Fabricating Reflectors for Displaying Multiple Images
KAISEI SAKURAI, DWANGO Co., Ltd., Dwango CG Research
YOSHINORI DOBASHI, Hokkaido University and Dwango CG Research
KEI IWASAKI, Wakayama University and Dwango CG Research
TOMOYUKI NISHITA, Dwango CG Research and Hiroshima Shudo University
structured raised stripes

our reflectors

ink

substrate(b)

structured raised stripes
our reflectors
our reflectors

cell

(a)

(c)

subcells

Fig. 1. (a) Examples of our reflectors printed on everyday personal objects: a pen holder, smartphone cases, toy ducks, and a ruler. The reflectors on thesmartphone cases are directly printed by a standard UV printer. Other reflectors are printed on stickers and pasted onto the objects. All the reflectors aredesigned to display SIGGRAPH logo with different colors depending on the viewing direction (see Fig. 6 for input images). (b) Microstructure of the printed
pattern consisting of raised stripes on an ink layer. (c) Close-up photograph of printed patterns.
A great deal of attention has been devoted to the fabrication of reflectors thatcan display different color images when viewed from different directions
not only in industry but also for the arts. Although such reflectors havepreviously been successfully fabricated, the number of images displayed
has been limited to two or they suffer from ghosting artifacts where mixed
images appear. Furthermore, the previous methods need special hardware
and/or materials to fabricate the reflectors. Thus, those techniques are notsuitable for printing reflectors on everyday personal objects made of different materials, such as name cards, letter sheets, envelopes, and plastic
cases. To overcome these limitations, we propose a method for fabricating reflectors using a standard ultraviolet printer (UV printer). UV printercan render a specified 2D color pattern on an arbitrary material and by
overprinting the printed pattern can be raised, that is, the printed pattern
Authors’ addresses: Kaisei Sakurai, DWANGO Co., Ltd., Dwango CG Research, kaisei_
sakurai@dwango.co.jp; Yoshinori Dobashi, Hokkaido University, Dwango CG Research,
doba@ime.ist.hokudai.ac.jp; Kei Iwasaki, Wakayama University, Dwango CG Research,iwasaki@sys.wakayama-u.ac.jp; Tomoyuki Nishita, Dwango CG Research, Hiroshima
Shudo University, nishita@shudo-u.ac.jp.
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© 2018 Association for Computing Machinery.
0730-0301/2018/8-ART158 $15.00
https://doi.org/10.1145/3197517.3201400

becomes a microstructure having color and height. We propose using thesemicrostructures to formulate a method for designing spatially varying reflections that can display different target images when viewed from different
directions. The microstructure is calculated by minimizing an objectivefunction that measures the differences between the intensities of the lightreflected from the reflector and that of the target image. We show several
fabricated reflectors to demonstrate the usefulness of the proposed method.
CCS Concepts: • Computing methodologies → Image manipulation; Reflectance modeling;
Additional Key Words and Phrases: digital fabrication, reflector, ultraviolet
printer, spatially varying anisotropic BRDF
ACM Reference Format:
Kaisei Sakurai, Yoshinori Dobashi, Kei Iwasaki, and Tomoyuki Nishita. 2018.
Fabricating Reflectors for Displaying Multiple Images. ACM Trans. Graph.
37, 4, Article 158 (August 2018), 10 pages. https://doi.org/10.1145/3197517.
3201400

1

INTRODUCTION

To address the demand for increasing variation in industrial design,
a number of techniques for displaying images have been developed,
such as reflectors [Regg et al. 2010; Snelgrove et al. 2013; Weyrich
et al. 2007], relief sculptures [Alexa and Matusik 2010; Belhumeur
et al. 1999; Mitra and Pauly 2009; Schüller et al. 2014; Song et al. 2007;Sýkora et al. 2014], and lenses [Baran et al. 2012; Papas et al. 2012;
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

