6.1 Introduction

Stereo vision involves the simultaneous use of two vision sensors and leads to the recovery of depth. It is based on the principal, first utilized by nature, that two spatially differentiated views of the same scene provide enough information to enable perception of the depth of the portrayed objects. This fact was first realized by Sir Charles Wheatstone, about two centuries ago, who stated: “…the mind perceives an object of three dimensions by means of the two dissimilar pictures projected by it on the two retinae…” [1].

Calculating the depth of various points, or any other primitive, in a scene relative to the position of a camera, is one of the important tasks of computer and robot vision systems. The most common method for extracting depth information from intensity images is by means of a pair of synchronized camera images acquired by a stereo rig, as the ones shown in Figure 6.1. Matching pixels of one image to pixels of the second image (also known as the stereo correspondence problem) results in the so called disparity map [2]. Disparity is defined as the difference of the coordinates of a matched pixel in the two images, which is proportional to its depth value. For rectified stereo pairs, the vertical disparity is always zero and, as a result, a disparity map consists of the horizontal disparity values for the matched image pixels. Nevertheless, the accurate and efficient estimation of disparity maps is a long-standing issue for the computer vision community [3].

The importance of stereo vision is apparent in the fields of machine vision [4], computer vision [5], virtual reality, robot navigation [6], simultaneous localization and mapping [7, 8], depth measurements [9] and 3D environment reconstruction [4]. The objective of this chapter is to provide a comprehensive survey of real-time stereo imaging algorithms and systems. The main characteristics of stereo vision algorithms are highlighted, with references to more detailed analyses where needed, and the state of the art concerning real-time implementations is given. The categorization followed in the rest of this Chapter follows the one shown in Figure 6.2.

6.2 Background

Even if stereo vision has been the focus of attention for many researchers during the past decades, it is still a very active and popular topic. The richness of the related literature and the number of newly published works every year reveal that there is room for improvement in the current state of the art. A consequence of this is that every attempt to capture the state of the art is destined to be outdated soon. However, what one can still deduce from the study of such surveys is the shift of focus and the new trends that arise over time.

Historically, the first attempts to gather, survey and compare stereo vision algorithms resulted in the review papers of Barnard and Fischler [10], Dhond and Aggarwal [11], and Brown [12]. However, the work that has had the biggest impact on the topic and defined clear directions is probably the seminal taxonomy and review paper of Scharstein and Szeliski [13]. In that work, apart from a detailed snapshot of the contemporary algorithms, a taxonomy framework, and an openly available objective test bench were proposed. The test bench consists a standard dataset of stereo images, metrics for evaluating the accuracy of the results, and also a website that hosts all the aforementioned, as well as an ever-updating list with the results of evaluated algorithms [14].

While the focus of Scharstein and Szeliski was the definition of a framework that would allow the computer vision community to pursue further accuracy of stereo algorithms, the rapid evolution of robotic applications and real-time vision systems in general, has indicated the execution speed as a factor of equal (if not greater) importance as the accuracy of the depth estimations. This trend is reflected and captured in subsequent reviews, such as in [15], in [16], where the branch of real-time hardware implementations is additionally covered, and in [17], where the focus is on real-time algorithms targeting resource-limited systems.

Apart from reviews and comparative presentations of complete stereo algorithms, there are also very useful works that compare the various alternatives for implementing each basic block of a stereo algorithm. As will be discussed in Section 3 of this chapter, the structure of stereo algorithms can be very well defined, and the various constituent parts can be clearly identified. Hirschmuller and Scharstein, in [18, 19] compare different (dis-)similarity measures (also referred to as matching cost functions) for local, semi-global, and global stereo algorithms.

Additionally, Gong and his colleagues, in [20], consider various alternatives for matching cost aggregation in real-time GPU-accelerated systems. Considering global methods for disparity assignment, Szeliski et al., in [21] propose some energy minimization benchmarks and use them to compare the quality of the results and the speed of several common energy minimization algorithms.

