(b, s)-BCSL : Structured Light Color Boundary Codingfor 3D Photography

Asla M. Sa, Paulo Cezar P. Carvalho, Luiz Velho

IMPA, Instituto de Matem´atica Pura e AplicadaEstrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil

{asla|pcezar|lvelho}@visgraf.impa.br

Abstract

In this paper we present(b, s)-BCSL, a new codedstructured light method for 3D photography. It em-ploys a sequence of complementary color stripepatterns to define correspondences within a cam-era/projector shape acquisition system. Uniquecodes are associated with the boundaries of suc-cessive stripes in time. The use of color fully ex-ploits current imaging hardware and makes possi-ble an effective trade-off between spatial and tem-poral coherence. Furthermore, alternating comple-mentary colors completely eliminates photometricrestrictions.

1 Introduction

Shape from structured light is an active stereo vi-sion technique. Measurement of depth values is car-ried out with a system that resembles a two-camerastereo system, except that a projection unit is usedinstead of the second camera. A very simple tech-nique to achieve depth information with the help ofstructured light is to scan a scene with a projectedlaser plane and detect the location of the reflectedstripe in the camera image. Assuming that the pro-jected laser can be seen by the camera, and both arecalibrated, the depth information can be computedby triangulation using the known correspondences.

In order to get dense range information, the laserplane has to be moved in the scene. In an attemptto alleviate the problem of capturing only one depthline per image, a slide containing multiple stripesis projected onto the scene. To distinguish betweendifferent stripes they must be coded appropriately,in such a way that their location in the projectorcan be identified. This type of encoding scheme iscalledcoded structured light (CSL).

There is a natural analogy between coded struc-tured light and a digital communication system [7].The projector coordinates are encoded by the slidepatterns and transmitted through the scene. At eachpoint of the camera image, a noisy transmission isreceived and needs to be decoded. The transmis-sion channel is the object’s surface and the transmit-ted message is the encoded position of the projectorpixel. Considering this analogy, two main issueshave to be studied:

1. Limitations of the transmission channel, re-lated to surface reflectance properties.

2. How to encode projector pixels, which willrestrict the class of objects suitable to bescanned.

In this paper, we propose a new coded structuredlight scheme based on time-varying complementarycolor stripe boundaries.

The original contributions of our work include:• A stripe boundary encoding that, through the

use of color supports high resolution depth tri-angulation, and, at the same time, guaranteesthe existence of stripe boundaries.

• An scheme that employs a sequence of com-plementary color patterns to completely elim-inate the restrictions on reflectivity propertiesof the scene.

Our scheme is very robust to image noise and canreliably detect both stripe projected color and stripeboundaries with subpixel accuracy. Moreover, it isalso able to recover surface reflectance using timeintegration of color.

In summary, these unique characteristics providean excellent trade-off between spatial, temporal andphotometric coherence for structured light coding.

VMV 2002 Erlangen, Germany, November 20–22, 2002

The paper is organized as follows. Section 2 re-views the literature of CSL methods and discusseshow our approach is related to previous researchin the area. Section 3 describes the design of our(b, s)-BCSL coding scheme. Section 4 presents themechanisms to minimize restrictions imposed onthe scene to be scanned. Section 5 shows someresults obtained with an experimental 3D photog-raphy setup using our scheme. Finally, section 6concludes with directions for future work.

2 Previous work

The research in structured light coding can be di-vided in two distinct periods. Early work startedwith the investigation of active vision systems and,more recently, a new impulse to the area was pro-vided by 3D-photography applications.

Classical research of CSL methods was done in80’s and 90’s [2, 10, 13, 15]. A good survey of thework developed in this period can be found in [1].The flavor of this work was to empirically createlight patterns to capture the geometry of scanned3D objects.

Three main approaches were used in the designof projected light coding systems: spatial chromaticmodulation; spatial intensity modulation, and tem-poral intensity modulation.

