Optimal edge detection using multiple operators for image understanding
- Stamatia Giannarou^{1}Email author and
- Tania Stathaki^{2}
https://doi.org/10.1186/1687-6180-2011-28
© Giannarou and Stathaki; licensee Springer. 2011
Received: 7 February 2011
Accepted: 23 July 2011
Published: 23 July 2011
Abstract
Extraction of features, such as edges for the understanding of aerial images, has been an important objective since the early days of remote sensing. This work aims at describing a new framework which allows for the quantitative combination of a preselected set of edge detectors based on the correspondence between their outcomes. This is inspired from the problem that despite the enormous amount of literature on edge detection techniques, there is no single technique that performs well in every possible image context. Two approaches are proposed for this purpose. The first approach is the well-known receiver operating characteristics analysis which is introduced for a sound quality evaluation of the edge maps estimated by combining different edge detectors. In the second approach, the so-called kappa statistics are employed in a novel fashion to amalgamate the above-mentioned selected edge maps to form an improved final edge image. This method is unique in the sense that the balance between the false detections (false positives and false negatives) is explicitly determined in advance and incorporated in the proposed method in a mathematical fashion. For the performance evaluation of the proposed techniques, a sample set of the RADIUS/DARPA-IU Fort Hood aerial image database with known ground truth has been used.
1 Introduction
Automatic detection of both geographical (natural) and man-made structures, such as vegetation, buildings, roads and vehicles, in aerial or satellite images has been an active research topic the last decade [1, 2]. Aerial images, with their highly detailed contents, are an important source of information for applications including GIS [3], traffic surveillance [4] and military applications [5]. When processing aerial images, the extraction of high-level features for object detection is an important field. Features of interest can be extracted using a variety of image-processing techniques, which analyze the image to detect characteristics, such as edges, texture and shape.
Edge-driven approaches have been extensively used in understanding remote sensing images and detecting man-made objects in them. In Noronha and Nevatia [6], extract edge points to build a system that detects and constructs 3D models of buildings using multiple aerial images. Tupin et al. [7] applied the ratio-of-averages (RoA) edge detector that was first presented by Touzi et al. [8] to identify linear structures, such as main axes in road networks in synthetic aperture radar (SAR) images. In [9], a framework for automatic change detection of linear features (e.g. roads and buildings) in aerial images is built, based on the edge maps which indicate pixels that segment areas with significantly different brightness values. Gamba et al. [10] proposed an approach to extract the map of urban areas exploiting edge information in very high-resolution images (VHR).
Edge detection in aerial images is a challenging task for many reasons. Aerial images differ in resolution, sensor type, orientation, quality, dynamic range, light conditions, different weather and seasons, factors that increase the complexity of the edge detection process. In several cases, some of the detected edges do not correspond to meaningful objects, while some edges that belong to objects are distorted, broken or missed. Furthermore, edges of different objects or edges of different layers of a structure are likely to stick to each other. Edge detection must be efficient and reliable because it is crucial in determining how successful subsequent processing stages will be. To fulfill the reliability requirement of edge detection, a great diversity of operators have been devised with differences in their mathematical and algorithmic properties.
Some of the earliest methods, such as the Sobel [11] and Roberts [12], are based on the so-called "Enhancement and Thresholding" approach [13]. According to that method, the image is convolved with small kernels which represent low-order high-pass filters and the result is thresholded to identify the edge points. Since then, more sophisticated operators have been developed. Marr and Hildreth [14] were the first to introduce the Gaussian smoothing as a pre-processing step in edge feature extraction. Their method detects edges by locating the zero-crossings of the Laplacian (second derivative) of the output of a Gaussian filtered image. Canny [15] developed an alternative Gaussian edge detector based on the optimizing three criteria. He employed Gaussian smoothing to reduce noise and the first derivative of the Gaussian to detect edges. Deriche [16] extended Canny's work to derive a recursively implemented edge detector. Rothwell [17] designed a spatially adaptive operator which is able to recover reliable topological information.
