Accurate and Efficient Stereo Processing by SemiGlobal Matching and Mutual Information


 Claude Fleming
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1 Accurate and Efficient Stereo Processing by SemiGlobal Matching and Mutual Information Heiko Hirschmüller Institute of Robotics and Mechatronics Oberfaffenhofen German Aerosace Center (DLR) P.O. Box 1116, Wessling, Germany Abstract This aer considers the objectives of accurate stereo matching, esecially at object boundaries, robustness against recording or illumination changes and efficiency of the calculation. These objectives lead to the roosed Semi Global Matching method that erforms ixelwise matching based on Mutual Information and the aroximation of a global smoothness constraint. Occlusions are detected and disarities determined with subixel accuracy. Additionally, an extension for multibaseline stereo images is resented. There are two novel contributions. Firstly, a hierarchical calculation of Mutual Information based matching is shown, which is almost as fast as intensity based matching. Secondly, an aroximation of a global cost calculation is roosed that can be erformed in a time that is linear to the number of ixels and disarities. The imlementation requires just 1 second on tyical images. 1. Introduction Accurate, dense stereo matching is an imortant requirement for many alications, like 3D reconstruction. Most difficult are often the boundaries of objects and fine structures, which can aear blurred. Additional ractical roblems originate from recording and illumination differences or reflections, because matching is often directly based on intensities that can have quite different values for corresonding ixels. Furthermore, fast calculations are often required, either because of realtime alications or because of large images or many images that have to be rocessed efficiently. An alication were all of the three objectives come together is the reconstruction of urban terrain, catured by an airborne ushbroom camera. Accurate matching at object boundaries is imortant for reconstructing structured environments. Robustness against recording differences and illumination changes is vital, because this often cannot be controlled. Finally, efficient (offline) rocessing is necessary, because the images and disarity ranges are huge (e.g. several 100MPixel with 1000 ixel disarity range). 2. Related Literature There is a wide range of dense stereo algorithms [8] with different roerties. Local methods, which are based on correlation can have very efficient imlementations that are suitable for real time alications [5]. However, these methods assume constant disarities within a correlation window, which is incorrect at discontinuities and leads to blurred object boundaries. Certain techniques can reduce this effect [8, 5], but it cannot be eliminated. Pixelwise matching [1] avoids this roblem, but requires other constraints for unambiguous matching (e.g. iecewise smoothness). Dynamic Programming techniques can enforce these constraints efficiently, but only within individual scanlines [1, 11]. This tyically leads to streaking effects. Global aroaches like Grah Cuts [7, 2] and Belief Proagation [10] enforce the matching constraints in two dimensions. Both aroaches are quite memory intensive and Grah Cuts is rather slow. However, it has been shown [4] that Belief Proagation can be imlemented very efficiently. The matching cost is commonly based on intensity differences, which may be samling insensitive [1]. Intensity based matching is very sensitive to recording and illumination differences, reflections, etc. Mutual Information has been introduced in comuter vision for matching images with comlex relationshis of corresonding intensities, ossibly even images of different sensors [12]. Mutual Information has already been used for correlation based stereo matching [3] and Grah Cuts [6]. It has been shown [6] that it is robust against many comlex intensity transformations and even reflections.