Chromatic modulation imposes restrictions onthe allowable colors in the scene. When spatialcodification is used, local continuity of the scene isnecessary for recovering the transmitted code. Con-versely, temporal codification restricts motion of thescene.

A characterization of CSL methods in terms ofthe underlying assumptions about reflectance, spa-tial and temporal coherence of the scene is givenin [6].

In recent years CSL systems have been revisitedby many researchers with the motivation to find atheoretical basis that could provide a better under-standing of known methods and make possible todevelop new improved systems [11,12,17].

Such a shift of paradigm can be seen in thework of Hall-Holt and Rusinkiewicz [6], who pro-posed a CSL scheme based on stripe boundaries. Intheir scheme, the codes are associated with pairs ofstripes, instead of with the stripes themselves as intraditional methods. Boundary coding has several

advantages: it gives higher spatial precision and re-quires less slides (better temporal coherence).

In order to allow the greatest possible varia-tions in scene reflectance, the scheme of Hall-Holt and Rusinkiewicz is based on black and whitestripes. This option leads to an undesirable prob-lem: ”ghost” boundaries must be allowed (i.e.,black to black and white to white transitions).Their scheme is also complicated by restrictions onboundary transitions because of the decoding algo-rithm, which is augmented by an additional step thatsolves the matching problem.

The most recent work done by Zhang, Curlessand Seitz [17] uses a dynamic programming tech-nique to compute an optimal surface given a pro-jected pattern and the observed image. The cameraand projector correspondence is obtained up to pixelresolution and a post-processing step is carried outto achieve sub-pixel accuracy.

The concept of using color boundary codificationis present in [17] but the option to use a one-shotcodification implies in considering a sub-sequenceof consecutive stripes to guarantee uniqueness ofcodification with the desired resolution. Augment-ing the size of the basis used in codification com-plicates the decodification step. The price of adopt-ing a one-shot codification is that requirements onspatial coherence cannot achieve its minimum, andsome information will be lost due to discontinuitiesin the scene. Some considerations to reduce colormisalignment and account for color crosstalk aredone in designing and processing correspondences,but the problem is not robustly solved, as shown intheir results. To increase resolution and robustness,the same pattern is shifted and projected, leading toa spacetime analysis of the color stripe pattern.

Our coding scheme adopts a stripe boundary timecoding as in [6], but uses color to augment the cod-ification basis. In this way, we avoid the need ofghost boundaries. The larger basis makes possi-ble to achieve higher resolution and better preci-sion, as discussed in [4]. This basis also makesthe decodification more efficient, consisting essen-tially of a table lookup [8]. We propose a pro-cedure that solves the reflectance restrictions im-posed by the traditional color coding methods, suchas in [17]. Our scheme employs alternating com-plementary colors. This makes the use of coloredstripes non-restrictive, in any sense, to reflectiveproperties of the scene.

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3 The (b, s)-BCSL Scheme

In designing a code for structuring light, the mainparameters to be considered are the number of dif-ferent stripe colors (i.e., the base lengthb of thecode) and the numbers of slides projected at eachstep of geometry acquisition. We call the resultingcoding, a(b, s) CSL scheme and leads to a totalof bs different patterns. A larger number of pat-terns can be obtained if one codes the transitionsbetween stripes, rather than the stripes themselves.For a boundary-coded scheme (i.e.(b, s)-BCSL),the number of different patterns is[b(b − 1)]s (weassume that two successive stripes may not have thesame color, to avoid ghost boundaries).

Increasing the number of colors and slides allowsusing a larger number of stripes, thus increasing theresolution of the acquired geometry. However, dis-tinguishing the stripes becomes harder and the re-strictions become more severe. The goal in design-ing a BCSL scheme is to be able to provide satis-factory resolution, without making stripe recogni-tion too complex and without imposing strong co-herence constraints on the scene.