Further to the above, an alternative approach to edge detection is the multiresolution one. In such a detection framework, the image is convolved with Gaussian filters of different sizes to produce a set of images at different resolutions. These images are integrated to produce a complete final edge map. Typical algorithms which follow this approach have been produced by Bergholm [18], Lacroix [19] and Schunck [20]. Another interesting category of edge detectors is the logical/linear operators [21] which combine aspects of linear operators' theory and Boolean algebra. Furthermore, the idea of mimicking the human vision function using mathematical models gave space to the development of feature detection algorithms based on the human visual system. A representative example is the edge detector developed by Peli [22].
Recent approaches have used supervised learning to detect edges and object boundaries. In [23], a data-driven statistical edge detection approach has been proposed, where the probability distributions of edge filter responses on and off edges are learnt from pre-segmented data sets, while edges are detected using the log-likelihood ratio test. In a similar spirit, Martin et al. [24] combine multiple local cues to detect local boundaries. Based on the brightness, color and texture features, a classifier is trained using pre-segmented data to model the true posterior probability of a boundary at every image location and orientation. Another supervised learning algorithm for edge detection is the boosted edge learning (BEL) [25]. In this approach, a large number of features across different scales are combined to learn a discriminative model using an extended version of the probabilistic Boosting tree classification algorithm.
Intuitively, the question that arises is which edge detector and detector parameter settings can produce optimal results. Despite the aforementioned volume of existing work, an ideal scheme able to detect and localize edges with precision in many different contexts, has not yet been produced. Depending on the application, pre-segmented data might not be available for supervised training. This is getting even more difficult because of the absence of an evident "correct" edge map (ground truth), on which the performance of an edge detector could be evaluated. Although an edge detector may be robust to noise, it may fail to mark corners and junctions properly. Another common issue with edge detection is the incomplete contour representation. Problems, such as the above, strongly motivate the development of a general method for combining different edge detection schemes to take advantage of their strengths, while overcoming their weaknesses.
Let us assume n original detectors, where a detector refers to a mathematical method that attempts to identify the presence (or absence) of an event. In our work, we are interested in edge detectors that investigate the presence of edges in a digital image signal. These original detectors are transformed to a new set of detectors, where each new detector is a function of all of the original detectors. This function is solely controlled by a parameter named correspondence threshold (CT) which will be explained in the main body of the paper. Each one of the new detectors is associated with a specific value of the CT parameter; this value identifies uniquely the detector. The new detectors vary with respect to their richness, starting from weak detectors that highlight only the strong edges and are basically noise free, to strong detectors that also highlight weak edges and fine details, but exhibit significant amount of noise. In this work, we are interested in selecting one of the new edge detectors as the final detection result. The method assumes that the best of the new edge maps is the one which is most consistent with the verity of the detections produced by the set of original edge detectors. We present two novel contributions.
The first contribution is based on the use of the so-called receiver operating characteristic (ROC) curve. The only related work was presented in [26]. However, in [26] the original edge maps are generated for different combinations of the parameter values of a single edge detector and more specifically the Canny edge detector. In this paper, the original edge maps are different popular edge detectors which although follow similar mathematical techniques, they still produce different results.
The second contribution is based on the employment of a normalized and corrected edge detection performance statistical metric known as kappa statistic. The kappa statistic has been used solely in medicine [27]. We are seeking at optimizing the kappa statistic which, in the specific framework, is a function of the available edge detectors and additionally a scalar parameter which controls the strength of the final detector and consequently the balance between false alarms and misdetections.
The later is the main novelty of this paper. It is an important research contribution to the edge detection problem, since it allows for the blind combination of multiple detectors and more importantly the pre-specified control of the type of preferred misclassifications.
The work presented in this paper is a significant extension of the preliminary work presented in [28, 29]. Two different contributions for optimal edge detection are studied in detail. Exhaustive experimental results are provided for assessing the relative strength of intrinsic technical merit of the proposed techniques for detecting edges in natural scenes. The proposed framework is compared against existing methods and their respective performance is evaluated on aerial images.