158:2 •Sakurai, K. et al

Yue et al. 2012]. In recent years, several methods have emergedthat improve the ability of metallic reflectors to display various
images [Pjanic and Hersch 2015a,b]. These methods enable the fabrication of metallic reflectors that can display specified images byspecular reflection. However, in these methods, the number of images to be displayed is limited to two since a phenomenon known asghosting, where images other than the specified image are displayed,arises.
One additional but important issue with the previous methods
is how the reflector is fabricated; these often need special devices
and/or materials for fabrication. Thus, the benefits of the method
cannot easily be obtained unless one has access to such devices/material. This prevents many interesting reflectors becoming widespread.
If the reflectors can be printed easily by anyone, they can be used for
different personal purposes. For example, we can print reflectors on
letters, name cards, envelopes, posters, or (curved) plastic cases, etc.,
for decoration, artwork, or advertisement. The previous methods
do not allow us to use the techniques for these personal purposeswith a reasonable printing cost.
Our goal is to address the above problems. We use a standard
ultraviolet (UV) printer, which enables anyone to print our reflectors
on their own everyday objects. A UV printer is capable of printing UV curable ink on different types of material, including paper,
plastic, metal, transparent sheet, etc. It can also print on curved
surfaces. UV printers are usually available via a printing agency or
one can simply buy one. Our method provides the user with a tool
to compute the printing pattern to be fed to the UV printer. Then,the printed reflector displays multiple images specified by the user
toward different viewing directions.
To achieve this goal, we make use of the ability of UV printers tofabricate tiny height fields by overprinting. The optimal microstructure is computed and represented as a height field printable by the
UV printer. In fact, with our method, the number of images displayed with less ghosting artifacts can be increased. Our method
renders spatially-varying diffuse reflections on the reflector by fabricating high-resolution microstructures in order to produce thedesired anisotropic reflections. The microstructure is computed byminimizing an energy function representing the sum of the differences between the desired intensities and the simulated results. This
problem is hard to solve in general because this energy function isa high-dimensional, nonlinear function of the lighting conditions,viewing directions, and the spatially varying BRDF (SVBRDF) of thereflector. We focus here on a SVBRDF that minimizes the energyfunction, assuming fixed lighting conditions and viewing directions.
The microstructure on our reflector is designed in the form of a
stripe pattern, as shown in Fig. 1(c). The stripe pattern consists of
two parts: an ink layer and small-scale vertical walls called structured raised stripes (SRSs), as shown in Fig. 1(b). We decompose the
SVBRDF into spectral and directional components, which correspond to the ink layer and the SRSs, respectively. The directional
component is derived from the SRSs that occlude the reflected light.
The spectral component, resulting from the combination of theprinted inks, controls the color of the SVBRDF.
Figure 1(a) shows an example of our reflectors printed on different
objects. The reflectors in this figure are designed to display the
SIGGRAPH logo with four different colors depending on the viewing
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

direction. Fig. 1(c) shows a close-up view of the reflector where we
can observe the SRSs (black line segments) and the inks making
up the reflector. In order to fabricate the reflector, we use a UVprinter. UV curable ink can be stacked by overprinting. The SRSs
are fabricated by overprinting black lines over the ink layer.
The contributions of this paper are summarized as follows.
• Our reflectors can be easily fabricated by printing patterns using a standard UV printer, which is readily accessible throughcommercial printer services. Anyone can easily reproduce
our reflectors.
• Our reflector does not require additional equipments (e.g. alens for lenticular and parallax barriers), nor electrical power(e.g. lightfield displays), to display multiple color images.
• We propose a novel method for designing microstructural
stripe patterns to display multiple color images with minimal
ghosting effects and better clarity.

2

RELATED WORK

Pjanic and Hersch [2015a] displayed a single image using a metallic
reflector that had different colors when viewed from two different directions. This was achieved by printing a grid pattern, or acollection of horizontal and vertical line segments on the reflector.
However, multiple images with sufficient quality could not be displayed using this method, because one image appeared slightly inthe other. The authors [Pjanic and Hersch 2015b] also proposed amethod using a metallic reflector, where two colored images are
printed on it. Although the purpose of this method is to hide one
image in the other, the number of displayable images is limited totwo.
Snelgrove et al. [2013] proposed a method for displaying an image
with different colors, depending on the elevation angle of the viewing direction. This method is based on the idea of a parallax barrier
by showing a portion of the diffuse colored surface. The difference
between the gradient of the target image and that of the presented
image is minimized in order to preserve the edges of the target
image. For this reason, the images are displayed with undesirable
colors. Furthermore, the viewing direction in this method is limited.
In our method, which can be considered to be a generalization ofhis method, the viewing direction can be chosen arbitrarily.
The work most closely related to ours is that by Sakurai et al. [2017],
who use a similar microstructure to ours. However, they do notprovide a method to determine the ’optimal’ microstructure; they
manually determine a good-looking microstructure via a trial anderror process. This may be fine for artwork but the technical contribution in their paper is minimal. Furthermore, their fabricatedreflector often produced ghosting artifacts. Our approach is fundamentally different; we propose a method to automatically computethe optimal microstructure comprising structured raised stripes.
We can use different optical phenomena to display the images,
such as self-shadows [Bermano et al. 2012], diffuse reflections [Alexaand Matusik 2010], refractions [Papas et al. 2012], shadows [Mitra
and Pauly 2009], and moiré imagings [Hersch and Chosson 2004].
Specular holograms can display a series of images by specular reflection with a collection of grooves [Brand 2011; Regg et al. 2010].

Fabricating Reflectors for Displaying Multiple Images • 158:3&'!

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Fig. 2. The basic idea of our method for the case in which N = 3. (a) We designed a SVBRDF f (x, ωo ) to display the target images I 0 , I 1 , and I 2 for theviewing directions ωo0 , ωo1 , and ωo2 , respectively. ϕ 0 , ϕ 1 , and ϕ 2 are the azimuthal angles for ωo0 , ωo1 , and ωo2 , respectively. (b) The cell at x is subdivided into
three subcells, and each subcell corresponds to f (x, ωo0 ), f (x, ωo1 ), and f (x, ωo2 ). (c) The directions of the SRSs (shown as black walls) are aligned to ϕ 0 , ϕ 1 ,and ϕ 2 . The SRSs block the light from ωo1 incident on subcells 0 and 2, making only subcell 1 is visible to viewing direction ωo1 . (d) Finally, the colors of the
dotcells are determined by optimization to render the three colors of the pixel for the three target images at x.

