SHADOWPIX: Multiple Images from Self Shadowingpeople.csail.mit.edu/ibaran/papers/2012-EG-ShadowPIX.pdf · EUROGRAPHICS 2012 / P. Cignoni, T. Ertl (Guest Editors) Volume 31 (2012),

EUROGRAPHICS 2012 / P. Cignoni, T. Ertl(Guest Editors)

Volume 31 (2012), Number 2

SHADOWPIX: Multiple Images from Self Shadowing

Amit Bermano1,2 Ilya Baran1 Marc Alexa3,5 Wojciech Matusk1,4,5

1Disney Research Zürich 2ETH Zürich 3TU Berlin 4MIT CSAIL 5Disney Research Boston

Figure 1: SHADOWPIX are diffuse white surfaces whose self-shadowing pattern appears in the form of multiple prescribedimages, depending on the illumination direction. The top row has seven input images. The bottom row has photographs of twoSHADOWPIX, lit from different sides. We present two methods for constructing SHADOWPIX. The left three photographs areof a surface constructed with the local method, and the right four photographs are of a surface constructed with the globalmethod. Due to poor downsampling in common software, we recommend viewing these images zoomed in to avoid aliasing.

Abstract

SHADOWPIX are white surfaces that display several prescribed images formed by the self-shadowing of the sur-face when lit from certain directions. The effect is surprising and not commonly seen in the real world. We presentalgorithms for constructing SHADOWPIX that allow up to four images to be embedded in a single surface. SHAD-OWPIX can produce a variety of unusual effects depending on the embedded images: moving the light can ani-mate or relight the object in the image, or three colored lights may be used to produce a single colored image.SHADOWPIX are easy to manufacture using a 3D printer and we present photographs, videos, and renderingsdemonstrating these effects.

Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-DimensionalGraphics and Realism—Shadowing

1. Introduction

The effect of illuminating a white diffuse surface from dif-ferent directions is usually easily predictable. In this paper,we show how to generate surfaces for which this is not thecase: depending on the light direction, the shadows the sur-face casts on itself form different prescribed images. Wecall these surfaces SHADOWPIX. While our work is primar-ily an exploration of the space of intelligent material struc-

tures, SHADOWPIX are a step towards passive dynamic dis-plays that interact with lighting (in the style of [NBB04]and [FRSL08]). Other potential applications include archi-tectural design: for example, they can be used as walls thatlook different depending on the time of day, or as a sophisti-cated sundial. SHADOWPIX can be used in a theater setting,where precisely controlled directional illumination is readilyavailable. Another possibility is a steganography-like appli-

c© 2012 The Author(s)Computer Graphics Forum c© 2012 The Eurographics Association and Blackwell Publish-ing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ,UK and 350 Main Street, Malden, MA 02148, USA.

Bermano et al. / SHADOWPIX: Multiple Images from Self Shadowing

cation, where an object reveals a secret image only underspecific illumination. Finally, SHADOWPIX can simply befascinating memorabilia.

To concentrate on the self-shadowing effect, we considerSHADOWPIX to be made of a diffuse, uniformly white ma-terial, with minimal subsurface scattering. We assume thesurface to be a grid of elements with square bases and vary-ing heights (Figure 8, top), and later slightly relax this as-sumption (Figure 6). The top of each element is planar andparallel to the ground plane and we assume that each el-ement is either fully lit or fully shadowed for a certainlight direction. Thus, the images generated by the surfacethrough self-shadowing for a certain light direction are bi-nary. Consequently, grayscale images can be reproducedonly by halftoning.

Since we need to halftone the input image, we necessarilyhave the competing goals of similarity to the input image andsupporting high-resolution images. In addition, we assumethe viewer sees all elements equally well. This is true foran idealized orthographic viewer or, more practically, if theheight variation is small. This additional requirement of lowheight variation also makes SHADOWPIX easier and cheaperto manufacture because tall thin rigid spikes are fragile.

While there are potentially more general models for SHAD-OWPIX, discrete heightfields have the advantage of beingeasy to manufacture. The resulting objects can be con-structed with a wide range of processes; while we createdthe prototypes with 3D printing, milling or creating moldswould also be easy. Similar models have been used for along time outside the computer graphics field, like the pin-screen, which was invented by Alexandre Alexeieff, ClaireParker, and Alexandra Grinevskaya in the 1930’s and used toproduce several short films using the self-shadowing effect.