2 3. SemiGlobal Matching 3.1. Outline The SemiGlobal Matching (SGM) method is based on the idea of ixelwise matching of Mutual Information and aroximating a global, 2D smoothness constraint by combining many 1D constraints. The algorithm is described in distinct rocessing stes, assuming a general stereo geometry of two or more images with known eiolar geometry. Firstly, the ixelwise cost calculation is discussed in Section 3.2. Secondly, the imlementation of the smoothness constraint is resented in Section 3.3. Next, the disarity is determined with subixel accuracy and occlusion detection in Section 3.4. An extension for multibaseline matching is described in Section 3.5. Finally, the comlexity and imlementation is discussed in Section Pixelwise Cost Calculation The matching cost is calculated for a base image ixel from its intensity I b and the susected corresondence I mq at q = e bm (,d) of the match image. The function e bm (,d) symbolizes the eiolar line in the match image for the base image ixel with the line arameter d. For rectified images e bm (,d) = [ x d, y ] T with d as disarity. An imortant asect is the size and shae of the area that is considered for matching. The robustness of matching is increased with large areas. However, the imlicit assumtion about constant disarity inside the area is violated at discontinuities, which leads to blurred object borders and fine structures. Certain shaes and techniques can be used to reduce blurring, but it cannot be avoided [5]. Therefore, the assumtion of constant disarities in the vicinity of is discarded. This means that only the intensities I b and I mq itself can be used for calculating the matching cost. One choice of ixelwise cost calculation is the samling insensitive measure of Birchfield and Tomasi [1]. The cost C BT (,d) is calculated as the absolute minimum difference of intensities at and q = e bm (,d) in the range of half a ixel in each direction along the eiolar line. Alternatively, the matching cost calculation is based on Mutual Information (MI) [12], which is insensitive to recording and illumination changes. It is defined from the entroy H of two images (i.e. their information content) as well as their joined entroy. MI I1,I 2 = H I1 + H I2 H I1,I 2 (1) The entroies are calculated from the robability distributions P of intensities of the associated images. Z 1 H I = P I (i)logp I (i)di (2) H I1,I 2 = 0 Z 1 Z P I1,I 2 (i 1,i 2 )logp I1,I 2 (i 1,i 2 )di 1 di 2 (3) For well registered images the joined entroy H I1,I 2 is low, because one image can be redicted by the other, which corresonds to low information. This increases their Mutual Information. In the case of stereo matching, one image needs to be wared according to the disarity image D for matching the other image, such that corresonding ixels are at the same location in both images, i.e. I 1 = I b and I 2 = f D (I m ). Equation (1) oerates on full images and requires the disarity image a riori. Both revent the use of MI as matching cost. Kim et al. [6] transformed the calculation of the joined entroy H I1,I 2 into a sum of data terms using Taylor exansion. The data term deends on corresonding intensities and is calculated individually for each ixel. H I1,I 2 = h I1,I 2 (I 1,I 2 ) (4) The data term h I1,I 2 is calculated from the robability distribution P I1,I 2 of corresonding intensities. The number of corresonding ixels is n. Convolution with a 2D Gaussian (indicated by g(i, k)) effectively erforms Parzen estimation [6]. h I1,I 2 (i,k) = 1 n log(p I 1,I 2 (i,k) g(i,k)) g(i,k) (5) The robability distribution of corresonding intensities is defined with the oerator T[], which is 1 if its argument is true and 0 otherwise. P I1,I 2 (i,k) = 1 n T[(i,k) = (I 1,I 2 )] (6) Kim et al. argued that the entroy H I1 is constant and H I2 is almost constant as the disarity image merely redistributes the intensities of I 2. Thus, h I1,I 2 (I 1,I 2 ) serves as cost for matching the intensities I 1 and I 2. However, if occlusions are considered then some intensities of I 1 and I 2 do not have a corresondence. These intensities should not be included in the calculation, which results in nonconstant entroies H I1 and H I2. Therefore, it is suggested to calculate these entroies analog to the joined entroy. H I = h I (I ), h I (i) = 1 n log(p I(i) g(i)) g(i) (7)
3 The robability distribution P I must not be calculated over the whole images I 1 and I 2, but only over the corresonding arts (otherwise occlusions would be ignored and H I1 and H I2 would be almost constant). That is easily done by just summing the corresonding rows and columns of the joined robability distribution, e.g. P I1 (i) = k P I1,I 2 (i,k). The resulting definition of Mutual Information is, MI I1,I 2 = mi I1,I 2 (I 1,I 2 ) mi I1,I 2 (i,k) = h I1 (i) + h I2 (k) h I1,I 2 (i,k). This leads to the definition of the MI matching cost. (8a) (8b) C MI (,d) = mi Ib, f D (I m )(I b,i mq )with q = e bm (,d) (9) The remaining roblem is that the disarity image is required for waring I m, before mi() can be calculated. Kim et al. suggested an iterative solution, which starts with a random disarity image for calculating the cost C MI. This cost is then used for matching both images and calculating a new disarity image, which serves as the base of the next iteration. The number of iterations is rather low (e.g. 3), because even wrong disarity images (e.g. random) allow a good estimation of the robability distribution P. This solution is well suited for iterative stereo algorithms like Grah Cuts [6], but it would increase the runtime