Poisson-Based Continuous Surface Generation for Goal-Based
Caustics
Yonghao Yue
Columbia University
Kei Iwasaki
Wakayama University and UEI Research
Bing-Yu Chen
National Taiwan University and UEI Research
Yoshinori Dobashi
Hokkaido University and UEI ResearchandTomoyuki Nishita
UEI Research and Hiroshima Shudo University
We present a technique for computing the shape of a transparent object
that can generate user-defined caustic patterns. The surface of the object
generated using our method is smooth. Thanks to this property, the resulting caustic pattern is smooth, natural, and highly detailed compared to theresults obtained using previous methods. Our method consists of two processes. First, we use a differential geometry approach to compute a smooth
mapping between the distributions of the incident light and the light reaching the screen. Second, we utilize this mapping to compute the surface ofthe object. We solve Poisson’s equation to compute both the mapping andthe surface of the object.
Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.8 [Computer Graphics]: Applications
General Terms: Algorithms, Design
Additional Key Words and Phrases: Caustics, surface generation, continuous, Poisson equation
ACM Reference Format:
Yonghao Yue, Kei Iwasaki, Bing-Yu Chen, Yoshinori Dobashi, Tomoyuki
Nishita. 2014. Poisson-based continuous surface generation for goal-based
caustics. ACM Trans. Graph. 33, 3, Article 31 (May 2014), 7 pages.
DOI: http://dx.doi.org/10.1145/2580946

1.

INTRODUCTION

In computer graphics, many methods have been proposed to compute the shape or structure of an object to obtain a desirable illumination effect, such as reflectance fields [Fuchs et al. 2008], surface reflections [Weyrich et al. 2009], shadows [Mitra and Pauly
2009; Baran et al. 2012; Bermano et al. 2012], subsurface scattering [Dong et al. 2010; Hašan et al. 2010], shading [Alexa and Matusik 2010], holography [Regg et al. 2010], and refractive steganography [Papas et al. 2012].
Among these methods, caustics provide us with some of the mostimpressive illumination effects. Our aim is to compute the shape ofthe transparent object that produces the caustic pattern specified by
the user (Figure 1). The setting is shown in Figure 2. We assume
that the light incident on the object (the corresponding surface iscalled the incident surface) is nearly parallel, and more or less normal to the incident surface, so that the light is in essence not refracted at the incident surface. On the opposite side of the object,
there is a surface called the refractive surface, where the light canbe refracted. When we place this object between the light source
and a planar screen, we want the refracted light reaching the screen
to form the desired caustic pattern. The problem dealt with in thisarticle is the computation of the shape of this refractive surface. In
Figure 1, we show a real example of an object constructed in thisway and the resulting caustic pattern.
Although previous methods [Finckh et al. 2010; Papas et al.
2011; Yue et al. 2012] have been proposed for realizing therequired caustics, they are somewhat restricted. Finckh et al.’s
method [2010] can generate only simple caustics. Papas et al.’s
method [2011] and Yue et al.’s method [2012] can generate more
complex caustics; however, these methods generate non smoothsurfaces, affecting the quality of the caustics. In this article, we
propose a new method with which both continuous surfaces andcaustics can be generated, with the aim of producing high-quality
caustic patterns.
For the production of high-quality caustic patterns, we want toconsider not only the case where the object is placed in focus, butalso the case where the object is placed out of focus. To produce
high-quality caustics in both cases, five criteria need to be met: (i)the need to produce complex patterns such as general images, (ii)the need to produce high-resolution images, and (iii) the need tohave a continuous dynamic range (with high quality even in regionsof low intensity). In addition, (iv) the focus range must be wide, and(v) the caustic pattern produced must be continuous even when theobject is not in focus. The criteria (iv) and (v) are needed because
we do not want the quality of the generated caustic pattern to besensitive to the locations and orientations of the light source, object,and screen (i.e., caustics can be produced even when the placement
is not perfect).
Our aim is to generate object surfaces that match all of the criteria listed before, whereas previous methods have at least one criterion which cannot be satisfied (see Section 2).

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Refraction
Incident light beamdSyp(x,y)
Intensity L(x,y)

dS ’q(u,v)zx
Coordinate system

Intensity L ’(u,v)
Transparent object

ScreenFig. 3. An illustration of the computation of the relationship.

