that has the same style as cubic
sculptures.
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

Discriminative Geometric Styles. The growing interest in understanding geometric styles has been inspiring recent works on building discriminative models for style analysis. One of the main challenges is to define a similarity metric aligned with human perception.
Many works propose to compare projected feature curves [Li et al.
2013; Yu et al. 2018], sub-components of a shape [Hu et al. 2017;
Lun et al. 2015; Xu et al. 2010], or using learned features [Lim et al.
2016]. These models enable users to synthesize style compatible
scenes [Liu et al. 2015] or transfer style components across shapes[Berkiten et al. 2017; Lun et al. 2016; Ma et al. 2014]. However, thesemethods are designed for discerning and transfering styles, insteadof generating 3D stylized shapes directly.
Generative Geometric Styles. Direct 3D stylization has been an
important topic in computer graphics. Many generative models havebeen proposed for producing specific styles, without relying on
identifying and transferring style components from other shapes.
This includes creating the collage art [Gal et al. 2007; Theobalt et al.
2007], voxel/lego art [Luo et al. 2015; Testuz et al. 2013], neuronalhomunculus [Reinert et al. 2012], the manga style shapes [Shen et al.
2012], shape abstraction [Kratt et al. 2014; Mehra et al. 2009; Yumer
and Kara 2012], and bas-relief sculptures [Bian and Hu 2011; Kerber
et al. 2009; Schüller et al. 2014; Song et al. 2007; Weyrich et al. 2007].
While not pitched as stylization techniques, many geometric flows
and filters can also be used for creating stylized geometry, such as
creating edge-preserving smoothing geometry [Zhang et al. 2018],
piece-wise planar [He and Schaefer 2013; Stein et al. 2018b] or developable shapes [Stein et al. 2018a], and stylized shapes prescribedby image filters [Liu et al. 2018] (see Fig. 6). Our method contributes
to the field of direct 3D stylization, focusing on the style of cubic
sculptures (Fig. 7).

Cubic Stylization •197:3

adapt the method of Manson and Schaefer [2011] to accelerate our
cubic stylization to meshes with millions of faces.

input[Zhang et al. 2018] [Stein et al. 2018a]ours

Fig. 6. Our energy-based deformation shares similarities with many energybased geometric flows and mesh filters, such as the methods of [Zhang
et al. 2018] and [Stein et al. 2018a].

Fig. 7. Cubic style sculptures are common throughout history, such as theDraped Seated Woman by Henry Moore (right). Our cubic stylization offers
an instrument to create cubic geometry (middle). We obtain the the photofrom flickr.com photographed by puffin11k under CC BY-SA 2.0.

input

[Liu et al. 2018][Huang et al. 2014]

ours

Fig. 8. Paparazzi [Liu et al. 2018] with image quantization and polycube
method (e.g., [Huang et al. 2014]) can create cubic style shapes (red, green),
but unlike our method (blue) they do not preserve geometric details.

Shape Deformation. Many works deal with the question of how to
deform shapes given modeling constraints. One of the most popular
choices is the arap energy [Chao et al. 2010; Igarashi et al. 2005; Liuet al. 2008; Sorkine and Alexa 2007], which measures local rigidityof the surface and leads to detail-preserving deformations. Not justdeformations, similar formulations to arap can also be extended toother tasks such as constrained shape optimization [Bouaziz et al.
2012], parameterization [Liu et al. 2008], and simulating mass-spring
systems [Liu et al. 2013]. Ever since, optimizing the arap energy
has been substantially accelerated by a large amount of work, suchas [Kovalsky et al. 2016; Peng et al. 2018; Rabinovich et al. 2017;
Shtengel et al. 2017; Zhu et al. 2018]. However, having nearly interactive performance on highly detailed meshes still remains a major
challenge. An alternative strategy to speed it up is to use the hierarchical deformation which optimizes arap on a low resolution modeland then recover the original details back afterwards [Manson and
Schaefer 2011]. This class of accelerations shares similar characteristics to multiresolution modeling (see [Garland 1999; Zorin 2006]).
We take advantage of the arap energy for detail preservation and

Axis-Alignment in Polycube Maps. Axis-alignment is an important
property for many geometry processing tasks, such as [Muntoni
et al. 2018; Stein et al. 2019]. Especially, this concept is one of themain instruments in the construction of polycube maps [Tarini et al.
2004], including defining polycube segmentations [Fu et al. 2016;
Livesu et al. 2013; Zhao et al. 2018] and the cost function for polycube
deformations [Gregson et al. 2011; Huang et al. 2014]. Althoughpolycube methods can obtain cubic geometry, they fail to preserve
detail (Fig. 8) because they are not desirable for intended applications
such as parameterization and hexahedral meshing [Cherchi et al.
2016; Fang et al. 2016; García Fernández et al. 2013; He et al. 2009;
Lin et al. 2008; Wang et al. 2007, 2008; Yu et al. 2014].
One tempting direction of creating cubic geometry is to use voxelization. However, voxelization fails to capture the details depictedby the artists and cannot capture the wide spectrum of cubenessacross cubic sculptures. Another tempting direction is to recovergeometric features from the polycube results. This would lead to a
multi-step algorithm and suffer from limitations of particular detail
encoding schemes (e.g., bump maps). Even if we stop the polycube
algorithm earlier such as the method of[Gregson et al. 2011] to maintain details,
it does not provide a satisfactory solution (see the inset for a comparison with
Fig. 5 in [Gregson et al. 2011]). More
importantly, many artistic controls in
Sec. 4 would be nontrivial to add on.
Building stylization on top of polycube
methods would also suffer from slow
performance. For instance, Huang et al.
[2014] propose a polycube method that minimizes the ℓ 1 -norm of
the normals on the deformed tetrahedral mesh with arap for regularization. Their formulation involves minimizing a complicatednon-linear function and requires minutes to hours to optimize. Thusa stylization built on top of this method would be even slower. In
contrast, our formulation is a single energy optimization which
can easily incorporate many artistic controls (Sec. 4). Our energy
is similar to the polycube energy of [Huang et al. 2014] in that we
also minimize the arap energy with a ℓ 1 regularization, but the key
difference is that we define the ℓ 1 -norm on the rotated normals of
the original mesh instead. This allows us to optimize our energy
much faster using the local-global approach with ADMM in only afew seconds (Table 1).