These methods, however, cannot take into account the reflectanceof the colors.
Multiple images can also be displayed by controlling the BRDF.
Lan et al. [2013] developed a method for producing a colored reflector that presents an arbitrary BRDF. The limitation of that method
is that the presented BRDF depends on the triangular mesh of thereflector. Although this could be used to present colored images,
the data size of the triangular mesh is too large to be printed. Matusik et al. [2009] proposed a method for printing a BRDF with a
combination of inks, with which a BRDF in a printable range isderived from a given BRDF; however, it is difficult to display multiple colored images by reflection because the method is limited
to isotropic BRDFs and it is impossible to change the colors at agiven position. There are also methods that use microstructures tocontrol the intensity of the reflectance [Glasner et al. 2014; Levinet al. 2013]. These approaches present anisotropic reflection, but are
not suitable for colored reflections; therefore, the method is limited
to grayscale images only.
Light field displays have also an ability to display different images depending on view directions. Some methods use stacks ofliquid-crystal displays (LCDs), each of which displays its own basis image [Lanman et al. 2011; Wetzstein et al. 2011, 2012]. Thesemethods can render images with high brightness and contrast but
ghosting artifacts could be produced due to the fact that all thelayers are visible. A parallax barrier display can be constructed byusing two LCDs [Lanman et al. 2010; Perlin et al. 2000]. The front
panel contains a uniform grid of slits or pinholes, through which
the viewer sees a subset of pixels in the rear panel depending on theviewer location. However, these devices need some gaps between
the LCD layers and they require electricity and special hardware
setups. Therefore, these are not suitable for our purpose, printing
reflectors on posters or stickers, etc.

3BASIC IDEA

Our goal is to fabricate a reflector using a SVBRDF, f , by printing
stripe patterns on an ink layer so that the reflector can display
N target images viewed from N directions ωok (k = 0, . . . , N − 1)
specified by the user. We assume that all the target images havethe same resolution, and the surface of the reflector is subdivided

into grid cells with the same resolution. Therefore, each square cellcorresponds to a pixel in each of the target images. We also assume
that the viewpoints and the light source are located sufficiently
far from the reflector so that the directions from each cell to theviewpoint and to the light source can be considered to be parallel,
and the SVBRDF, f , is assumed to be the average reflectance overthe cell.
We first explain the basic idea of the method using Fig. 2 where
the number of target images N is three. Figs. 2(b) through (d) illustrate the construction of a microstructure used to display threecolors at x in Fig. 2(a). Fig. 2(b) shows the cell at x, which is a collection of a number of grid cells, each of which corresponds to adot for the printer. We refer to these cells as dotcells. Our method
subdivides the cell into subcells. These subcells correspond to eachtarget image. SRSs are then generated by overprinting with black
ink as shown in Fig. 2(c). The direction of SRSs in each subcell k isaligned to the viewing direction ωok . The height of the SRSs and thegaps between them are determined so that each subcell k is visible
from the corresponding viewing direction ωok only. In Fig. 2(c), forexample, rays from viewing direction ωo1 can reach subcell 1 butother rays reaching subcells 1 and 2 are blocked by the SRSs. Finally,
the colors of the dotcells are determined by optimization to replicate
each target image.
Problem Definition. SVBRDF f is a function of the cell x, the incident direction ω i , and the outgoing direction ω o , that is, f (x, ω i , ωo ).
To make the design feasible, we design f for incident light with ω iparallel to the normal n of the cell with unit intensity, though our
method can display multiple target images in general illumination
cases as shown in Fig. 1. Since incident light parallel to n is not
occluded by the SRSs and the cosine factor calculated by the dot
product of ω i and n is simplified to 1, the reflected radiance fromcell x illuminated by unit intensity then depends on two parameters,i.e., f (x, ω o ). The reflected radiance L(yj , ω o ) from the j-th dotcell
yj in x is calculated from the geometric structure G of SRSs in thecell as:L(yj , ω o ) = v(yj , ωo ; G)ρd (yj ),(1)where v(yj , ω o ; G) is a binary visibility function that returns 1 if
a ray from yj in direction ω o does not intersect the SRSs G, and
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

158:4 •Sakurai, K. et al

returns 0 otherwise. ρd (yj ) ∈ R3 is the diffuse reflectance at yj ,which is calculated from the color of the ink printed on yj . Since
the reflected radiance from cell x is the average of those from thedotcells, f is calculated from G and ρd as follows:ÍM
j=1 v(y j , ω o ; G)ρ d (y j ),(2)f (x, ω o ) =M
where M is the number of dotcells in the cell.
For each cell x, we define the following objective function
e(x) =N−1Õ∥ f (x, ωok ) − pk (x)/N ∥,

(3)

k =0

where pk ∈ R3 is the pixel intensity associated with cell x in thek-th target image. The division by N is necessary since f (x, ωok )is proportional to the area of the corresponding subcell, which
decreases as N increases.
By substituting Eq. (2) into Eq. (3), the objective function for cellx can be represented by the following matrix:
arg min ∥V(G) · C(x) − P(x)∥ ,(4)

G,C(x)

where V(G) is an N × M visibility matrix given by:v(y1 , ωo0 ; G)
1 ©­..
V(G) =.
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N −1« v(y1 , ωo ; G)

···
..
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···

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..
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N
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v(yM , ωo ; G) ¬(5)

C(x) and P(x) are M × 3 and N × 3 matrices, respectively.
T© ρd (y1 )­..
C(x) = ­.
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DESIGNING THE SVBRDF