For producing SHADOWPIX, we have developed a local anda global algorithm. The algorithms roughly correspond tosimilar halftoning strategies. For the local algorithm, a gray-value image pixel is represented by a small square of surfaceelements. We separate the elements inside each square intoshadow-casters and receivers a priori. A certain percentageof the square will be in shadow, corresponding to a grayscalevalue in the image. This scenario yields an explicit construc-tion algorithm. However, at most three images can be pro-duced this way and they are not ideally exploiting the avail-able printer resolution.

In the global scenario, every element can both cast and re-ceive shadows, allowing us to produce four images and usethe full printer resolution by having a printer element repre-sent an image pixel. The complexity of the objective functionrequired us to use stochastic global optimization techniquesover a reasonable subset of all possible height vectors. Fig-ure 1 shows SHADOWPIX produced by both methods.

We present manufactured results for both algorithms, show-ing that our techniques work in practice. We demonstrate

that the encoded images can be unrelated, they can be framesof a video, lighting configurations, or color channels of a sin-gle image.

2. Related Work

Computing shape from shadows (darkness) is a classic prob-lem studied by computer vision researchers. In this problem,a 3D surface is reconstructed from a set of shadow images.The initial work of Shafer and Kanade [SK83] describes ge-ometrical constraints imposed by shadows. The first algo-rithms [KS86, RPL89] assume that the scene is a heightfield(terrain), the camera is fixed and orthographic, and the lightis directional and is moving along a known arc. These algo-rithms use a shadowgram data structure to encode shadowdependencies, but they typically reconstruct only a singlecross-section of the surface. Daum and Dudek [DD98] ex-tend the shadowgram algorithms to compute a complete 2Dheightfield. Yu and Chuang [YC02] propose shadow graphs,a more efficient representation for encoding constraints in aterrain surface. The general setting for the problem of shapefrom shadows appears to be similar to ours at a high level.However, there are key differences. First, our input imagesare grayscale and not binary, as typical shadow images. Fur-thermore, our goal is different: we do not attempt to re-construct the actual 3D scene geometry but only produce aheightfield that induces the desired images. Finally, we alsotry to keep the height of the terrain as small as possible inorder to reduce view-dependent effects and to make the fab-rication process more robust.

Our work is also related to shadow sculptures—3D struc-tures that can generate up to three desired shadow im-ages on planar surfaces when illuminated from differentdirections. These shadow images are effectively silhou-ettes of the shadow sculpture from different views. Shadowsculptures have been typically assembled by trial and er-ror by artists (e.g., Tim Nobel and Sue Webster, ShigeoFukuda). Elber [Elb02] has constructed sculptures withprescribed views from different directions. More recently,computational methods [KvWW09, MP09] have consideredautomatically constructing shadow sculptures. Kieren etal. [KvWW09] give algorithms to determine whether a con-nected 3D shape can have 3 given rectilinear silhouettes, butprovide no hint for how to adjust the silhouettes to be at-tainable by a 3D object. Shadow Art [MP09] partially ad-dresses this problem, showing how to continuously warp sil-houettes to “match” in 3D. Shadow sculptures display shad-ows on external planar surfaces, whereas a SHADOWPIX

serves as both the shadow caster and the shadow reciever(hence, self-shadowing). The main advantages of using self-shadowing are simplicity of manufacturing (SHADOWPIX

are just uniformly white heightfields) and the ability to dis-play halftoned images using a single connected object: Mitraand Pauly [MP09] demonstrate an example with images, but

c© 2012 The Author(s)c© 2012 The Eurographics Association and Blackwell Publishing Ltd.


Figure 2: Physically-based renderings of a relief generatedby Alexa and Matusik’s method [2010] from the first two in-put images in Figure 1 (top) and our local method (bottom)under ideal viewing and lighting conditions. Our methodscapture more detail and attain significantly higher quality,while simultaneously embedding more images.

it requires embedding the sculpture into a transparent cubeand only a simulated result with perfect lighting is shown.