3

METHOD

The input of our method is a manifold triangle mesh with/withoutboundaries. Our method outputs a cubified shape where each subcomponent has the style of an axis-aligned cube. Meanwhile, our
stylization will maintain the geometric details of the original mesh.
Let V be a |V| × 3 matrix of vertex positions at the rest state andeV be a |V| × 3 matrix containing the deformed vertex positions. We
denote di j = [vj − vi ]⊤ and edi j = [evj − evi ]⊤ be the edge vectors
between vertices i, j at the rest and deformed states respectively.
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

197:4• Hsueh-Ti Derek Liu and Alec Jacobson

input meshFig. 9. One can control the cubeness by changing the λ parameter in Eq. 1.

The energy for our cubic stylization is as follows

minimizeeV, {Ri }Õ Õ wi j∥Ri di j − edi j ∥F2 + λai ∥Ri n̂i ∥1 .
2
i ∈V j ∈N(i) |{z} | {z }(1)

cubenessarap

The first term is the arap energy [Sorkine and Alexa 2007], where
Ri is a 3-by-3 rotation matrix, w i j is the cotangent weight [Pinkall
and Polthier 1993], and N (i) denotes the “spokes and rims” edges of
the ith vertex [Chao et al. 2010] (see the inset). In the second term,n̂i denotes the unit area-weighted normal vector of a vertex i in R3 .
The ai ∈ R+ is the barycentric area of vertex i, which is crucial for
λ to exhibit the similar cubeness across different mesh resolutions.
Intuitively, minimizing the ℓ 1 -norm of
the rotated normal encourages Ri n̂ito align with one of coordinate axes
because ℓ 1 -norm encourages sparsity.
Combining the two, the optimal rotation {Ri⋆ } would simultaneously preserve the local structure (arap) and encourage axis alignment (cubeness).
We adapt the standard local-global
update strategy to optimize our energy [Sorkine and Alexa 2007](see Alg. 1). Our global step, updating e
V, is achieved by solving alinear system, the same as the Equation 9 in Sorkine and Alexa[2007]. Our local step, finding the optimal rotation, is however
different from the previous literature due to the ℓ 1 term.

Fig. 10. We turn 3D shapes into the cubic style (blue) with Alg. 1. ©Angelo
Tartanian (top left), Splotchy Ink (top), Dan Slack (top right) under CC BY.

Eq. 3 is a standard ADMM formulation. We solve this local stepusing the scaled-form ADMM updates [Boyd et al. 2011]:k1
ei ∥ 2 + ρ ∥Ri n̂i − zk + uk ∥ 2 (4)∥Ri Di − D2Wi2
Ri ∈SO(3) 2

Rki +1 ← arg min

zk+1 ← arg min λai ∥z∥1 +3.1

zLocal-Step

k +1
ũk +1 ← uk + Rk+1
i n̂i − z

Our local step for each vertex i can be written asρ
1ei ∥ 2 + λai ∥Ri n̂i ∥1 ,Ri⋆ = arg min ∥Ri Di − DWiRi ∈SO(3) 2(2)where Wi is a |N (i)| × |N (i)| diagonal matrix of cotangent weights,
ei are 3 × |N (i)| matrices of rim/spoke edge vectors at the
Di and D
rest and deformed states respectively. We denote ∥X∥Y2 = Tr(XYX⊤ )for notational convenience. By setting z = Ri n̂i , we can rewrite
Eq. 2 asminimizez,Ri ∈SO(3)subject to1ei ∥ 2 + λai ∥z∥1
∥Ri Di − D
Wi2z − Ri n̂i = 0.

ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

ρ k k +1
∥Ri n̂i − z + uk ∥22
2

(3)

k +1

k +1

,u

(5)
(6)

k

← update (ρ )(7)where ρ ∈ R+ is the penalty and u is the scaled dual variable.
Eq. 4 is an instance of the orthogonal Procrustes [Gower et al. 2004]
Rki +1 ← arg max Tr(Ri Mi )Ri ∈SO(3)Mi = Di


n̂i


 Wi


ρk


e⊤
D
i
.
(zk − uk )⊤One can derive the optimal Ri from the singular value decompositionof Mi = Ui Σi Vi⊤ :Rk+1← Vi Ui⊤ ,i(8)

up to changing the sign of the column of Ui so that det(Ri ) > 0.

Cubic Stylization •Fig. 11. We can also turn meshes with boundaries (red) into the cubic style.
©Takeshi Murata (left) under CC BY.

(rescaled) log energy(rescaled) log energy

small cubeness

197:5

Fig. 13. The global orientation of the shape influences the ℓ 1 term in Eq. 1.
Applying different rotations to the mesh lead to different results. ©My Dog
Justice under CC BY.

Algorithm 1: Cube Stylization (λ)
Input : A triangle mesh V, F
Output : Deformed vertex positions eVe
V←Vwhile not converge do
3.
R ← local-step (V, e
V, λ)
e
4.
V ← дlobal-step (R)1.