Schemes havings = 1 are purely spatial codings,imposing no restrictions on object movement. Fors > 1, we have spatial-temporal codings. Amongthese, the cases = 2 reduces the need for time co-herence to a minimum, namely that objects in thescene move slowly enough so that the displacementbetween the two captures is smaller than the pro-jected stripe.

In our scheme, we only use the primary colorsR, G and B and the corresponding complementarycolors C, M and Y, respectively. This is equivalentto using a binary code (0 or 1) for each color chan-nel. We avoid using Black (0 in all channels), sinceblack stripes may be confused with shadow areas.For symmetry reasons, we do not use White, either.Thus, the maximum number of different colors is 6.Usingb = 6 ands = 2, leads to 900 coded bound-aries for (6, 2)-BCSL. This is more than what canbe processed by most cameras or projectors.

3.1 Coding

We turn now to the problem of generating a partic-ular sequence ofb-colored stripes for each of thesslides. This sequence should be generated in such away that, given the stripe transitions at some pixel

for thes slides, we can efficiently recover the posi-tion of the corresponding projected boundary.

The problem of generating a set of stripe se-quences can be modeled as a the problem of findinga eulerian path in a suitable graphG. G hasbs ver-tices, each corresponding to a possible assignmentof theb colors at a given position for each of thesslides. For instance, forb = 3 ands = 2, G has9 vertices, each labeled by a 2-digit number in baseb, as shown in figure 1(a). For example, vertex01corresponds to projecting color 0 in the first slideand color 1 in the second, at a given stripe position.

00 01 02

10 11 12

20 21 22

(a) graph (3,2)

d(3) d(2)

d(1) d(0)

(b) neighborhood

Figure 1: (3,2)-BCSL Encoding.

The edges of G correspond to the possible tran-sitions for two consecutive stripe positions. Theforbidden transitions are those that repeat the samecolor at the same slide, in order to disallow ghostboundaries. For instance, in 1(a), there is not anedge connecting vertex 01 to vertex 02, since thatwould mean that two consecutive stripes in the firstslide would use color 0. On the other hand, thereis an edge connecting vertex 20 to vertex 01. Thissituation is illustrated in figure 2 and 3: at the sameborder position, we go from color 2 to color 0 in thefirst slide and from color 0 to color 1 in the second.

2

0

0

1

slide 1

slide 2

Figure 2: Example of boundary code for dashededge in Figure 1(a).

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For the (3, 2) case, the neighborhood structure isshown in figure1(b), with each vertex having 4 pos-sible neighbors. For the general (b, s) scheme, thereare (b−1)s possible neighbors for each vertex, lead-ing to a regular graph where each of the bs verticeshas degree (b−1)s. It is more appropriate, however,to think of G as a directed graph where each vertexhas (b − 1)s incoming arcs and (b − 1)s outgoingarcs, since the same pair of vertices correspond totwo distinct transitions, one for each ordering.

Possible color stripe schemes correspond to pathswith no repeated edges (meaning that each multi-slide transition occurs only once) in the directedgraph G. The maximum number of stripes isachieved for a eulerian path, i.e., a path that vis-its once each edge of G. This path certainly existssince the every vertex in G has even degree and Gis connected (for b ≥ 3) ( [16]).

In fact, there is a very large number of differentEulerian paths in G, and an optimization problemcan be formulated to search for the best path accord-ing to some desired criteria. The computation of op-timal paths is an open and beautiful problem fromthe coding theory point of view, and several heuris-tics have been proposed to tackle it. Horn et al. [7]formulated the problem as one of designing optimalsignals for digital communication. They considermore important to encode with very distinct code-words points which are spatially distant, than to en-code neighbor points with similar codewords. Theyuse this criterion to evaluate the path quality. Wecould also adopt an image processing perspective,using information about the photometric propertiesof the scene to be scanned as the criteria to generatean adaptive best code.

In most cases, there is no need to use the com-plete Eulerian path, since it suffices to use a path oflength equal to the maximum resolution handled bythe cameras or projectors used.