of noniterative algorithms unnecessarily. Therefore, a hierarchical calculation is roosed, which recursively uses the (uscaled) disarity image, that has been calculated at half resolution, as initial disarity. If the overall comlexity of the algorithm is O(W HD) (i.e. width height disarity range), then the runtime at half resolution is reduced by factor 2 3 = 8. Starting with a random disarity image at a resolution of 1 16th and initially calculating 3 iterations increases the overall runtime by the factor, (10) 163 Thus, the theoretical runtime of the hierarchically calculated C MI would be just 14% slower than that of C BT, ignoring the overhead of MI calculation and image scaling. It is noteworthy that the disarity image of the lower resolution level is used only for estimating the robability distribution P and calculating the costs C MI of the higher resolution level. Everything else is calculated from scratch to avoid assing errors from lower to higher resolution levels Aggregation of Costs Pixelwise cost calculation is generally ambiguous and wrong matches can easily have a lower cost than correct ones, due to noise, etc. Therefore, an additional constraint is added that suorts smoothness by enalizing changes of neighboring disarities. The ixelwise cost and the smoothness constraints are exressed by defining the energy E(D) that deends on the disarity image D. E(D) = C(,D ) + P 1 T[ D D q = 1] q N + P 2 T[ D D q > 1] q N (11) The first term is the sum of all ixel matching costs for the disarities of D. The second term adds a constant enalty P 1 for all ixels q in the neighborhood N of, for which the disarity changes a little bit (i.e. 1 ixel). The third term adds a larger constant enalty P 2, for all larger disarity changes. Using a lower enalty for small changes ermits an adatation to slanted or curved surfaces. The constant enalty for all larger changes (i.e. indeendent of their size) reserves discontinuities [2]. Discontinuities are often visible as intensity changes. This is exloited by adating P 2 to the intensity gradient, i.e. P 2 = P 2 I b I bq. However, it has always to be ensured that P 2 P 1. The roblem of stereo matching can now be formulated as finding the disarity image D that minimizes the energy E(D). Unfortunately, such a global minimization (2D) is NPcomlete for many discontinuity reserving energies [2]. In contrast, the minimization along individual image rows (1D) can be erformed efficiently in olynomial time using Dynamic Programming [1, 11]. However, Dynamic Programming solutions easily suffer from streaking [8], due to the difficulty of relating the 1D otimizations of individual image rows to each other in a 2D image. The roblem is, that very strong constraints in one direction (i.e. along image rows) are combined with none or much weaker constraints in the other direction (i.e. along image columns). This leads to the new idea of aggregating matching costs in 1D from all directions equally. The aggregated (smoothed) cost S(, d) for a ixel and disarity d is calculated by summing the costs of all 1D minimum cost aths that end in ixel at disarity d (Figure 1). It is noteworthy that only the cost of the ath is required and not the ath itself. Let L r be a ath that is traversed in the direction r. The cost L r(,d) of the ixel at disarity d is defined recursively as, L r(,d) = C(,d) + min(l r( r,d), L r ( r,d 1) + P 1,L r ( r,d + 1) + P 1, min i L r ( r,i) + P 2). (12) The ixelwise matching cost C can be either C BT or C MI. The remainder of the equation adds the lowest cost of the
4 (a) Minimum Cost Path L r (, d) d x, y (b) 16 Paths from all Directions r Figure 1. Aggregation of costs. revious ixel r of the ath, including the aroriate enalty for discontinuities. This imlements the behavior of equation (11) along an arbitrary 1D ath. This cost does not enforce the visibility or ordering constraint, because both concets cannot be realized for aths that are not identical to eiolar lines. Thus, the aroach is more similar to Scanline Otimization [8] than traditional Dynamic Programming solutions. The values of L ermanently increase along the ath, which may lead to very large values. However, equation (12) can be modified by subtracting the minimum ath cost of the revious ixel from the whole term. L r (,d) = C(,d) + min(l r ( r,d), L r ( r,d 1) + P 1,L r ( r,d + 1) + P 1, min i L r ( r,i) + P 2 ) minl r ( r,k) k y x (13) This modification does not change the actual ath through disarity sace, since the subtracted value is constant for all disarities of a ixel. Thus, the osition of the minimum does not change. However, the uer limit can now be given as L C max + P 2. The costs L r are summed over aths in all directions r. The number of aths must be at least 8 and should be 16 for roviding a good coverage of the 2D image. S(,d) = L r (,d) (14) r The uer limit for S is easily determined as S 16(C max + P 2 ) Disarity Comutation The disarity image D b that corresonds to the base image I b is determined as in local stereo methods by selecting for each ixel the disarity d that corresonds to the minimum cost, i.e. min d S(,d). For subixel estimation, a quadratic curve is fitted through the neighboring costs (i.e. at the next higher or lower disarity) and the osition of the minimum is calculated. Using a quadratic curve is theoretically justified only for a simle correlation using the sum of squared differences. However, is is used as an aroximation due to the simlicity of calculation. The disarity image D m that corresonds to the match image I m can be determined from the same