On the optical flow model selection through metaheuristics
 Danillo R Pereira^{1}Email author,
 José Delpiano^{2} and
 João P Papa^{1}
https://doi.org/10.1186/s1364001500665
© Pereira et al.; licensee Springer. 2015
Received: 17 December 2014
Accepted: 14 April 2015
Published: 9 May 2015
Abstract
Optical flow methods are accurate algorithms for estimating the displacement and velocity fields of objects in a wide variety of applications, being their performance dependent on the configuration of a set of parameters. Since there is a lack of research that aims to automatically tune such parameters, in this work, we have proposed an optimizationbased framework for such task based on socialspider optimization, harmony search, particle swarm optimization, and NelderMead algorithm. The proposed framework employed the wellknown large displacement optical flow (LDOF) approach as a basis algorithm over the Middlebury and Sintel public datasets, with promising results considering the baseline proposed by the authors of LDOF.
Keywords
Optimization methods Evolutionary algorithms Optical flow methods1 Introduction
Optical flow estimation is one of the most important research areas in computer vision, and it aims at identifying the patterns of motion of objects and surfaces in a visual scene, i.e., to approximate the motion field from a timevarying image intensity. The literature is wide, being some very recent works related to optical flow estimation using Laplacian mesh structures [1], total generalized variation [2], probabilistic motion detection [3], and as an optimization problem in a highdimensional motion field [4], just to name a few. The importance of optical flow estimation can be evidenced in image segmentation [5], rigid object reconstruction [6], cell tracking [7], video stabilization [8], among others. Some parallelbased implementations can be found in [911] as well.
Recently, Sun et al. [12] stressed that the theoretical foundations of a broad range of optical flow methods have changed little since the seminal work of Horn and Schunck [13]. Basically, they argued that, although the results have improved over the past years, the vast majority of optical flow methods rely on the same basis of the work proposed by Horn and Schunck. Another shortcoming related to the optical flowbased techniques relies on the estimation of their parameters, which poses a big challenge to the field. Since most of techniques are parameterdependent, a gridsearch for a set of nearoptimal/optimal parameters in video content may not be a viable task [14]. Therefore, many works often set the parameters by hand, which may limit our understanding about how well the considered optical flow method can generalize unseen data. As a matter of fact, the problem of estimating the parameters of optical flow techniques may be seen as a largescale learning problem. Albeit, we usually need to estimate a few parameters only, the huge amount of data to be processed for such estimation in video datasets demands a high computational effort.
Although the reader can face several works that cope with the problem of estimating/calibrating camera parameters, only a few of them deal with the problem of parameter estimation in optical flow techniques. Heas et al. [15] and Krajsek and Mester [16], for instance, employed a Bayesian optimization framework for such purpose, and Li and Huttenlocher [17] presented an interesting stochastic optimization approach based on Markov random fields for optical flow parameter estimation. The authors state several arguments concerning the advantages of optimizing an error criterion instead of using a maximum likelihood approach for that, as employed by the work of Roth and Black [18]. The reader can refer to a few other works that model the task of optical flow parameter estimation as an optimization task by means of metaheuristic techniques. Delpiano et al. [19], for instance, proposed a multiobjective approach for parameter estimation aiming at optimizing both the training loss and the computational load. Later on, Pereira et al. [20] applied metaheuristic optimization algorithms for the same task considering the large displacement optical flow (LDOF) technique [21], being the results of socialspider optimization (SSO) [22], harmony search (HS) [23], and particle swarm optimization (PSO) [24] compared against each other in the wellknown Middlebury dataset. Although one can find several other optical flowbased implementations out there [2527], we opted to use LDOF due to its simplicity, reliability, and good rank in Sintel website. LDOF implementation is very accurate, but computationally expensive, thus being an interesting choice for applications that require high accuracy, but does not require very short execution times.
In order to fill the lack of research regarding model selection in optical flow environments, we extended the work of Pereira et al. [20] by adding two more optimization techniques, being one of them based on exact computations called NelderMead (NM) [28] and the other one a ‘baseline’ using the parameters proposed by the authors of the LDOF technique [21], as well as we added one more dataset to the experimental section. Additionally, the work of Pereira et al. [20] proposed a local approach to estimate the parameters of LDOF: in short, the idea of their work is to optimize each sequence separately and then to employ the best set of parameters to optimize the remaining sequences. We propose here to optimize the techniques globally, which means we consider all sequences from a given dataset for parameter optimization, being the results more accurate than the ones reported by Pereira et al. [20]. The remainder of this paper is organized as follows: Section 2 presents a brief theoretical background about optical flow, and Section 3 revisits the techniques employed in this paper for comparison purposes. The proposed methodology and experimental results are discussed in Sections 4 and 5, respectively. Section 6 states the conclusions and future works.
