Multi-resolution stereo matching using genetic algorithm

Multi-resolution Stereo Matching Using Genetic Algorithm Minglun Gong Yee-Hong Yang Department of Computer Science University of Alberta Abstract In this paper, a new genetic-based stereo matching algorithm is presented. Our motivation is to improve the accuracy of the disparity map generated by removing the mismatches caused by both occlusions and false targets. In our approach, the stereo matching problem is considered as an optimization problem. The algorithm first takes advantage of multi-view stereo images to detect occlusions, therefore, removes mismatches caused by visibility problems. A genetic algorithm is then used to optimize both the compatibility between corresponding points and the continuity of the disparity map, which removes mismatches caused false targets. In addition, the quadtree structure is used to implement a multiresolution framework. Since nodes at different level of the quadtree cover different number of pixels, selecting nodes at different levels gives similar effect as adjusting the window size at different locations of the image. The experimental results show that our approach can generate more accurate disparity maps than two existing approaches. Keywords: Multi-resolution image, stereo vision, genetic algorithm, Markov random fields. 1 Introduction A stereo vision system normally takes two or more images taken by parallel cameras as inputs and produces a disparity map. A desired disparity map should be smooth and should contain sufficient detail for subsequent processing. Previous work in stereo vision can be coarsely classified into feature-based methods [4; 5; 7] and intensity-based methods [1; 9; 10; 16]. Feature-based methods first detect features, e.g. edges and corners, in the source images. The matching process is then conducted on these features. Generally, feature-based approaches can provide robust, but sparse, depth information. A complex interpolation process is needed to obtain a complete disparity map. Intensity-based methods select a window centered at the each pixel. Pixels within the window are used to compute correlation or sum of squares differences between input images. The disparity that produces the best match is set as the disparity of the pixel. The selection of window size is a critical problem for intensity-based methods. The window size should be large enough to include enough intensity variation for reliable matching, but small enough to avoid disparity variation inside the window. Levine, et al. [10] change the window size locally according to the intensity pattern. Kanade and Okutomi [9] propose an iterative algorithm to select optimal window sizes for different parts of the image. The advantage of intensity-based methods is that dense disparity maps can be estimated. Most of the intensity-based methods compute disparities based on local information only, and therefore, are sensitive to noise. To address this problem, cooperative algorithms apply global constraints in the matching process. The disparity of a pixel is influenced by the disparities of its neighbors. Cooperative algorithms can be formulated as the process of extracting a surface from a threedimensional u-v-d volume [1], i.e., the so-called disparity space. The value of any voxel in the disparity space indicates the probability of that the disparity of pixel (u,v) is d. Consequently, the desired disparity map should be a surface that satisfies two constraints (1) the surface is smooth, (2) it which passes through voxels of high probabilities. In previous approaches [1; 16], three-dimensional functions are used to update the disparity space iteratively until the process converges. Recovering disparity information from two views has its inherent limitations. First of all, the length of the baseline has to be chosen carefully. Using a short baseline cannot estimate distance accurately due to the triangulation procedure. On the other hand, to increase the baseline means that a larger disparity range must be searched to find the match, and therefore, has a higher probability of finding a false match. Hence, a tradeoff has to be made between the precision in estimating distance and the correctness in finding a match. The multiple-baseline approach [11] is an attempt to address this tradeoff problem. In this approach, multiple Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE cameras are placed along a line with their optical axes perpendicular to the line. Matches between different stereo pairs are considered together to improve the accuracy of the disparity map. Another commonly known issue of analyzing binocular disparity is that some part of the scene may be visible in only one of the images. As a result, the disparity information is not recoverable since it is not possible to find a correct match. The Stereo by Eye Array (SEA) approach [13] is proposed to solve this problem. In this approach, nine cameras are placed in a 3×3 matrix on a plane with the same baseline length between neighboring cameras along both the vertical and horizontal directions. The matching algorithm uses the geometric relation among captured images to detect occlusions. 