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Image corner detection is a fundamental task in computer vision. Many applications require reliable detectors to accurately detect corner points, commonly achieved by using image contour information. The curvature definition is sensitive to local variation and edge aliasing, and available smoothing methods are not sufficient to address these problems properly. Hence, we propose Mean Projection Transform (MPT) as a corner classifier and parabolic fit approximation to form a robust detector. The first step is to extract corner candidates using MPT based on the integral properties of the local contours in both the horizontal and vertical directions. Then, an approximation of the parabolic fit is calculated to localize the candidate corner points. The proposed method presents fewer false-positive (FP) and false-negative (FN) points compared with recent standard corner detection techniques, especially in comparison with curvature scale space (CSS) methods. Moreover, a new evaluation metric, called accuracy of repeatability (AR), is introduced. AR combines repeatability and the localization error (_{e}

Feature detection is a fundamental issue in image processing and computer vision that is directly related to interest points. Corner points are considered important features for feature extraction [

Numerous corner detection methods have been introduced over the last several decades. These methods can be divided into three main categories: intensity-based detectors [

Robustness to noise is an important issue for contour-based detectors [

Over the last two decades, curvature scale space (CSS) methods have been widely used as corner detectors in the literature due to their high performance. CSS-based detectors exhibit some weaknesses, which are considered in this paper. They generally use second-order derivatives, which can cause an increase in the FP rate because of contour variation. Additionally, they require a Gaussian scale selection to smooth the curve area, which is application based and a difficult task. The basic idea was introduced by Rattarangsi and Chin [_{e}

In this paper, a new projection transform, called mean projection transform (MPT), is proposed to extract the corner candidates and address the aliasing problem. Next, a parabolic fit approximation is used to determine the corner points in the extracted candidates. This method reduces problems related to the existing CSS-based algorithms. The proposed method is compared to the detectors presented in [

This paper is organized as follows: Section 2 discusses the MPT method for selecting corner candidates. Section 3 presents the parabolic fit approximation to confirm corner points from the MPT candidates and localize them. Section 4 discusses the evaluation results and proposes a new corner detection evaluation method called AR, which addresses the limitations of the current evaluation metrics for FP and FN points; the proposed corner detector is then assessed using _{e}

A new projection transform based on the mean of integral values in both the horizontal and vertical directions is proposed. Contour-based detectors use contour information to extract the corner candidates and corner points. Based on CSS problems regarding contour aliasing and variation, the MPT method is proposed to extract the corner candidates. MPT representation guarantees that the detector only selects candidates that have high curvature, and it addresses the aforementioned problems.

MPT is a transform that consists of the integrals over straight lines in a digital image. If ^{2}, then MPT is a transform of L, where the mean of the integrals in vertical and horizontal directions is calculated using

The arc-length ^{2} for all lines, and MPT can be represented in the aforementioned coordinates according to

This equation can also be written as:

The MPT that considers the multi-directional integral can be formulated as

MPT calculates the mean of the integrals in an input image in both the vertical and horizontal directions of line L. The parameters of MPT can detect the available angular contours from a straight contour on the edge map of the objects in an image. The MPT of the sample image is shown in

MPT calculates the integral of

A projection of universal corner model (PUCM) describes all corner types. The basic corner model (BCM) of a curve is the area in which the horizontal and vertical integrals are significantly different than the non-corner model (NCM). In the polar coordinate system, the BCM and NCM can be defined by _{low} is the lower band, and _{high} is the upper band of θ. In _{1}, θ _{2}, θ _{3}] denotes the polar angles, and R = [r_{1}, r_{2}, r_{3}] is the radius of three points of the curve. _{3} is assumed as the middle point in the polar coordinate system in terms of θ. The value r in the same coordinate system follows

The MPT representation of the BCM is presented in

As shown in

Curvature extraction and angle estimation are the key features of the contour-based corner detection methods. CSS detectors extract the curve, analyze the curvature properties of the contour map, and then detect the corner points. Γ is considered the curvature at a point, as presented in _{1}, p_{2}, …., p_{n}〉is the n points on the curve Γ(t) = (x(t), y(t)) with a given distance function d(p_{i}, p_{j}), ρ(p, r) = {q|(p, q) ≤ r} is a parabola with the radius _{i} are the points inside the area, as shown in

Orthogonal lines meet at the center point of the parabola ρ, which are defined as D. If ρ_{i} denotes ρ (p_{i}, ε), then p_{i}p_{j} is a segment δ_{h}(p_{i}p_{j}) ≤ ε if p_{i}p_{j} intersect at d_{i+1},…,d_{j−1}, and the parabola radius passing the points is:
_{i} to P_{j} points inside the parabola area. Additionally, the proposed method is adjustable for detecting the low- and high-order corners in different image scaling by adjusting the values of ϑ as the focal control parameter. The general definition for ε is:

Generally, the approximation of the parabolic fit is robust to the local variation [

To evaluate the proposed method, a dataset called “_{e}_{x}, s_{y} in [0.9,1.3], are used for assessment.