Fig. 1. The fabricated object (right) and its caustics (left) on the screen.
Please see Figure 7 for the shape of the refractive surface of the object.
Nearlyparallel light

Incident surface:
plane

Refraction surface

Light sourcee.g. sunlight, projector

Transparent object
Screen Caustic pattern
Screen: plane

Fig. 2. An illustration of the proposed display system.

2.

PREVIOUS APPROACHES

The approaches taken in the previous methods [Finckh et al. 2010;
Papas et al. 2011; Yue et al. 2012] can be classified into the following two types. The first approach, taken by Finckh et al. [2010],
randomly perturbs the refractive surface to find the solution. The
second approach, taken by Papas et al. [2011] and Yue et al. [2012],
first determines the relationship between the light on the incident
surface and that on the screen, and then computes the shape ofthe refractive surface. In the first approach, the solution space iscomplex, and there can be many solutions that are optimal locally.
Papas et al. [2011] pointed out that the result generated using thisapproach can converge to a solution that does not span the full contrast range of a complex target image (criterion (i) in Section 1).
In the second approach, the key element is the way in which therelationship between the light on the incident surface and that on
the screen is found.With previous methods only a discrete relationship could be found. In those previous methods, the incident surface (or the refractive surface) and the screen are subdivided into
sets of small regions, and the relationships between these small regions are established. Papas et al.’s method [2011] subdivides theincident surface into a grid and the caustics into a set of Gaussian
kernels, and uses simulated annealing to find the relationship. In
Yue et al.’s method [2012], the object can be rearranged using a setof discrete sticks.
In these methods [Papas et al. 2011; Yue et al. 2012], the lightincident on neighboring regions generally arrives at distant regions
on the screen. Due to this discontinuity, the following problems ocACM Transactions on Graphics, Vol. 31, No. 3, Article 31, Publication date: May 2014.

cur. First, each small region usually needs to be of a certain size,
which restricts the resolution of the caustic pattern (criterion (ii) in
Section 1). Second, the dynamic range of the caustics is somewhat
restricted (criterion (iii) in Section 1). This is because the intensity of the caustics is basically determined by the number of overlapping regions on the screen, and since each small region on theincident surface is of a similar size, the intensity of the caustics becomes discrete. Furthermore, the range over which the caustics arein focus is short (criterion (iv) in Section 1), and if the object isplaced out of focus, the resulting caustic pattern would be a set of
discrete spots (criterion (v) in Section 1). Our goal is to eliminate
discretization of the resulting image and to enhance the stability ofthe projected image to perturbations in the physical display setting.
In this article, we propose a new method for computing the shapeof the object. Our method can be classified as the second approachdescribed previously. The distinct aspect of our method is that wecompute a continuous relationship. Owing to this property, the refractive surface becomes continuous, and the distribution of thelight leaving the refractive surface is also continuous. The problems described earlier can be resolved as a result of this.

3.

OUR METHOD

Our method consists of two steps. In the first step, we compute therelationship between the light reaching the incident surface and thatreaching the screen. In the second step, we compute the shape ofthe refractive surface from the relationship.

3.1

Computing the RelationshipAs shown in Figure 3, the z-axis is aligned with the direction of theincident parallel light (the light is coming from +z toward −z), andthe incident surface is a plane perpendicular to the z-axis. Points on
the incident surface can be represented using x and y coordinates.
As the light is perpendicular to the incident surface, it enters theobject without refraction and reaches the refractive surface. The z
coordinate of any point on the refractive surface can be represented
as a single-valued function of x and y. The light refracts at therefractive surface and then reaches the screen. To distinguish thecoordinates on the screen from those on the incident surface, we
use u and v to describe the x and y coordinates on the screen. For
simplicity, we use (x, y) or (u, v) and drop the z coordinate todescribe the points on the incident surface and the screen.
The light path can be described by the relationship between apoint p(x, y) on the incident surface and a point q(u, v) on thescreen. In our method, we consider the case when this relationshipis one-to-one, and represent the caustics by changing the densityof the light reaching the screen. Thus, we need to find from which
regions of the incident surface we need to gather light in order toform the intensity distribution of the desired caustic pattern. This
problem can be formulated as the problem of finding a mappingfrom q(u, v) to p(x, y) satisfying the following two conditions.