2 Genetic-based Stereo Matching The genetic algorithm (GA) proposed by John Holland [8] is a general-purpose global optimization technique based on randomized search and incorporates some aspects of iterative algorithms. GA is often regarded as an alternative method for solving complex optimization problems, especially for combinatorial optimization problems or problems whose derivatives cannot be computed numerically. The research on GA and its applications are a topic of active research [15]. Some researchers have applied GA to stereo vision [6; 12]. In the approach proposed by Saito and Mori [12], different window sizes are used to calculate several candidate disparity values at each pixel location. The GA is then applied to select the disparity of each pixel from these candidates. To simplify the calculation, the disparity map is partitioned into 8×8 blocks and the GA is applied to analyze each block. This approach is limited because it assumes that the correct disparity value is one of the candidates calculated with different window sizes. This assumption may not hold when the images are noisy or when part of the scene has a plain color and lacks in features for matching. Han et al. [6] divide the reference image into regions using the modified nonlinear Laplace (MNL) filter. Chromosomes are then determined adaptively based on the extracted regions. Eventually, disparities are calculated by selecting a fitness value among several candidate genes using a genetic algorithm. The quality of the disparity map generated by this approach highly depends on the region extraction result of the MNL filter. The approach presented in this paper is an intensitybased approach. Our motivation is to improve the accuracy of the disparity map generated by removing the mismatches caused by both occlusions and false targets. Therefore, multiple-view stereo images are used to detect occlusions, and global constraints are applied in the matching process to eliminate false targets. A problem associated with using global constraints is how to process around the boundaries. We notice this problem when we experimented with the iterative algorithm implemented by Zitnick and Kanade, which involves summation of a three-dimensional local support area [16]. The boundary problem not only manifests itself at the boundary of the disparity maps, which shows up as a black border, it also introduces errors when the true disparity is close to the given minimum or maximum disparity because the summation is also conducted in the disparity dimension. A possible solution to the second problem is to enlarge the disparity search range. However, it will increase the memory and the time needed for calculation and also raise the possibility of false matches. In this paper, to avoid the above -mentioned boundary problem, a novel approach is used to apply the global constraints in the matching process. Here, we borrow the idea from neighborhood-based segmentation and use the Markov Random Fields (MRFs) to model the interactions between neighboring pixels. Within the MRFs framework, the stereo matching process is equivalent to finding the optimum state of the MRF. Because of the Gibbs’ equivalence, the probability that the MRF is in a particular state can be calculated using local energies over the entire image. Consequently, a given stereo matching result can be evaluated by modeling local interactions, and the problem of finding the best stereo matching can be viewed as finding the solution to a combinatorial optimization problem. The outline of the approach is as follows: First the three-dimensional disparity space is populated with dissimilarity values based on the source images. In the case of multi-view stereo images, the space will be filled with the output of an occlusion detection function proposed in SEA. A fitness function is defined based on the MRF to evaluate a given disparity map. A genetic algorithm is used to extract the best surface from the disparity space with respect to the fitness measure. In addition, the quadtree structure is used to represent possible disparity maps. Since nodes at different level of the quadtree contain different number of pixels, selecting nodes at different resolution levels gives similar effect to that of adjusting the window size at different locations of the image. In the following subsections, detailed issues are discussed first, which include initialization of the disparity space, the encoding mechanism for all possible disparity maps, the formulation of the fitness function, and the appropriate crossover and mutation Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE operators. An outline of the optimization process is given latter. 