Surface Reconstruction from Feature Based Stereo

Camillo J. TaylorGRASP Laboratory

University of PennsylvaniaPhiladelphia, PA, 19104

Abstract

This paper describes an approach to recovering surfacemodels of complex scenes from the quasi-sparse data re-turned by a feature based stereo system. The method can beused to merge stereo results obtained from different view-points into a single coherent surface mesh. The techniqueproceeds by exploiting the freespace theorem which pro-vides a principled mechanism for reasoning about the struc-ture of the scene based on quasi-sparse correspondences inmultiple image. Effective methods for overcoming the diffi-culties posed by missing features and outliers are discussed.Results obtained by applying this approach to actual imagesare presented.

1 Introduction

This paper addresses the problem of recovering represen-tations for the surface structure of 3D scenes such as theone shown in figure 1 from a set of images. The pro-posed scheme takes as input the results of applying a fea-ture based, multi-baseline stereo algorithm to clusters of im-ages taken from various vantage points around the scene.Based on this information, the system constructs a meshrepresentation of the surface like the one shown in Figure1b. Importantly, the approach is able to handle scenes thatinvolve multiple disconnected surfaces which may occludeeach other in interesting ways as opposed to scenes that caneffectively be modeled as 2.5D surfaces.

a. b.

Figure 1: a. Image of a typical scene b. Surface mesh re-covered by the proposed procedure.

When accurate dense range scans are available, several

excellent techniques are available for merging this informa-tion into coherent surfaces [12, 15, 2, 7, 16]. Unfortunately,there are situations in which it can be very difficult to obtainthe dense, accurate depth maps required by these techniquessolely from image data.

Passive image based ranging techniques like stereo pro-ceed by correlating pixels or features between images. Inany non trivial scene we can expect to encounter regionswhich cannot be readily matched between images. Thiscan occur either because certain portions of the structureare occluded in one or more of the viewpoints under con-sideration or because the images contain homogenous re-gions which admit multiple possible matches. In eithercase, dense stereo systems are obliged to deal with theseproblems by appealing to a’priori expectations about scenestructure such as smoothness or continuity rather than directmeasurements.

It would seem, then, that there is a certain mismatchbetween the character of the data returned by most stereosystems and the accuracy requirements of the previouslymentioned surface reconstruction techniques. This paperargues that, given the nature of the stereo matching pro-cess, it may be more appropriate to view each stereo re-construction as providing information about the freespacein the scene rather than measurements of surface structure.In this framework, the stereo results obtained from variousvantage points can be combined by considering the unionof the freespace volumes they induce. This procedure canbe made to work even when the individual reconstructionscontain interpolation artifacts.

The proposed scheme proceeds from the position thatour multi-baseline stereo system can be expected to find ac-curate correspondences with high confidence in areas wherethere is sufficient intensity variation in the image. Theseregions typically correspond to intensity edges or texturedregions in the scene. Although the stereo system may onlyobtain depth information for a small fraction of the pixels inthe input images, these results can be used to reason aboutthe scene freespace in a principled manner and, hence, toconstruct an appropriate 3D model.

The model of the stereo process used in this work issomewhat extreme since it is usually possible to produce

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reasonable disparity values for a fairly large fraction of thepixel locations. The point of choosing such an impover-ished model is to demonstrate that it is still possible to de-rive useful information about the structure of the scene froma small number of high confidence correspondences. Whenmore correspondences are available they will only serve toimprove the quality of the reconstructions.

Several techniques have been proposed to construct3D models from the sparse 3D data returned by featurebased structure from motion algorithms. Faugeras, Lebras-Mehlman and Boissonnat [4] describe a technique for build-ing 3D models by triangulating a set of 3D point featuresand eliminating surfaces which contradict feature visibilityconstraints. Manessis et al [9] proposed a related methodfor triangulating sparse feature sets. Morris and Kanade[11] describe a scheme that searches for reasonable trian-gulations of a point set by considering the implied motionof textures in the image. The method described in this pa-per differs from these algorithms in that it generates a sur-face description by considering the freespace volumes as-sociated with the stereo results instead of tesselating the re-covered 3D points.

Several effective techniques for recovering the structureof a scene based on the voxel carving algorithm proposed byKutulakos and Seitz [8] have been proposed. The methodproposed in this paper differs from these schemes in twoimportant ways. Firstly it does not make use of the view-point consistency constraint directly, it makes determina-tions about the structure of the scene based on the 3D re-sults returned by the stereo system. Secondly the proposedapproach does not involve discretizing the scene into vox-els.