Poisson-Based Continuous Surface Generation for Goal-Based Caustics•

3in order to compute the pressure field φ (see Figure 4(b)), and update the points p(x, y) along the gradient of φ. That is, we let
∂p
= ∇φ.
∂t

(a)

(b)

(c)

Fig. 4. (a) The difference D (red and blue regions indicate positive andnegative portions); (b) pressure field φ; (c) vector field ∇φ.

The first condition is that the mapping is continuous over theentire domain. If this mapping is continuous, we can obtain a continuous refractive surface. The second condition is that this mapping preserves the light energy. Imagine the light beam shown in
Figure 3. If the light incident on an infinitesimal region dS aroundpoint p(x, y) on the incident surface has intensity L(x, y) (let us
use L for simplicity), and the intensity of this light on an infinitesimal region dS  around point q(u, v) on the screen is L (u, v) (let us
use L for simplicity), then LdS = L dS  . The aim of our method
is to find this mapping. Since it is difficult to analytically compute
this mapping, we use a geometric flow approach, as described next.
The mapping from q(u, v) to p(x, y) can be regarded as parameterization of x and y using u and v. That is, we start from an initial
parameterization x = u and y = v, and then maintain the continuity of the parameterization while modifying it so that it increasingly
satisfies the conservation of light energy.
Under the parameterization of u and v, the ratio dS/dS  is equivalent to the determinant J of the Jacobian given by ∂x
J =  ∂u
∂x
∂v

∂y
∂u
∂y
∂v



.


(1)

Hence, L = LJ. If the parameterization, or the mapping, satisfies
LJ = C at an arbitrary point, where C is the intensity of the caustics at the corresponding point on the screen, then that mapping is
the solution to the problem.
During computation, LJ = C is not always satisfied, and there
is a difference. Let this difference D(x, y) (again, we use D for
simplicity) be
D = LJ − C.

(2)

We reduce this difference (see Figure 4(a)) by updating the parameterization to continuously modify the point p(x, y) on theincident surface, while fixing the corresponding point q(u, v) on
the screen. Although we could consider the gradient ∇D (where∂∂∇ = ( ∂x, ∂y)) and modify the points as ∂p= ∇D (where t is a∂tvirtual time to represent the computation process), ∇D is usually
not continuous and may take an infinite value. Thus it is difficult tostably perform the computation using ∇D.
Instead, inspired by a concept in computational fluid dynamics,
we modify the parameterization as follows (our method is related
to one that generates area-preserving parameterization [Zou et al.
2011]). First, there are regions where the difference D is positive or
negative. Analogous to those in computational fluid dynamics, we
regard the positive and negative regions as source and sink, respectively. To relax the source and sink, we solve Poisson’s equation∇2 φ = −D,

(3)

(4)

Unlike ∇D, ∇φ (see Figure 4(c)) is a much smoother vector field,
thus we can stably update the points p(x, y). In our method, we
repeatedly update the parameterization by computing the difference
using Eq. (2), solving the Poisson’s equation (3), and updating p
according to Eq. (4).
A physical interpretation of the preceding technique is as follows. If we want a region on the screen to be brighter/darker,
the corresponding light beam needs to collect light from awider/narrower region (i.e., dS should be larger/smaller). This implies that we have to locally “expand” or ‘shrink” dS. By solving
Eq. (3) and computing ∇φ, we get a “compressible” flow field thatrealizes this expansion or shrinkage. By iteratively updating the parameterization, the contrast of the resulting caustics increases andapproaches the contrast of the input pattern.
In our implementation, we use a triangular mesh to represent theparameterization. To compute the light energy, instead of computing the Jacobian, we compute a median dual mesh (where we havea face for each vertex in the triangular mesh, each of which is generated by connecting the midpoints of the edges and the centroids ofthe triangles adjacent to the corresponding vertex in the triangularmesh). We then integrate the intensities in each face in the mediandual mesh to compute the light energy. Poisson’s equations (3) and(7) are solved using a first-order Galerkin finite-element method.
To update the vertices of the triangular mesh, we discretize Eq. (4)using the equation p(tn+1 ) = p(tn ) + ∇φΔt, where n indicatesthe n-th computation step, and Δt is determined so that there will
be no flipped triangles. We obtain Δt by first regarding p(tn+1 ) asa function of Δt. Next, for each triangle, we compute the maximum
Δti that the (i-th) triangle can take such that the area assigned to it
remains nonnegative. Then, we compute the minimum value Δtm
among Δti (i.e., Δtm = mini Δti ). Finally, we set Δt to 0.5Δtm .