2.1 Initialization of the Disparity Space The disparity space is initialized based on the input stereo images. If only two stereo images, I and Iref, are available, this space will be filled directly by the dissimilarity measure between corresponding pixels of I and Iref, which is calculated by: S (r , c , d ) = e ref (r , c , d ) = ∑ ρ (I (r + i, c + j), I (r + i, c + j + d )) ref (1) −w≤i , j ≤ w where w is the window radius, I(x,y) the color of pixel (x,y), and ρ the color dissimilarity function. The color dissimilarity function could be any function that produces low values for correct matches, such as the squared difference or the absolute difference. In this paper, the absolute difference function is used and the window radius is set to 1. If multi-view stereo images captured by a camera matrix are available, occlusion detection functions proposed in SEA can be applied. Basically, eight stereo pairs are formed using the central camera, I(0,0), and each of the eight peripheral cameras, I(m,n). For each pixel in the center image, the dissimilarity values of all eight stereo pairs are computed. The disparity space is then filled using the following function: S (r , c , d ) =  e (−1, −1 ) (r , c, d ), e (−1, 0 ) (r , c , d ), e (−1 ,1) (r , c , d ),    (2)  σ  e (0 , −1 ) (r, c, d ), e (0 ,1) (r , c, d ),  (1 ,−1 )  (1, 0 ) (1 ,1)  e (r , c , d ), e (r , c, d ), e (r , c , d )  where e(m,n)(r,c,d) is the dissimilarity between corresponding pixels of I(0,0) and I(m,n), and σ the occlusion detection function. Different occlusion detection functions can be applied. Satoh and Ohta give a systematic comparison of different functions [14]. The “sorting summation” function is used in this paper since it can detect most of mismatches caused by occlusions. This algorithm normally does not perform as good as some other functions in removing mismatches caused by false targets. However, our algorithm will rely on the genetic-based optimization process to get rid of these mismatches. 2.2 Encoding Scheme for Disparity Maps Fundamental to all GA’s is the encoding scheme for representing the solutions of the corresponding optimization problems. Normally, the method to encode the solutions depends on the applications to which the GA is applied. In our approach, the quadtree structure is used to provide the multi-resolution scheme. Therefore, the possible solutions, i.e. the disparity maps, are encoded by encoding the corresponding quadtrees. The same encoding scheme has been recently applied to color image segmentation [3]. The quadtrees used to represent the disparity maps must satisfy the following two constraints: • Every leaf (i.e. a node with no child) k of the quadtree has an associated disparity value x, which implies that the disparity values of all the pixels covered by k are x. A leaf k is said to cover pixel p if k contains p. • Any interior node in the quadtree cannot have all its descendents assigned to the same disparity value; otherwise, all the descendents of the node should be removed and the node itself should be selected as a leaf. Now we need to find a way to encode all the possible quadtrees. In this paper, an array representation of a complete quadtree is used so that we can encode different quadtrees using a one-dimensional integer string. In such a representation, the index of node k in the string can be computed by:  h −1  O (k ) =  4 i  + y × 2 h + x  i =0  ∑ (3) where h, x, and y are the height and the x-, ycoordinates of node k, respectively. The content at location O(k) in the integer string is determined by:  x if node k is a leaf and its disparity is x ; D[O(k )] =  otherwise − 1 2.3 Fitness Evaluation The fitness of a given chromosome, which is represented as a string, controls the evolution process. The fitter the chromosome, the greater is its probability to survive from one generation to the next. In the context of stereo matching, we need a way to evaluate different disparity maps. Here we use we use an energy function, which is based on the MRF, as the fitness evaluation tool. The smaller the value of the energy function, the better the disparity map. The format of the function is: f (D ) =   ∑  ( ∑) S (r, c, D[k ]) + λT  k k ∈P (4) r ,c ∈k where D denotes the disparity map, P the set that contains all the leaves in the quadtree, D[k] the disparity value of leaf k. λ is the weight of the penalty term with a small value of λ favors a detailed disparity map and a large value encourages a coarser result. Tk is the length of the boundary of leaf k, which is defined as: Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE Tk = ∑ (1 − δ (D[k ] == D[g]))× t k ,g (5) g∈P ∩Qk v where δ(true)=1, δ(false)=0, Qk the set that contains all the nodes that share boundary with node k, tk,g the length of the shared boundary between leaf k and leaf g. u (a) 2.4 Initial Population Generation The GA needs a number of initial disparity maps as the initial population to start with. The choice of the population size is very important. If the selected population size is too small, the algorithm may result in premature convergence without finding an appropriate solution. On