To design a SVBRDF to display N target images, we calculate thegeometric structure G of the SRSs and the matrix C that minimizes
the objective function in Eq. (4). Although the objective function
is represented by a simple matrix, it is difficult to optimize thissince even if an optimal visibility matrix Vopt is obtained, finding
the geometric structure G opt = V−1 (Vopt ) is infeasible due to theextensive solution space. There are more than 2M possible patternsof the geometric structures in the cell, resulting in a prohibitive
computation time to find the optimum one. On the other hand, if G
is determined, it is easy to compute the visibility matrix V by ray
casting.
To tackle this, we design the geometric structures G that determine the visibility matrix V in the first step, then in the second step,
C is calculated to minimize Eq. (4) for each cell x. The first step
jointly optimizes the geometric structures of all the SRSs in the cell.
The direction of the SRSs printed on the k-th subcell is the sameand set to be parallel to the azimuthal angle of ωok . Other geometricparameters of the SRSs such as their height and the gaps between
them are calculated to minimize the sum of the objective functions
representing the amount of light occluded by the SRSs, the ghosting
effects, and the deviation of the reflected radiances described inSec. 4.1.
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

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Fig. 3. (a) Dotcells and the parameters for the SRSs. The offsets, o 1 and o 2 ,and the gaps between the SRSs, d 1 and d 2 , are shown in subcell 1 (dotcells
shaded in blue) and subcell 2 (dotcells shaded in green). The dotcells shadedin black represent the SRSs. (b) When the cell is viewed from direction ωo1(i.e. the azimuthal angle of ωo1 is aligned to the direction of the SRSs insubcell 1), the dotcells in subcell 1 are expected to be visible, and those insubcell 2 are not visible (dotcells shaded in gray are shadowed/masked by
SRSs). However, reflected light from some dotcells in subcell 2 (surroundedby red lines) can reach the viewpoint. This light leakage results in ghosting
effects.

4.1

Geometric Structure of the SRSs

The geometric structure of the SRSs printed on the k-th subcell
depends on the height of the SRSs, hk , the gap between them, dk ,and the offset, ok , which is the vertical distance of the nearest SRS
from the upper left dotcell as shown in Fig. 3(a). The SRSs depend on
the area of the k-th subcell as well, which is controlled by the ratioof the area of the k-th subcell to that of the entire cell, ak . Of theseparameters, our method uses a common height parameter h for all
the subcells to reduce the number of parameters. We define theparameter set of the SRSs in the k-th subcell as G k = {ak , dk , ok },and define G = {h, G 0 , · · · , G N −1 }.
We determine the parameter set G by minimizing an objectivefunction that measures the ghosting effects. As shown in Fig. 3,
since the masking effects at the dotcells in the k-th subcell can beaffected by the SRSs in neighboring subcells (especially dotcells at
the boundary of two subcells), our method solves the minimization
problem simultaneously for the whole parameter set G. The sameeight cells are placed around the cell to take into account occlusions
from SRSs in neighboring subcells. The same parameter set G isused for all the cells (i.e. the SRSs in each cell are identical), making
fabrication simple and optimization computationally efficient.
4.1.1 Objective function for G. To display multiple target images
faithfully, we want the reflector (1) to reflect the light as strongly as
possible for better visibility, (2) to reduce the ghosting effects, and(3) to display each of the N target images with as similar reflectedradiances as possible. The objective function E(G) comprises threeterms that correspond to the above three demands and is given by:
E(G) = wo Eo (G) + wд Eд (G) + w s Es (G),(7)

where Eo , Eд , and Es are the occlusion term, the ghosting effectterm, and the standard deviation term, respectively. wo , wд , and w s
are user-specified weights for each term, respectively. To formulate
the three terms, we define a function D k (ωo ; G) that represents theratio of dotcells in the k-th subcell visible from direction ωo as:Íj ∈Jk v(y j , ω o ; G)D k (ωo ; G) =,(8)
MFabricating Reflectors for Displaying Multiple Images • 158:5
where Jk is the set of dotcell indices in the k-th subcell.
To make the reflected radiances from the cells for N directions
large, the occlusion term Eo is calculated as the sum of the amountof light occluded by the SRSs using the following equation:
Eo (G) =N−1
Õ

Algorithm 1 Evaluation of E(G). G = {h, G 0 , · · · , G N −1 } is theparameter set for all subcells.
1: procedure computeE(G)2:3:(1 − D k (ωok ; G)).

(9)

k =0

4:
5:

The ghosting effect viewed from the direction ωok is caused bylight leaking from subcells other than the k-th subcell and thus theghosting effect for ωok can be measured by the sum of Dl except forl = k as follows:N−1ÕDl (ωok ; G).
(10)
l =0,l ,kThe ghosting effect term Eд is calculated using the following equation:
N
−1 N
−1
Õ
Õ
Eд (G) =Dl (ωok ; G).
(11)
k =0 l =0,l ,kTo make the reflected radiances of the N target images as similar
as possible, our method uses the standard deviation term Es :v
ut N −11 Õ(D k (ωok ; G) − D̄)2 ,(12)
Es (G) =N