3.2

Computing the Refractive Surface

As we have assumed the refractive surface to be a single-valued
function, its z coordinate can be represented as z = h(x, y). A
normal vector N to the refractive surface can be represented by
N=(

∂h ∂h
,
, −1),∂x ∂y(5)where N is normalized so that its z coordinate is −1. Considering
only the x and y components, we have
Nxy = ∇h,(6)

where Nxy = (Nx , Ny ) is a vector composed of the x and y components of N. Taking the divergence of both sides of Eq. (6), we
obtain another Poisson’s equation (similar to Yu et al. [2004])∇2 h = ∇ · Nxy .

(7)

Thus, if the normal vector at an arbitrary point on the refractive
surface is known, we can use Eq. (7) to compute the shape of therefractive surface. To compute the normal vectors, we use Snell’s
law based on the parameterization obtained from Section 3.1, and
get
Nxy =

1
(q − p),k(8)

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Y. Yue et al.

(a)

(b)

(c)

(d)

c
Fig. 5. Simulated caustic patterns. (a) The input pattern (Playboy
Magazine Nov. 1972); (b) the result using Finckh et al.’s method [2010]; (c) the result
using Papas et al.’s method [2011]; (d) the result using our method. The images of (b) and (c) are reprinted from Papas et al. [2011].

+200%

-200%

(a)

(b)

(c)

(d)

(e)

Fig. 6. Computation of the relationship. (a) to (e): Results after 1, 3, 5, 7 and 25 steps, respectively. The top row shows the isoparametric lines for displaying
the parameterization (please enlarge the figure to see the details). The bottom row shows the difference D (red and blue portions indicate positive and negative
regions, and the difference is normalized such that the average intensity of the desired caustics corresponds to 100%).

where (q − p) means (u − x, v − y),k=η||q − p||2 + (H − h)2 − (H − h).

(9)η is the refractive index (1.49 for an acrylate basin), and H is the
distance between the screen and the transparent object.
Since the computation of the right-hand side of Eq. (7) needsh, we use an iteration method as follows. First, we initialize therefractive surface as a plane. Then, we iterate the following two
steps. In the first step, we compute Eqs. (8) and (9) using the current
refractive surface. In the second step, we solve Eq. (7) and update
the refractive surface. Usually, only after a few iterations, we obtain
convergence.

3.3

A Remark on the Uniqueness of the Solution

Let us briefly consider the relationship between a point p(x, y) on
the incident surface and a point q(u, v) on the screen, obtained using our method. When we take the difference (u−x, v−y) between
the coordinates for all the points, this difference can be regarded asa vector field. According to the Helmholtz decomposition, a vector field can generally be decomposed into an incompressible field
ACM Transactions on Graphics, Vol. 31, No. 3, Article 31, Publication date: May 2014.

(divergence-free field) and a compressible field (curl-free field). In
our method, the initial parameterization is curl free, and since ∇φalways satisfies ∇ × ∇φ = 0, only curl free components are added
into the parameterization. Thus, the parameterization obtained using our method contains no divergence-free components. On theother hand, any divergence-free flow would preserve the area. Thus,(when the incident light distribution is uniform) any divergencefree mapping, which preserves the area and hence preserves thelight energy, can be combined with the mapping obtained using
our method to produce the same desired caustic pattern (i.e., take adivergence-free mapping which maps (x , y  ) to (x, y) and define
a composite mapping from (x , y  ) to (u, v)). In the sense that our
mapping does not contain divergence-free components, our mapping can be regarded as the “minimum” mapping.

4.

RESULTSWe show the results when using Figure 5(a) as the input. The resolution of the input image is 512 × 512. The progress of the computation of the relationship is shown in Figure 6, which shows thatit almost converges after seven steps. Based on the computed rela-Poisson-Based Continuous Surface Generation for Goal-Based Caustics•

5

Fig. 7. The refractive surface computed using our method.
Fig. 10. Real caustic patterns produced by fabricated objects using Papas
et al. [2011] (left) and our method (right). The left image is reprinted from
Papas et al. [2011]. Note that the photography environments are different,
altering the impression of the caustic patterns.

Fig. 8. Simulated results with different distances between the object andthe screen. Left to right: the distances are 0.5×, 1.0×, 1.5×, and 2.0× the
focusing length.