2.

large cubeness
075#iteration

150

0

75

#iteration

150

Fig. 12. We show the convergence behavior of different meshes in Fig. 10(left, blue), Fig. 16 (left, green), and different cubenesses in Fig. 9 (right).
Note that the dotted line imply the optimization has stopped.

Algorithm 2: Fast Cube Stylization (λ, m)Input : A triangle mesh V, F
Output : Deformed vertex positions eVEq. 5 is an instance of the lasso problem [Boyd et al. 2011; Tibshirani 1996], which can be solved with a shrinkage step:// pre-processing
1. m ← target number of faces
2. Vc , Fc ← edge collapses (V, F, m)// cubic stylization
3. e
Vc ← Vc
4. while not converge do
5.
R ← local-step (Vc , e
Vc , λ)e6.
Vc ← дlobal-step (R)7. e
V, F ← affine vertex splits (eVc , Fc )kzk+1 ← Sλai/ρ k (Rk+1i n̂i + u )(9)Sκ (x)j = (1 − κ/|x j | )+ x j
We update the penalty ρ (Eq. 7) according to Sec. 3.4.1 in [Boyd et al.
2011] where u needs to be rescaled accordingly after updating ρ.
In short, local fitting is performed by running Eq. 8, 9, 6, and 7
iteratively until the norm of primal/dual residuals are small. Warm
starting the local-step parameters from the previous iteration cansignificantly speed up the optimization. Specifically, we initialize z, uwith zeros, and set the initial ρ = 10−4 , ϵ abs = 10−5 , ϵ rel = 10−3 , µ =10, and τ incr = τ decr = 2 (the same notation as used in Sec. 3 of [Boydet al. 2011]). Then z, u, ρ are reused in consecutive iterations. Notethat for extremely large λ one may need to increase the initial valueof ϵ abs accordingly in order to avoid bad local minima. We stop the
optimization when the relative
displacement, the infinity norm of
relative per vertex displacements,
is lower than 3 × 10−3 (see Fig. 12
for the convergence plots). Moreelaborate stopping criteria, suchas the method of [Zhu et al. 2018],
could also be used.
At this point we have com- Fig. 14. Our method canpleted the cubic stylization algo- cubify non-orientable surfaces
such as the Klein bottle.
rithm summarized in Alg. 1, enabling us to efficiently create cubified shapes (see Fig. 10). In Fig. 11

and 14 we show that this formulation is applicable to meshes withboundaries and non-orientable surface respectively. As the cubeness is dependent to the orientation of the mesh, one can apply
different rotations to control how the stylization runs (Fig. 13). We
expose the weighting λ to be a design parameter controlling the
cubeness of a shape (Fig. 9).
However, the “vanilla” cube stylization shares the same caveat asother distortion minimization algorithms: having slow runtime onhigh resolution meshes.

3.2

Affine Progressive Meshes

Manson and Schaefer [2011] propose a hierarchical approach toaccelerate arap deformations. The main idea is to deform a lowresolution model and recover the details back after convergence.
Specifically, Manson and Schaefer [2011] propose a progressive
mesh [Hoppe 1996] representation which first simplifies a given
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

197:6• Hsueh-Ti Derek Liu and Alec Jacobson

mesh via a sequence of edge collapses, and then represents the meshas its coarsest form together with a sequence of vertex splits. After
applying some deformations tothe coarsest mesh, each “deformed” vertex split is computedby fitting the best local rigid transformation. This approach is suitable for deformations that are loedge collapse
cally rigid (e.g., arap), but our
cubic stylization is less rigid forlarger λ.
So we fit the best affine transformation in each vertex split,
rather than rigid transformations. Specifically, in each edge collapse we store the displacement vectors from the newly inserted
vertex pi to the endpoints pj , pk (see the inset) together with amatrix A:|F|: 2,018,232|F|: 345,944|F|: 810,990

Fig. 16. With the affine progressive meshes, we can scale the cubic stylization to meshes with millions of faces. The Nefertiti mesh (left) was scannedby Nora Al-Badri and Jan Nikolai Nelles from the Nefertiti bust.

A = (Qi Qi⊤ )−1 Qi .
Qi is a 3 × |N (i)| matrix where each column is the vector frompi to one of its one-rings neighbors N (i). If (Qi Qi⊤ ) is singular(e.g., in planar regions), we remedy the issue with the Tikhonov
regularization [Tikhonov et al. 2013]. Then A is used to computed
the deformed displacements for each vertex split as
ei A⊤ (pj − pi ),epj − epi = Q
where epi denotes the position of vertex i in the cubified coarsened
ei is a 3 × |N (i)| matrix containing vectors from e
shape, and Q
pi to
its one-rings neighbors.
Affine progressive meshes allows us to losslessly recover theoriginal meshes undergoing affine transformations. For smooth
non-affine transformations such as our cube stylization, it could still
be approximately recovered (see Fig. 15). We summarize our cubic
stylization with the affine progressive mesh in Alg. 2. Note thatthe edge collapses is just a pre-processing step. In the online stage,
one only needs to run cubic stylization on the coarsest mesh andthen apply a sequence of vertex splits to visualize the result on theoriginal resolution. This offers a huge speed-up when interacting
the parameter λ on highly detailed models (see Fig. 16).
An interesting observation is that the number of faces m in thecoarsest mesh not only controls the runtime, but implicitly controls

decimation

cubic stylization

vertex splits

Fig. 15. Affine progressive meshes allow us run cubic stylization on a lowresolution model and then recover original details when converged. ©Colin
Freeman under CC BY.
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

input meshFig. 17. The number of faces m used in the decimated mesh not only controls the runtime but also the frequency level of details that get preserved.
©Joseph Larson under CC BY.

the frequency level of geometric details that gets preserved. In
Fig. 17 we show that, under the same λ, a smaller m keeps details
across a wider frequency range; in contrast, a larger m only keeps
details at higher frequencies. Therefore one can manipulate the levelof preserved features by playing with m.