6:7:

subdivide cell into subcells according to ak in G
for l = 0 to N − 1 dofor k = 0 to N − 1 do
calculate Dl (ωok ; G) by ray casting
calculate Eo (G), Eд (G), Es (G), and E(G)return E*,%!)!"#$% #*%!+!&'(% ('%!"#$%&''',%!)!". /,% !) - !+!" - !&

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Fig. 4. Examples of subdivision of a cell into four subcells according to (a)the ratio of the area of each subcell to the area of the cell, a i . (b) two groups
are partitioned vertically based on the sum of the ratio between a 0 + a 2
and a 1 + a 3 . (c) each group is further partitioned based on the ratios but
the split direction is selected perpendicular to the direction of the longer
edge of the subcell. (d) each group is subdivided according to the ratio a i .

k =0

Í N −1

D (ω k ;G)where D̄ is the average of D k , calculated by D̄ = k =0 Nk o .
Our method uses the standard deviation instead of the variance inorder to equalize the degree for D to that of the other two terms Eo
and Eд (the degree of the variance in D is two, while that for Eo andEд is one).
Algorithm 1 shows the procedure for evaluating the energy function. First the cell is subdivided into subcells according to the parameter ak in G using the method described in the next section.
Dl (ωok ; G) is then evaluated by casting rays from each of the dotcells
in subcell l to the viewing direction ωok for the visibility function v
in Eq. (8).
4.1.2 Cell Subdivision. In the optimization process, each cellis subdivided according to the ratio ak . We prefer to make theminimum edge length of the subcell not too small, since the subcell
size can be smaller than the gap between the SRSs, which makes it
difficult to draw the line segments for the SRSs using a UV printer.
To avoid this, we want to maximize the smallest length of theedge of the subcells. Fig. 4 illustrates our subdivision algorithm for
N = 4. Our algorithm first calculates a 0 + a 2 and splits the cell into
two groups using the ratio Ía30 +a2 . Next, each of the two groups are
i =0 a ifurther subdivided in the same way but we choose the direction thatis perpendicular to the longer edge of the subcell for the subdivision.
These processes are repeated recursively.
4.1.3 Optimization of E(G). The parameter set G is calculated
by solving the minimization problem for E(G). The objective function E(G) is minimized by using a genetic algorithm. We follow thestandard approach [Goldberg 1989]. Each parameter is quantizedand converted into a corresponding binary bit string. The range ofeach parameter is specified by the user. The bit strings for all theparameters are connected together to form a single longer bit stringthat is used as the gene of an individual. Initially, the GAs generaten individuals using random numbers, where n is specified by theuser. The GAs iteratively generate a new set of n individuals basedon a fitness function that evaluates the quality of each individual.
We use the inverse of our objective function, 1/(E + 1), as the fitness
function so that the maximum is one. New individuals are generated
through two genetic operators, one called a crossover operator andthe other a mutation operator. We use a two-point crossover. We
also employ a so-called elitist selection strategy, by which the bestindividual at a certain iteration is carried over to the next iteration.
We chose these genetic operators experimentally. Although there
are different operators such as a one-point crossover, we have notobserved any significant differences in the convergence of the objective function. The elitist selection, however, is important to increase
the convergence speed. For more details about GAs, there are many
good textbooks such as the one by Goldberg [1989].
The search process terminates if the fitness function becomes
smaller than a specified threshold ϵGA or the number of iterations
exceeds a specified number nmax . ϵGA = 0.999 and nmax = 500
worked fine for all cases. We used the typical values mentioned in[Goldberg 1989] for the other parameters of the GAs.

4.2

Diffuse reflectanceOnce the geometric structures for G are obtained, the visibility matrix V(G) is calculated by ray-casting. Our method then determines
C, which comprises the diffuse reflectance for each dotcell by solving the minimization problem for Eq. (4). A naive solution, suchas the least squares method or the psuedoinverse of V, cannot be
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

158:6 •Sakurai, K. et alapplied to obtain C due to the two constraints for printing: nonnegativity and the smoothness of the ink. Since each component of
C is the diffuse reflectance of each dotcell, C should be non-negative.
In addition, the printer cannot print significantly different colors inneighboring dotcells since the colors of the inks are created using
the dithering method and the error diffusion method.
Based on these two constraints, our method employs nonsmoothnonnegative matrix factorization (nsNMF) proposed by PascualMontano et al. [2006]. nsNMF approximately decomposes P into
the product of three non-negative matrices VSC. S is a smoothing
M × M matrix given by S = (1 − θ )I + θ J, where θ is a parameter
controlling the smoothness, I is the M × M identity matrix, and all
the elements of M ×M matrix J are equal to 1/M. C is a M × 3 matrix.
The original algorithm for nsNMF iteratively updates both V and
C to decompose the input matrix P. Since the visibility matrix V is
fixed in our case, we only update C. For the smoothing matrix, we
use θ = 0.1 for all the examples shown in this paper. When we usedlarger values for θ , C was too blurry and we could not obtain goodresults.
We have further found an additional issue when we actuallyfabricate a reflector with the colors computed by the above method.
The method should theoretically work fine but we find that thecolors on the actual reflector became darker than expected. Thisis due to the fact that the printed inks in neighboring dotcells are
mixed, resulting in decreased reflectances. To avoid this, we apply an
additional averaging process to the colors of the dotcells. That is, we
subdivide a cell into regular groups of m × m dotcells and computethe average color for each group. We use m = 3 for all the examples
shown in this paper. By incorporating this averaging process into
the nsNMF algorithm, we can obtain much better results. Our final
procedure to compute C is:(1) initialize C with random numbers ranging from zero to one.
(2) update C by Ci j = Ci j ∗((VS)T P)i j /((VS)T (VS)C)i j , where thesubscript ij indicates the (i, j) element of the corresponding
matrix.
(3) compute the average colors of the groups of m×m dotcells andsubstitute the average colors into the corresponding elementsof C.
(4) clamp each element of C between zero and one.
The above procedure is repeated until the energy is less than aspecified threshold or the specified number of iterations is reached.
To update C in step (2), we simply follow the method described in[Pascual-Montano et al. 2006].