(a)

(a)

(b)
(b)

(c)

Fig. 11. From left to right: photographs of the fabricated objects and theircaustics, the simulated caustic patterns, and the caustic patterns of the fabricated objects.

(d)

(e)

(f)

Fig. 9. Other examples. From left to right: input patterns, isoparametriclines of the mapping, computed surfaces, and simulated caustic patterns.

tionship, we calculated the refractive surface as shown in Figure 7(the computation described in Section 3.2 took three iterations toconverge). When computing the surface, we assumed the dimensions of the object to be 100mm × 100mm in the x− and y− axes,
and the focal length to be 100mm. The maximum difference in thedepth (z-axis) of the object was 14.4mm. In Figure 5(d), we show
the simulated caustics using an inverse ray-tracing method. We cansee that the texture of the hat and the details of the hair are nicely reproduced. Comparison to the simulated results obtained using previous methods [Finckh et al. 2010; Papas et al. 2011] is shown in
Figure 5. Compared to the previous methods, our method is able toreproduce the detail.

Fig. 12. From left to right: the parameterization computed using remeshing, the refractive surface, the simulated caustics without remeshing, thesimulated caustics with remeshing, and the caustics produced by the fabricated object.

In Figure 8, we show the simulated results when the distance
between the refractive surface and the screen is changed. When we
set the distance to half or 1.5 times the focal length, the caustic
pattern is slightly distorted; nevertheless, we can still recognize thepattern. Moreover, the caustic pattern changes smoothly with the
distance.
Some additional results are shown in Figure 9. Using our
method, we can also generate vector-graphics-like caustic patterns
as shown in Figure 9(a). Figure 9(e) shows an example containing
Chinese characters.
In Figure 1, we show the fabricated object for Figure 5(a) and its
caustic pattern. This object was fabricated using a CNC machine(Okuma FMR-40) with a ball end mill. We applied three passes tomill the surface: two rough milling passes and a finishing milling
pass. In the first rough milling pass, we used a ball end mill of
3mm radius and the spacing (pitch) between neighboring milling
paths was 0.5mm. In the second rough milling pass, the radius
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was 0.5mm and the pitch was 0.05mm. The purpose of these roughmilling passes is to progressively mill the surface and to reduce themechanical load during the finishing pass. In the finishing milling
pass, the radius was 0.5mm and the pitch was 0.02mm. The fabrication quality is essentially dependent on the setting of the finishingmilling pass. The total fabrication time for these passes was approximately six days. We can vaguely perceive the vertical stripes,
which are due to the quality of the path of the end mill (see Section 5 for a detailed discussion). In Figure 10, we compare the realcaustic patterns produced by the fabricated object using our method
against one fabricated using that of Papas et al. [2011]. Since withour computation, the relationship between the light reaching theincident surface and that reaching the screen is continuous, we canobtain caustic patterns with continuous tone representation also fordark regions, whereas the dark regions in Papas et al.’s caustic pattern tend to be too dark.
In Figure 11, we show the fabricated objects for Figures 9(a) and(e). As shown in Figure 11(a), we can see that the sharp features,such as the thin dark regions and the crisp corners, are also reproduced in the caustics generated by the fabricated object. We werealso able to represent the characters in the caustics of the fabricatedobject, as shown in Figure 11(b).
The settings used in fabricating the object in Figure 11(a) are
the same as those used for the Lena example. For the fabricationof the object in Figure 11(b), we used an NC machine (Roland
MDX-40A) with a ball end mill. We applied three passes to millthe surface as well, and the settings of the radius and the pitch are
the same as those used for the Lena example.

5.

z

pitch

Fig. 13. Left: an illustration of the milling path. Right: a top view. Paths
shown in black color are for milling the surface and those shown in graycolor are “subsidiary” paths, each of which is used to position the end millto the next section without touching the surface (the end mill goes up, thenmoves to the next starting position straightly and horizontally, and then goesdown).

Fig. 14. Left: an exaggerated illustration of a cross-section through therefractive surface (black line) and the fabricated surface (red line) due tothe finite radius of the end mill (the materials milled out by the end mill are
shown as dashed red lines). Right: an exaggerated illustration of the lightrefraction with respect to the fabricated surface.

