Computational Intelligence and Neuroscience

Volume 2016, Article ID 3629174, 12 pages

http://dx.doi.org/10.1155/2016/3629174

## A Multiobjective Approach to Homography Estimation

^{1}Sciences Division, Centro Universitario de Tonalá of Universidad de Guadalajara, 45400 Guadalajara, JAL, Mexico^{2}Electronic Division, Centro Universitario de Ciencias Exactas e Ingenierías of Universidad de Guadalajara, 44430 Guadalajara, JAL, Mexico^{3}Computer Sciences Department, Tecnológico de Monterrey Campus Guadalajara, 45201 Guadalajara, JAL, Mexico

Received 4 June 2015; Accepted 4 October 2015

Academic Editor: Yufeng Zheng

Copyright © 2016 Valentín Osuna-Enciso et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In several machine vision problems, a relevant issue is the estimation of homographies between two different perspectives that hold an extensive set of abnormal data. A method to find such estimation is the random sampling consensus (RANSAC); in this, the goal is to maximize the number of matching points given a permissible error (Pe), according to a candidate model. However, those objectives are in conflict: a low Pe value increases the accuracy of the model but degrades its generalization ability that refers to the number of matching points that tolerate noisy data, whereas a high Pe value improves the noise tolerance of the model but adversely drives the process to false detections. This work considers the estimation process as a multiobjective optimization problem that seeks to maximize the number of matching points whereas Pe is simultaneously minimized. In order to solve the multiobjective formulation, two different evolutionary algorithms have been explored: the Nondominated Sorting Genetic Algorithm II (NSGA-II) and the Nondominated Sorting Differential Evolution (NSDE). Results considering acknowledged quality measures among original and transformed images over a well-known image benchmark show superior performance of the proposal than Random Sample Consensus algorithm.

#### 1. Introduction

A homography is a transformation that maps points of interest by considering movements as translation, rotation, skewing, scaling, and projection among image planes, all of them contained into a single, invertible matrix. In general terms, those displacements could be considered to be belonging to three cases: an object moving in front of a static camera, a static scene captured by a moving camera, and multiple cameras from different viewpoints. In either case, those approximations simplify the utilization of image sequences to construct panoramic views [1–3], to increment resolution in low quality imagery [4–6], to remove camera movements when studying the motion of an object into a video [7], and to control the position of robots [8–10], among other uses [11–13].

Taking a set of experimental data as a base, in a modeling problem there exist two data types: those that can be adjusted to a model with a certain probability (also known as inliers) and those that are not related to the model (e.g., outliers). There are several algorithms specialized in solving this classification problem; one of such techniques is the Random Sample Consensus (RANSAC) [14].

In the algorithm, minimum subsets of experimental data are randomly taken, and a model is proposed and evaluated according to a permissible error (Pe), in order to determine how well the model adjusts to the data [15]. This process is repeated until a number of iterations are completed, and the model with the maximum number of inliers is taken.

Even considering that RANSAC is a robust and simple algorithm, it has some drawbacks [16–18], two of which are the high dependency between the number of matching points (model quality) and the permissible error. In this work, it is considered that those disadvantages belong to a multiobjective optimization problem. On the one hand, due to the random nature of RANSAC, achieving improvements in the quantity of inliers implies more iterations in order to discard unadjusted data to the proposed model. On the other hand, the number of matching points conflictingly depends on the permissible error (Pe). A low Pe value increases the accuracy of the model but degrades its generalization ability to tolerate noisy data (number of matching points). By contrast, a high Pe value improves the noise tolerance of the model but adversely drives the process to false detections. The main error source in the model estimation procedure arises from defining the Pe value with no consideration of the relationship between the dataset and the model.

In order to make the RANSAC algorithm more efficient, some improvements have been suggested; for instance, in the algorithm called MLESAC [19] it is considered that the inliers into the images will follow a Gaussian distribution whereas the outliers are considered as uniformly positioned; according to that, the voting process is achieved through maximizing the likelihood and the original RANSAC. The SIMFIT method [20] proposed the forecasting of the permissible error, through an iterative reestimation of that value, until the model is adjusted to the experimental data. Some other variants to the original RANSAC are the projection-pursuit method, the Two-Step Scale Estimator, and the CC-RANSAC [15, 21, 22], all of them focused on maximizing the number of inliers by making more searches into the data and therefore making the complete process more expensive, computationally speaking. In such sense, an algorithm that tries to reduce the computational cost is the one proposed in [17], where the maximization of the inliers is achieved by using a metaheuristic technique, called Harmony Search.