costs, by traversing the eiolar line, that corresonds to the ixel q of the match image. Again, the disarity d is selected, which corresonds to the minimum cost, i.e. min d S(e mb (q,d),d). However, the cost aggregation ste does not treat the base and match images symmetrically. Therefore, better results can be exected, if D m is calculated from scratch. Outliers are filtered from D b and D m, using a median filter with a small window (i.e. 3 3). The calculation of D b as well as D m ermits the determination of occlusions and false matches by erforming a consistency check. Each disarity of D b is comared with its corresonding disarity of D m. The disarity is set to invalid (D inv ) if both differ. D = { D b if D b D mq 1, q = e bm (,D b ), D inv otherwise. (15) The consistency check enforces the uniqueness constraint, by ermitting one to one maings only Extension for MultiBaseline Matching The algorithm could be extended for multibaseline matching, by calculating a combined ixelwise matching cost of corresondences between the base image and all match images. However, valid and invalid costs would be mixed near discontinuities, deending on the visibility of a ixel in a match image. The consistency check (Section 3.4) can only distinguish between valid (visible) and invalid (occluded or mismatched) ixels, but it can not searate valid and invalid costs afterwards. Thus, the consistency check would invalidate all areas that are not seen by all images, which leads to unnecessarily large invalid areas. Without the consistency check, invalid costs would introduce matching errors near discontinuities, which leads to fuzzy object borders. Therefore, it is better to calculate several disarity images from individual image airs, exclude all invalid ixels by the consistency check and then combine the result. Let the disarity D k be the result of matching the base image I b against a match image I mk. The disarities of the images D k are scaled differently, according to some factor t k. For rectified images, this factor corresonds to the length of the baseline between I b and I mk. The robust combination selects the median of all disarities D k t k for a certain ixel. Additionally, the accuracy
5 is increased by calculating the weighted mean of all correct disarities (i.e. within the range of 1 ixel around the median). This is done by using t k as weighting factor. D = k V D k, V = {k D k k V t k t k med i D i t i 1 } (16) t k This combination is robust against matching errors in some disarity images and it also increases the accuracy Comlexity and Imlementation The calculation of the ixelwise cost C MI starts with collecting all alleged corresondences (i.e. defined by an initial disarity as described in Section 3.2) and calculating P I1,I 2. The size of P is the square of the number of intensities, which is constant (i.e ). The subsequent oerations consist of Gaussian convolutions of P and calculating the logarithm. The comlexity deends only on the collection of alleged corresondences due to the constant size of P. Thus, O(WH) with W as image width and H as image height. The ixelwise matching costs for all ixels at all disarities d are scaled to 11 bit integer values and stored in a 16 bit array C(,d). Scaling to 11 bit guarantees that all aggregated costs do not exceed the 16 bit limit. A second 16 bit integer array of the same size is used for the aggregated cost values S(, d). The array is initialized with 0. The calculation starts for each direction r at all ixels b of the image border with L r (b,d) = C(b,d). The ath is traversed in forward direction according to equation (13). For each visited ixel along the ath, the costs L r (,d) are added to S(, d) for all disarities d. The calculation of equation (13) requires O(D) stes at each ixel, since the minimum cost of the revious ixel (e.g. min k L r ( r,k)) is constant for all disarities and can be recalculated. Each ixel is visited exactly 16 times, which results in a total comlexity of O(W HD). The regular structure and simle oerations (i.e. additions and comarisons) ermit arallel calculations using integer based SIMD 1 assembler instructions. The disarity comutation and consistency check requires visiting each ixel at each disarity a constant number of times. Thus, the comlexity is O(WHD) as well. The 16 bit arrays C and S have a size of W H D, which can exceed the available memory for larger images and disarity ranges. The suggested remedy is to slit the inut image into tiles, which are rocessed individually. The tiles overla each other by a few ixels, since the ixels at the image border receive suort by the global cost function only from one side. The overlaing ixels are ignored for combining the tiles to the final disarity image. 1 Single Instruction, Multile Data This solution allows rocessing of almost arbitrarily large images. 