The most relevant prior work [AM10] constructs relief sur-faces that can form two images for different light directions.This method only uses diffuse shading and assumes no self-shadowing or other shading effects. Focusing only on shad-ing makes the problem simpler due to its local nature, but itresults in significant limitations. In particular, creating morethan two images using only shading is impossible since nor-mals have only two degrees of freedom. Even with two im-ages, the requirement that the normal field be integrable isan additional constraint, which results in severe ghosting ar-tifacts between the two images (Figure 2). In contrast, shad-ing due to self-shadowing is not a local effect: a surface pointcan cast a shadow on another point that is potentially faraway. This non-locality makes it more challenging to controlself-shadowing, but in return, it allows us to encode moreimages into one surface and obtain sharper results.

Our global method is structurally similar to someoptimization-based halftoning methods [AA92, PQW∗08].These halftoning methods maximize the similarity to a sin-gle given image and can directly set the pixels of the binaryimage. In contrast, we are maximizing similarity to four im-ages simultaneously, we can only influence the binary im-ages by changing heightfield element heights, and have addi-tional manufacturing considerations, such as height and sur-face smoothness.

3. Method

We offer two different methods for generating SHADOW-PIX. The local method uses a grid corresponding to the pix-els of the input images and distinguishes between shadow

receivers, making up most of each pixel, and walls castingshadows. The local method can handle up to three differentinput images to be displayed for three light directions, whichare aligned with the grid of elements, when projected to thexy plane.

The global method individually assigns height values to thesmallest printable elements. Each element is either shad-owed or not. This binary pattern can be used to dither theinput images. For this method we can potentially use an ar-bitrary number of input images as well as arbitrary light di-rections, but for reasons of efficiency in the optimization, itis beneficial if up to four light directions are considered, allbeing aligned with the grid of elements.

h

+x light −x light

h/s

θ

SHADOWPIX

−z

Figure 3: A single heightfield element casting differentshadows depending on the light direction.

For both methods we use the following variables (Figure 3):the ground plane for the SHADOWPIX is parallel to the xyplane and the view direction is along the z-axis. We assumethat all light sources are directional. For clarity of exposi-tion, we assume that the light directions all form the sameangle θ with the z-axis (i.e., have the same elevation over theground plane). Letting s = cotθ, an element of height h castsa shadow of length h/s onto a plane parallel to xy. In the fol-lowing discussion, we will make use of the proportionalityconstant s, which depends only on the setup of the lights.

3.1. Local Algorithm

The shadow receivers are arranged as a grid of unit squaresto match the pixels of the images. The shadow casters are thewalls between the squares. As shown in Figure 4, the heightsof the receivers are ri, j and the heights of the x and y castersare ui, j and vi, j, respectively. The three light directions aretowards +x, −x, and +y. We admit that this makes the for-mulation asymmetric, but aligning the light directions withthe pixel grid keeps the shadow geometry simple and leadsto a very efficient algorithm.

Let the grayscale values of the images be I1i, j, I2

i, j , and I3i, j.

We need the unshadowed fraction of each square from eachlight direction to be the prescribed grayscale value. The con-straints this requirement imposes on the element heights are:

ui, j− ri, j = sI1i, j

ui+1, j− ri, j = sI2i, j (1)

vi, j− ri, j = sI3i, j.



ri, j ri+1, ju i,j

u i+

1,j

u i+

2,j

vi, j vi+1, j

vi, j+1 vi+1, j+1

y

x

Figure 4: A view from above of the grid and heights usedby the local method.

Figure 5: Without chamfering, the local method producesSHADOWPIX that are too tall to be usable. The height in-crease is necessary to prevent a part of the surface closer tothe +y light direction from shadowing a part of the surfacefarther from the light direction (see Equation (3)). Chamfer-ing produces SHADOWPIX whose maximum height is about1 cm for an image that is 20 cm on each side.

Note that this system of equations provides necessary condi-tions for the solution. In addition we require inequality con-straints that make sure that only the walls are casting shad-ows and that these shadows are contained only in the pixelsadjacent to the walls. The first condition simply means ri, j <min(ui, j,ui+1, j,vi, j,vi, j+1). With this condition in place, thelatter requirement imposes a restriction on the height differ-ence between adjacent walls. For example, in the +x direc-tion, if ui, j−ui+1, j > s then the shadow cast by ui, j may spillover onto the pixel at (i+ 1, j). For the −x and +y light di-rections, this leads to the requirements |ui, j−ui+1, j|< s andvi, j− vi, j+1 < s, respectively. We note that these restrictionsdo not limit the space of images we can represent.