In the present work we generate the completepath, given a base b and the desired number of slidess, without considering any optimization criteria. Toachieve the desired number of stripes, we simplytruncate the path.

The encoding algorithm is described below:Input: A connected digraph G with din(v) =

dout(v) for all v in V (G).

Procedure BCSL Code(G)Step 1: Choose an initial vertex v0 in V (G).

Construct an spanning in-tree1 T starting fromv0 using a breadth-first search algorithm.

Step2: Specify an arbitrary ordering of theedges that leave each vertex v, with therestriction that the edge in T has to be the lastone.

Step3: Construct an eulerian circuit startingfrom v0 as follows: whenever u is the currentvertex, exit along the next unused edge in theordering specified for edges leaving u, whereδin(v) is the number of in-edges of v in graphG, and δout(v) is the number of its out-edges.

Figure 3 shows a particular (3,2)-BCSL gener-ated by this algorithm; this stripe coloring schemeis used in the first example presented in Section 5.

slide1 RGB

RGB

slide2

Figure 3: (3,2)-BCSL using R, G, B as base. Theoutlined boundary has code 20|01 (see figure 2).

3.2 Decoding

We now consider the problem of recovering a bor-der position, given the color transitions at eachslide. This is equivalent to find the position of agiven edge in the eulerian path found in the previ-ous algorithm.

We adopt a decoding algorithm presented in [8]to obtain the position of a identified codeword. Inour case, to identify the codeword means to find thecolors on both sides of the projected boundary thatare reflected by the object. A specific image pro-cessing method, described in section 4, is used toachieve this goal.

1An in-tree starting from v0, is a tree consisting of a uniquepath from each vertex u in V(G) to v0 .

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The decoding algorithm employs a decoding ta-ble that allows computing, in constant time, the pro-jector position of a given stripe boundary found inthe images. This table is shown in 1 for the (3, 2)case. Each row of the table corresponds to a ver-tex v of G (i.e., to a stripe color assignment for allslides), ordered according to its expansion in baseb. Each column corresponds to one of its neigh-bors, ordered according to the pattern shown in fig-ure 1(b). Each one of the neighbors can be conve-niently expressed by means of arithmetic operationsmodulo-b, exploiting the regularity of the adjacencyrelationships, as shown in detail in [8].

The entry in a given row and column of the ta-ble gives the position, in the eulerian path, of thearc corresponding to the transition from the vertexassociated with the row to the neighbor associatedwith the column. For instance the entries 0, 3, 6 and9 in the first row mean that vertex v0 = 00 occursfour times in the path, at those positions. The corre-sponding transitions are to neighbors d0, d1, d2 andd3, which are vertices 11, 12, 21 and 22.

Suppose now that a given transition, from vertexvi to vertex vj has been detected. To find its posi-tion in the path, it suffices to determine the locationof vj in the neighborhood of vi and recover the en-try for that column in the row corresponding to v0.For instance, suppose that the transition illustratedin Figure 2, from codeword 20 to 01, has been de-tected. Since 20 expands to 6 in base 3, the corre-sponding row is V (6). Subtracting the digits of 01and 20 module-3 yields 11. To find the position of01 in the neighborhood of 20 it suffices to subtract1 from each digit and expand the resulting patternin base 2. In this case, we find 00 = 0, meaning that11 is the d0 neighbor of 20. Since the table entry atthe (V (6), d(0)) position is 16, this means that thefound transition corresponds to the 16th boundary,as marked in Table 1 and in Figure 3.

4 Uncompromising Use of Color

Considering the proposed codification and the de-coding algorithm discussed in the previous section,the problem of corresponding camera pixels to pro-jector pixels is reduced to an image processing task,responsible to identify in camera images the transi-tions and colors of the projected stripes.

As we have adopted a vertical stripe boundarycoding, scan lines can be treated independently.

vertices d(0) d(1) d(2) d(3)V(0) 0 3 6 9V(1) 14 17 19 11V(2) 28 34 22 24V(3) 26 29 18 21V(4) 1 31 33 35V(5) 15 4 8 13

V(6) 16 23 32 12V(7) 27 5 7 25V(8) 2 10 20 30

Table 1: Decoding table for (3,2)-BCSL.