The stereo correspondence problem can be efficiently addressed by considering the geometry of the two stereo images and by applying image rectification to them. In the general case, the image planes of the two capturing cameras do not belong to the same plane. While stereo algorithms can deal with such cases, the demanded calculations are considerably simplified if the stereo image pair has been rectified. The process of rectification, as shown in Figure 6.3, involves the replacement of the initial image pair by another projectively equivalent pair [5, 2]. The initial images are reprojected on a common plane that is parallel to the baseline joining the optical centers of the initial images.

Epipolar geometry provides tools in order to solve the stereo correspondence problem, i.e., to recognize the same feature in both images. If no rectification is performed, the matching procedure involves searching within two-dimensional regions of the target image. However, this matching can be done as a one-dimensional search, if accurately rectified stereo pairs are assumed in which horizontal scan lines reside on the same epipolar line, as shown in Figure 6.4. A point in one image plane may have arisen from any of points in the line C₁P₁, and may appear in the alternate image plane at any point on the epipolar line [4]. Thus, the search is theoretically reduced within a scan line, since corresponding pair points reside on the same epipolar line. The difference of the horizontal coordinates of these points is the disparity value. The disparity map consists of all the disparity values of the image.

6.3 Structure of Stereo Correspondence Algorithms

The majority of the reported stereo correspondence algorithms can be described using more or less the same structural set [13]. The basic building blocks are the following:

1. Computation of a matching cost function for every pixel in both the input images.
2. Aggregation of the computed matching cost inside a support region for every pixel and every potential disparity value.
3. Finding the optimum disparity value for every pixel of one picture.
4. Refinement of the resulted disparity map.

Every stereo correspondence algorithm makes use of a matching cost function, so as to establish correspondence between two pixels, as discussed in Section 6.3.1. The results of the matching cost computation comprise the Disparity Space Image (DSI). DSI can be visualized as a 3D matrix containing the computed matching costs for every pixel and for all its potential disparity values [22]. The structure of a DSI is illustrated in Figure 6.5.

Usually, the matching costs are aggregated over support regions. These regions could be 2D or even 3D [23, 24] ones within the DSI cube. The selection of the appropriate disparity value for each pixel is performed afterwards. It can be a simple Winner-Takes-All (WTA) process or a more sophisticated one. In many cases, this is an iterative process, as depicted in Figure 6.6. An additional disparity refinement step is frequently adopted. It is usually intended to filter the calculated disparity values, to give sub-pixel accuracy, or to assign values to not calculated pixels. The general structure of the majority of stereo correspondence algorithms is shown in Figure 6.6. Each block is discussed in more detailed in the rest of this section.

6.3.1 Matching Cost Computation

Detecting conjugate pairs in stereo images is a challenging research problem known as the correspondence problem, i.e., to find for each pixel in the left image the corresponding pixel in the right one [25]. The pixel to be matched without any ambiguity should be distinctly different from its surrounding pixels. To determine that two pixels form a conjugate pair, it is necessary to measure the similarity of these pixels, using some kind of a matching cost function. Among the most common ones are the absolute intensity differences (AD), the squared intensity differences (SD), and the normalized cross correlation (NCC). Evaluation of various matching costs can be found in [13, 18, 26].

AD is the simplest measure of all. It involves simple subtractions and calculations of absolute values. As a result, it is the most commonly used measure found in literature. The mathematical formulation of AD is:

6.1

where and denote the intensity values for the left and right image respectively, is the value of the disparity under examination ranging from 0 to and are the coordinates of the pixel on the image plane.

SD is somewhat more accurate in expressing the dissimilarity of two pixels. However, the higher computational cost of calculating the square of the intensities difference is not usually justified by the accuracy gain. It can be calculated as:

6.2

The normalized cross correlation calculates the dissimilarity of image regions instead of single pixels. It produces very robust results, on the cost of computational load. Its mathematical expression is:

6.3

where is the image region under consideration.