LIMITATIONS AND DISCUSSIONS

Since we are considering fully continuous parameterization, we
cannot handle regions with zero intensity. To handle zero intensity, we need to extend our method to incorporate singular points
or lines in the parameterization, in order to enable the splitting oflight beams and to prevent light reaching such regions.
A limitation of the current implementation is that the mesh could
get stuck (i.e., we cannot advect the mesh vertices). This could
happen when the input pattern contains “localized high-contrast regions”, that is, when there is a very dark region adjacent to a brightregion. In such a case, the triangles (in the mesh) corresponding to
the bright/dark regions tend to expand/shrink greatly. Hence, triangles between these regions tend to be very thin and to impose severe restriction on Δt (see Section 3.1). It is interesting to note thatwhen the mesh gets stuck, if we use the corresponding parameterization to compute the object surface, the resulting caustic patternis still recognizable. The drawback is that the contrast is lower thanrequired. The examples in Figures 9(c) and (f) are such cases.
Introducing a remeshing technique could prevent the mesh getting stuck. We implemented a simple remeshing by first detecting
triangles which admit small Δt, then removing the vertices of thetriangles, and inserting random vertices using dart throwing. In thisway, thin triangles will be automatically removed, and we will beable to advect the mesh with large Δt. As shown in Figure 12, thecontrast of the resulting caustics increases and reaches the samelevel as that of the input pattern. (The results shown in Section 4
are all computed without remeshing.) One drawback of this simple remeshing technique is that the isoparametric lines could get“dirty”, and the edges appearing in the caustics could become wavy.
We think we could further introduce a mesh smoothing technique tokeep the mesh “clean”.We leave this improvement for future work.
Under the current fabrication settings, we obtain slightly blurredcaustics and striped artifacts. A possible reason for the blurriness is
ACM Transactions on Graphics, Vol. 31, No. 3, Article 31, Publication date: May 2014.

(a)

(b)

(c)

Fig. 15. Simulated caustics taking into account bumps in the refractive
surface. The radius of the end mill was 0.5mm in each case. The pitch of
the milling path was set to: (a) 0.02mm, (b) 0.005mm, and (c) 0.02mm. In(c), deviations in the paths in the z direction are taken into account and aremodeled using white noise.

as follows. For milling, the paths of the ball end mill are taken as
illustrated in Figure 13 (for simplicity, we only consider the finishing milling pass). Since the pitch between consecutive paths is finite (0.02mm) and the ball end mill has a finite radius (0.5mm), themilled surface will have tiny bumps on it (see the exaggerated illustration in Figure 14). These bumps will cause some of the refracted
light to deviate slightly, creating blurred caustics. In Figure 15(a),we show our simulation of the caustics taking these bumps into
account. We find that if we set the pitch to 0.005mm and use thesame ball end mill, we can significantly reduce this blurriness as
simulated in Figure 15(b). Note that using a smaller pitch meansthat the total length of the milling path will be longer, hence theend mill will gall more during the milling process, which could be
another cause of the artifact. A highly strengthened end mill would
be needed for this fine setting.
A possible reason for the striped artifact is as follows. With thecurrent setting, the fabrication proceeds by repeatedly milling thesurface from one side to the other (as in Figure 13), and each timePoisson-Based Continuous Surface Generation for Goal-Based Caustics
the ball end mill enters the material, a physical collision takes place
between the end mill and the surface. The impact of this sometimes
causes the end mill to slightly deviate from its intended coordinates, and this deviation will be carried over the remaining milling
process. In Figure 15(c), we model this deviation as white noise(in the z direction), and obtain similar striped artifacts in the simulated caustics (in this example, the pitch is set to the same as our
fabrication setting, 0.02mm). Hence, we believe these stripe artifacts can be reduced if we are able to lessen this physical impact(for example, by using a single smoothly connected milling path,
which traces mostly along the contour lines of the surface, in orderto keep the end mill moving within the surface as much as possible
and to reduce the events where the end mill leaves and reenters thesurface).

6.

CONCLUSIONS AND FUTURE WORKIn this article, we have presented a new method for computing thesmooth refractive surfaces that generate user-specified caustics. Using our method, the relationship between the light reaching the incident surface and that reaching the screen is computed as a continuous function. As a result, the computed refractive surface issmooth. We have demonstrated that we can generate continuousand high-quality caustics through the simulated results, and haveshown the effectiveness of our method. For future work, we would
like to develop a clean remeshing technique which enables betteredge representation for high-contrast input patterns.