5

RESULTSIn this section, we present the simulated and fabricated results ofour method and compare them to previous work. The simulated
results of the reflector are visualized by rendering each dotcell withthe evaluated value of f in Eq. (2) taking into account the occlusion
due to the SRSs.
The UV printer used for fabrication is a Roland VersaUV LEF12, which is an inkjet printer that propels droplets of UV curable inks from print head nozzles and emits UV light to dry them.
The printer resolution is 720 dots per inch (dpi) and one dot is

ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

(a)

(b)

Fig. 5. Photographs of the fabricated reflector displaying two different
images. The input images are shown in the insets.

Fig. 6. Color changing effect. Photographs of the fabricated reflector displaying the SIGGRAPH logo with four different color variations. The input
images are shown in the inset.

25.4/720=0.03527mm. The SRSs and the ink layer are printed separately. We first print the ink layer on a substrate, then fabricate
the SRSs by overprinting black line segments on the ink layer. We
found experimentally that the width of the black lines are larger
than the size of the dotcell. We measured the actual size of the blacklines and take this into account in computing the SRSs. Our method
calculates the colors of the reflector in the RGB color space while
the UV printer uses CMYK colors. We used GRACoL2006 Coated(ISO 12647-2:2004) for conversion from RGB to CMYK.
We used a standard PC with Intel Core i7-5820K 3.30Ghz (CPU)and 32.0GB RAM for creating the following examples. The computation of the SRSs usually finishes within one minute. However, thecomputation for the diffuse reflectances of the dotcells takes a few
hours. The computation time mainly depend on the size of the celland the longest computation time among our examples was threehours. All the reflectors in our experiments were fabricated in 3 to
4 hours.

5.1

Experimental Results

This section reports results of some experiments we made to investigate the ability and characteristics of our method.
Displaying two different images. Fig. 5 shows an example ofa reflector displaying two different images; the image changes depending on the angle of rotation. We set the two azimuthal angles
for Figs. 5(a) and (b) to zero and 90 degrees, respectively. The size of
this reflector is 108.37×108.37mm2 , and the resolution of the target
image is 128×128 pixels (each cell consists of 242 dotcells). Although
the input images are completely different, the reflector can showeach image clearly, and the ghosting effect is negligible.

Fabricating Reflectors for Displaying Multiple Images • 158:7

Color changing effect. Fig. 6 shows an example of a color changing effect; the hue of the SIGGRAPH logo changes depending on
angle of rotation. Four input images whose hues are 0, 90, 180, and270 degrees are used to fabricate the reflector. Although the previous
method [Pjanic and Hersch 2015a] achieved the color changing effect, the number of colors is limited to two. Our method can produce
a color changing effect with more than three colors. The size of thisreflector is 73.92×73.92mm2 , and the resolution of the target image
is 25×25 pixels (each cell consists of 482 dotcells). Depending on theviewing direction, mixed colors appear on the logo.
Validation of Our Prediction Model. We measured the reflectances of the fabricated reflector and compared them with the simulated results. The comparison is done as follows using a KONICAMINOLTA CS-150 that can measure the intensity of an object. This
device measures the intensity in XYZ color space, so that we convert
our simulated reflectance from RGB to XYZ color space. We firstmeasure the intensity of the substrate before printing the ink and the
SRSs. After printing them, the intensity of the reflector is measured
under the same lighting environment. Since our method assumes thereflectance of the substrate is one (which corresponds to the color
white in the target images), we normalize the measured intensity of
the reflector by the intensity of the substrate for comparison.
We fabricated two monotonic reflectors, each of which can display three colors, RGB and CMY, respectively. We chose these two
reflectors since RGB and CMY are the vertices of the gamut represented by additive and subtractive color mixing. So, we consider
that these reflectors correspond to cases difficult to reproduce. The
three colors are displayed for three viewing directions, 0, 60, and 120
degrees. The elevation angle of the viewing directions is the same,45 degrees. We obtained the six reflectances of the reflectors viewed
from these three viewing directions. The averages of the absolutedifference between the measured and the simulated reflectances for
XYZ components are 0.01, 0.02, and 0.04. Our method faithfully predicts the reflectances of the real, fabricated reflectors. The relatively
higher error in Z component is due to the fact that the UV printerwe use is less accurate in printing colors with higher Z values. We
confirmed this fact by printing a color chart.
Experiments on weights wд and w s . Fig. 7 demonstrates the effect of weights wд and w s in Eq (7), by setting wд or w s to zero. We
omit the case wo = 0 since in this case, E yields the minimum value
zero when D k are all zero, resulting in black images. Note that theimages in this figure are created by rendering a virtual reflector bycomputer simulation. The input images are the same as those used
for Fig. 11. Fig. 7(a) shows the results with wд = 0. Our method thentries to maximize the intensity by making the height of the SRSs
minimum, resulting in the obvious ghosting effects. Fig. 7(b) shows
the results with w s = 0 (i.e. the standard deviation term is ignored).
The average intensities of the images differ considerably, resulting
in an unnatural appearance when changing viewing direction. The
ghosting effects are also visible. Fig. 7(c) shows the results without
optimization of the SRSs. The parameters of the SRSs are randomly
determined but the diffuse reflectance for each dotcell is optimized.