3.3

Implementation

We implement the cubic stylization in C++ using libigl [Jacobson
et al. 2018] and evaluate our runtime on a MacBook Pro with an Inteli5 2.3GHz processor. Table 1 lists the parameters and the runtimeof our stylization in Fig. 10 (top) and Fig. 16. We test our methods
on meshes in the Thingi10K [Zhou and Jacobson 2016] and showthat we can obtain stylized geometry within a few seconds. This isimportant for users to receive quick feedback on their parameterchoices and iterate on their designs, such as the cubeness λ in Fig. 9
and the the level of details m in Fig. 17.
User study. We prototype auser interface (see the inset)
to conduct an informal user
study with six participants (4male, 2 female) between theages of 24 and 29. Participant
3D modeling experience ranged
from none (complete novice)to three years of hobbyist use.
Each participant was instructed
for three minutes on how to use our software to load a mesh andcontrol the cubeness parameter λ. Then we asked them to cubify ashape of their choosing from a collection of ten shapes. The resultsof their work is show in Fig. 18. All users reported that they weresatisfied with the cubeness of their resulting shape. One user saidCubic Stylization •Table 1. For each example in Fig. 10 and Fig. 16, we report the numberof faces in the original model ( |F |), l 1 weight (λ), number of faces of thecoarsest mesh (m), number of iterations (Iters.), pre-processing time (Pre.),and runtime at the online stage (Runtime).

Model
Fig. 10, left
Fig. 10, mid.
Fig. 10, rightFig. 16, left
Fig. 16, mid.
Fig. 16, right

|F |
39K
41K
21K
2018K
346K
811K

λ
0.20
0.20
0.4
0.20
0.40
0.30

m
n/a
n/a
n/a
20K
20K
40K

Iters.
106
93
86
83
222
173

Pre.
n/a
n/a
n/a
64.19s10.69s30.44s

Runtime
5.08s
4.50s
2.26s
3.93s
4.59s
8.38s

197:7

high

small
Fig. 19. We vary λ across the surface to have different cubeness for different
parts. We apply higher λ on the red region and smaller λ for the blue region
to create an ottoman-like shape (middle). ©pmoews under CC BY.

highGauss map

input scene

smallFig. 20. We can paint the λ function on the Gauss map to have non-uniform
λ over the surface. In the figure, we have higher λ for the original normals
pointing towards left, and vice versa. ©Tom Cushwa under CC BY.

cubified scene

Fig. 21. We can vary λ for different axes to turn inputs into the biased cubic
style (blue) towards x,y,z axes respectively. ©MakerBot (right) under CC BY.
Fig. 18. Even non-professional users can effortlessly turn an input scene(top) into a cubified scene (bottom). Different colors are results createdby different users. From left to right, ©Peter Leppik, Cleven, TerenceKing,
MakerBot, TerenceKing, PerryEngel, and Christina Chun under CC BY.

that controlling the cubeness of their resulting shape is very easy
because it only requires tuning a single parameter.

4

ARTISTIC CONTROLSIn addition to the two parameters λ, m, we expose many variantsof our stylization to incorporate artistic controls. As a non-realistic
modeling tool, this is important for users to realize their creativity.
We first focus our discussion on a variety of artistic controlsthat are related to the cubeness parameter λ. Although Eq. 1 only
has a single λ for an entire shape, we can actually specify different
λi for each vertex independently to have non-uniform cubeness,
which leads to the expression λi ai ∥Ri n̂i ∥1 . In Fig. 19, we use thisapproach to make the back of the sheep much more cubic than therest of the shape to create an ottoman-like geometry. We can also
specify the non-uniform cubeness λi in a different way, insteadof painting on the surface directly. In Fig. 20 we paint a function
on the Gauss map in which the surface normal pointing towards
left has higher cubeness. When we map this function back to thesurface, we can have a cubified owl that is more cubic when initialnormals pointing towards the left and less cubic when pointing
towards the right. Similarly, we can have different λ x , λy , λz for
different axes. In Fig. 21, we replace the cubeness in Eq. 1 withai (λ x |(Ri n̂i )x | + λy |(Ri n̂i )y | + λz |(Ri n̂i )z |) and specify different
values for each λ x , λy , λz to have the style of a rectangular prism.
If one wants to fix certain parts of the shape, we can easily add
constraints in the global step, the same way as the method of Sorkine
and Alexa [2007]. In Fig. 4 we add the parts constraint by fixing theposition of some vertices when solving the linear system; we add
the points constraint by specifying some deformed vertices e
Vi atuser-desired positions. We can also use the same methodology to
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

197:8• Hsueh-Ti Derek Liu and Alec Jacobson

Fig. 22. We constrain certain parts of the geometry lying on certain planes
to create a 3D printed table clinger (right). ©Morena Protti under CC BY.

Fig. 25. We apply non-symmetric coordinate transformations inside theℓ 1 -norm to create irregular polyhedral stylization. ©Johannes under CC BY.

Fig. 23. We can define the ℓ 1 -norm on different coordinate systems fordifferent parts of the shape, instead of using the world coordinates. In the
figure the hands and the body use different coordinate systems (left). Bychanging them, we can vary the cube orientations for different parts. ©David
Hagemann under CC BY.