A generic way to solve this problem would be to use linearprogramming. However, we note that the structure offers aparticularly efficient assignment of heights in scan line order,i.e. the system is almost triangular.

First, we combine the first two linear equations to get a con-

ui, j

ui+1, j

ui+2, jri, j

ri+1, j

+x light −x light

c−i, j

c+i, j

c−i+1, jc+i+1, j

x

z

y

Figure 6: Top: A single chamfered pixel produced by thelocal method. Bottom: The x-z cross-section of two cham-fered pixels, illustrating the parameters. In this case, be-cause there is no chamfer on the right of the second pixel,c+i+1, j = 0. Our algorithm would never actually produce this

configuration because c−i, j could be increased further, allow-ing ui+1, j and therefore ri+1, j to increase.

dition that determines heights along scan lines:

ui, j = u0, j + si

∑x=1

(I2x, j− I1

x, j

). (2)

We note that this directly satisfies the inequalities among theui, j. Thus, we just walk along a scan line, starting at u0, j anddetermine the values of ui, j. Now, the interesting observationis that the inequality ri, j < vi, j can be rewritten in terms ofthe ui, j by exploiting their inequalities with ri, j as

ui, j+1 ≥ ui, j + s(−I1

i, j + I1i, j+1− I3

i, j+1

), (3)

leading to a condition to be satisfied when moving to the nextscan line. We can always satisfy this condition by makingu0, j+1 sufficiently larger than u0, j. As we would like to min-imize the height, we pick u0,0 = 0 and incrementally picku0, j to be the smallest nonnegative value that still satisfiesthis inequality for all i. In this setup, no degrees of freedomare left for a fourth image.

This method can reproduce any three images for the ide-alized case of an orthographic viewer and unconstrainedheight differences. However, with the viewer at a finite dis-tance and considering practical limitations of the construc-



tion process, we need to limit variation in height, becauseotherwise, the results are unusable (Figure 5). While it wouldbe possible to add additional linear and non-linear con-straints to limit the height variation, the number of degrees offreedom in the pixel model is too small to improve the heightsignificantly. Instead, we address the problem of height vari-ation by adjusting the geometry of the shadow receivers,which we have so far assumed to be flat. The simplest con-figuration we have found to be satisfactory works as follows:we add chamfers to the +x and −x edges of each shadowreceiver, as shown in Figure 6. While these chamfers contra-dict our assumption that the surface is parallel to the groundplane, we have found that when they are produced with the3D printer, their reflectance behaves as expected.

The chamfers are always parallel to the opposite light direc-tions, so the chamfer at the +x edge is never lit by the lighttowards −x and vice versa. The heights of the chamfers atthe walls, given by c−i, j and c+i, j, are therefore additional de-grees of freedom that affect the +x and −x images inde-pendently. The chamfers are potentially lit by their light di-rections, and are brighter than the base of the pixel becausetheir normals are closer to the light directions. They mayalso shorten the shadows of their respective lights. There-fore, increasing c−i, j always brightens the pixel (i, j) when litby the −x light and has no effect on the +x light configu-ration. The brightness of the pixel when lit by the −x lightdepends monotonically and piecewise-linearly on c−i, j. Thesituation is symmetric for c+i, j. We note that the piecewiselinear function would be difficult to incorporate into a gen-eral linear system, but our incremental construction makes iteasy to solve for appropriate values, as we explain below.

Suppose we computed row j using Equation (2), assum-ing u0, j = 0. We then greedily introduce non-zero chamferheights to adjust the r’s to vary as little as possible. Specifi-cally, we walk over the pixels in the row in order of increas-ing i and for each pixel, compare ri, j to ri+1, j. If ri+1, j ishigher, we increase c+i+1, j, which brightens the pixel (i+1, j)lit by the +x light, allowing us to lower ri+1, j and all u’sand r’s to the right of it and thus keep the illumination un-changed, while reducing height variation. We increase c+i+1, juntil either ri+1, j = ri, j or c+i+1, j = ui+2, j− ri+1, j. If, on the

other hand, ri+1, j is lower than ri, j, we increase c−i, j and raiseui+1, j, ri+1, j and all of the r’s and u’s to their right, whilekeeping illumination unchanged. In this case, we stop whenri+1, j = ri, j, or when c−i, j = ui, j− ri, j, or when the chamfers

collide and c−i, j + c+i, j = s.

