- Research
- Open Access
A new construction of a fractional derivative mask for image edge analysis based on Riemann-Liouville fractional derivative
- Peter Amoako-Yirenkyi^{1, 2}Email authorView ORCID ID profile,
- Justice Kwame Appati^{1} and
- Isaac Kwame Dontwi^{1, 2}
https://doi.org/10.1186/s13662-016-0946-8
© Amoako-Yirenkyi et al. 2016
- Received: 8 February 2016
- Accepted: 16 August 2016
- Published: 15 September 2016
Abstract
We present a new way of constructing a fractional-based convolution mask with an application to image edge analysis. The mask was constructed based on the Riemann-Liouville fractional derivative which is a special form of the Srivastava-Owa operator. This operator is generally known to be robust in solving a range of differential equations due to its inherent property of memory effect. However, its application in constructing a convolution mask can be devastating if not carefully constructed. In this paper, we show another effective way of constructing a fractional-based convolution mask that is able to find edges in detail quite significantly. The resulting mask can trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges. The experiments conducted on the mask were done using some selected well known synthetic and Medical images with realistic geometry. Using visual perception and performing both mean square error and peak signal-to-noise ratios analysis, the method demonstrated significant advantages over other known methods.
Keywords
- convolution
- fractional integral
- fractional derivative
- edge detection
- Riemann-Liouville
1 Introduction
Image edge analysis constitutes a set of mathematical methods which aim at identifying points in a digital image at which the image brightness changes sharply (point of discontinuities). The organization of these points into a set of curved line segments then becomes the edge. With no doubt, detecting these points and subsequently constructing an edge map is one of the most common and fundamental operations in image processing and analysis since it is consistent with the human perception and serves as the first step in image understanding and interpretation. They provide useful structural [1, 2] information, which can be used for feature extraction, object identification and region segmentation. This information, by common practice, is extracted by developing a convolution mask mostly known as the gradient operator which is a relatively smaller two dimensional array where each pixel value of the original image is modified according to the value of the neighborhood around the pixel of interest (POI) [3]. However, there are other forms, such as the segmentation- and transform-based operators, which could also be used for edge extraction. The definitions of particular operators in any of these three categories have their own pros and cons. Although several studies [4–7] have been done to improve these methods for edge detection, they still produce edges with some compromise in accuracy, completeness, complexity, connectedness, and smoothness. A recent study [8] by Guo and Lai confirmed how gradient operators resist just low-level noise but tend to mistakenly detect fake edges in the presence of excessive noise or artifacts. In an attempt to resolve some of these issues other operators [9–12] mostly based on fractional calculus with improved characteristics over the classical methods like Canny and Prewitt have been proposed.
In particular the fractional-based operators have been used in image quality enhancement, image texture enhancement [13], image denoising [14], and image edge analysis. Among such operators [15], which are a generalization of the concept of an integer-order derivative to real order and the n-fold integral operator, are the Caputo, Erdélyi-Kober, Srivastava-Owa, and Weyl-Riesz operators [16], the Riemann-Liouville operator, and the Grünwald-Letnikov operators. Typically, the operators generate both high and low frequencies with the high frequencies characterizing a large change in pixel intensity value over a small distance including noise and edges. In contrast, the low frequency is characterized by a small change in pixel intensity value where background and texture in the image can be found [17]. This means that, in the presence of high-level noise, some of these operators proposed, if not carefully constructed, will tend to mistakenly detect fake edges.
In this paper, we present a new construction of fractional-based convolution mask for image edge analysis with equivalent complexity (\(\bigcirc(mn \log(mn)) \)) as the standard gradient operators but with significant robustness to noise. We also show that it is able to detect edges very well as a result of the memory (kernel) function embedded in the fractional derivative. These interesting characteristic allows the operator to describe systems with memory phenomena. The paper is organized as follows: we start by reviewing some edge detection operators in Section 2. Section 3 discusses the generalized fractional calculus operator adopted for this study and subsequently in Section 4 we show how the proposed mask is constructed. In Section 5.2, we compare results from the proposed mask with two methods known to perform well. Finally, conclusions are drawn in Section 6.