Fig. 26. Once we have the cubic geometry (blue), standard deformation
techniques (e.g., [Sorkine and Alexa 2007]) can be used to manipulate the
cubified shape (yellow).

Fig. 24. We apply a coordinate transformation inside the ℓ 1 -norm to generalize cubic stylization to polyhedrons. ©Proto Paradigm (middle), Ola
Sundberg (right) under CC BY.

constrain some parts of the geometry lying on certain planes. For
instance, setting (eVi )x = 0 can force vertex i lying on the yz-plane.
In Fig. 22 we use this plane constraint to create a table clinger.
In addition, one can utilize the property of the ℓ 1 -norm to havedifferent artistic effects. Because the cubeness term is orientation
dependent, in Fig. 13 we can apply different rotations to the meshbefore the stylization to control the results. Rather than rotating
the mesh, another way is to encode the normal vector in a different
coordinate system λai ∥Ri n̂ilocal ∥1 , where we use n̂ilocal to denotethe user-desired coordinate system for vertex i. This perspective
allows us to define the ℓ 1 -norm on different coordinate systems fordifferent parts of the shape to obtain different cube orientations(Fig. 23). Beyond the cubic stylization, in Fig. 24, 25 we apply acoordinate transformation B inside the ℓ 1 -norm λai ∥BRi n̂i ∥1 toachieve polyhedral stylization, for which we provide the details inApp. A. Once we obtain the stylized shapes, they are ready to be
used by standard deformation techniques in animations (Fig. 26).
ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

input mesh

|F|: 12,812|F|: 35,924|F|: 64,650

Fig. 27. Although exhibiting similar cubenesses, our stylization is still notinvariant to different resolutions.

5

LIMITATIONS & FUTURE WORK

Accelerating the stylization to real-time would enable faster iterations between designs. Developing a more robust stylization tofor bad quality triangles, non-manifold meshes, or even point cloud
could be useful for stylizing real-world geometric data. Guaranteeing results to be self-intersection free would be desirable fordownstream tasks. Extending our energy to be invariant to discretizations could achieve more consistent results across different
resolutions (see Fig. 27). Extending to quadrilateral meshes and
NURBS surfaces could benefit existing modeling or engineering design softwares. Generalizing to volumetric meshes could have a better volume preservation. Exploring different deformation energies
and ℓp -norm could lead to novel stylization tools for non-realistic
modeling. Beyond generating stylized shapes, the mathematicalexpression of the cubic geometry could offer insights toward understanding more intricate styles. For instance, Cubism has beenCubic Stylization •considered as a revolutionized artistic style for paintings and sculptures. Cubism has appeared since the early 20th century. Since then,several attempts have tried to describe [Henderson 1983] and generate Cubist art [Corker-Marin et al. 2018; Wang et al. 2011], butmore efforts still required to offer scientific explanations to a wide
variety of Cubist art. Our cubic stylization only focuses on a specificstyle. We hope this could inspire future attempts to capture different
sculpting styles such as those presented in African art, or even a
generic approach to create different styles in an unified framework.

ACKNOWLEDGMENTS
Our research is funded in part by New Frontiers of Research Fund(NFRFE–201), the Ontario Early Research Award program, NSERC
Discovery (RGPIN2017–05235, RGPAS–2017–507938), the Canada
Research Chairs Program, the Fields Centre for Quantitative Analysis and Modelling and gifts by Adobe Systems, Autodesk and MESH
Inc. We thank members of Dynamic Graphics Project at the University of Toronto; Michael Tao and Wen-Hsiang Tsai for project
motivations; David I.W. Levin and Yotam Gingold for ideas on theartistic controls and the user study; Oded Stein for sharing results;
Rahul Arora for fabricating stylized shapes; Leonardo Sacht and Silvia Sellán for proofreading; Omid Poursaeed, Rahul Arora, Whitney
Chiu, Yang Zhou, Yifan Wang, Youssef Alami Mejjati, and Zhicong
Lu for participating in the user study.

REFERENCES
Sema Berkiten, Maciej Halber, Justin Solomon, Chongyang Ma, Hao Li, and Szymon
Rusinkiewicz. 2017. Learning detail transfer based on geometric features. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 361–373.
Zhe Bian and Shi-Min Hu. 2011. Preserving detailed features in digital bas-relief making.
Computer Aided Geometric Design 28, 4 (2011), 245–256.
Sofien Bouaziz, Mario Deuss, Yuliy Schwartzburg, Thibaut Weise, and Mark Pauly. 2012.
Shape-up: Shaping discrete geometry with projections. In Computer Graphics Forum,Vol. 31. Wiley Online Library, 1657–1667.
Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. 2011.
Distributed optimization and statistical learning via the alternating direction methodof multipliers. Foundations and Trends® in Machine learning 3, 1 (2011), 1–122.
Isaac Chao, Ulrich Pinkall, Patrick Sanan, and Peter Schröder. 2010. A simple geometricmodel for elastic deformations. ACM transactions on graphics (TOG) 29, 4 (2010), 38.
Gianmarco Cherchi, Marco Livesu, and Riccardo Scateni. 2016. Polycube simplification
for coarse layouts of surfaces and volumes. In Computer Graphics Forum, Vol. 35.
Wiley Online Library, 11–20.
Quentin Corker-Marin, Alexander Pasko, and Valery Adzhiev. 2018. 4D Cubism: Modeling, Animation, and Fabrication of Artistic Shapes. IEEE computer graphics andapplications 38, 3 (2018), 131–139.
Xianzhong Fang, Weiwei Xu, Hujun Bao, and Jin Huang. 2016. All-hex meshing using
closed-form induced polycube. ACM Transactions on Graphics (TOG) 35, 4 (2016),
124.
Xiao-Ming Fu, Chong-Yang Bai, and Yang Liu. 2016. Efficient volumetric polycube-map
construction. In Computer Graphics Forum, Vol. 35. Wiley Online Library, 97–106.
Ran Gal, Olga Sorkine, Tiberiu Popa, Alla Sheffer, and Daniel Cohen-Or. 2007. 3D
collage: expressive non-realistic modeling. In Proceedings of the 5th international
symposium on Non-photorealistic animation and rendering. ACM, 7–14.
Ismael García Fernández, Jiazhi Xia, Ying He, Shi-Qing Xin, and Gustavo Patow. 2013.
Interactive Applications for Sketch-Based Editable Polycube Map. © IEEE Transactions on Visualization and Computer Graphics, 2013, vol. 19, núm. 7, p. 1158-1171(2013).
Michael Garland. 1999. Multiresolution modeling: Survey & future opportunities. Stateof the art report (1999), 111–131.
Bruce Gooch and Amy Gooch. 2001. Non-photorealistic rendering. AK Peters/CRC
Press.
John C Gower, Garmt B Dijksterhuis, et al. 2004. Procrustes problems. Vol. 30. Oxford
University Press on Demand.
James Gregson, Alla Sheffer, and Eugene Zhang. 2011. All-hex mesh generation via
volumetric polycube deformation. In Computer graphics forum, Vol. 30. Wiley Online
Library, 1407–1416.