The dot product of the +y direction with the normal of achamfer is smaller than with the normal of the pixel floor(the z direction), so chamfers make a pixel potentially lessbright when lit by the +y light. We compensate for this whencomputing the v’s, but in some cases even when the pixel iscompletely unshadowed, it ends up insufficiently bright dueto the chamfers. In this case, we uniformly reduce the bright-

Figure 7: A binary image produced by the global methodwithout the gradient term (left) and with the gradient term(right). The four target images are those in Figure 1, right.The gradient term leads to better reproduction of thin fea-tures, like the clock hands and reduces the ghosting fromMona Lisa’s mouth.

ness of I3. In practice we have not had to reduce the bright-ness by more than 10%, and the effect is not noticeable.

This chamfering dramatically reduces the overall height ofSHADOWPIX and improves quality. We have experimentedwith several variations of chamfering and they performedsimilarly.

3.2. Global Algorithm

For the global algorithm, we make no assumptions about thestructure of the surface: each heightfield element can bothcast and receive shadows. We assume that the light direc-tions are aligned with the grid of elements. This simplifiesthe shadow computations, as each element can only shadowother elements along the grid directions. Moreover, we as-sume that the heights are multiples of s so that each elementis either completely lit or completely in shadow. This allowsus to treat the resulting images as binary. For the moment,we consider the four different images for light sources alongthe ±x and ±y directions. Our goal is to make these four bi-nary images good dithered representations of the four inputimages while ensuring that the heightfield can be manufac-tured.

Objective Given a heightfield hi, j, let Ldi, j(h) be the binary

function indicating whether the element at (i, j) is lit whenthe surface is illuminated from direction d. Our goal is tofind h such that Ld

i, j(h) looks as close to Idi, j as possible, for

all d. This goal is similar to image halftoning, but we havemore than one image and only indirect control over indi-vidual pixels. Many halftoning methods measure the differ-ence between the target grayscale image and binary imageby modeling the human visual system as a convolution witha low-pass filter p. This leads to the following objective:

4

∑d=1‖p∗Ld(h)− p∗ Id‖2, (4)



Figure 8: Physically-based renderings of an image and thesurface produced by the global method without the smooth-ing term (left) and with the smoothing term (right). Smooth-ing slightly reduces contrast, but significantly reduces theheight variation.

where ∗ denotes a convolution. Unlike in regular imagehalftoning, SHADOWPIX have lower contrast than the origi-nal image, so preserving edges is an important concern. Wetherefore do not convolve the target images with p, leadingto increased sharpness. To further improve quality, we incor-porate a term to match the gradients, and not just the absolutepixel values:

4

∑d=1‖G∗ p∗Ld(h)−G∗ p∗ Id‖2, (5)

where G is the finite difference gradient filter. Figure 7 il-lustrates the effect of this term. We have also experimentedwith other saliency maps, but found that using the gradi-ent yields nearly identical results. An additional concern formanufacturing is that the heightfields should be as smooth aspossible—otherwise, fragile heightfield elements may breakoff and low heightfield elements surrounded by high onescan be difficult to paint (Figure 8). To improve smoothness,we add a term that penalizes the gradient of the heightfield.Our final objective is:

4

∑d=1‖p∗Ld

i, j(h)− Idi, j‖2+

+wG

4

∑d=1‖G∗ p∗Ld

i, j(h)−G∗ p∗ Idi, j‖2+

+wS‖G∗h‖2,

(6)

where wG is the importance of the gradient term and wS isthe importance of the smoothness term. For all our experi-

ments, we used the values wG = 1.5 and wS = 0.001, withimage dimensions measured in pixels, intensities between 0and 1, and the height measured in units of s. For the pointspread function p, we simply use a 5-by-5 pixel Gaussiankernel; a more sophisticated perceptually-motivated p couldbe used instead.

Optimization Earlier optimization-based halftoning meth-ods (for instance, [AA92]) typically used “direct binarysearch,” which is essentially greedy objective descent. How-ever, for more complicated objective functions [PQW∗08]and for our problem, the greedy approach tends to get stuckin local minima. We therefore decided to use simulated an-nealing [KGJVM83] to optimize the objective. Our configu-ration is the heightfield hi, j, which we initialize to all 0’s.To transition between configurations, we change a singleheightfield element up or down by up to 5s. The transitionis made if either the new objective function is lower or withprobability e( ft−1− ft )/T , where ft−1 and ft are the previousand current objective function values and T is the currenttemperature. We restrict the maximum length of a shadowthat’s cast by one element in one image to 10. This amplifiesthe smoothness effect and helps create a binary pattern that ismore random, which is a desirable feature when halftoningis concerned. For the 20cm-by-20cm SHADOWPIX, we use109 steps and a linear cooling schedule. In our experiments,simulated annealing converged to a point whose objectivewas roughly half of the local minimum found by greedysearch and the image fidelity was significantly higher (Fig-ure 14).