2 Review of edge operators
In this section, some existing edge operators or detectors are reviewed. Edge detection is an important field in image processing and an effective edge detector is expected to reduce a large amount of data, while keeping most of the important features contained in an image. These operators are usually categorized as the gradient-based edge detectors [18–20], segmentation-based edge operators [21, 22] and the transform-based edge operators [23, 24]. The first category of edge detectors are mostly based on either first-order gradient operators or second-order operators, sometimes called Laplacians. According to [20], although higher orders are more accurate compared with first-order operators, it is relatively sensitive to noise when extracting relatively more information. For example gradient-based edge detection operators, such as the Roberts, Sobel, and Prewitts, Laplacian of Gaussian (LoG) and their improvements [25–30] uses 2-D linear filters to process vertical and horizontal edges separately in order to approximate the first-order derivative of pixel values of an image. The work of [19] also presents a classified and comparative study of edge detection operators. In the study, the Canny operator proved to be better than LoG experimentally while LOG was better than Prewitt and Sobel in handling noisy images. A 2-D gamma distribution in the work of [18] demonstrated the efficiency of their method but, however, suffered from the drawback of big time complexity as a result of the constructed masks.
Aside from the segmentation-based edge detection, we as well have the transform-based edge detectors. Current research in this area is the wavelet approach with box spline tight framelets in the eighth direction (\(B_{8}\)) as proposed by [8]. These operators have predefined properties such as compactness and smoothness, which makes it possible to approximate various edges and features better [23, 24, 33]. Nonetheless, from their study, it was concluded that the operator was efficient in tracking edges more accurately but is eight and five times more computationally expensive than that of the wavelet- and shearlet-based methods, respectively. However, it is comparable to that of the Chan-Vese method [8].
Among other edge operators used we have: the morphological gradient [34], the high-order and variable-order total variation [35], and the Mumford-Shah Green function [36] and fractal geometry-based methods [37].
3 Generalized fractional calculus operator
Fractional calculus generalizes the concept of classical calculus by taking into account non-integer orders. These classes of operators become useful when handling natural problems with memory effect since by definition these operators already possess the memory kernel [38–40] As edges need to be close and complete, based on the edge points, it is common to make use of these operators in constructing the edge map.
In this section, we provide the definitions with some preliminary concept of fractional calculus which will be the bases for the construction of a fractional mask.
Definition 3.1
(Gamma function)
Definition 3.2
(Beta function)
From Definition 3.1 and Definition 3.2, we define the fractional derivative and integral operator as follows in Definition 3.3 and Definition 3.4, respectively.
Definition 3.3
Definition 3.4
Theorem 3.1
Proof
3.1 Previous methods for constructing fractional mask
In 2014, Gao et al. wrote a paper on ‘Edge detection based on the Newton interpolation’s fractional differentiation’ and made use of a generalized Grunwald-Letnikov definition as in equation (21). In their approach, the summation term was expanded to a number of terms equivalent to the size of the mask required. Here, it was believed that these discrete points were not precise enough and needed to be improved using the Newton interpolation method.
We note that although the Riemann-Liouville definition theoretically provides an exact value for the purpose of calculus it is practically difficult when used to evaluate an integral or a derivative. Theoretically, the Riemann-Liouville definition is equivalent to the Grunwald-Letnikov definition but one question always arises as to what number of terms should be computed and summed for the Grunwald-Letnikov definition of fractional derivative to be as accurate as that of the Riemann-Liouville definition. In an attempt to answer this question Loverro et al. [44] used up to 171 terms to obtain an error of \(1\mathrm{e}\mbox{-}3\%\). This in a way implies that a mask size of \(171\times171\) is required for that accuracy to be achieved. Unfortunately, the bigger the mask size, the more computationally expensive it becomes and hence this theoretical equivalence is mostly not achievable in practice.
Interestingly, the masks are extracted without evaluating the entire derivative and therefore, when carefully constructed, it can produce more desired masks results.