•

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PAPAS , M., H OUIT, T., N OWROUZEZAHRAI , D., G ROSS , M., AND
JAROSZ , W. 2012. The magic lens: refractive steganography. ACM
Transactions on Graphics (Proc. of SIGGRAPH Asia 2012) 31, 6, 186:1–
186:10.
PAPAS , M., JAROSZ , W., JAKOB , W., RUSINKIEWICZ , S., M ATUSIK , W.,
AND W EYRICH , T. 2011. Goal-based caustics. Computer Graphics Forum (Proc. of Eurographics 2011) 30, 2, 503–511.
R EGG , C., RUSINKIEWICZ , S., M ATUSIK , W., AND G ROSS , M. 2010.
Computational highlight holography. ACM Transactions on Graphics(Proc. of SIGGRAPH Asia 2010) 29, 6, 170:1–170:12.
W EYRICH , T., P EERS , P., M ATUSIK , W., AND RUSINKIEWICZ , S. 2009.
Fabricating microgeometry for custom surface reflectance. ACM Transactions on Graphics (Proc. of SIGGRAPH 2009) 28, 3, 32:1–32:6.
Y U , Y., Z HOU , K., X U , D., S HI , X., BAO , H., G UO , B., AND S HUM , H.Y. 2004. Mesh editing with poisson-based gradient field manipulation.
ACM Transactions on Graphics (Proc. of SIGGRAPH 2004) 23, 3, 644–
651.
Y UE , Y., I WASAKI , K., C HEN , B.-Y., D OBASHI , Y., AND N ISHITA , T.
2012. Pixel art with refracted light by rearrangeable sticks. Computer
Graphics Forum (Proc. of EUROGRAPHICS 2012) 31, 2, 575–582.
Z OU , G., H U , J., G U , X., AND H UA , J. 2011. Authalic parameterization ofgeneral surfaces using lie advection. IEEE Transactions on Visualization
and Computer Graphics 17, 12, 2005–2014.
Received February 2013; revised December 2013; accepted January 2014

Acknowledgments
This work is supported by JSPS Grant-in-Aid for Research Activity
Start-up (23800011). The stay of Y. Yue at Columbia University isfunded by JSPS Postdoctoral Fellowships for Research Abroad. We
are deeply grateful to MTM Corporation (Japan) for fabricating thetransparent objects and for useful discussions.
REFERENCESA LEXA , M. AND M ATUSIK , W. 2010. Reliefs as images. ACM Transactions on Graphics (Proc. of SIGGRAPH 2010) 29, 4, 60:1–60:7.
BARAN , I., K ELLER , P., B RADLEY, D., C OROS , S., JAROSZ , W.,
N OWROUZEZAHRAI , D., AND G ROSS , M. 2012. Manufacturing layered
attenuators for multiple prescribed shadow images. Computer Graphics
Forum (Proc. of EUROGRAPHICS 2012) 31, 2, 603–610.
B ERMANO , A., BARAN , I., A LEXA , M., AND M ATUSK , W. 2012. Shadowpix: Multiple images from self shadowing. Computer Graphics Forum(Proc. of EUROGRAPHICS 2012) 31, 2, 593–602.
D ONG , Y., WANG , J., P ELLACINI , F., T ONG , X., AND G UO , B. 2010.
Fabricating spatially-varying subsurface scattering. ACM Transactions
on Graphics (Proc. of SIGGRAPH 2010) 29, 4, 62:1–62:10.
F INCKH , M., DAMMERTZ , H., AND L ENSCH , H. P. A. 2010. Geometry
construction from caustic images. In Proceedings of the 11th European
Conference on Computer Vision. 464–477.
F UCHS , M., R ASKER , R., S EIDEL , H.-P., AND L ENSCH , H. 2008. Towardpassive 6D reflectance field displays. ACM Transactions on Graphics(Proc. of SIGGRAPH 2008) 27, 3, 58:1–58:8.
H A ŠAN , M., F UCHS , M., M ATUSIK , W., P FISTER , H., AND
RUSINKIEWICZ , S. 2010. Physical reproduction of materials withspecified subsurface scattering. ACM Transactions on Graphics (Proc.
of SIGGRAPH 2010) 29, 4, 61:1–61:10.
M ITRA , N. J. AND PAULY, M. 2009. Shadow art. ACM Transactions on
Graphics (Proc. of SIGGRAPH Asia 2009) 28, 5, 156:1–156:7.
ACM Transactions on Graphics, Vol. 31, No. 3, Article 31, Publication date: May 2014.