The luminosity-compensated dissimilarity measure (LCDM), as introduced in [27], provides stereo algorithms with tolerance to difficult lighting conditions. The images are initially transformed from the RGB to the HSL colorspace. The HSL colorspace inherently expresses the lightness of a color and demarcates it from its qualitative characteristics [28]. That is, an object will result in the same values of H and S regardless of the environment's illumination conditions. According to this assumption, LCDM disregards the values of the L channel in order to calculate the dissimilarity of two colors. The omission of the vertical (L) axis from the colorspace representation leads to a 2D circular disk, defined only by H and S. Thus, any color can be described as a planar vector with its initial point being the disk's center. As a consequence, each color can be described as a polar vector or, equivalently, as a complex number with modulus equal to and argument equal to . As a result, the difference, or equivalently the luminosity-compensated dissimilarity measure (LCDM), of two colors and can be calculated as the difference of the two complex numbers:

6.4

On the other hand, the rank transform replaces the intensity of a pixel with its rank among all pixels within a certain neighborhood, and then calculates the absolute difference between possible matches [29]. Another matching cost used for pixel correlation is Mutual Information. This can express how similar two parts of an image are by calculating and evaluating the entropies of the joint and the individual probability distributions. The probability distributions are derived from the histograms of the corresponding image parts. Finally, there are also phase-based methods that treat images as signals and perform matching of some phase-related function.

A useful review and evaluation of various matching costs for establishing pixel correlation can be found in [19].

6.3.2 Matching Cost Aggregation

Usually, the matching costs are aggregated over support regions. Those support regions, often referred to as support or aggregating windows, could be square or rectangular, of fixed size or adaptive ones. The aggregation of the aforementioned cost functions leads to the core of most of the stereo vision methods, which can be mathematically expressed as follows, for the case of the sum of absolute differences (SAD):

6.5

And for the case of the sum of squared differences (SSD):

6.6

where are the intensity values in the left and right image, are the pixel's coordinates, is the disparity value under consideration and is the aggregated support region.

More sophisticated aggregation is traditionally considered as time-consuming. Methods based on extended local methodologies sacrifice some of their computational simplicity in order to obtain more accurate results [30]. Methods based on adaptive support weights (ASW) [31, 32] achieved this by using fixed-size support windows whose pixels' contribution in the aggregation stage varies, depending on their degree of correlation to the windows' central pixel. Despite the acceptance that these methods have enjoyed, the determination of a correlation function is still an open topic.

An ASW-based method for correspondence search is presented in [31]. The support-weights of the pixels in a given support window are adjusted based on color similarity and geometric proximity to reduce the image ambiguity. A similar approach is also adopted in [27], where stereo vision in non-ideal lighting conditions is examined, and in [33], where a psychophysically inspired stereo algorithm is presented.

On the other hand, some matching cost functions do not need an additional aggregation step, since this is an integral part of the cost calculation. As an example, the NCC and the rank transform require the a priori definition of a region where they are to be calculated.

An in-depth review and evaluation of different matching cost aggregation methods, suitable for implementation in programmable GPUs and for use in real-time systems can be found in [20].

6.4 Categorization of Characteristics

As already mentioned, there are numerous papers about stereo vision algorithms and systems published every year. This ever-growing list needs to be categorized according to some meaningful criteria to become tractable. Traditionally, the most common characteristics to divide the stereo algorithms literature has been the density of the computed disparity maps and the strategy used for the final assignment of disparity values. The rest of this Section examines the classes of such a categorization and discusses some indicative algorithms.

6.4.2 Optimization Strategy

Stereo matching algorithms can be divided in three general classes, according to the way they assign disparities to pixels. First, there are algorithms that decide the disparity of each pixel according to the information provided by its local, neighboring pixels – thus called local or area-based methods. They are also referred to as window-based methods, because disparity computation at a given point depends only on intensity values within a finite support window. Second, there are algorithms that assign disparity values to each pixel, depending on information derived from the whole image – thus called global algorithms. These are also sometimes referred to as energy-based algorithms, because their goal is to minimize a global energy function, which combines data and smoothness terms, taking into account the whole image. Finally, the family of algorithms called semi-global select disparity values minimizing an energy function along scan-lines [41]. Of course, there are many other methods [42] that are not strictly included in either of these broad classes. The issue of stereo matching has recruited a variation of computation tools. Advanced computational intelligence techniques are not uncommon and constitute interesting and promiscuous directions [43, 44].