2 Optical flow
Optical flow (OF) is a vector field representing ‘the distribution of apparent velocities of movement of brightness patterns in an image’ [13]. The idea contains two basic assumptions: the ‘grey value constancy’ and the ‘smooth flow of the intensity values’ between two successive images. Some articles still maintain the grey value constancy (as an example, see [29]), while other works report the necessity to loosen this assumption [30].
The early work presented in [13] stated the need for an extra constraint to compute the optical flow field from an image sequence and proposed one ad hoc constraint based on the assumption of flow smoothness. Another research work [31] proposed to consider the OF equation for several neighboring pixels in order to avoid the need for an extra constraint. More than 10 years later, Barron et al. [32] provided a comparison of several OF methods, mainly with respect to their average angular error (AAE) when applied to some image sequences. The experiments showed that the method in [31] was one of the most reliable methods at the moment.
Recently, several image datasets have been compiled for a more precise evaluation and comparison of OF methods [33,34]. Many shortcomings of the original methods have been overcome, and the accuracy of OF methods on the top of the rankings has grown continuously. Additionally, several researchers have tried to preserve the discontinuity of natural motion fields [35], overcoming the original assumption of OF smoothness in [13]. After the work by [32], there have been further attempts to compare different methods. Liu et al. [36], for instance, showed a tradeoff between computational time and angular error using operation curves to compare different OF techniques. It is also interesting to consider the time comparison among OF algorithms given by [37], since the authors provide a picture of the computational load of some OF algorithms. More recently, a group of researchers presented a series of real image sequences and their respective ground truth obtained by tracking hidden fluorescent textures [33]. The authors also suggest a method to evaluate OFbased algorithms.
2.1 Large displacement optical flow
where the term E _{color} represents the common assumption of grey value or color constancy; E _{gradient} represents gradient constancy, which is invariant to a uniform illumination change; E _{smooth} enforces regularity of the resulting optical flow; E _{match} stands for an energy related to point correspondences; and the minimization of E _{desc} assures descriptor matching. The quantity ϕ _{ 1 } is an auxiliary variable which allows integrating descriptor matching into a continuous approach. The implementation available for LDOF [38] has a reduced number of parameters, which means we can consider all of them for optimization purposes. Such implementation allows the user to finetune four parameters: (i) σ is related to the Gaussian presmoothing of the images (preprocessing parameter), (ii) α controls the importance attributed to smoothness of the resulting optical flow, (iii) β enforces the matching of points in both images, and (iv) γ regulates the penalization of violations to the gradient constancy assumption. It is important to highlight that this set of parameters influences significantly the accuracy (consequently the error metrics) and the computational load.
3 Optimization background
In this section, we describe the techniques employed in this paper for comparison purposes. The methods can be divided in two classes: (i) metaheuristic algorithms and (ii) exact methods. Concerning the former approaches, we used socialspider optimization, particle swarm optimization, and harmony search, and with respect to exact methods, we employed the NelderMead, which is a deterministic algorithm for convex functions that employs a simplex for optimization purposes.
3.1 Socialspider optimization
Socialspider optimization is based on the cooperative behavior of social spiders [22], and it takes into account two genders of search spiders: males and females. Depending on the gender, each agent is conducted by a set of different operators emulating a cooperative behavior in a colony. The search space is assumed as a communal web, and a spider’s position represents an optimal (near optimal) solution.
where fitness_{ i } is the fitness value obtained by the evaluation of the ith spider’s position i=1,2,…,N. The worst and best mean the worst fitness value and best fitness value of the entire population, respectively.

The vibrations V _{ i,c } are perceived by the spider i as a result of the information transmitted by the member c who is the nearest member to i and possesses a higher weight ϕ _{ c }>ϕ _{ i };

The vibrations V _{ i,b } perceived by the spider i as a result of information transmitted by the spider b holding the best weight of the entire population;

The vibrations V _{ i,f } perceived by the spider i as a result of the information transmitted by the nearest female f.
where θ,α,β,γ, and rand are uniform random numbers between [0,1], PF is an input parameter, and s _{ c } and s _{ b } represent the nearest member to i that holds a higher weight and the best spider of the entire population, respectively.
where s _{ f } represents the nearest female spider to the male spider i and \(\tilde {\phi }\) is the median weight of male spider population. Thus, the reader can observe that we have distinct movement equations for male and female spiders. Notice that we are using \(\phi _{N_{f}+i}\) to denote the male spiders, since we consider ϕ as a vector containing the fitness of every spider within the web, being the first N _{ f } spiders the female ones.
where n is the dimension of the problem, and \(l_{j}^{\text {high}}\) and \(l_{j}^{\text {low}}\) are the upper and lower bounds, respectively. Once the new spider is formed, it is compared to the worst spider of the colony. If the new spider is better, the worst spider is replaced by the new one.