4.1 Detecting Stripe Boundaries

The crucial step for range image accuracy is themeasurement of stripe transition positions in thecamera image.

It is desirable to determine the stripe edge po-sition with subpixel accuracy. A known methodto obtain sub-pixel accuracy on boundary positionconsists of projecting the positive and the negativeslides of the same pattern.

In this methodology the position of the stripeedge P is computed as the intersection between theline AB and the line EF of positive and negativepatterns respectively. (See Figure 4.)

184 186 188 190 192 194 1960

20

40

60

80

100

120

140

160

A

E

F

B

P

negativesignal

positive signal

Figure 4: Boundary from complementary patterns.

The above technique assumes that the consideredsampling points lie in the linear part of the edge pro-file. This is true in general, except for very narrowedges whose width is less than two pixels.

The linear interpolation technique with inversepatterns is also very robust against slightly differentwidth of black and white stripes. This phenomenonis caused by a small isolating gap between adjacentstripes in the LCD array.

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Trobina [14] did a comparison of the techniquesfor detecting stripe position and concluded that lin-ear interpolation with inverse pattern is more accu-rate and robust. The study considered only blackand white patterns (one channel). We extend thistechnique for color patterns (three channels) and ex-ploit the information redundancy to make it morerobust with respect to projected shadows.

Shadow regions are viewed by the camera, butare not illuminated by the projector. These areasare characterized by having a very small brightnessdifference in both positive and negative slides. Forthese areas, the stripe pattern is not processed andno range values are obtained.

Our boundary detection technique is based onpairs of complementary color stripes. Two subse-quent slides are projected with stripes of comple-mentary colors. Each color channel is processed asin the one-channel inverse pattern edge detection,except that we merge boundaries of distinct chan-nels which are within 2 pixel distance.

Shadow regions are detected by analyzing themagnitude and derivative of the complementarycolor difference image. A point is in shadow ifthe absolute value in the three channels of the dif-ference image are all bellow the threshold, and thederivative in the horizontal direction is large.

4.2 Recovering Color

The intensity p of a projector light beam is scatteredfrom the object surface into a camera pixel. If thecamera characteristic is linear, the sensor clips in-tensity at a maximum value. The digitized intensityper channel is given by:

IR = min(uR + rRpR, IRmax)

IG = min(uG + rGpG, IGmax)

IB = min(uB + rBpB, IBmax)

where (uR, uG, uB) is the ambient light compo-nent, (rR, rG, rB) is the local intensity transfer fac-tor mainly determined by local surface propertiesand (pR, pG, pB) is the projector intensity for eachchannel [9]. Supposing that IRmax , IGmax andIBmax are never achieved we have: (IR, IG, IB) =(uR + rRpR, uG + rGpG, uB + rBpB).

We can estimate parameters u and r if we fix pro-jector, sensor and object in relative positions, andproduce sequential projected patterns varying p. Asmentioned previously, our option was to project two

complementary slides, that is, if pi = 0 on first slidethen pi = 1 on second. We have:

Ii =

{ui when pi = 0ui + ri when pi = 1

If we just take the maximum value per pixel foreach channel between the complementary slides, itwill be equivalent to recovering the value of eachpixel as if it were illuminated with white light com-ing from the projector, that is , p = (1, 1, 1). Equiv-alently, if we take the minimum value per pixel foreach channel, we are recovering the ambient light,that is, p = (0, 0, 0).

One problem in the use of color coding is thecross-talk between the RGB sensors [3]. In that re-spect, color fidelity can be improved by a color cor-rection pre-processing step that takes into accountthe response of the projector-camera system.

The traditional use of color in coding restrictsthe object surface reflectivity, because we would notwant to modify the projected color when acquiringthe surface. By projecting complementary slides,the reflectivity restrictions are eliminated.