The paper is organized as follows. Section 2 concerns the brief analysis of a set of popular edge detectors that will be used in this work. Section 3 presents two novel approaches for the quantitative combination of multiple edge detectors. Section 4 contains experimental results yielded using our implementation of the automatic edge detection algorithms together with a comparative study of the methods' performance. Conclusions are given in Section 5.
2 Operators implemented for this work
Several approaches to edge detection focus their analysis on the identification of the best differential operator necessary to localize sharp changes of the image intensity. These approaches recognize the necessity of a preliminary filtering step, as a smoothing stage, since differentiation amplifies all high-frequency components of the signal, including those of the textured areas and noise. The most widely used smoothing filter is the Gaussian one which has been shown to play an important role in detecting edges.
where σ^{2} denotes the variance of the Gaussian filter and controls the degree of smoothing. After this process, candidate edge pixels are identified as the pixels that survive an additional thinning process known as nonmaximal suppression[15]. Then, the candidate edges are thresholded to keep only the significant ones. Moreover, Canny suggests hysteresis thresholding to eliminate streaking of edge contours.
where a, c and ω are positive real numbers. This filter is sharper than the derivative of the Gaussian and is efficiently implemented in a recursive fashion. The procedure that follows in Deriche's method is the same as the one used in Canny's edge detection; nonmaximal suppression and hysteresis thresholding are applied as described previously.
Although Canny's detector performs well in localizing edges and suppressing noise; yet in several cases, it fails to provide a complete boundary in objects. Rothwell's [17] operator is an improvement to earlier edge detectors, capable of recovering sound topological descriptions. It follows a line of work similar to Canny's. The uniqueness of this algorithm originates in the use of a dynamic threshold which varies across the image [17].
In general, it is very difficult to find a single scale of smoothing which is optimal for all the edges in an image. One smoothing scale may keep good localization while giving detections sensitive to noise. Thus, multiscale edge detection is introduced as an alternative. In this approach, edge detectors with different filter sizes are applied to the image to extract edge maps at different smoothing scales. This information is then combined to result in a more complete final edge image.
Bergholm [18] introduced the coarse-to-fine tracking as an approach to multiscale edge detection. The initial steps of this method are based on Canny's approach. This algorithm relies on the fact that edge detection at a coarse resolution yields significant edges, while their accurate location is detected at a finer resolution. Therefore, the main idea is to initially detect the edges applying a strong Gaussian smoothing and then focus on these edges by tracking them over decreasing smoothing scale.
In Lacroix [19] introduces another algorithm for multiscale detection based on Canny's method. Contrary to Bergholm [18] who proposed the tracking of edges from coarse-to-fine resolution, in Lacroix's method the edge information is combined moving from fine-to-coarse resolution aiming at avoiding the problem of splitting edges. Schunck's work [20] is another study that advocates the use of derivatives of Gaussian filters with different variances to detect intensity changes at different resolution scales. The gradient magnitudes over the selected range of scales are multiplied to amplify significant edges, while suppressing the weak ones. Hence, a composite edge image is formed.
In this work, we use the six edge detectors mentioned in this section. The use of convolution-based methods is justified by the fact that they are simple to implement, while producing accurate detection results.
3 Automatic edge detection
In this paper, we intend to throw light on the uncertainty associated with the parametric edge detection performance. The statistical approaches described here attempt to automatically form an optimum edge map, by combining edge images emerged from different detectors.
We begin with the assumption that N different edge detectors will be combined. The first step of the algorithm comprises the correspondence test of the edge images, E_{ i } for i = 1, ..., N. A correspondence value is assigned to each pixel and is then stored in a separate array, V, of the same size as the initial image. The correspondence value is the frequency of identifying a pixel as an edge by the set of detectors. Intuitively, the higher the correspondence associated with a pixel, the greater the possibility for that pixel to be a true edge. Hence, the above correspondence value can be used as a reliable measure to distinguish between true and false edges [26].