(a) wo = 0.5, wg = 0.0, ws = 1.0(b) wo = 0.5, wg = 1.5, ws = 0.0(c) without optimization of SRSs

Fig. 7. Effect of weights w д and w s , and the effect of optimization.

Without the optimization, the ghosting effects are clearly visible.
The optimized result can be seen in Fig. 11.

5.2

Evaluation of the contrast displayed

We evaluated the contrasts displayed by our reflectors. We conducted a set of experiments to reveal the relationships between the
maximum contrast displayed by a single cell and the following threeparameters:• the azimuthal angle between two viewing directions• the number of equally distributed viewing directions• the elevation angle of the viewing direction
We focused on the maximum contrast displayed by our reflector
from a theoretical point of view. That is, we did not take into accountthe physical constraints imposed by a real UV printer. In particular,
we assumed the height of the SRSs to be 1.0mm and that no color
mixing between adjacent dotcells occurred. These assumptions are
not applicable to any currently available UV printer; the maximum
height of the curable ink is usually less than 1.0mm and the ink
between neighboring dotcells is easily mixed. In addition to therelationships described above, we investigated the dependence of thecontrast on the resolution of the dotcell by measuring the radiances
for dotcells with three different resolutions: 50 × 50, 100 × 100, and150 × 150.
In order to measure the maximum contrast, we used white (1.0)
and black (0.0) images as input images. We measured the radiancesfrom a single cell towards two adjacent viewing directions and computed the contrast as the ratio between the higher radiance andthe lower radiance. More specifically, the contrast K was computed
using the formula K = (Lw − Lb )/Lw , where Lw and Lb are theradiances for the directions corresponding to the white and black
images, respectively. K ranges from zero to one; where zero signifies that there is no contrast. Figs. 8, 9, and 10 show the results ofour experiments, showing the relationships between the maximum
contrast and the above three parameters, respectively. We describe
the details of the individual experiments below.
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

Sakurai, K. et al

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Dependence on the difference in azimuthal angle between
two viewing directions: Fig. 8 was created by measuring the radiances for two viewing directions where the azimuthal angle of one
viewing direction is fixed at zero degrees and the other one is varied
from 0 to 180 degrees. Note that the SRSs in the two subcells corresponding to the two viewing directions become parallel when the
difference in azimuthal angle between the two viewing directions is
0 or 180 degrees. In this case, the same color is displayed, resulting
in no contrast. Fig. 8 indicates that our reflector can display a certain
amount of contrast even if the difference is only five degrees. The
contrast is relatively higher and almost constant when the azimuthal
angle difference is between 20 and 160 degrees. However, there issome perturbation in the contrast, which is caused by the fact thatthe optimal SRS changes discretely with respect to the azimuthal
angle.
Dependence on the number of equally distributed viewing
directions: In this experiment, we changed the number of viewing
directions whose azimuthal angles are sampled at a regular interval. The input images are one white image and Nb black images.
The contrast is measured by the difference between the reflectedradiance corresponding to the white image and that to the adjacent
black image. Fig. 9 shows the relationship between the contrast andthe number of black images Nb . The contrast slightly decreases inproportion to the number of input images.
Dependence on the elevation angle of the viewing directions:Fig. 10 was created by using the two target images and by varying
the elevation angle of the corresponding two viewing directions
from 5 to 90 degrees. The azimuthal angles are fixed to 0 and 90
degrees. Fig. 10 shows the contrast for the different elevation angles.
All the colors are mixed when the viewing direction is normal to
the surface, i.e., the elevation angle is 90 degrees. Thus, there is no
ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

ŽŶƚƌĂƐƚ

Ϭ͘ϴThe measured radiances are actually different from the expectedideal radiances. We computed Eq. (3), the error function indicating
the degree of difference for all the experiments shown in this section. The average error was 0.097. One common finding we obtained
from these experiments is that the resolution of a cell has little effect on the contrast. This is observed in the figures; the three plots
corresponding to dotcells with the three different resolutions are all
similar (see Figs. 8, 9, and 10).

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contrast. While a certain amount of contrast is observed for elevation angles up to approximately 70 degrees, the contrast decreases
considerably for elevation angles larger than 70 degrees.