An interesting and effective technique for recoveringsurface models from multiple images was proposed byFaugeras and Keriven [5]. This work reformulates thestereo reconstruction process as a surface evolution prob-lem and solves it using variational principles. In contrast,the approach taken in this paper directly recovers scene oc-cupancy from the available disparity measurements.

Another important class of techniques for recovering sur-face models from image data draw inspiration from thework of Baumgart [1]. These methods work by consider-ing the intersection of the volumes obtained by extrudingthe object silhouette into the working volume. Variants ofthis scheme have been proposed by Sullivan and Ponce [17],Wong and Cipolla [18] and Matusik et al [10]. The methodproposed in this paper is in some sense a dual to these tech-niques in that it considers the union of the freespace vol-umes associated with each image as opposed to the inter-section of the volumes associated with each silhouette. It isalso important to note that the technique does not involvethe use of distinguished contours like silhouettes which canbe difficult to obtain.

pq

r PQ

R

ImagePlane

Figure 2: If the space triangle �PQR was occluded by an-other surface in the scene then the corresponding trianglein the image, �pqr, would contain points correspondingto the boundary of the occluding surface which would pre-clude the triangle �pqr from the Delaunay triangulation ofthe image.

2 Reasoning About Scene Structure

Figure 8 shows the typical input provided to the surfacereconstruction algorithm by the feature based stereo algo-rithm. In this case the results were obtained from a setof five images taken from slightly different vantage points.Note that the stereo system only returns depth informationin regions of the image corresponding to intensity disconti-nuities. Since this particular scene contains numerous ho-mogenous regions, the fraction of pixels with depth valuesis very small. The disparities at other locations in the im-age can be estimated by linearly interpolating the disparityvalues of the recovered features based on a Delaunay tri-angulation of their projections in the image. Typically, thevast majority of the triangles produced by this scheme willcorrespond to actual surface regions in the scene. However,at occlusion boundaries the scheme will naively produce tri-angles that connect foreground surfaces to background sur-faces. While it is true that these interpolated disparity mapsmay not yield accurate information about the surfaces in thescene they do yield reliable information about the freespacein the scene. The following theorem states this observationmore concisely:Freespace Theorem : Suppose that three recovered spacepoints P , Q, and R project to the pixel locations p, q, and rrespectively in a given input image, and suppose further that�pqr is one of the Delaunay triangles formed in that image.Then the tetrahedron formed by the camera center and thespace triangle �PQR must consist entirely of free space,i.e. no surface in the scene intersects this tetrahedron.Proof: Assume, to the contrary, that there is a surfacewithin this tetrahedron. Then the surface cannot occludeany of the points P, Q, or R (or else the occluded pointwould not be visible in the image). Therefore the bound-ary of this surface must lie at least partially inside the tetra-hedron. Points from the boundary would then show up as

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edges inside the image triangle �pqr (see Figure 2). Thiscontradicts our assumption that p, q, and r are part of a sin-gle Delaunay triangle, since the interiors of Delaunay trian-gles cannot contain other vertices1. Hence, the tetrahedronmust consist entirely of free space. ♣ 2

Note that this argument rests on two assumptions. Firstlyit assumes that we have a procedure that accurately recon-structs the depths of all of the salient edges visible from agiven vantage point. If the reconstruction system only re-turned the depths of isolated corner features in the image,this property would not hold. Secondly the argument pre-sumes that the surface in question can be adequately repre-sented by a polygonal mesh where the vertices correspondto intensity discontinuities on the surface. Note that thisdoes not preclude the reconstruction of curved surfaces solong as there are sufficient surface markings to provide ver-tices for an approximating mesh. It would, however, pose aproblem in the case of curved objects with no surface mark-ings. Consider, for example, the problem of applying a fea-ture based stereo algorithm to images of a featureless cylin-der. In this case, the only structural features that could berecovered would correspond to points along the occludingedges of the cylinder. Triangulations that simply connectedthese points together in the images would provide inappro-priate approximations of the object. In the sequel we willdescribe how both of these assumptions can be relaxed toaccount for problems that may occur in practice.

The practical consequence of the freespace theorem isthat we can think of each interpolated disparity map asdefining a star shaped region of freespace and we can rea-son about the 3-D structure of the scene by reasoning aboutthe union of these freespace volumes as shown in Figure 3.