However, these data require specific statistical methods to assess the accuracy of the resulted edge images-accuracy here being the extent to which detected edges agree with true edges. Correspondence values ranging from 0 to N produce N + 1 thresholds which correspond to edge detections with different combinations of true-positive and false-positive rates. The threshold that corresponds to correspondence value 0 is ignored. Hence, the main goal of the method is to estimate the correspondence threshold CT (from the set CT_{ i } where i = 1, ..., N) which results in an accurate edge map that gives the finest fit to all edge images E_{ i } . In this section, we describe two different approaches for this purpose.
3.1 ROC analysis
In our case, the classification task is a binary one including the actual classes {e, ne}, which stand for the edge and non-edge event, respectively and the predictive classes, predicted edge and predicted non-edge, denoted by {E, NE}. Traditionally, the possible outcomes obtained by an edge detector are displayed graphically in a 2 × 2 matrix, the confusion matrix.
Relying on the value of only one of the above metrics for edge detection accuracy estimation would be an oversimplification and will possibly lead to misleading inferences. Based on this idea, the ROC analysis [32, 33] can be introduced to quantify detection accuracy. In fact, a ROC curve provides a view of all the true-positive/false-positive rate pairs emerged from varying the correspondence over the range of the observed data. In this work, the ROC curve is used to select the correspondence threshold CT that would provide an optimum trade-off between the TP_{rate} and the FP_{rate} rate of edge detectors.
where represents the average number of true edges in M_{ j } .
Equation 9 defines a line that connects the points (0, 1) and (P, P) in the ROC plane, known as diagnosis line. Therefore, the optimum CT occurs at the intersection (or close to that) of the ROC curve and the diagnosis line. The value of the selected CT determines how detailed the final edge image, EGT, will be. In the case of a noisy environment, the selection of CT should give a trade-off between the increase in information provided by the final edge image and the decrease in noise.
3.2 Weighted kappa coefficient
In edge detection, it is prudent to consider the relative seriousness of each possible disagreement between true and detected edges when performing accuracy evaluation. This section is confined to the examination of an accuracy measure which is based on the acknowledgement that in detecting edges, depending on the specific application, the consequences of a false positive may be quite different from the consequences of a false negative. For this purpose, the weighted kappa coefficient[34, 35] is introduced for the estimation of the correspondence threshold that results in an optimum final edge map.
In edge detection, A_{0} may be defined as a measure of agreement between true and detected edges. The definition of A_{ c } and A_{ a } is obvious.
Probabilities for legitimate and random edge detection
Legitimate edge detection | random edge detection | |
---|---|---|
TP | d_{1,1} = P · SE | c_{1,1} = P · Q |
FP | d_{1,2} = P' · SP' | c_{1,2} = P' · Q |
FN | d_{2,1} = P · SE' | c_{2,1} = P · Q' |
TN | d_{2,2} = P' · SP | c_{2,2} = P' · Q' |
and r' is the complement of r. The weighted kappa coefficient k(r, 0) indicates the quality of the detection as a function of r. It is unique in the sense that the balance between the false detections is determined in advance and then is incorporated in the measure.
From Equation 17 it is deduced that the index r is indicative of the relative importance between false negatives and false positives. Its value is dictated by which type of error carries the greatest importance for a particular application and ranges from 0 to 1. If we focus on the elimination of false positives in edge detection, W_{2} will predominate in Equation 15 and consequently r will be close to 0 as it can be seen from Equation 17. On the other hand, a choice of r close to 1 signifies our interest in avoiding false negatives since W_{1} will predominate in Equation 15. A value of r = 1/2 reflects the idea that both false positives and false negatives are equally unwanted. No standard choice of r can be regarded as optimum because the balance between the two errors shifts according to the application.
Thus, for a selected value of r, the weighted kappa coefficient k_{ j } (r, 0) is calculated for each edge map as in Equation 16. The optimum CT is the one that maximizes the weighted kappa coefficient.
3.3 Geometric approach for the weighted kappa coefficient
It is obvious that the point (k_{ j } (r, 0),k_{ j } (r, 0)) lies also on the r-projection line and also on the main diagonal described by equation k(0, 0) = k(1, 0).