Nevertheless the mentioned improvements, the search strategy used in the mentioned articles (with exception of [17]), are more close to random walking, and therefore those approaches are computationally expensive. Moreover, in all the cases only one objective function is considered, usually related to the number of matching points, while the permissible error is left behind. In accordance with that, and in order to overcome the typical RANSAC problems, we propose to visualize the RANSAC operation as a multiobjective problem solved by an evolutionary algorithm. Under such point of view, at each iteration, new candidate solutions are built by using evolutionary operators taking into account the quality of the previously generated models, rather than purely random, reducing significantly the number of iterations. Likewise, new objective functions can be added to incorporate other elements that allow an accurate evaluation of the quality of a candidate model.

When an optimization problem involves more than one objective function, the procedure of finding one or more optimum solutions is known as multiobjective optimization (MO) [23]. Under MO, different solutions produce conflicting scenarios among the objectives [24]. Contrary to single objective optimization, in MO it is usually difficult to find one optimal solution. Instead, algorithms for optimizing multiobjective problems try to find a family of points known as the Pareto optimal set [25]. These points verify that there is no different feasible solution which strictly improves one component of the objective function vector without worsening at least one of the remaining ones. Evolutionary algorithms (EAs) are considered the most adequate methods for solving complex MO problems, due mainly they are many times capable of maintaining a good diversity [26], can extend to multiple populations [27], as well as can work with a variety of problems such as discrete ones [28]. Several variants of nondominated sorting as well as new methods have been proposed in recent years in order to solve problems related to feature selection [29], community detection [30], among other issues [24, 31]; however, the Nondominated Sorting Genetic Algorithm II (NSGA-II) [32] and the Nondominated Sorting Differential Evolution (NSDE) [31] are some of the most representative.

In this paper, the estimation process is considered as a multiobjective problem where the number of matching points and the permissible error (Pe) are simultaneously optimized. In order to solve the multiobjective formulation, two different evolutionary algorithms have been explored: the Nondominated Sorting Genetic Algorithm II (NSGA-II) and the Nondominated Sorting Differential Evolution (NSDE). Results considering acknowledged quality measures among original and transformed images over a well-known image benchmark show superior performance of the proposal than Random Sample Consensus algorithm on the problem being assessed, giving good results even with high outliers levels.

The remainder of the paper is organized as follows: Section 2 explains the problem of image homography considering multiple views. Section 3 introduces the fundamentals of the RANSAC method. Section 4 briefly explains the evolutionary approaches that are used in this paper in order to solve the multiobjective problem while Section 5 presents the proposed method. Section 6 exhibits the experimental set and its performance. Finally, Section 7 establishes some final conclusions.

#### 2. Homography between Images

For the case where two images are taken of the same scene from different perspectives, a problem consists in finding a transformation that permits the matching among the pixels belonging to both images. This denominated the image matching problem. The search of a geometric transformation is achieved by utilizing corresponding points from image pairs [33, 34], which enable forming feature vectors, also called image descriptors. Even when considering that such descriptors are not completely reliable, so they can produce erroneous results for the image matching, in this paper they are used to find the geometric relations between images by using the homography, which is explained in the next paragraphs.

Consider a set of pointssuch that and are the positions with respect to a given image pair.

By means of a plane, a homography establishes a geometric relation between two images taken under different perspectives, as can be seen in Figure 1; this allows for a projection of the points from the plane to a pair of images, through or . Conducive to find the homography between an image pair, a set with four point matches is only required, to construct a linear system which must be solved [35]. Concerning evaluation of the quality of the candidate homography, it is necessary to calculate the distance among the point positions of the first image with respect to the second image; that distance is labeled as the Mismatch Error and is defined byas long as and are the respective errors from each image.