Given the coordinates of a point in space, P , we can eas-ily test whether that point is in the union of the freespacevolumes by projecting the point into each of the originalimages and determining whether the depth of the point Pwith respect to the image center is less than the depth of thecorresponding entry in the interpolated depth map for thatimage. This operation can be performed in constant time bydividing each image into rectangular regions and cachingreferences to the triangles that intersect those regions.

This freespace procedure can be represented by an indi-cator function Φ(P ) : R3 → [0, 1] which returns 0 when-ever the point P lies within the freespace volume and 1otherwise. This function can be thought of as an implicitrepresentation of the total freespace volume.

Given this freespace indicator function Φ(P ) : R3 →[0, 1], it is relatively straightforward to construct an explicit

1In fact, the defining property of Delaunay triangulations is that theinterior of the circumcircle of every triangle does not contain any othervertices; however, we do not require such a strong property for our argu-ment.

2This argument has appeared in a previous publication it is repeatedhere for ease of reference [6].

P

Figure 3: The first row of figures depicts the freespace vol-umes associated with the triangulations of each of the 3 in-put images. The union of these volumes provides a more ac-curate approximation of the 3D structure of the scene thanany of the original depth maps. It is a simple matter to con-struct a function Φ(P ) which indicates whether a particularpoint P lies within the union of the freespace volumes.

representation for the freespace boundary in the form of atriangular mesh by invoking an isosurface algorithm suchas Marching Cubes [14]. The boundary of the freespacevolume can be thought of as a fair approximation for thesurface geometry in the sense that it will correspond to asurface which is consistent with all of the available informa-tion. Notice, however, that the actual surface may containregions that are not visible from any of the camera positions.The freespace union method will still produce reasonableresults in these situations.

2.1 Handling Missing Features

The procedure described in the previous subsection willwork perfectly so long as the feature based stereo algorithmcan be relied upon to produce depth values for all of thesalient edges in the scene. Unfortunately, there are casesin practice where the feature based stereo may fail in thisregard. Problems can arise in situations where a feature isoccluded in one or more of the images in the stereo clusteror when a salient feature does not have sufficient contrast toproduce a reliable correspondence.

Figure 4 shows a situation in which the stereo results ob-tained from stereo cluster 1 do not reflect the presence of thepoint P. This means that the freespace volume constructed

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a. Stereo Cluster 1

Stereo Cluster 2 Stereo Cluster 3

Feature Points

P

b. Stereo Cluster 1

Stereo Cluster 2 Stereo Cluster 3

P

c. Stereo Cluster 1

Stereo Cluster 2 Stereo Cluster 3

PP

c. Stereo Cluster 1

Stereo Cluster 2 Stereo Cluster 3

PP

Figure 4: (a) This figure shows a situation where the featurelabeled P is not recovered by stereo cluster 1. (b) Withoutfeature point P, the freespace volume obtained by triangulat-ing the remaining points will eliminate a significant portionof the surface (c) Since the point P was among the points re-constructed by stereo cluster 2 one could consider combin-ing the two stereo results by simply adding all of the featurepoints that appear inside the freespace volume to the mixand recomputing the triangulation for stereo cluster 1. Un-fortunately this policy is too liberal and the resulting surfacecan still have problems as shown here. (d) These problemscan be overcome by adding the points to the triangulationincrementally based on their height above the current trian-gulated surface.

for that cluster will eliminate a significant portion of thestructure. This is clearly undesirable.

In practice, it is usually the case that multiple stereo clus-ters are used to reconstruct a given scene and these are typ-ically arranged with a signifcant region of overlap. Thismeans that features that may be missing from one stereoreconstruction are often recovered by a neighboring stereosystem. In the example shown in Figure 4 we assume thatthe point P was recovered by stereo cluster 2. Since thereconstruction system is aware of this point, P, it can de-termine that its initial estimate for the freespace volume as-sociated with cluster 1 is flawed since it includes a knownfeature point, P. This observation that the freespace volumecannot contain any of the recovered feature points is referedto as the minimum disparity constraint since it effectivelyconstrains the freespace volumes associated with each van-tage point.

The most straightforward approach to fixing thefreespace volume associated with stereo cluster 1 so thatit satisfies this constraint would be to add all the featurepoints that violated the constraint to the mix and recomputethe triangulation associated with stereo cluster 1. Figure 4cshows the results of such a process. The resulting freespace

volume contains all of the known feature points but it stillincludes regions of the object that should not have been in-cluded.