3.2 Harmony search
Harmony search is a metaheuristic technique based on the improvisation process of musicians searching for a good harmony [39]. The main idea is to generate a new harmony \(h_{\text {new}} = (h^{1}_{\text {new}}, h^{2}_{\text {new}},..., h^{N}_{\text {new}})\) at each iteration, based on memory considerations and pitch adjustment. In this case, N stands for the number of decision variables to be optimized.
where τ is an arbitrary distance (bandwidth) for the continuous design variable, and \(\delta _{j}~\sim {\mathcal {U}}(0,1)\) is an ad hoc parameter.
Recently, several researches have focused on developing variants of traditional HS. In our implementation, we employed the novel global harmony search (NGHS) [40], which has demonstrated better results than vanilla HS in our experiments. The NGHS does not employ PAR and HMCR parameters, but it introduces a new parameter P that denotes the probability of occurring an improvisation schema during a new harmony’s creation, and therefore modifies the improvisation process. Another difference between NGHS and the HS is that a new harmony always replaces the worst one, even when the new one does not improve the worst harmony.
3.3 Particle swarm optimization
Particle swarm optimization can be seen as a search algorithm based on stochastic processes [24], where the learning of social behavior allows each possible solution (particle) ‘fly’ onto that space (swarm) looking for other particles that have the best features and thus minimizing or maximizing the objective function.
Each particle has a memory that stores its best local solution (local maxima or minima) and the best global solution (global maximum or minimum). Besides, each particle has the ability to imitate others that provide the best positions in the swarm. This mechanism can be summarized in three principles: (i) evaluation, (ii) comparison, and (iii) imitation. Each particle can evaluate others within your neighborhood through some objective function; it can compare with your own value and finally decide whether it is a good choice to imitate it or not.
where Ψ is the inertia force that controls the interaction power between particles, and r _{1},r _{2}∈[0,1] are random variables that give the idea of stochasticity concerning PSO. The constants c _{1} and c _{2} are also used to guide the particles (input parameters for the algorithm) onto good solutions.
3.4 NelderMead method
The NelderMead is an iterative heuristic of direct search approach (it does not compute derivatives) used to find stationary points (minimum or maximum) in multidimensional unconstrained functions [28]. This approach is commonly used in problems where the derivative is not known, or when the computational cost to compute it is prohibitive.
where s is the step size that determines the simplex size and \(e = \{1, 1,..., 1\} \in \mathbb {R}^{n}\) is a diagonal vector with size \(\sqrt {n}\). Thus, the initial simplex \({\mathcal {S}}^{0}\) has all edges with the same size s.
After the construction of simplex \({\mathcal {S}}^{i}\), the NelderMead starts the iterative process to find a stationary point x ^{∗}. The first step is to compute all sample values f _{ j }=f(p _{ j })∀0≤j≤n. Next, we determine the indices w, v, and b, which represent the worst, second worst, and best samples’ indexes, respectively. Soon after, we compute the centroid \(c = \frac {1}{n} \sum _{j \neq w} p_{j}\) of all sample points except the worst once.
where j=1,2,…,n. The iterative process is repeated until the maximum number of iterations is reached, or some convergence criterion is met. Notice that the NelderMead algorithm has the following parameters: 𝜗,φ,ρ, and ς.
4 Methodology
We employed the LDOF technique (Section 2.1) together with our implementation of SSO, NGHS, PSO, and NM. The main reason behind the use of such techniques is to alleviate the high computational burden often required by optimization techniques. In light of such shortcoming, we opted to use techniques with easy implementation, which usually reflects in their complexity. For the sake of comparison, we computed the average of ‘end point error’ (EPE) [44] values obtained over five runnings for each optimization technique, which is basically the difference between the ground truth and estimated optical flow.