197:9

Lei He and Scott Schaefer. 2013. Mesh denoising via L 0 minimization. ACM Transactions
on Graphics (TOG) 32, 4 (2013), 64.
Ying He, Hongyu Wang, Chi-Wing Fu, and Hong Qin. 2009. A divide-and-conquer
approach for automatic polycube map construction. Computers & Graphics 33, 3(2009), 369–380.
Linda Dalrymple Henderson. 1983. The Fourth Dimension and Non-Euclidean Geometry.
Princeton, Princeton University Press.
Aaron Hertzmann, Carol O’Sullivan, and Ken Perlin. 2009. Realistic human body
movement for emotional expressiveness. In ACM SIGGRAPH 2009 Courses. ACM,
20.
Hugues Hoppe. 1996. Progressive meshes. In Proceedings of the 23rd annual conference
on Computer graphics and interactive techniques. ACM, 99–108.
Ruizhen Hu, Wenchao Li, Oliver Van Kaick, Hui Huang, Melinos Averkiou, Daniel
Cohen-Or, and Hao Zhang. 2017. Co-locating style-defining elements on 3d shapes.
ACM Transactions on Graphics (TOG) 36, 3 (2017), 33.
Jin Huang, Tengfei Jiang, Zeyun Shi, Yiying Tong, Hujun Bao, and Mathieu Desbrun.
2014. l1-Based Construction of Polycube Maps from Complex Shapes. ACM Transactions on Graphics (TOG) 33, 3 (2014), 25.
Takeo Igarashi, Tomer Moscovich, and John F Hughes. 2005. As-rigid-as-possible shape
manipulation. In ACM transactions on Graphics (TOG), Vol. 24. ACM, 1134–1141.
Alec Jacobson, Daniele Panozzo, et al. 2018. libigl: A simple C++ geometry processing
library. http://libigl.github.io/libigl/.
Jens Kerber, Art Tevs, Alexander Belyaev, Rhaleb Zayer, and Hans-Peter Seidel. 2009.
Feature sensitive bas relief generation. In 2009 IEEE International Conference on
Shape Modeling and Applications. IEEE, 148–154.
Shahar Z Kovalsky, Meirav Galun, and Yaron Lipman. 2016. Accelerated quadratic
proxy for geometric optimization. ACM Transactions on Graphics (TOG) 35, 4 (2016),
134.
Julian Kratt, Ferdinand Eisenkeil, Sören Pirk, Andrei Sharf, and Oliver Deussen. 2014.
Non-realistic 3D Object Stylization. In Proceedings of the Workshop on Computational
Aesthetics (CAe ’14). ACM, New York, NY, USA, 67–75. https://doi.org/10.1145/
2630099.2630102
Jan Eric Kyprianidis, John Collomosse, Tinghuai Wang, and Tobias Isenberg. 2013. Stateof the “Art”: A Taxonomy of Artistic Stylization Techniques for Images and Video.
IEEE transactions on visualization and computer graphics 19, 5 (2013), 866–885.
Honghua Li, Hao Zhang, Yanzhen Wang, Junjie Cao, Ariel Shamir, and Daniel Cohen-Or.
2013. Curve style analysis in a set of shapes. In Computer Graphics Forum, Vol. 32.
Wiley Online Library, 77–88.
Isaak Lim, Anne Gehre, and Leif Kobbelt. 2016. Identifying style of 3D shapes using
deep metric learning. In Computer Graphics Forum, Vol. 35. Wiley Online Library,
207–215.
Juncong Lin, Xiaogang Jin, Zhengwen Fan, and Charlie CL Wang. 2008. Automatic
polycube-maps. In International Conference on Geometric Modeling and Processing.
Springer, 3–16.
Hsueh-Ti Derek Liu, Michael Tao, and Alec Jacobson. 2018. Paparazzi: Surface Editing
by way of Multi-View Image Processing. In SIGGRAPH Asia 2018 Technical Papers.
ACM, 221.
Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven J Gortler. 2008. A local/global
approach to mesh parameterization. In Computer Graphics Forum, Vol. 27. Wiley
Online Library, 1495–1504.
Tiantian Liu, Adam W Bargteil, James F O’Brien, and Ladislav Kavan. 2013. Fast
simulation of mass-spring systems. ACM Transactions on Graphics (TOG) 32, 6(2013), 214.
Tianqiang Liu, Aaron Hertzmann, Wilmot Li, and Thomas Funkhouser. 2015. Stylecompatibility for 3D furniture models. ACM Transactions on Graphics (TOG) 34, 4(2015), 85.
Marco Livesu, Nicholas Vining, Alla Sheffer, James Gregson, and Riccardo Scateni. 2013.
PolyCut: monotone graph-cuts for PolyCube base-complex construction. ACM
Transactions on Graphics (TOG) 32, 6 (2013), 171.
Zhaoliang Lun, Evangelos Kalogerakis, and Alla Sheffer. 2015. Elements of style:learning perceptual shape style similarity. ACM Transactions on Graphics (TOG) 34,
4 (2015), 84.
Zhaoliang Lun, Evangelos Kalogerakis, Rui Wang, and Alla Sheffer. 2016. Functionality
preserving shape style transfer. ACM Transactions on Graphics (TOG) 35, 6 (2016),
209.
Sheng-Jie Luo, Yonghao Yue, Chun-Kai Huang, Yu-Huan Chung, Sei Imai, Tomoyuki
Nishita, and Bing-Yu Chen. 2015. Legolization: optimizing LEGO designs. ACM
Transactions on Graphics (TOG) 34, 6 (2015), 222.
Chongyang Ma, Haibin Huang, Alla Sheffer, Evangelos Kalogerakis, and Rui Wang.
2014. Analogy-driven 3D style transfer. In Computer Graphics Forum, Vol. 33. Wiley
Online Library, 175–184.
Josiah Manson and Scott Schaefer. 2011. Hierarchical deformation of locally rigid
meshes. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 2387–2396.
Jacob Mattingley and Stephen Boyd. 2012. CVXGEN: A code generator for embedded
convex optimization. Optimization and Engineering 13, 1 (2012), 1–27.

ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

197:10• Hsueh-Ti Derek Liu and Alec Jacobson

Ravish Mehra, Qingnan Zhou, Jeremy Long, Alla Sheffer, Amy Gooch, and Niloy J
Mitra. 2009. Abstraction of man-made shapes. In ACM transactions on graphics(TOG), Vol. 28. ACM, 137.
Alessandro Muntoni, Marco Livesu, Riccardo Scateni, Alla Sheffer, and Daniele Panozzo.
2018. Axis-aligned height-field block decomposition of 3d shapes. ACM Transactions
on Graphics (TOG) 37, 5 (2018), 169.
Yue Peng, Bailin Deng, Juyong Zhang, Fanyu Geng, Wenjie Qin, and Ligang Liu. 2018.
Anderson acceleration for geometry optimization and physics simulation. ACM
Transactions on Graphics (TOG) 37, 4 (2018), 42.
Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces andtheir conjugates. Experimental mathematics 2, 1 (1993), 15–36.
Michael Rabinovich, Roi Poranne, Daniele Panozzo, and Olga Sorkine-Hornung. 2017.
Scalable locally injective mappings. ACM Transactions on Graphics (TOG) 36, 2(2017), 16.
Bernhard Reinert, Tobias Ritschel, and Hans-Peter Seidel. 2012. Homunculus Warping: Conveying importance using self-intersection-free non-homogeneous meshdeformation. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 2165–2171.
Christian Schüller, Daniele Panozzo, and Olga Sorkine-Hornung. 2014. Appearancemimicking surfaces. ACM Transactions on Graphics (TOG) 33, 6 (2014), 216.
Liang-Tsen Shen, Sheng-Jie Luo, Chun-Kai Huang, and Bing-Yu Chen. 2012. SD Models:Super-Deformed Character Models. In Computer Graphics Forum, Vol. 31. Wiley
Online Library, 2067–2075.
Anna Shtengel, Roi Poranne, Olga Sorkine-Hornung, Shahar Z Kovalsky, and Yaron
Lipman. 2017. Geometric optimization via composite majorization. ACM Trans.
Graph. 36, 4 (2017), 38–1.
Wenhao Song, Alexander Belyaev, and Hans-Peter Seidel. 2007. Automatic generationof bas-reliefs from 3d shapes. In IEEE International Conference on Shape Modelingand Applications 2007 (SMI’07). IEEE, 211–214.
Olga Sorkine and Marc Alexa. 2007. As-rigid-as-possible surface modeling. In Symposium on Geometry processing, Vol. 4. 109–116.
Oded Stein, Eitan Grinspun, and Keenan Crane. 2018a. Developability of Triangle
Meshes. ACM Trans. Graph. 37, 4 (2018).
Oded Stein, Eitan Grinspun, Max Wardetzky, and Alec Jacobson. 2018b. Natural Boundary Conditions for Smoothing in Geometry Processing. ACM Trans. Graph. 37, 2,
Article 23 (May 2018), 13 pages. https://doi.org/10.1145/3186564
Oded Stein, Alec Jacobson, and Eitan Grinspun. 2019. Interactive design of castable
shapes using two-piece rigid molds. Computers & Graphics (2019).
Marco Tarini, Kai Hormann, Paolo Cignoni, and Claudio Montani. 2004. Polycube-maps.
In ACM transactions on graphics (TOG), Vol. 23. ACM, 853–860.
Romain Testuz, Yuliy Schwartzburg, and Mark Pauly. 2013. Automatic Generation ofConstructable Brick Sculptures. In Eurographics 2013 - Short Papers. The Eurographics Association. https://doi.org/10.2312/conf/EG2013/short/081-084
Christian Theobalt, Christian Roessl, Edilson de Aguiar, and Hans-Peter Seidel. 2007.
Animation Collage. In Eurographics/SIGGRAPH Symposium on Computer Animation,
Dimitris Metaxas and Jovan Popovic (Eds.). The Eurographics Association. https://doi.org/10.2312/SCA/SCA07/271-280
Robert Tibshirani. 1996. Regression shrinkage and selection via the lasso. Journal of
the Royal Statistical Society: Series B (Methodological) 58, 1 (1996), 267–288.
Andreï Nikolaevitch Tikhonov, AV Goncharsky, VV Stepanov, and Anatoly G Yagola.
2013. Numerical methods for the solution of ill-posed problems. Vol. 328. Springer
Science & Business Media.
Hongyu Wang, Ying He, Xin Li, Xianfeng Gu, and Hong Qin. 2007. Polycube splines.
In Proceedings of the 2007 ACM symposium on Solid and physical modeling. ACM,
241–251.
Hongyu Wang, Miao Jin, Ying He, Xianfeng Gu, and Hong Qin. 2008. User-controllable
polycube map for manifold spline construction. In Proceedings of the 2008 ACM
symposium on Solid and physical modeling. ACM, 397–404.
Yu-Shuen Wang, Min-Wen Chao, Chin-Chueng Yi, and Chao-Hung Lin. 2011. Cubist
Style Rendering for 3D Polygonal Models. Journal of Information Science andEngineering 27, 6 (2011), 1885–1899.
Tim Weyrich, Jia Deng, Connelly Barnes, Szymon Rusinkiewicz, and Adam Finkelstein.
2007. Digital bas-relief from 3D scenes. In ACM transactions on graphics (TOG),
Vol. 26. ACM, 32.
Kai Xu, Honghua Li, Hao Zhang, Daniel Cohen-Or, Yueshan Xiong, and Zhi-Quan
Cheng. 2010. Style-content separation by anisotropic part scales. In ACM Transactions on Graphics (TOG), Vol. 29. ACM, 184.
Fenggen Yu, Kai Xu, Ali Mahdavi-Amiri, Hao Zhang, et al. 2018. Semi-Supervised
Co-Analysis of 3D Shape Styles from Projected Lines. ACM Transactions on Graphics(TOG) 37, 2 (2018), 21.
Wuyi Yu, Kang Zhang, Shenghua Wan, and Xin Li. 2014. Optimizing polycube domain
construction for hexahedral remeshing. Computer-Aided Design 46 (2014), 58–68.
Mehmet Ersin Yumer and Levent Burak Kara. 2012. Co-abstraction of shape collections.
ACM Transactions on Graphics (TOG) 31, 6 (2012), 166.
Juyong Zhang, Bailin Deng, Yang Hong, Yue Peng, Wenjie Qin, and Ligang Liu. 2018.
Static/dynamic filtering for mesh geometry. IEEE transactions on visualization andcomputer graphics (2018).