This problem can be solved by modifying the update pro-cedure to add features to the triangulation in order based ontheir height above the current estimate for the freespace sur-face. On each iteration the update procedure would projecteach of the recovered scene points into the viewpoint ofcluster 1 and determine its height above or below the cur-rent triangulated disparity surface associated with that view-point. The point with the greatest height above the disparitysurface would be used to update the Delaunay triangulationincrementally. The process would continue until all of thefeature points were safely contained within the triangulatedsurface associated with the cluster. Using this procedurethe system would update its original flawed estimate for thefreespace surface by adding point P and retriangulating. Atthis point it would determine that there were no further ille-gal points and terminate.

This procedure for updating the triangulation to includeall of the recovered points can be implemented efficiently bymodifying a standard incremental Delaunay triangulationalgorithm such as the one described in [3]. Typically, onlya small fraction of the points need to be added to to fix thetriangulation. Furthermore, the process of finding the pointwith the greatest height above the triangulation can be opti-mized very effectively by caching references to the Delau-nay triangles that enclose the projected feature points. Theworst case computational complexity of this update proce-dure would be n2 log n where n denotes the total numberof feature points, this corresponds to the case where all nfeatures need to be added to the triangulation.

This update procedure also provides an avenue for deal-ing with featureless curved surfaces such as the one consid-ered in section 2. Figure 5 shows the case where multiplestereo clusters are viewing a featureless cylinder, each ofthe stereo clusters locates two feature points correspondingto the occluding contour visible from that vantage point. Byapplying the minimum disparity constraint decribed previ-ously, the reconstruction system would recover the polygo-nal estimate for the curved surface shown in Figure 5b. Thisestimate contains all of the recovered feature points. Clearlythe fidelity of such an approximation would depend uponthe numer of stereo clusters used to reconstruct the sceneand their distribution in space. Nonetheless, even in thisworst case scenario it is possible to recover some informa-tion about the scene structure from the sparse data providedby a feature based stereo system.

2.2 Recovering Surfaces

Once the minimum disparity constraint has been applied toupdate the freespace volumes associated with each of the

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a. Stereo Clusters

CylinderRecoveredFeature Points

b. Stereo Clusters

Figure 5: In this worst case scenario we consider a set ofstereo clusters viewing a featureless cylinder. a. Each of thestereo clusters reconstructs a pair of feature points corre-sponding to points on the occluding contour of the cylinder.b. The approximation constructed based on these featuresby applying the minimum disparity constraint.

stereo clusters, the only remaining task is to recover an ex-plicit, triangulated representation for the boundary of thefreespace defined by the implicit function Φ(P ). Actu-ally, the reconstruction procedure recovers the boundary ofa slightly different indicator function, Φ ′(P ) which evalu-ates to 1 iff P is inside the freespace defined by Φ(P ) oroutside the convex hull of the points. This modification re-flects the fact that the reconstructed surface should not ex-tend beyond the convex hull of the recovered feature points.It is a very simple matter to determine whether a point, P,lies within the convex hull of our point set. One need onlyevaluate a set of linear equations which correspond to thefacets of the convex hull.

The boundary of this function, Φ ′(P ) can be obtainedby invoking a modified version of the well known march-ing cubes algorithm 3. Recall that the marching cubes pro-cedure finds isosurfaces by sampling the given function ata set of grid points in the specified volume (see Figure6). When the algorithm detects a transition in the functionvalue between two neighboring grid points it inserts a ver-tex along that edge in the grid. In this situation it is possibleto refine the location of the transition along the segment be-tween the two grid points by applying standard bisectionsearch [13] to the freespace indicator function Φ ′(P ). Thismakes it possible to use a fairly large sampling distance be-tween the grid points while still accurately determining thelocation of surface vertices.

One issue to be mindful of is that isolated outliers canhave a significant impact on the recovered surface since incertain situations. Problems with outliers can typically beovercome by insisting that the recovered features have arequisite amount of support in a local neighborhood of theimage.

3The same arguments can also be applied to modify the marching tetra-hedra algorithm

GRID LOCATIONS

TRANSITIONSLOCALIZED USING BISECTIONSEARCH

Figure 6: The marching cubes algorithm proceeds by sam-pling grid points in the specified volume in search of tran-sitions in the given implicit function. Once a transition hasbeen detected it can be further localized by applying stan-dard bisection search techniques.

a. b.