From the signal transmission point of view weare introducing redundancy on the transmitted mes-sage replicating it on complementary slide. This isa good procedure as it reduces the probability of er-rors and the codeword read can be checked to assurethe correctness of the received message.

Note that, the use of complementary colors al-lows us to recover both the stripe and scene colorsusing only two slides.

The considerations above are valid for all pixels,except at stripe boundaries. At those locations, werecover color by interpolating the information fromneighbor pixels at both sides of the boundary.

5 Experimental results

Now we demonstrate the application of our (b, s)-BCSL scheme with some examples.

We start with an overview of the method usinga virtual acquisition pipeline in which the camera–projector setup is simulated with BMRT [5]. Themodel in this example is the Stanford Bunny.Color Plates 1(a) to (d) show the input images forthe two slide pairs of a (3,2) code. Plate 1(e) showsthe computed stripe boundaries. Plates 1(f) and(g) show the stripe codes for slide pairs (a)(b) and(c)(d), respectively. Plate 1(h) shows the recoveredtexture.

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We continue with results generated using a realacquisition setup, consisting of a FujiPix 2400Zdigital camera and a Sony VPL-CS10 projector.The acquired images have a resolution of 1024 ×768 in JPEG format. Three objects with differentcharacteristics were used in the following examples.

The first object is a Ganesh statue, which has adetailed geometry and homogeneous surface prop-erties, as can be seen in Figure 5(a). We usedthree (b, s) codes with increasing resolution: (3, 2),(4, 2) and (6, 2). The stripe width is 18 pixels forthe (3,2) code, 10 pixels for the (4,2) code and 5 pix-els for the (6,2) code. The (b, s) slides were gener-ated with 600×480 pixels. Note that the maximumresolution of our codification was achieved only forthe (3,2) code, and we could achieve the 5 pixel res-olution using the (4,2) code (we used the (6,2) codejust to exemplify its use).

Figures 5(b), (c) and (d) show the computedstripes for the (3,2), (4,2) and (6,2) codes, respec-tively, and Figures 5(f), (g) and (h) show the corre-sponding stripe boundaries. Figure 5(e) shows themask for background and shadow areas.

The second object is a furry frog which has afuzzy geometry. Figures 6(a) and (b) show one ofthe slide pairs. Figure 6(c) shows the stripe bound-aries and Figure 6(d) shows the recovered texture.

The third object is a Babuska doll. It has a simpleshape, a complex painted texture and also a highlyreflective surface. Color Plates 2(a) and (b) showone of the slide pairs, Color Plate 2(c) shows thestripe codes and Color Plate 2(d) shows the recov-ered texture.

6 Current Work

We are working on an extension of the method thatcan process slow moving objects. In this way, wewill be able to generalize the (n, 2) version of ourcodification to dynamic scenes. The main idea isto motion blurr the frames, perform tracking of thestripe boundaries and warp the images accordingly.

We can also formulate a one shot version of(b, s)-BCSL. In this case, instead of projectingstripe coded patterns, we generate a checkerboardpattern alternating vertically the 2 temporal pat-terns. A vertical “ghosts” restriction has to be for-mulated. In this one shot version, vertical pixels arenot independent and the image processing phase hasto be revised.

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(a) original (b) (3, 2) code (c) (4, 2) code (d) (6, 2) code

(e) mask (f) (3, 2) edges (g) (4, 2) edges (h) (6, 2) edges

Figure 5: Ganesh.

(a) slide i (b) slide i+1 (c) boundaries (d) texture

Figure 6: Frog.

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(a) Slide i (b) Slide i+1 (c) Slide i+2 (d) Slide i+3

(Slide pair A) (Slide pair B)

(e) Boundaries (f) Stripe code A (g) Stripe code B (h) Texture

Color Plate 1: Bunny.

(a) Slide i (b) Slide i+1 (c) Stripe code (d) Texture

Color Plate 2: Babuska.

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