2.2 Global Algorithms
2.2.3 Optimization
The next step of global-based disparity map estimation methods is the minimization of the energy function which can be solved by an optimization framework. As the name implies, global algorithms use the entire image in the minimization processi.e.,disparity value assigned to a single pixel influences the disparity values of all other pixels of the image. The main disadvantage of the global methods is that they are computationally complex. This is due to the fact that the computational complexity increases with the disparity range and the image dimension.
2.2 Global Algorithms
2.2.3.1 Dynamic programming
Dynamic programming is one of the optimization techniques which minimizes the energy function along the horizontal scanline. This method breaks down a complex problem into a set of subproblems.
After the solution for each subproblem is computed, they are finally combined to obtain the solution of a complex problem. In [45], a series of prior models are formed for stereo matching. The resulting objective function is optimized using dynamic programming. The generated disparity map suffers from horizontal streaks as the matching is performed along the horizontal scanline only. To overcome this, a heuristic method is proposed which incorporates vertical smoothness in the optimization technique [45]. Furthermore, this step introduces additional streaks in the vertical direction. To overcome the problem of horizontal scanline optimization, vertical consistency is introduced by applying dynamic programming on a pixel-tree structure [46]. Pixels form the nodes and the most important edges in a four-connected neighborhood are retained in the tree structure. This tree structure gives a good approximation of the two-dimensional grid. This algorithm is fast, and hence suitable for real- time implementation. The reduction in the computational complexity is achieved by choosing an appropriate tree structure. The tree structure only contains some selected edges. But this reduction in edges affects the quality of the computed disparity map.
Gong and Yang proposed an orthogonal reliability dynamic programming algorithm for dispar- ity map computation [47, 48]. Inter-scanline consistency is accomplished by applying the reliability dynamic programming process along both the horizontal and vertical scanlines.
In order to reduce horizontal streaks, Chang et al. proposed a one-dimensional optimization technique which uses output from Winner-Take-All as a prior information for dynamic optimization [49].
2.2.3.2 Graph cut
Boykov et al. proposed two different algorithms that allows large moves in order to obtain the local minimization of the multidimensional energy function using iterative graph cuts [41]. The allowed moves are named as α−β swap and α−expansion. In the α−β swap algorithm, a large number of pixels with a labelα are changed to label β, while another set of pixels with label β are changed to label α. In the α−expansion algorithm, a group of pixels having different labels are assigned a label ofα. These moves are performed until moves are found to get a minimum energy value. Among these two algorithms,α−expansion performs better thanα−β swap but requires linear time for execution.
A new approach termed as fusion move is proposed, which is used in binary graph cut for opti-
mization of continuous disparity values [50, 51]. Graph cut is applied to the image in the bit-level representation. It is applied in a sequential order from the most significant bit to the least significant bit, which leads to different numbers of solutions. These solutions are merged in an optimal way to obtain the final disparity map. That is why this algorithm is termed as fused move algorithm.
As opposed to linear complexity, this approach has logarithmic complexity. When the graph con- tains non-submodular edges then obtaining the optimal fusion move turns out to be an NP-complete problem.
Zureiki et al. proposed a reduced graph cut algorithm for disparity map computation [52]. A simplified graph is constructed by only considering some potential values selected from the disparity range for each of the pixels. This reduction increases the coarseness of the computed disparity map [53].
A hierarchical bilateral disparity structure algorithm is presented in [53]. This hierarchical ap- proach divides the disparity range within the stereo images into lower level bilateral disparity struc- tures. Disparity values are classified as a foreground disparity set and a background disparity set within this bilateral structure. The disparity map specific to this bilateral structure is computed by minimizing the energy function using a graph cut method.
2.2.3.3 Belief propagation
Belief propagation is an iterative optimization process which works on the principle of sending messages to its four-connected neighbors in a graph [54]. Each message is a vector whose dimension is equal to the maximum disparity range. A message from pixel p to pixel q encodes the belief of p about q i.e., the probability of q at a particular disparity value. New messages are computed and updated at each iteration. This finally leads to a belief vector that minimizes the energy function.
This method is very slow for practical implementation.
To reduce memory requirements, Yuet al. applied a compression technique to efficiently represent the message [55]. This reduces the memory requirement by 12%, but the compression and decompres- sion techniques makes the algorithm slow. A novel Envelope Point Transform method using principal component analysis (PCA) is proposed for compressing the messages. The extra belief propagation pass required to achieve this task further makes it slower than the conventional belief propagation technique.
Wanget al. classified pixels as reliable and unreliable using bi-labelling process [56]. Consequently, this reduces the search range of reliable pixels. The disparity values of reliable pixels are then propa- gated to the unreliable pixels to obtain an accurate disparity map. This method is slow as compared to the conventional belief propagation as the classification of the reliable pixels and also the joint