Figure 7: Two of the 30 images acquired of this scene

The marching cubes procedure also provides a certainamount of robustness to isolated outliers since these typesof features usually induce thin spikes in the freespace vol-ume. These thin volumes typically do not contain manygrid points and, hence, will not induce many surface ver-tices. This amounts to a form of “structural aliasing” whichworks to our advantage in this case.

3 Experimental Results

In order to test the effectiveness of the proposed recon-struction procedure, a series of image data sets was ob-tained. Each data set consisted of 30 images divided into6 stereo clusters. The stereo clusters were positioned atevenly spaced intervals all around the target scene. The cal-ibration grid visible in the images shown in Figure 7 wasused to determine the intrinsic and extrinsic parameters as-sociated with each of the images in the data set.

Each stereo cluster consisted of 5 images taken inroughly the pattern of a cross as shown in Figure 9. Theimages were taken with a Foveon SD9 Digital SLR cam-era outfitted with a 50mm lens. For each stereo cluster twotrinocular stereo procedures were performed. The first pro-cedure used the three images corresponding to the horizon-tal axis of the cross while the second used the three imagescorresponding to the vertical axis of the cross. The cen-

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Figure 8: This figure shows typical results obtained fromour feature based stereo algorithm on this scene. a. Thecenter image of one of the stereo clusters b. Features recov-ered in that stereo cluster

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Figure 9: Figure showing the arrangement of viewpointsused in this experiment. The images are divided into 6stereo clusters each of which consists of five views takenin a cross configuration.

Figure 10: Novel view of the set of feature points recoveredby the stereo clusters.

a. b.

Figure 11: Various novel views of the 3D mesh obtainedwith the reconstruction algorithm.

a. b.

Figure 12: Various novel views of the 3D mesh renderedusing a view dependent texture mapping technique

ter image was used twice and served as the base frame ofreference for the stereo cluster. This scheme was able toreconstruct edges in the scene at all orientations.

The trinocular stereo algorithm employed in this workrectifies the three images to a common orientation, extractsedge tokens in all of the images, and matches these tokensacross the views. Image windows on either side of the ex-tracted edge features are examined in order to verify pro-posed correspondences. Significantly, this approach is ableto deal with occlusion boundaries in the scene which areoften mishandled by correlation based stereo methods.

Figure 8 shows an example of the feature points recov-ered from a stereo cluster projected into the viewpoint ofthe central image of that cluster. Figure 10 shows all ofthe recovered feature points viewed from a novel vantagepoints. Feature points were typically recovered with mil-limeter precision.

The results of applying the surface reconstruction pro-cedure are shown in Figure 11. They demonstrate that themethod is able to recover reasonable approximations for thesurface struture of a fairly complex 3 dimensional scene. Inthis case, some of the volume between the blocks was notvisible from any of the vantage points used in the recon-struction. As a result the surface reconstruction procedureproduced some “extra” surface elements corresponding tothis volume. This problem could be fixed by adding stereoresults from additional vantage points.

4 Conclusions

This paper presents a novel scheme for reconstructing the3D structure of a scene based on the information returnedby a feature based stereo algorithm. The approach exploitsthe fact that even though feature based stereo methods mayonly return disparity values for a small fraction of the pixelsin the input images, the features that they do return tend tocorrespond to salient structural features in the environment.This allows us to reason about the structure of the scene ina principled manner.

The entire reconstruction procedure is summarized be-low:

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1. Acquire imagery

2. Run feature based stereo on each cluster

3. Construct a triangulated disparity map for each stereocluster taking into account the minimum disparity con-straint

4. Recover an estimate for the surface structure by ap-plying marching cubes to the union of the freespacevolumes and the complement of the convex hull of thefeature points.

This work deliberately considers a fairly extreme form ofthe reconstruction problem, the case where the stereo sys-tem can only provide information about the depth of edgefeatures in the input images. The intent has been to showthat it is still possible to determine quite a lot about thestructure of the scene in this case. In many cases, it is rea-sonable to expect that a stereo system would be able to pro-vide much more information about the structure of the sceneand these additional correspondences will only serve to im-prove the quality of the reconstruction.

It is also interesting to note that for some applicationslike obstacle avoidance and ray tracing the implicit repre-sentation for scene freespace, Φ(P ), is very useful since itprovides us with a direct method for answering questionsabout the occupancy structure of the scene.

Acknowledgments

This material is based upon work supported by the NationalScience Foundation under Grant No. IIS-98-75867. Anyopinions, findings, and conclusions or recommendations ex-pressed in this material are those of the author(s) and do notnecessarily reflect the views of the National Science Foun-dation.

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