Parameters used for each optimization technique
Technique  Parameters 

SSO  PF=0.5 
NGHS  P=0.1 
PSO  c _{1}=c _{2}=2.0 and ψ=0.9 
NM  𝜗=1, φ=ρ=0.5 and ς=2 
The methodology employed in this paper differs from the one used by Pereira et al. [20], which optimized each dataset image individually, i.e., they aimed at finetuning LDOF for each image, being the final result the average over all images considering the AAE metric. In this work, we conducted the optimization process over the whole dataset, i.e., we aimed at finetuning LDOF considering all images of the dataset at the same time. Therefore, the fitness function adopted in this work was the one given by the average of EPE values of all dataset images.
5 Experimental results
This section presents the results obtained by SSO, NGHS, PSO, and NM for optical flow parameter optimization purposes. We would like to stress that we did not consider the runtime (computational load), since our goal is to minimize the EPE metric only. Furthermore, the parameters to be optimized have a strong influence on both EPE and runtime.
Results obtained by SSO, NGHS, PSO, NM, and LDOF baseline [ 21 ] over Middlebury dataset
Sequences  SSO  NGHS  PSO  NM  Baseline 

Dimetrodon  0.108±0.02  0.114±0.01  0.114±0.01  0.135±0.03  0.115 
Grove2  0.194±0.00  0.205±0.01  0.191±0.01  0.224±0.03  0.176 
Grove3  0.670±0.04  0.747±0.02  0.685±0.01  0.761±0.05  0.701 
Hydrangea  0.211±0.00  0.211±0.02  0.202±0.01  0.235±0.05  0.184 
RubberWhale  0.126±0.01  0.157±0.00  0.117±0.00  0.149±0.04  0.129 
Urban2  0.401±0.04  0.426±0.06  0.346±0.01  0.459±0.12  0.381 
Urban3  0.614±0.06  0.756±0.11  0.561±0.14  0.867±0.14  0.818 
Venus  0.319±0.02  0.756±0.01  0.381±0.00  0.403±0.03  0.377 
Mean  0.330±0.01  0.372±0.00  0.325±0.01  0.384±0.02  0.360 
Results obtained by SSO, NGHS, PSO, NM, and LDOF baseline [ 21 ] over Sintel dataset
Sequences  SSO  NGHS  PSO  NM  Baseline 

alley_1  0.214±0.02  0.263±0.00  0.209±0.01  0.226±0.00  0.212 
ambush_2  15.532±0.50  13.947±1.57  16.210±1.39  14.53±2.05  19.64 
bamboo_1  0.246±0.01  0.275±0.02  0.236±0.01  0.239±0.00  0.238 
bandage_1  0.740±0.06  0.990±0.04  0.915±0.20  0.817±0.04  0.785 
cave_2  1.857±0.07  1.992±0.10  1.819±0.04  1.899±0.04  1.820 
market_2  0.846±0.00  0.928±0.03  0.972±0.16  0.853±0.02  0.852 
mountain_1  0.865±0.06  0.798±0.04  1.076±0.33  0.741±0.06  0.891 
shaman_1  0.230±0.02  0.297±0.03  0.249±0.01  0.278±0.01  0.256 
sleeping_1  0.114±0.01  0.122±0.01  0.110±0.01  0.105±0.01  0.102 
temple_2  1.536±0.05  1.612±0.05  1.485±0.01  1.563±0.03  1.523 
Mean  2.218±0.05  2.122±0.12  2.293±0.16  2.185±0.12  2.632 
6 Conclusions
In this paper, we have validated the optimization algorithms in the context of model selection in optical flowbased applications, which play an important role in computer vision systems. The experimental section compared the baseline parameters obtained by Brox and Malik [21] against with four optimization techniques: SSO, NGHS, PSO, and NM. Two rounds of experiments have been conducted over the wellknown Middlebury and Sintel datasets: (i) the first round aimed at learning the best set of parameters (i.e., the ones that minimizes the end point error criterion) over the Middlebury dataset and (ii) the second phase performed the same over the Sintel dataset. In the first round, two optimization algorithms (SSO and PSO) achieved better results than the baseline parameters, and in the second round, all optimization algorithms achieved better results than the baseline. Therefore, this paper highlighted the need for an automatic finetuning of the parameters of optical flow techniques. In addition, the computational load of the compared techniques have been assessed in terms of the number of calls to the LDOF technique, evidencing the lower computational burden of NGHS and NM techniques.
7 Endnote
^{a} The number of agents and iterations have been chosen based on previous experiments [20].
Declarations
Acknowledgements
The authors are grateful to FAPESP grants #2013/203877 and #2014/162509, CNPq grants #303182/20113, #470571/20136, and #306166/20143, and Universidad de los Andes FAI grant #05/2013.
Authors’ Affiliations
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