3.4 An alternative to the selection of the r parameter value
In the previous section, parameter r is evaluated according to Equation 17. By assigning more weight to the false detection, we want to eliminate the ratio in Equation 17 yields the appropriate value of r. However, a more efficient analysis is necessary. An alternative analysis that justifies the previously described selection of r is presented in this section.
The measures σ_{ p } , σ_{ q } are positive as they express standard deviations. The correlation coefficient, ρ, is positive, as well. Thus, it becomes obvious that the sign of the the derivative, , is determined by the value of Q relative to P.
A level, Q, greater than the prevalence, P corresponds to an edge detection that eliminates the misdetections by favoring the false positives. In this case, according to Equation 21, the derivative of the Weighted Kappa Coefficient is positive for any value of r and the quality measure k(r, 0) is an increasing function of r. This means in applications where we are more interested in the elimination of false negatives, a higher value of r in the interval [0, 1] will result in the selection of a more accurate edge map.
Equivalent conclusions are derived for the elimination of false positives, i.e. detections where the level is smaller than the prevalence. According to Equation 21 the derivative, , will be negative and the weighted kappa coefficient will be a decreasing function of r. Therefore, small values of r in the interval [0, 1] will yield CT s that correspond to more accurate edge maps.
4 Experimental results and discussion
Using the above framework, six edge detectors, proposed by Canny [15], Deriche [16], Bergholm [18], Lacroix [19], Schunck [20] and Rothwell [17] were combined to produce optimum edge maps. The selection of the above edge detectors relies on the fact that they basically follow the same mathematical approach.
The performance evaluation study of the proposed approach is carried out on a set of 10 images, selected from the RADIUS/DARPA-IU Fort Hood aerial image set [36]. Some of the images contain mainly vertical views and others contain more oblique views as well. All the images are 8-bit/pixel and their size ranges from 476 × 477 to 645 × 667 pixels. The ground truth for the aerial images is provided in the data set [36], in the form of specified image points that should be identified as edges and specified image regions where no edges should be detected.
where N_{GT} is the number of edge points on the ground truth and d_{ i } is the minimum distance between the i th edge point on the ground truth and the estimated edge map. Intuitively, the lower the detection error and the higher the similarity to the ground truth, the better is the performance of the operator.
Specifying the value of the edge detection operators' input parameters was a crucial step. In fact, the parameter selection depends on the implementation and intends to maximize the quality of the detection. In our work, we were consistent with the parameters proposed by the authors of the selected detectors. The Bergholm algorithm was slightly modified by adding hysteresis thresholding to allow a more detailed result. In Lacroix technique, we applied non-maximal suppression by keeping the size k × 1 of the relative window fixed at 3 ×1 [19]. For simplicity, in the case of Schunck edge detection, the non-maximal suppression method we used is the one proposed by Canny [15] and hysteresis thresholding was applied for a more efficient thresholding.
For all the images on the data set, the standard deviation (sigma) of the Gaussian filter in Canny's algorithm [15] was set to sigma = 1, whereas, the low and high thresholds were automatically calculated by the image histogram. In Deriche's technique [16], the parameters' values were set to a = 2 and w = 1.5. The Bergholm [18] parameter set was a combination of starting sigma, ending sigma and low and high threshold and these where starting sigma = 3.5, ending sigma = 0.7 and the thresholds were automatically determined as previously. For the Primary Rater in Lacroix's method [19], the coarsest resolution was set to σ_{2} = 2 and the finest one to σ_{0} = 0.7. The intermediate scale σ_{1} was computed according to the expression proposed in [19]. The gradient and homogeneity thresholds were estimated by the histogram of the gradient and homogeneity images, respectively. For the Schunck edge detector [20], the number of resolution scales was arbitrarily set to three as: σ_{1} = 0.7, σ_{2} = 1.2, σ_{3} = 1.7. The difference between two consecutive scales was selected not to be greater than 0.5 to avoid edge pixel displacement in the resulted edge maps. The values for the low and high thresholds were calculated by the histogram of the gradient magnitude image. In the case of Rothwell method [17], the alpha parameter was set to 0.9, the low threshold was estimated by the image histogram again and the value of the smoothing parameter, sigma, was equal to 1. It is important to stress out that the selected values for all of the above parameters fall within the ranges proposed in the literature by the authors of the individual detectors.
