Mathematical Problems in Engineering

Volume 2015, Article ID 721842, 15 pages

http://dx.doi.org/10.1155/2015/721842

## Research on the Fundamental Principles and Characteristics of Correspondence Function

^{1}Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada N9B 3P4^{2}School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China^{3}Department of Electrical Engineering & Computer Science, University of Kansas, Lawrence, KS 66045, USA

Received 17 April 2015; Revised 19 August 2015; Accepted 20 August 2015

Academic Editor: Erik Cuevas

Copyright © 2015 Xiangru Li 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

The correspondence function (CF) is a concept recently introduced to reject the mismatches from given putative correspondences. The fundamental idea of the CF is that the relationship of some corresponding points between two images to be registered can be described by a pair of vector-valued functions, estimated by a nonparametric regression method with more flexibility than the normal parametric model, for example, homography matrix, similarity transformation, and projective transformations. Mismatches are rejected by checking their consistency with the CF. This paper proposes a visual scheme to investigate the fundamental principles of the CF and studies its characteristics by experimentally comparing it with the widely used parametric model epipolar geometry (EG). It is shown that the CF describes the mapping from the points in one image to their corresponding points in another image, which enables a direct estimation of the positions of the corresponding points. In contrast, the EG acts by reducing the search space for corresponding points from a two-dimensional space to a line, which is a problem in one-dimensional space. As a result, the undetected mismatches of the CF are usually near the correct corresponding points, but many of the undetected mismatches of the EG are far from the correct point.

#### 1. Introduction

Finding point correspondences between two images is a fundamental problem in computer vision [1, 2]. In two given images, the corresponding points (CPs) are the projections of the same point in a scene. Many computer vision algorithms and applications rely on the successful identification of point correspondences between two images, for example, tracking, stereo vision, motion analysis, object recognition, remote sensing, image mosaicing, and automatic quality control, among others [3–7].

Point correspondences are usually established by the following procedures: extracting salient points, calculating their descriptors based on a small and local area around them, establishing putative correspondences by comparing the descriptors, and refining the putative correspondences. Compared with the representations based on a large spatial area, local feature descriptors are usually more robust to brightness variation, deformation, and occlusion but have less distinctiveness. This typically results in a high percentage of mismatches/outliers among the computed putative correspondences, which are very likely to ruin traditional estimation methods [8–10]. Therefore, an essential problem in computer vision is rejecting mismatches from given putative correspondences in a refining stage [11–13].

Correspondence function (CF) is a model recently introduced based on point set mapping theory to reject mismatches from putative correspondences [2]. The fundamental idea of CF is that, for two given images and of a scene, the relationships between their CPs can be described by a pair of vector-valued functions, which are estimated by a nonparametric regression method. Mismatches are then detected by checking whether they are consistent with the CFs. The key of the model is that the relationships between CPs are represented with more flexibility than the usual parametric model, for example, homography matrix [14], similarity transformation [15], and projective transformations [15]. The flexibility is to account for nonrigidity of objects or to reduce the undue influence from outliers.

In this work, we propose a visual scheme to investigate the fundamental principles of CF (Figures 1 and 2) and to study its characteristics by experimentally comparing it with the widely used parametric model epipolar geometry (EG) (some rudimentary comparisons were made between CF and EG in a conference paper ([16])). It is shown that the CF describes the mapping from a point in one image to its CP(s) in another image, which enables us to directly estimate the positions of the CPs. However, the EG acts by reducing the search space for CPs from a two-dimensional space to a line, which is a problem in one-dimensional space. In applications, the result of the difference between EG and CF is that the undetected mismatches by CF are usually near the correct CPs, but many of the undetected mismatches of EG are far from the correct correspondence.