This procedure is fairly time consuming, so we implementeda simple optimization to take advantage of multicore paral-lelism. Every heightfield element of height hs, will cast ashadow whose length is at most h pixels. Because of this,each height change affects the blurred image only within aradius r, where r is the maximum of the height before andafter the change (but at most 10) plus the radius of the blurkernel p. The effects of changing heightfield elements morethan 2r apart are therefore guaranteed to be independent.Therefore, at every step, we pick multiple heightfield ele-ments at distances at least 2r from each other. We pick asmany elements as we have cores and in parallel process theeffect on the objective function of changing each of them.

4. Results and Evaluation

To test our construction methods, we designed and printedseveral SHADOWPIX, using both the local and globalmethod. The methods were run on a 1.6GhZ Core i7 mobileCPU, the global method taking advantage of all eight cores.Our algorithms require a negligible amount of memory. Tomanufacture the SHADOWPIX we used an Objet Connex350printer and an opaque black material (VeroBlack) with min-imal subsurface scattering, which was then painted whitewith an airbrush. Based on the printer resolution, for the



Figure 9: Input images (top row), renderings (middle row),and photographs of a single SHADOWPIX produced by theglobal method (bottom row). This case is particularly chal-lenging due to the extreme brightness differences required bythe different black-and-white chess pieces.

Figure 10: Input images (top row) and photographs of asingle SHADOWPIX produced by the local method (bottomrow).

local method we chose a pixel size of 2.5x2.5mm and awall thickness of 0.25mm, which means that 81% of thetotal surface area is covered with receiving elements, andthe shadowing of the other 19% is arbitrary. For the globalmethod, we used an element size of 0.5x0.5mm becausethinner heightfield elements tend to bend or break. Our sam-ples varied between 10 and 20 centimeters in each dimen-sion.

We compared the printed results with renderings producedusing a physically-based renderer; the viewing and light di-rections are slightly perturbed against the idealized assump-tions to simulate the error in a real setup. Figure 9 showsthat the rendering captures all of the qualitative features ofthe physical result, up to color balance. The ability to testwith renderings considerably sped up the design process.

Figure 11: An input image (left) and the image pro-duced with the local method (middle) and the global method(right). The SHADOWPIX are the same size and were printedwith the same 3D printer resolution. The global method im-age has higher resolution, higher contrast, and shares theSHADOWPIX with three other images (as compared to twofor the local method), but it is noisier, has more ghosting,and is more sensitive to light and viewing directions.

Figure 12: Input images (top row) and photographs of asingle SHADOWPIX produced by the global method (bottomrow). When this SHADOWPIX is lit from a direction, the im-age appears as if lit from the opposite direction, for an un-usual effect.

We experimented with distinct portrait photographs (Fig-ure 10) and paintings (Figure 1). We found that the local andglobal methods have their advantages and disadvantages.Figure 11 shows a side-by-side comparison of the same Lin-coln’s image produced by both methods. The local methodproduces a cleaner image, has less ghosting, while the globalmethod is higher-resolution, has higher contrast, and can ac-comodate up to four images. We also compared their ro-bustness with respect to deviations from ideal viewing andlighting conditions and found that the local method is sig-nificantly more tolerant (Figure 16). For many applications,this is an advantage, but the more startling effect of the im-age forming on a global SHADOWPIX may sometimes bepreferred.

While the local method generates the surface nearly instan-taneously (taking 6 seconds on our largest examples), theglobal method requires stochastic optimization and takes acouple of hours to run (ranging from from 29 minutes forthe smaller chess pieces example, up to 104 minutes for



Figure 13: Top Left: An input photograph of a bird. TopRight: A white SHADOWPIX that encodes the red, green, andblue channels of the input as separate images, illuminated bycolored lights from three directions, using the global method.Bottom Left: A lower res input photograph of a lizard. LowerRight: A white SHADOWPIX that encodes the RGB channels,using the local method. These images were obtained by cap-turing three photographs in raw mode, each with a singlecolor light, and adding them in software.