When the optimum correspondence threshold is estimated using the maximization of the 'Weighted Kappa Coefficient' approach, the value of the r parameter needs to be set. The cost, r, is initially determined according to the particular quality of the detection (FP or FN) that is chosen to be optimized. For example, as far as target object detection in military applications is concerned, missing existing targets in the image (misdetections) is less desirable than falsely detecting non-existing ones (false alarms). This is as well the scenario we assume in this piece of work; namely, we are primarily concerned with the elimination of FN at the expense of increasing the number of FP. Therefore, according to the analysis in the previous section, the cost value r should range from 0.5 to 1.
The above examples emphasize the ability of the approaches to combine high accuracy with good noise reduction in the final edge detection result. Insignificant information is cleared, while the one preserved allows for easy, fast and accurate object recognition. Furthermore, it is interesting to note that objects missed by one of the selected edge detectors are included in the final edge image. This is particularly obvious in the "Large Building" image set when comparing the final edge maps in Figure 4c,d with the Bergholm detection in Figure 5c. Finally, edges due to texture are suppressed in the final edge maps.
Comparing the edge maps produced by applying the above two approaches it is observed that the edge maps for the "weighted kappa coefficient" approach have better quality than those for the "ROC analysis". The objects detected by the "weighted kappa coefficient" approach are better defined regarding their shape and contour and the number of detected edges is greater. This is expected since the selected value of r is 0.7. The performance of the "weighted kappa coefficient" approach for the particular choice of r, seems to be superior to "ROC analysis" since it is more sensitive to minor details.
Detection error
Rothwell | Bergholm | Canny | Schunck | Lacroix | Deriche | ROC | Weighted kappa coeff. (r= 0.7) | |
---|---|---|---|---|---|---|---|---|
Buildings | 0.5506 | 0.6047 | 0.5593 | 0.5640 | 0.5736 | 0.5500 | 0.5425 | 0.4885 |
Baseball | 0.5710 | 0.5968 | 0.5668 | 0.5689 | 0.5865 | 0.5726 | 0.5505 | 0.5183 |
Airfield | 0.4707 | 0.4643 | 0.5057 | 0.5147 | 0.5340 | 0.5276 | 0.4634 | 0.4126 |
Homes | 0.5991 | 0.5703 | 0.5712 | 0.5743 | 0.4851 | 0.6302 | 0.5177 | 0.4402 |
Large building | 0.5292 | 0.5970 | 0.5328 | 0.5459 | 0.5180 | 0.5194 | 0.5084 | 0.4440 |
Main building | 0.5608 | 0.5651 | 0.6180 | 0.6130 | 0.5193 | 0.6677 | 0.5439 | 0.4561 |
Pool tennis | 0.5273 | 0.5639 | 0.5086 | 0.5367 | 0.5396 | 0.4852 | 0.5029 | 0.4343 |
School | 0.5638 | 0.5844 | 0.5571 | 0.5647 | 0.5701 | 0.5928 | 0.5444 | 0.5001 |
Series | 0.4321 | 0.5297 | 0.4143 | 0.4276 | 0.4522 | 0.5303 | 0.4139 | 0.3619 |
Woods | 0.5634 | 0.5846 | 0.5526 | 0.5695 | 0.5274 | 0.6856 | 0.5241 | 0.4602 |
Edge map similarity
Rothwell | Bergholm | Canny | Schunck | Lacroix | Deriche | ROC | Weighted kappa coeff. (r= 0.7) | |
---|---|---|---|---|---|---|---|---|
Buildings | 0.6886 | 0.5895 | 0.6892 | 0.6804 | 0.6570 | 0.6871 | 0.6946 | 0.7355 |
Baseball | 0.6815 | 0.6359 | 0.6961 | 0.6848 | 0.6579 | 0.6934 | 0.6995 | 0.7249 |
Airfield | 0.7692 | 0.7651 | 0.7593 | 0.7382 | 0.7162 | 0.7516 | 0.7760 | 0.8492 |
Homes | 0.6332 | 0.5978 | 0.6851 | 0.6762 | 0.7143 | 0.6169 | 0.7064 | 0.7599 |