ACM Trans. Graph., Vol. 38, No. 6, Article 197. Publication date: November 2019.

Hui Zhao, Na Lei, Xuan Li, Peng Zeng, Ke Xu, and Xianfeng Gu. 2018. Robust edgepreserving surface mesh polycube deformation. Computational Visual Media 4, 1
(01 Mar 2018), 33–42. https://doi.org/10.1007/s41095-017-0100-x
Qingnan Zhou and Alec Jacobson. 2016. Thingi10K: A Dataset of 10,000 3D-Printing
Models. arXiv preprint arXiv:1605.04797 (2016).
Yufeng Zhu, Robert Bridson, and Danny M Kaufman. 2018. Blended cured quasi-newton
for distortion optimization. ACM Transactions on Graphics (TOG) 37, 4 (2018), 40.
Denis Zorin. 2006. Modeling with multiresolution subdivision surfaces. In ACM SIGGRAPH 2006 Courses. ACM, 30–50.

A

POLYHEDRAL GENERALIZATION

Fig. 28. By specifying different coordinate transformations B inside theℓ 1 -norm, we can encourage polyhedral style.

Simply applying a coordinate transformation B : Rn → Rm
inside the ℓ 1 -norm can encourage polyhedral results, instead of
cubic results (see Fig. 28). The ℓ 1 -norm of a vector is defined as thesummation of its magnitudes along each basis vector. Thus applying
a coordinate transformation inside the ℓ 1 -norm changes its bahavior
because the basis vectors are different. Following the notation in
Eq. 1, polyhedron energy can be written asÕ Õ wi j
ei j ∥ 2 + λai ∥BRi n̂i ∥1 .
minimize∥Ri Di j − DF2eV, {Ri }i ∈V j ∈N(i)In our case, B is a m-by-3 coordinate transformation matrix forshapes embedded in R3 . Again by setting z = Ri n̂i we can reachalmost the same optimization procedures, except the Eq. 5 now
becomes (we ignore the iteration superscript for clarity)ρ
zk +1 ← arg min λai ∥Bz∥1 + ∥Ri n̂i − z + u∥22 .
(10)2z
Similar to common techniques for solving the Basis Pursuit problem,
we introduce a variable t ⪰ ∥Bz∥1 to transform Eq. 10 into a small
quadratic program subject to equality constraints  ρ/2 · I3 0 z
minimize z⊤ t⊤
0
0 t
z,t
 

 z
⊤
+ −ρ(Ri n̂i + u)⊤ λai 1mt 
B −Im zsubject to⪯ 0,−B −Im t
where Ix and 1x denote the identity matrix with size x and a column
vector of 1 with size x respectively. We then solve this efficiently
using cvxgen [Mattingley and Boyd 2012]. Note that the results in
Fig. 24 and Fig. 25 use m = 4.