the other hand, a large population size will lead to long computation time. Our experiments show that premature convergence is likely to occur when the population size is smaller than 30. In this paper, the population size is set to 40, which appears to be sufficient to avoid this problem. The initial disparity maps are generated purely randomly. First, a recursive procedure is invoked using the root of the quadtree as the input parameter. Inside the procedure, whether or not the given parameter, the node k, is selected as a leaf depends on a random number. If the decision is to select then the best disparity, which can minimize the sum of the dissimilarities between all the pixels in the node and their corresponding pixels, is assigned to node k. Otherwise, the procedure invokes itself recursively using the four child nodes of k until the bottom level is reached. 2.5 Crossover Operator In our scenario, it will be inefficient to apply the commonly used crossover operators, such as the two point crossover, because they do not guarantee that the crossover results of the two quadtrees will still be legal quadtrees. To address this problem, a recently proposed crossover operation for color image segmentation, called graft crossover, is applied [3]. For completeness, details of this operator are discussed in the following: Given two strings, which represent two quadtrees, we want to compare them and find out all the leaves that appear in only one of the two quadtrees. The crossover process will be terminated if no such leaves are found. Otherwise, we randomly select one of these leaves as a seed node. For example, after comparing the two quadtrees shown in Figure 1(a) and (b), we can pick leaf u since it appears only in the left quadtree. Then the cover node that is the predecessor of the seed node and appears in both quadtrees is determined (it is node v in our example). Finally, we swap all the nodes that are the descendents of the cover node. Figure 1(c) and (d) show the results after we do the graft crossover at cover node v. v (b) v v u (c) (d) Figure 1: Graft crossover for quadtrees, (a) and (b) the parents; (c) and (d) the offspring. By construction, this algorithm guarantees that the results of crossover will still be legal quadtrees. After crossover, the energies of the two quadtrees may change and one of the offspring’s strings may have a lower energy than either of its parents. 2.6 Mutation Operator The mutation operation is important for the GA since the crossover operator cannot generate offspring that have genes that do not appear in the initial population. In our approach, three mutation operations, which are the splitting, the merging, and the alteration are adopted in this paper [3]. The mutation operator randomly selects a number of pixels from the original image. In our experiments, the number of pixels selected is equal to half of the perimeter of the image. For each pixel, we search for leaf k in the quadtree that covers this pixel. One of the splitting, merging, and altering operations will be applied with equal chance. When trying to merge leaf k with its siblings, we need to find out all of its siblings first. The merging operation will be inhibited if any of its siblings has children. Otherwise, the local energy is computed, and the merging operation will happen only if the energy is lower after merging than that before merging. When trying to split leaf k, the process is similar. The local energies are computed for disparity maps before and after splitting. The splitting operation will occur only when the energy decreases after splitting. When trying to alter leaf k, all the neighbors of k are found first. The alteration operation will change the label of leaf k to the label of one of the labels of its neighbors, which can reduce the energy. 2.7 Minimization Process After the above issues are addressed, the GA can be implemented as an iterative procedure. When the population evolves from one generation to the next, two Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE stereo, which contains two grayscale images and is provided by R. Szeliski of Microsoft Research. The last dataset, “random dot stereogram”, is a synthetic stereo image pair with 50 percent density, which was used in [16]. The SEA approach [13] and the cooperative algorithm [16] are used for comparison. All the cooperative algorithm results shown below are generated using the “ZK Stereo” program written by Zitnick [17]. The default values are used for most of the parameters. In particular, the window radius for initial match value is set to 1, the local support radius for row and column dimensions is set to 2, and the local support radius for disparity dimension is set to 1. The values for the minimum disparity and maximum disparity depend on the particular stereo dataset. In order to make sure that the algorithm converges, the number of iterations is changed to 15, which is the upper bound according to the document of the program [17]. The program