In addition to the simple images, the proposed method indicates good performance in complex shapes.

In detection theory, the receiver operating characteristic, or ROC, is a graphical plot that illustrates the performance of the system based on detection rates to provide a more appropriate comparison [

When comparing the performance of the detectors considering the ROC plot, a detector is better when its plot points are located on the top-left side of the plot area, which shows higher sensitivity and specificity. To determine the FNs and FPs, human judges generate the ground truth [

Among the detectors, CPDA attains comparable detection performance with the proposed method. The proposed method concentrates in the top-left of the graph, which indicates higher TPs and few FPs, indicating higher performance. JUDOCA provides the lowest FPs in comparison with the others, whereas ANDD shows many FPs and fewer FNs.

The _{e}_{e}_{oi} and y_{oi} are the ground truth coordinates of the corners, x_{ti} and y_{ti} are the coordinates of the _{r} is the total detected points of the detector. _{e}_{e}_{e}_{e}_{e}_{e}

Average repeatability (_{avg}_{e}_{o} is the number of corners in the ground truth, and N_{t} is the number of detected corners. N_{r} is the repeated corners between two results within a maximum three-pixel error rate.

To compare the proposed method with other methods, the same dataset with the same conditions is used to evaluate the results. The proposed method outperforms the other methods in different conditions. For the combined rotation and scale transform, after the proposed method, the best results are achieved by CPDA [

When comparing the effect of the transformation on the results, the repeatability and localization parameters are not sufficient because they do not directly consider FPs and FNs. FPs and FNs are quite important in corner detection methods and should directly affect the evolution result. Moreover, average repeatability does not consider the ground truth information, which means that it does not determine whether the detected points in the original image are localized correctly. Therefore, a new comparison method based on both the _{e}

In the proposed comparison method, each corner point is analyzed to provide a probability of P_{i}, and the mean of probability for all points generates the AR, as defined in:
_{j}, and the result points are R_{i}. Each R_{i} has a corresponding P_{i} ∈ [0,1]. The value ‘1’ exhibits the highest probability that is a TP, and the value ‘0’ exhibits the lowest probability of the corner point that is either an FN or FP. The number of points in the ground truth and result image is M and M′ respectively. For the two points in the ground truth and result image, the distance is calculated by:
_{oj} and y_{oj} are the ground truth corner coordinates, and x_{ti} and y_{ti} are the coordinates of the

In the first step, δ_{xy} = min(D) = d_{xy} is obtained. Then, column y and row x corresponding to d_{xy} are eliminated. Therefore, matrix D is M−1 × M′−1. This process continues until all elements in matrix D are eliminated. In each step, δ_{xy} = min(D) = d_{xy} is the closest distance between the ground truth G_{x} and the result point R_{y}. For each point, the probability is calculated using the maximum size of the ground truth or result image, as defined in _{ij}

This paper introduced a new corner detection method based on contour information. Candidate selection using a new image transformation called MPT was the basic approach of this paper. MPT calculates the mean of the integral of the image contour in both the horizontal and vertical directions. After selecting the corner candidates by MPT, an efficient curvature estimation based on parabolic approximation was used to confirm and localize the corner points in the candidates. The results were evaluated by _{e}_{e}

The authors would like to thank the Center for Artificial Intelligence Technology (CAIT), Faculty of Information Science and Technology, National University of Malaysia (UKM) and the anonymous reviewers for their constructive comments. This research was, partially funded by the ERGS/1/2012/STG07/UKM/02/9 grant from the UKM.

All authors contributed extensively to the work presented in this paper. Seyed Mostafa Mousavi Kahaki conceived the basic idea of the paper. Md Jan Nordin supervised the project. Amir Hossein Ashtari designed and performed AR evaluation metric. Seyed Mostafa Mousavi Kahaki conducted the experiment and wrote the main paper, and then Md Jan Nordin and Amir Hossein Ashtari wrote the supplementary information. The critical revision was done by Seyed Mostafa Mousavi Kahaki and Amir Hossein Ashtari.

The authors declare no conflict of interest.

Chessboard image as (

(

Contour models: (

Different MPT representation peaks in (

Approximation of the parabolic fit estimation technique.

Results of the different corner detection techniques. (

Corner detection results (

ROC plot comparison of the proposed method, ANDD, JUDOCA, and CPDA.

Comparative results of the different methods under _{e}

Average repeatability under rotation, uniform scale change, non-uniform scale change, and the combined rotation and scale effect of the different methods.

Comparative results of the different methods under the