6.4.2.1 Local

After the computation and aggregation (if necessary) of matching costs, the actual selection of the best candidate matching pixel (and, thus, disparity value) has to be performed. Most local algorithms just choose the candidate match that exhibits the minimum matching cost. This simple winner-takes-all approach is very efficient in computational terms, but is also often inaccurate, resulting in faulty and incoherent disparity values.

Ogale and Aloimonos, in [45–47], propose a compositional approach to unify many early visual modules, such as segmentation, shape and depth estimation, occlusion detection, and local signal processing. The dissimilarity measure used is the phase differences from various frequency channels. As a result, this method can process images with contrast, among other mismatches.

6.4.2.2 Global

Contrary to local methods, global ones produce very accurate results at the expense of computation time. They formalize the disparity assignment step as a labeling problem, and their goal is to find the optimum disparity function that minimizes a global cost function which combines data and smoothness terms.

6.7

where takes into consideration the pixel's value throughout the image, provides the algorithm's smoothening assumptions and is a weight factor.

The minimization of the energy function is performed using a suitable iterative algorithm. Commonly used such algorithms include, among others, graph cuts [48, 49] and loopy belief propagation [50]. However, the main disadvantage of the global methods is that they are more time-consuming and computationally demanding.

An informative review and comparison of various energy minimization methods for global stereo matching can be found in [21].

6.4.2.3 Semi-global

The most popular semi-global stereo algorithms are based on Dynamic Programming (DP). DP has been used widely as an optimization method for the estimation of disparity along image scanlines [41, 51], and this is the reason for being called semi-global. The general idea behind a DP stereo algorithm is to treat the correspondence problem again as an energy minimization problem, but limited on scanlines. Thus, an energy function is formulated, again as in Equation 6.7, where the smoothness term can be formulated to handle depth discontinuities and occlusions along each scanline.

DP stands somewhere between local and global algorithms, providing good accuracy of results in acceptable frame rates. Moreover, recent advances in DP-based stereo algorithms seem to be able to significantly improve both of these two aspects. Hardware implementations of DP have been reported [52, 53] that provide high execution speeds. Additionally, the incorporation of adaptive support weight aggregation schemes has been shown to further improve the accuracy and detail of the produced depth maps [54].

6.5 Categorization of Implementation Platform

Many stereo vision algorithms can achieve real-time or near-real-time operation. Such execution speeds can be obtained either by optimizing local stereo algorithms, or by employing custom hardware for speeding up the computations. In this section, implementations based on central processing unit (CPU), graphics processing unit (GPU), as well as field-programmable gate array (FPGA), or application-specific integrated circuits (ASIC), are covered that can be used in real-time systems.

6.6 Conclusion

Stereo vision remains an attractive solution to the 3D imaging issue. This chapter has covered the basics of the theory behind stereo vision algorithms and has tried to categorize them according to their main characteristics and implementation platforms. A snapshot of the state of the art has been given.

One of the conclusions that can be deduced from this work is that the center of attention regarding stereo vision seems to have shifted; it is not only accuracy that is pursued, as it used to be, but also real-time performance that is now demanded. The solution to this issue is coming, either by simple algorithms implemented on powerful contemporary CPU systems, or by exploiting the power of programmable GPU co-processors, or by developing hardware-friendly implementations for FPGAs. The two latter options seem to be gaining popularity, as they can combine real-time execution speeds with very accurate depth estimations.

Another interesting observation is that local algorithms are not any more the only ones able to be implemented and executed in real time. Semi-global, or even purely global stereo algorithms, are reported to achieve acceptable frame rates. This trend seems to be growing, due to the fact that more powerful platforms are becoming constantly available, and more efficient and optimized versions of those algorithms are regularly being proposed.

The maturity of stereo vision techniques and their capability to operate in both indoor and outdoor environments give stereo vision a solid foundation in the harsh competition against other 3D sensing technologies. As a result, real-time implementations of stereo vision algorithms find their way in modern human-machine interaction systems, home entertainment systems, and autonomous robots, just to name a few.

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