provides two different ways to compute the initial match values. The “sums of absolute differences” is used because it gives better results and also because we use the same way to fill the disparity space when only two stereo images are available. strings are picked randomly each time to do the crossover and mutation until all the strings in the population are processed. The elitist strategy [2] is applied when selecting strings for the next generation. The energy value f(Dim ) of the best string Dim of generation m is compared with the energy value f(Djm+1 ) of the worst string Djm+1 of generation m+1. If f(Dim ) < f(Djm+1 ), then string Dim is substituted for Djm+1 . By means of this strategy, the minimum energy value of the population will not increase during the process of evolution. The above process is repeated until the termination condition is satisfied. In our approach, the evaluation process will be terminated if the energy difference between the best string and the worst string in the population is smaller than 0.01 percent of the average energy value. 3 Experimental Results Four datasets are used to test the presented algorithm. Each of the first two datasets, “head and lamp” and “plant”, contains nine color images, which are sampled using a camera matrix by University of Tsukuba. The third one, “slanted plane”, is a binocular (a) (b) (c) (d) (e) (f) Figure 2: “head and lamp” multi-view stereo (a) source image (b) ground truth (c) our approach (d) SEA (3×3 window) (e) SEA (5×5 window) (f) cooperative algorithm. Figure 2 shows the experimental result for the “head and lamp” dataset. The true disparity range for this stereo dataset is [5, 14]. For comparison, the results of using SEA with different window sizes are shown as Figure 2 (d) and (e). Figure 2 (f) is generated by the cooperative algorithm, which is quite good considering it uses two original images only. However, in order to generate this result, we have to enlarge the disparity range to [2, 17]. Poor results will be produced if a tighter bound is used. Since the disparity range is expanded by 60 percents, it is reasonable to believe that the computational time and the space required are also Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE increased by 60 percents. (a) (b) (c) (d) Figure 3: “plant” multi-view stereo (a) source image (b) our approach (c) SEA (3×3 window) (d) cooperative algorithm (disparity range [12, 51]) Figure 3 shows the experimental results for the “plant” dataset. The true disparity range for this stereo dataset is [15, 48]. Many occlusions exist in this stereo dataset since the scene is very complex and the length of the baseline is quite large. Figure 3(c) shows that the SEA approach can detect most of the occlusions. However, many mismatches caused by f alse targets still exist. For instance, incorrect depth variations show up in the blue rectangle region. As shown in Figure 3(b), our approach successfully removed these mismatches. The result generated by the cooperative algorithm shown in Figure 3(d) looks smooth. However, upon closer inspection of the picture shows that many details are lost. For example, the leaves are broken, “tears” shows up in the red rectangle area, and part of the background is covered in the green oral area. These problems might be caused by occlusions, which is not recoverable using only two images. Figure 4 shows the experimental results for the “slanted plane” dataset. The true disparity range for this stereo dataset is [4, 28]. Pixels of red color in Figure 4(b) are occluded in the other source image. Although mismatches caused by occlusions exist in the result of our approach, comparing with the result using simple window-based matching (Figure 4(d)), it shows that most of the mismatches caused by false targets have been eliminated. Figure 4(e) shows the boundary problem of the cooperative algorithm when a tight disparity range is given. Figure 4(f) shows the result of the cooperative algorithm using an enlarged disparity range. Since interpolation is applied, the result looks even smoother than the ground truth. Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE (a) (b) (c) (d) (e) (f) Figure 4: “slanted plane” binocular stereo (a) source image (b) ground truth (c) our approach (d) 3×3 window match (e) cooperative algorithm (disparity [4, 28]) (f) cooperative algorithm (disparity [1, 31]). (a) (b) (c) (d) (e) (f) Figure 5: “random dot stereogram” binocular stereo (a) reference (left) image (b) right image (c) ground truth (d) cooperative algorithm (disparity [0, 20]) (f) cooperative algorithm (disparity range [-3, 23]) Figure 5 shows the experiment results for the synthetic scene. The true disparity range for this stereo dataset is [0, 20]. Again, red color is used in Figure 5(c) to mark areas that are not visible in the reference Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE image. The result of our approach shows that, in area without occlusion, most of the disparities given are accurate. Figure 5(e) shows