Figure 14: This plot compares the greedy optimization con-vergence (magenta) to that of simulated annealing on thechess pieces example (other colors). Several simulated an-nealing runs are shown, demonstrating that the qualitativebehavior is stable with respect to the stochastic choices. Notsurprisingly, the greedy algorithm converges much faster,but its solution is significantly (almost 3dB) worse than thatfound by simulated annealing.

the larger paintings example) per SHADOWPIX. It took 2–6 hours to print each SHADOWPIX and about two hoursfor cleaning and painting; therefore, we did not expend toomuch effort to get the fastest possible convergence.

We then experimented with using related images in SHAD-OWPIX. We achieved an unusual lighting effect by encoding

Figure 15: This plot shows how the embedded image qual-ity decreases as more images are embedded into a singleSHADOWPIX using the global method. The different curvesshow four random embedding orders of the faces and thepaintings.

four photographs of a face lit from different sides and as-sociating them with the opposite light direction (Figure 12),so that when the SHADOWPIX is lit from a direction, it ap-pears as if lit from the opposite one. As the next example,we encoded the three color channels of a single image intoa SHADOWPIX, so that when the image is lit by coloredlights from different directions, the white surface displaysa color image. This experiment was done both for the localand global method (Figure 13). Alexa and Matusik [AM10]have done a similar experiment, but their method is limitedto two color channels, which is insufficient for most colorimages. We also encoded the animation frames of a runninghorse, so that when the light moves around the SHADOW-PIX, an animation can be seen. Our video shows this andother results.

Quantitative error analysis We have performed severalexperiments to analyze the global method. For image qualityevaluation, we used peak signal-to-noise ratio (PSNR) be-tween the input image and the output bitmap convolved withthe point spread function p. Figure 14 shows how simulatedannealing converges on the chess pieces example and whywe did not use greedy optimization. It also demonstrates thatour result quality is largely independent of the initial condi-tion and random choices made during a run.

We also looked at how encoding more images into a singleSHADOWPIX affects the image quality. For this experiment,we added support for encoding images for the four diago-nal light directions towards (±x,±y); this involves only achange to the objective function: the shadows are calculatedon four quarters of each pixel. Figure 15 shows the qualitysteadily decreasing as more images are added. While fourimages are not the absolute limit of the global method, em-bedding more images reduces the result quality below whatwe think is acceptable. The different curves in the figure


Bermano et al. / SHADOWPIX: Multiple Images from Self ShadowingPe

rspe

ctiv

eV

iew

Lig

ht

Figure 16: We use physically-based rendering to explorehow sensitive SHADOWPIX are to deviations from idealviewing conditions, showing the local method on the left andthe global method on the right. For the top images, we use anorthographic camera, orthogonally positioned, and a lightat the correct angle. In the next row, the camera has beenchanged to have a field of view of 50 and 100 degrees. In thethird row, we keep the camera orthographic, but change theview direction by 15 and 30 degrees. In the bottom row, weshow the effect of changing the light direction by 20 and 40degrees. These experiments show that the image produced bythe local method is much more stable.

show that adding images in different order has a signifi-cant influence on the effect, confirming that some imagesare more difficult to embed.

5. Conclusions

In this paper, we have explored the use of surface self-shadowing as a means of embedding multiple images intoa surface when it is lit from different directions. We haveshown two different methods that accomplish this task. Thefirst method, the local method, uses geometric primitives thatconsist of thin walls placed on a rectangular grid. Each prim-itive represents a light-dependent pixel that can reproducedifferent grayscale values when lit from three different di-rections. We have shown a deterministic algorithm that al-lows building a surface from these primitives while mini-

mizing the overall surface height. Our second method, theglobal method, represents a surface as an arbitrary height-field. Our key contribution for this method is an appropriateobjective function that preserves perceptually important im-age features and keeps the surface height small. This objec-tive function is optimized using simulated annealing to pro-duce heightfields that embed up to four different grayscaleimages into a single heightfield. We have evaluated bothmethods by producing many different surfaces that generatemultiple images when lit from different directions.