5.3

Practical Applications

This section show several applications of our reflector. The following
reflectors require no special devices nor electricity. The user canobserve the images displayed by the reflectors under the natural
illumination or a standard indoor illumination condition.
The first example shown in Fig. 1 demonstrates the ability of ourreflector to be fabricated on different material and/or objects. In
this example, a set of reflectors displaying the SIGGRAPH logo are
fabricated and we use them for decorating personal objects. For
smartphone cases, the reflectors are directly fabricated on them by
the UV printer. For other objects, the reflectors are fabricated on
stickers and we paste them onto the objects. The reflectors can beput even on curved surfaces such as the pen case or duck toys. These
reflectors exhibit different color variations of the logo dependingon the viewing direction.
Next, in Fig. 11, we use our reflector for an artistic purpose; wecall it mirror art. We fabricated the reflector using four input images
shown in the inset of Fig. 11(b). Fig. 11(a) shows four reflected imagesof our reflector in the four mirrors placed around the reflector. In
spite of the fact that all the mirrors reflect the same object (ourreflector), we see a different image in each mirror.
In Fig. 12, unlike the other examples, we fix the viewing direction
and change the direction of light illuminating the reflector. The
reflector can then display different images depending on the lighting
direction. We rendered six images of a teapot illuminated fromsix directions, respectively, and use them as the input images toour method. The fabricated reflector displays one of the images

Fabricating Reflectors for Displaying Multiple Images • 158:9

mirrors

mirrorsreflector(a)(b)

Fig. 11. Mirror art. (a) By placing mirrors around our reflector and viewing them from an appropriate position, the user sees different images in the mirrors
even though there is only a single sheet on which our reflector is printed. (b) shows the same reflector viewed from four different directions. The input images
are shown in the insets.

Fig. 12. Our reflector is used to display images of a teapot illuminated by the real light. The direction of light is indicated by the yellow arrows. The shading
and shadows of the teapot change depending on the lighting direction as if there was a 3D teapot in the reflector. We use six input images for this example
shown in the insets.

Fig. 13. Holographic reflector. By designing our reflector with the images ofa teapot rendered from different view points, appropriate image is displayed
according to the viewing direction as if there was a 3D teapot in the reflector.

according to the direction of real light illuminating the reflector.
The user can learn how the shading and shadows change by altering
the direction of real light by him/herself.
Finally, Fig. 13 demonstrates the use of our reflector for displaying
a 3D object, holographic reflector. We prepared four input imagesof the teapot rendered from four different viewing directions (seethe insets in the figure). The reflector can display the appropriate
image from them according to the viewing direction. The user can
see the 3D teapot through the reflector. Note that the shading of theteapot changes as well depending on the viewing direction.
Our method designs the SVBRDF only for N discrete viewing
directions, and thus the reflectances for other directions are not defined. However, as shown in the accompanying video, our designed
SVBRDF does not produce weird appearances for other viewpoints.
The proposed objective function in Eq. (7) does not consider the
durability of SRSs. Therefore, short line segments can peel off fromthe reflector. In our experiments, the lengths of the peeled line
segments are less than 0.85mm, and these are isolated (i.e. they do
not touch other line segments).
Currently, the number of displayable images in our method islimited to six mainly due to the resolution of the UV printer andthe huge computation time required to optimize Eq. (7) for N > 6.
We would like to optimize E(G) efficiently to increase the numberof displayable images, but this is left for future work.
One limitation of our method is that the images displayed tend
to be darker than the original images. This is because they aredisplayed such that the SRSs subtract some portion of the reflected
light. A simple solution to this problem is making the cell size larger
so that each cell can receive a larger amount of light. However, since
the maximum height of the SRSs is limited, the elevation angle ofthe viewing direction becomes lower, making it difficult to viewthe target images. This is a trade-off between the intensity and thevisibility of the images. Another possible solution would be to use
backlight behind the reflector. Combining a reflector printed on atransparent sheet with a backlight would address this problem.

6

CONCLUSION AND FUTURE WORK

We have proposed a method for fabricating a reflector that candisplay multiple color images. Our method designs a SVBRDF thatreflects the incident light in multiple directions, which makes it
possible to increase the number of displayable images compared tostate-of-the-art methods. These complex reflections are reproduced
by printing an ink layer and SRSs using a standard UV printer.
The SRSs are optimized to minimize ghosting effects for better
appearance. The color of the ink layer is calculated to match the

ACM Trans. Graph., Vol. 37, No. 4, Article 158. Publication date: August 2018.

158:10 •Sakurai, K. et alreflected light from the reflector with the target images, taking into
account the gamut of the UV printer through the color reproduction
process. We demonstrated that our simulation system can predict
the appearance of the reflector faithfully and the fabricated reflector
can display multiple color images without ghosting effects.
In the future, we would like to get faster computation of the microstructural stripe patterns and the color pattern of the ink layer byprecomputation. The most time-consuming part is the computationof the color pattern of the ink layer, whereas computation of the SRS
is rather efficient. We would like to precompute a set of color patterns for possible combinations of pixel colors in the input images.
Moreover, to obtain better contrast of the displayed images, the SRS
could be adapted to the content of the desired images; we can use
a different SRS for each pixel. However, we need to construct the
SRSs taking into account the coherence of the reflected radiancesfrom neighboring cells. Otherwise, the appearance would be noisy.
Moreover, we would like to incorporate durability of the SRSs into
the objective function.

ACKNOWLEDGEMENTS
The authors would like to thank Pierre Poulin for proofreading of
the first draft. The authors would also like to thank the reviewers
for their constructive comments to improve our paper. This work
was supported by JSPS KAKENHI Grant Numbers JP15H05924.

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