4. Exerimental Results 4.1. Stereo Images with Ground Truth Three stereo image airs with ground truth [8, 9] have been selected for evaluation (first row of Figure 2). The images 2 and 4 have been used from the Teddy and Cones image sequences. All images have been rocessed with a disarity range of 32 ixel. The MWMF method is a local, correlation based, real time algorithm [5], which has been shown [8] to roduce better object borders (i.e. less fuzzy) than many other local methods. The second row of Figure 2 shows the resulting disarity images. The blurring of object borders is tyical for local methods. The calculation of Teddy has been erformed in just 0.071s on a Xeon with 2.8GHz. Belief Proagation (BP) [10] minimizes a global cost function (e.g. equation (11)) by iteratively assing messages in a grah that is defined by the four connected image grid. The messages are used for udating the nodes of the grah. The disarity is in the end selected individually at each node. Similarly, SGM can be described as assing messages indeendently, from all directions along 1D aths for udating nodes. This is done sequentially as each message deends on one redecessor only. Thus, messages are assed through the whole image. In contrast, BP sends messages in a 2D grah. Thus, the schedule of messages that reaches each node is different and BP requires an iterative solution. The number of iterations determines the distance from which information is assed in the image. The efficient BP algorithm 2 [4] uses a hierarchical aroach and several otimizations for reducing the comlexity. The comlexity and memory requirements are very similar to SGM. The third row of Figure 2 shows good results of Tsukuba. However, the results of Teddy and esecially Cones are rather blocky, desite attemts to get the best results by arameter tuning. The calculation of Teddy took 4.5s on the same comuter. The Grah Cuts method 3 [7] iteratively minimizes a global cost function (e.g. equation (11) with P 1 = P 2 ) as well. The fourth row of Figure 2 shows the results, which are much better for Teddy and Cones, esecially near object borders and fine structures like the leaves. However, the comlexity of the algorithm is much higher. The calculation of Teddy has been done in 55s on the same comuter. The results of SGM with C BT as matching cost (i.e. the same as for BP and GC) are shown in the fifth row of Figure 2 htt://eole.cs.uchicago.edu/ ff/b/ 3 htt://
6 Tsukuba (384 x 288, Dis. 32) Teddy (450 x 375, Dis. 32) Cones (450 x 375, Dis. 32) Left Images Local, correlation (MWMF) Belief Proagation (BP) Grah Cuts (GC) SGM with BT (i.e. intensity based matching cost) SGM with HMI, (i.e. hierarchical Mutual Information as matching cost) Figure 2. Comarison of different stereo methods.
7 Left Image Modified Right Image Resulting Disarity Image (SGM HMI) Figure 3. Result of matching modified Teddy images with SGM (HMI). 2, using the best arameters for the set of all images. The quality of the result comes close to Grah Cuts. Only, the textureless area on the right of the Teddy is handled worse. Slanted surfaces aear smoother than with Grah Cuts, due to sub ixel interolation. The calculation of Teddy has been erformed in 1.0s. Cost aggregation requires almost half of the rocessing time. The last row of Figure 2 shows the result of SGM with the hierarchical calculation of C MI as matching cost. The disarity image of Tsukuba and Teddy aear equally well and Cones aears much better. This is an indication that the matching tolerance of MI is beneficial even for carefully catured images. The calculation of Teddy took 1.3s. This is just 30% slower than the nonhierarchical, intensity based version. The disarity images have been comared to the ground truth. All disarities that differ by more than 1 are treated as errors. Occluded areas (i.e. identified using the ground truth) have been ignored. Missing disarities (i.e. black areas) have been interolated by using the lowest neighboring disarities. Figure 4 resents the resulting grah. This quantitative analysis confirms that SGM erforms as well as other global aroaches. Furthermore, MI based matching results in even better disarity images. Errors [%] MWMF BP GC SGM (BT) SGM (HMI) Tsukuba Errors of different methods Teddy Cones Figure 4. Errors of different stereo methods. The ower of MI based matching can be demonstrated by manually modifying the right image of Teddy by dimming the uer half and inverting the intensities of the lower half (Figure 3). Such an image air cannot be matched by intensity based costs. However, the MI based cost handles this situation easily as shown on the right. More examles about the ower of MI based stereo matching are shown by Kim et al. [6] Stereo Images of an Airborne Pushbroom Camera The SGM (HMI) method has been tested on huge images (i.e. several 100MPixel) of an airborne ushbroom camera, which records 5 anchromatic images in different angles. The aroriate camera model and nonlinearity of the flight ath has been taken into account for calculating the eiolar lines. A difficult test object is Neuschwanstein castle (Figure 5a), because of high walls and towers, which result in high disarity changes and large occluded areas. The castle has been recorded 4 times using different flight aths. Each flight ath results in a multibaseline stereo image from which the disarity has been calculated. All disarity images have been combined for increasing robustness. Figure 5b shows the end result, using a hierarchical, correlation based method [13]. The object borders aear fuzzy and the towers are mostly unrecognized. The result of the SGM (HMI) method is shown in Figure 5c. All object borders and towers have been roerly detected. Stereo methods with intensity based ixelwise costs (e.g. Grah Cuts and SGM (BT)) failed on these images comletely, because of large intensity differences of corresondences. This is caused by recording differences as well as unavoidable changes of lighting and the scene during the flight (i.e. corresonding oints are recorded at different times on the flight ath). Nevertheless, the MI based matching cost handles the differences easily. The rocessing time is one hour on a 2.8GHz Xeon for matching 11MPixel of a base image against 4 match images with an average disarity range of 400 ixel.