The problem presented in the paper is new and thereforethere are many opportunities for future work in this researcharea. In particular, in the context of the local method, wewould like to explore different geometric primitives, for ex-ample, primitives based on a hexagonal grid. For the globalmethod, it would be desired to have a deterministic algo-rithm that is fast and computes surfaces of a quality similarto that of our optimization-based approach. In our methodswe have not accounted for the effects due to the global illu-mination and we suspect that interreflections between partsof the white surface are responsible for some of the ghostingand contrast reduction. We believe that accounting for theseeffects would improve the quality of the resulting images.At least in the case of the local method, indirect illumina-tion effects could be precomputed for each primitive con-figuration. We have only used rapid prototyping methods tomanufacture our surfaces, but we would like to explore dif-ferent manufacturing methods including injection moldingand CNC milling.

Looking into the future, we believe that it would be desir-able to exploit both geometry (e.g., due to self-shadowingand surface normals) and reflectance (e.g., diffuse, glossy,and specular effects) in order to obtain additional degreesof freedom and encode even more images in a single sur-face. Furthermore, extending our methods to non-planar sur-faces can broaden the range of possible applications. Finally,one can envision building a dynamic heightfield display thatwould exploit the concepts described in this paper and allowchanging the displayed images over time.

Acknowledgements

We thank Bernd Bickel for help with the 3D printer andChristian Regg for help with painting.

References[AA92] ANALOUI M., ALLEBACH J.: Model-based halftoning

using direct binary search. In Society of Photo-Optical Instru-mentation Engineers (SPIE) Conference Series (1992), vol. 1666,pp. 96–108. 3, 6

[AM10] ALEXA M., MATUSIK W.: Reliefs as images. ACMTransactions on Graphics 29, 4 (July 2010), 60:1–60:7. 3, 8

[DD98] DAUM M., DUDEK G.: On 3-d surface reconstructionusing shape from shadows. In IEEE Computer Vision and PatternRecognition (1998), pp. 461–468. 2



[Elb02] ELBER G.: “Beyond Escher for real” project, 2002. 2

[FRSL08] FUCHS M., RASKAR R., SEIDEL H.-P., LENSCH H.P. A.: Towards passive 6d reflectance field displays. ACM Trans-actions on Graphics 27, 3 (Aug. 2008), 58:1–58:8. 1

[KGJVM83] KIRKPATRICK S., GELATT JR C., VECCHI M.,MCCOY A.: Optimization by Simulated Annealing. Science220, 4598 (1983), 671–679. 6

[KS86] KENDER J. R., SMITH E.: Shape from darkness: Deriv-ing surface information from dynamic shadows. In AAAI (1986),pp. 664–669. 2

[KvWW09] KEIREN J., VAN WALDERVEEN F., WOLFF A.:Constructability of trip-lets. In Abstracts from the 25th Euro-pean Workshop on Computational Geometry, Brussels, Belgium(2009), pp. 251–254. 2

[MP09] MITRA N. J., PAULY M.: Shadow art. ACM Transactionson Graphics 28, 5 (Dec. 2009), 156:1–156:7. 2

[NBB04] NAYAR S. K., BELHUMEUR P. N., BOULT T. E.:Lighting sensitive display. ACM Transactions on Graphics 23,4 (Oct. 2004), 963–979. 1

[PQW∗08] PANG W.-M., QU Y., WONG T.-T., COHEN-OR D.,HENG P.-A.: Structure-aware halftoning. ACM Transactions onGraphics 27, 3 (Aug. 2008), 89:1–89:8. 3, 6

[RPL89] RAVIV D., PAO Y.-H., LOPARO K.: Reconstruction ofthree-dimensional surfaces from two-dimensional binary images.IEEE Transactions on Robotics and Automation 5, 5 (1989), 701–710. 2

[SK83] SHAFER S. A., KANADE T.: Using shadows in findingsurface orientations. Computer Vision, Graphics, and Image Pro-cessing 22, 1 (1983), 145 – 176. 2

[YC02] YU Y., CHANG J. T.: Shadow graphs and surface re-construction. In Proceedings of the 7th European Conference onComputer Vision-Part II (2002), pp. 31–45. 2


Documents

SHADOWPIX: Multiple Images from Self Shadowingpeople.csail.mit.edu/ibaran/papers/2012-EG-ShadowPIX.pdf · EUROGRAPHICS 2012 / P. Cignoni, T. Ertl (Guest Editors) Volume 31 (2012),