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Improved Sub-pixel Stereo Correspondences

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Improved Sub-pixel Stereo Correspondences through Symmetric Refine mentDiego Nehab1Szymon Rusinkiewicz1James Davis21Princeton University2University of California at Santa CruzAbstractMost dense stereo correspondence algorithms start by es-tablishing discrete pixel matches and later refine thesematches to sub -pixel precision. Traditional sub -pixel refine-ment methods attempt to determine the precise locatio n ofpoints, in the secondary image, that correspond to discretepositions in the reference image. We show that this strategycan lead to a systematic bias associated with the violation ofthe general symmetry of matching cost functions. Th is biasprod uces random or coherent noise in the final reconstruc-tion, but can be avoided by refining both image coordinatessimultaneou sly, in a symmetric way. We demonstrate thatthe symmetric sub-pixel refinement strategy results in moreaccurate correspondences by avoiding bias while preserv-ing detail.1. IntroductionThe compu tation of precise sub- pixel stereo correspo n-dences is vital to areas such as 3D scanning and image basedmodeling and rendering. Most dense stereo corre spondencealgorithm s start by determining discrete pixel matches andlater refine these matche s to sub-pixel precision [11]. Theinitial set of correspondences is usually computed b y mini-mization of a matching cost function that has been laid outas a disparity space image (DSI) [2, 14].Sub-pixel refinement of correspond ences can be per-formed over a finely sampled or continuously reconstru c te dneighborhood of the DSI around the initial integer match.The continuous reconstruction strategy has the advantageof being simple and efficient. On the other hand, althoughcomputationally more expensive, the supe rsampling alter-native tends to be more accurate. Efforts have been madeboth to im prove the quality of reconstruction-based refine-ment [1 2, 13] and to improve the efficiency of sup ersam-pling [6].In this paper, we identify a new source of bias forreconstruction-based sub-pixel refinement stra tegies (sec-tion 2). It can be observed when one image is consideredas reference and the refinement is performe d on the corre-sponding coo rdinate in the matching image . It arises fromthe sensitivity of this “traditional” approach to the varyingconfidence of the matching cost function when evaluated atneighboring pixels. In the final reconstruction, the bias can(a) (b) (c) (d)Figure 1: Examples of matching cost functions. (a) Sum of squareddifferences. (b) Normalized cross-correlation. (c) Birchfield andTomasi [1]. (d) Sum of absolute differences. Note the matchingridge and how the functions are symmetric with regard to it.be experienced as random or coherent n oise, as the “textureembossing” ad dressed by Curless a nd Levoy [4], or as the“striping effect” addressed by Zhang et al. [16].To avoid bias, o ur sym metric sub-pixel refinement strat-egy refines both the reference and the ma tc hing ima ge co-ordinates simultane ously, in a symmetric way, by lookingfor the m inimum of the matching c ost function along a di-rection that is insensitive to its confidence varia tions ( sec-tion 3). We present results on both synthetic data and realscans o btained using active stereo (section 4), which showthat this new method significantly reduc es bias in high-variance situations. Additionally, we demonstrate that oneof its variants avoids the “pixel locking” effect ad dressedby Shimizu and Okutomi [12].2. The Symmetry of Matching CostConsider two rectified cameras C1and C2, producing im-ages I1and I2of an object, such th at the scan-lin es in eachimage are corresponding epipolar lines [7]. In this setup,Yang et al. [14] reduc e d the problem of stereo matchingto that of finding a surface in the disparity-space imageΞ(x1, y, d), which mea sures the cost of matching points(x1, y) in I1and (x1+ d, y) in I2. The ma tching cost is de-fined by a metric M that compar es neighborhoods of pixelvalues, so that Ξ(x1, y, d) ≡ M (I1(x1, y), I2(x1+ d, y)).Note that for a given scan-line y, the problem simplifieseven further to that of find ing a matching ridge, which isthe extremum curve in Ξy(x1, d).Instead of working in disparity space, we prefer to workdirectly with image coordinates. The concept of disparityimplies taking one camera as reference and, as we shallsee, this is a source of bias. The direct parameterizationFy(x1, x2) ≡ Ξy(x1, x2− x1) is more symm etric and sim-plifies our analysis. Figure 1 shows examp le s of popular1q2C1C2p2Pp1ZZQX2X1r2(t)r1(t)Uuq1Oy(t)Figure 2: The slope of the matching ridge. The geometry of thesetup yields an expression for the slope dr2/dr1, as given by equa-tion 6. Each intersection U between the object tangent and thebaseline of the cameras produces a different slope.matching cost functions und er this direct para metrization.In eac h case, the matching ridge is clea rly visible. We alsonotice a certain symmetry of the matching cost, which weexplain below.Consider the intersection between the object being im -aged and a given epipolar plane, as shown in figure 2. Itdefines a curve Oy(t) that is projected into I1and I2. Ifr1(t) and r2(t) are the corresponding parametrizatio ns forthese projections, the matching ridge is simply the curve de-fined by Ry(t) = (r1(t), r2(t)). G iven a per fect matchingpair (x1, x2), it is clear that Rygoes through (x1, x2) forsome t. If r1and r2are continuous and smooth at t, the n(x1+ dr1(t), x2+ dr2(t)) is a first order approximationfor Ry. I t follows that (x1± dr1, x2± dr2) are also on thematching ridge and therefore are also matching pairs.Comparing the values of Fy(x1+ dr1, x2− dr2) andFy(x1−dr1, x2+dr2), we notice that they must be similar:Fy(x1+ dr1, x2− dr2) ≡≡ M (I1(x1+ dr1, y), I2(x2− dr2, y)) (1)= M (I2(x2− dr2, y), I1(x1+ dr1, y)) (2)≈ M (I1(x1− dr1, y), I1(x1+ dr1, y)) (3)≈ M (I1(x1− dr1, y), I2(x2+ dr2, y)) (4)≡ Fy(x1− dr1, x2+ dr2) (5)Steps (1) and (5) are by definition. Step (2) follows fromthe symmetry of M. Steps (3) and (4) come first from thefact that, since x1± dr1matches x2± dr2, I1(x1± dr1, y)must be similar to I2(x2± dr2, y). The continuity of Mthen leads to the approximations.We have thus shown that Fyis locally skew-symmetricabout Ry. The symmetry is such that, if a segment of thematching ridge is the diagonal of a rectangle, then symmet-ric pairs can be found along symmetric lines parallel to theother diagonal. These may or may not be


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