that for the cooperative algorithm, the boundary problem still exists. Figure 5(f) shows that even when the disparity range is increased to [-3, 23], the brightest vertical bar still does not show up. This might be caused by the boundary problem since this area has the highest disparity. Table 1: Comparison of accuracy for “head and lamp” dataset. % of correct disparities % of the disparities within truth value ±1 SEA (3×3 window) 78.048 SEA (5×5 window) 82.314 Our Approach 87.855 92.544 94.679 97.452 Table 2: Comparison for accuracy for “slanted plant” dataset (in area without occlusion only) % of correct disparities % of the disparities within truth value ±1 Cooperative Algorithm 69.132 Our Approach 82.774 97.235 98.547 Table 1 gives the percentage of accurate disparities found by the SEA algorithm and our approach for the “head and lamp” dataset. Table 2 gives the percentage of accurate disparities found by the cooperative algorithm and our approach for the “slanted plane” dataset. The result shows that our approach gives more accurate results for both stereo datasets. The computational costs of genetic-based algorithms are relatively high. Our experiments show that the new algorithm takes about 500-700 generations to converge. However, since the dissimilarity values are pre-calculated and stored in the disparity space, the computation time is reduced. On our 1.5GHz Pentium 4 computer with 256MB RAM running Windows 2000, about 4 minutes are needed to calculate the disparity map for the “head and lamp” example, which is a 384×288 multi-view stereo. 4 Conclusions In this paper, we proposed a novel genetic-based stereo matching algorithm. The algorithm can be applied to both binocular stereo data and multi-view stereo data that are captured by a camera matrix. When multi-view stereo images are available, they can be fully utilized to detect occlusions and to improve accuracy. A fitness function, which considers both intensity similarity and disparity smoothness, is introduced to evaluate a given disparity map. The genetic algorithm is used to find the best disparity map with respect to the fitness function. The quadtree structure is used to implement the multi-resolution framework, which gives similar effect as adjusting the window size at different locations of the image. To apply the genetic algorithm under the quadtree structure, an encoding mechanism for the quadtree structure is used. Graft crossover, splitting mutation, merging mutation, and alteration mutation, which are suitable for the quadtree structure, are applied. Comparing with the SEA algorithm, our approach is more flexible since it can also work with binocular stereo datasets and produces acceptable results. Our approach adopts the idea of detecting occlusions using the geometric relation among captured images. However, different from the SEA algorithm, rather than using the output of the occlusion detection function to determine disparities directly, our approach uses the output to fill the disparity space, which makes it possible to further remove mismatches. Comparing with the iterative -based cooperative algorithm, our approach uses the MRF framework to apply the global constraints in the matching process. The boundary problem is solved since no summation is needed in the process. In addition, our approach can fully utilize multi-view stereo images to remove mismatches caused by collusions. In summary, the proposed algorithm naturally integrates together the idea of occlusion detection using a camera matrix, matching with adaptive window size, and incorporating support from neighboring pixels. The experimental results show that the algorithm can generate better disparity maps than two existing approaches. Future research in this direction is warranted. Acknowledgments The authors would like to acknowledge financial support from NSERC and the University of Alberta. The authors would like to thank Dr. Y. Ohta of the University of Tsukuba for supplying the “head and lamp” and “plant” stereo images, Dr. R. Szeliski of Microsoft Research for the “slanted plane” stereo images, and Mr. L. Zitnick of Carnegie Mellon University for the “random dot stereogram” stereo images. We also thank Dr. L. Zitnick for sharing his software on the web. References [1] Q. Chen & G. Medioni, "Volumetric Stereo Matching Proceedings of the IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV’01) 0-7695-1327-1/01 $17.00 © 2001 IEEE Method: Application to Image-Based Modeling," In: Proceedings of the Conference on Computer Vision and Pattern Recognition Vol. 1, IEEE Computer Society, pp. 29-34, Fort Collins, CO, USA, June 23-25, 1999. [2] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, MA, USA, 1989. [3] M. Gong & Y.-H. Yang, "Genetic -Based Multiresolution Color Image Segmentation," In: Vision Interface Proceedings, Canadian Image Processing and Pattern Recognition Society, pp. 71-80, Ottawa, Ontario, Canada, June 7-9, 2001. [4] W. E. L. 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