Large building | 0.6844 | 0.5401 | 0.6976 | 0.6730 | 0.6823 | 0.7082 | 0.7014 | 0.7506 |
Main building | 0.6967 | 0.6191 | 0.6612 | 0.6499 | 0.6988 | 0.6088 | 0.6998 | 0.8357 |
Pool tennis | 0.6722 | 0.5923 | 0.7208 | 0.6596 | 0.6527 | 0.7100 | 0.7046 | 0.7639 |
School | 0.6495 | 0.5846 | 0.6743 | 0.6474 | 0.6419 | 0.6458 | 0.6669 | 0.7835 |
Series | 0.7577 | 0.6084 | 0.7682 | 0.7656 | 0.7259 | 0.6810 | 0.7767 | 0.8090 |
Woods | 0.6581 | 0.5912 | 0.6849 | 0.6748 | 0.6760 | 0.5667 | 0.6922 | 0.7448 |
Without code optimization, the MATLAB implementation of each of the proposed approaches ("ROC analysis" and the "weighted kappa coefficient") comfortably runs at around 2.8 s on a Pentium 2.8 GHz desktop for a 476 × 477 size image.
5 Conclusion
The selection of an edge detector operator is not a trivial problem, since different edge detectors often produce essentially varying edge maps, even if they follow similar mathematical approaches. In this paper, we propose two techniques for the automatic statistical analysis of the correspondence of edge images that have emerged from different operators; the ROC analysis and the weighted kappa coefficient method. Both techniques integrate efficiently the pre-selected set of edge detectors in terms of both the quality of the highlighted features and the elimination of noise and texture. However, the weighted kappa coefficient approach can be considered superior in the sense that the trade off between detection of minor edges and noise reduction can be quantified in advance as part of the problem specifications.
In future work, we intent to incorporate in the identification of edges information from their surrounding pixels. That means, the probability of a pixel being an edge will be affected by the state (edge/non-edge) of its neighbors. Furthermore, the possibility of using soft values for the final edge maps instead of hard edge extractors will be explored.
Appendix
Correlation coefficient between the probability of a pixel being a true edge and being detected as edge
The parameter ρ in Equation 1 denotes the correlation coefficient between the probability of a pixel being a true edge and being detected as edge. A positive correlation coefficient between two random variables indicates that these variables follow the same trend. In our case, the random variables of interest are the true edge image and the detected edge image. Therefore, a positive correlation coefficient indicates that if the probability of a pixel f(x_{1}, y_{1}) being a true edge is higher as compared to the same probability for the pixel f(x_{2}, y_{2}), then the probability of the pixel f(x_{1}, y_{1}) detected as edge pixel is also higher compared with the same probability for the pixel f(x_{2}, y_{2}).
Analytic form of probabilities for legitimate and random edge detection
Legitimate edge detection | Random edge detection | |
---|---|---|
TP | P · Q + ρ · σ_{ p } · σ_{ q } = P · SE | P · Q |
FP | P' · Q - ρ · σ_{ p } · σ_{ q } = P' · SP' | P' · Q |
FN | P · Q' - ρ · σ_{ p } · σ_{ q } = P · SE' | P · Q' |
TN | P' · Q' + ρ · σ_{ p } · σ_{ q } = P' · SP | P' · Q' |
Declarations
Acknowledgements
The investigations which are the subject of this paper were initiated by Dstl under the auspices of the United Kingdom Ministry of Defence Systems Engineering for Autonomous Systems Defence Technology Centre.
Authors’ Affiliations
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