8 (a) To View of Neuschwanstein (b) Result of Correlation Method (c) Result of SGM (HMI) Figure 5. Neuschwanstein castle (Germany), recorded by an airborne ushbroom camera. 5. Conclusion It has been shown that a hierarchical calculation of a Mutual Information based matching cost can be erformed at almost the same seed as an intensity based matching cost. This oens the way for robust, illumination insensitive stereo matching in a broad range of alications. Furthermore, it has been shown that a global cost function can be aroximated efficiently in O(W HD). The resulting SemiGlobal Matching (SGM) method erforms much better matching than local methods and is almost as accurate as global methods. However, SGM is much faster than global methods. A near realtime erformance on small images has been demonstrated as well as an efficient calculation of huge images. 6. Acknowledgments I would like to thank Klaus Gwinner, Johann Heindl, Frank Lehmann, Martin Oczika, Sebastian Pless, Frank Scholten and Frank Trauthan for insiring discussions and Daniel Scharstein and Richard Szeliski for makeing stereo images with ground truth available. References [1] S. Birchfield and C. Tomasi. Deth discontinuities by ixeltoixel stereo. In Proceedings of the Sixth IEEE International Conference on Comuter Vision, ages , Mumbai, India, January [2] Y. Boykov, O. Veksler, and R. Zabih. Efficient aroximate energy minimization via grah cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(11): , [3] G. Egnal. Mutual information as a stereo corresondence measure. Technical Reort MSCIS0020, Comuter and Information Science, University of Pennsylvania, Philadelhia, USA, [4] P. F. Felzenszwalb and D. P. Huttenlocher. Efficient belief roagation for early vision. In IEEE Conference on Comuter Vision and Pattern Recognition, [5] H. Hirschmüller, P. R. Innocent, and J. M. Garibaldi. Realtime correlationbased stereo vision with reduced border errors. International Journal of Comuter Vision, 47(1/2/3): , ArilJune [6] J. Kim, V. Kolmogorov, and R. Zabih. Visual corresondence using energy minimization and mutual information. In International Conference on Comuter Vision, [7] V. Kolmogorov and R. Zabih. Comuting visual corresondence with occlusions using grah cuts. In International Conference for Comuter Vision, ages , [8] D. Scharstein and R. Szeliski. A taxonomy and evaluation of dense twoframe stereo corresondence algorithms. International Journal of Comuter Vision, 47(1/2/3):7 42, Aril June [9] D. Scharstein and R. Szeliski. Highaccuracy stereo deth mas using structured light. In IEEE Conference for Comuter Vision and Pattern Recognition, volume 1, ages , Madison, Winsconsin, USA, June [10] J. Sun, H. Y. Shum, and N. N. Zheng. Stereo matching using belief roagation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(7): , July [11] G. Van Meerbergen, M. Vergauwen, M. Pollefeys, and L. Van Gool. A hierarchical symmetric stereo algorithm using dynamic rogramming. International Journal of Comuter Vision, 47(1/2/3): , ArilJune [12] P. Viola and W. M. Wells. Alignment by maximization of mutual information. International Journal of Comuter Vision, 24(2): , [13] F. Wewel, F. Scholten, and K. Gwinner. High resolution stereo camera (hrsc)  multisectral 3ddata acquisition and hotogrammetric data rocessing. Canadian Journal of Remote Sensing, 26(5): , 2000.
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