4 Construction of the proposed mask
Definition 4.1
Theorem 4.1
Definition 4.2
Definition 4.3
4.1 Edge mask representation
Proposed x -directional fractional mask
\(\frac{2\alpha\sqrt{8^{\alpha}}}{8\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt {5^{\alpha}}}{5\Gamma(1-\alpha)}\) | 0 | \(\frac{-\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{-2\alpha\sqrt{8^{\alpha}}}{8\Gamma (1-\alpha)}\) |
\(\frac{2\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt {2^{\alpha}}}{2\Gamma(1-\alpha)}\) | 0 | \(\frac{-\alpha\sqrt{2^{\alpha}}}{2\Gamma(1-\alpha)}\) | \(\frac{-2\alpha\sqrt{5^{\alpha}}}{5\Gamma (1-\alpha)}\) |
\(\frac{2\alpha\sqrt{4^{\alpha}}}{4\Gamma(1-\alpha)}\) | \(\frac{\alpha }{\Gamma(1-\alpha)}\) | 0 | \(\frac{-\alpha}{\Gamma(1-\alpha)}\) | \(\frac {-2\alpha\sqrt{4^{\alpha}}}{4\Gamma(1-\alpha)}\) |
\(\frac{2\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt {2^{\alpha}}}{2\Gamma(1-\alpha)}\) | 0 | \(\frac{-\alpha\sqrt{2^{\alpha}}}{2\Gamma(1-\alpha)}\) | \(\frac{-2\alpha\sqrt{5^{\alpha}}}{5\Gamma (1-\alpha)}\) |
\(\frac{2\alpha\sqrt{8^{\alpha}}}{8\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt {5^{\alpha}}}{5\Gamma(1-\alpha)}\) | 0 | \(\frac{-\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{-2\alpha\sqrt{8^{\alpha}}}{8\Gamma (1-\alpha)}\) |
Proposed y -directional fractional mask
\(\frac{2\alpha\sqrt{8^{\alpha}}}{8\Gamma(1-\alpha)}\) | \(\frac{2\alpha \sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{2\alpha\sqrt{4^{\alpha}}}{4\Gamma(1-\alpha)}\) | \(\frac{2\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha )}\) | \(\frac{2\alpha\sqrt{8^{\alpha}}}{8\Gamma(1-\alpha)}\) |
\(\frac{\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt {2^{\alpha}}}{2\Gamma(1-\alpha)}\) | \(\frac{\alpha}{\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt{2^{\alpha}}}{2\Gamma(1-\alpha)}\) | \(\frac{\alpha\sqrt {5^{\alpha}}}{5\Gamma(1-\alpha)}\) |
0 | 0 | 0 | 0 | 0 |
\(\frac{-\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{-\alpha \sqrt{2^{\alpha}}}{2\Gamma(1-\alpha)}\) | \(\frac{-\alpha}{\Gamma(1-\alpha )}\) | \(\frac{-\alpha\sqrt{2^{\alpha}}}{2\Gamma(1-\alpha)}\) | \(\frac {-\alpha\sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) |
\(\frac{-2\alpha\sqrt{8^{\alpha}}}{8\Gamma(1-\alpha)}\) | \(\frac{-2\alpha \sqrt{5^{\alpha}}}{5\Gamma(1-\alpha)}\) | \(\frac{-2\alpha\sqrt{4^{\alpha}}}{4\Gamma(1-\alpha)}\) | \(\frac{-2\alpha\sqrt{5^{\alpha}}}{5\Gamma (1-\alpha)}\) | \(\frac{-2\alpha\sqrt{8^{\alpha}}}{8\Gamma(1-\alpha)}\) |
4.2 Performance metric of proposed mask
In order to justify the performance of the proposed mask, we use the following standard measures as detailed in the subsections below.
4.2.1 Mean squared error
4.2.2 Peak signal-to-noise ratio
4.2.3 Structural similarity index
5 Result and analysis of proposed mask
5.1 Data source
5.2 Numerical experiments
This section consists of three sets of experimental results. In Section 5.2.1, we compared the proposed fractional edge detector with some selected edge detectors: Canny [46], and Tiansi [9], using the mean square error and the peak signal-to-noise ratio. In Section 5.2.2, we also tested the performance of the proposed method in detecting edges of an images under various noise conditions using the structure similarity index measure, and in Section 5.2.3 we focused on object segmentation of some selected Medical image. The choice of these methods is based on earlier studies like [9, 18, 19, 41].
5.2.1 Experimental result 1: performance test
In this section, we demonstrate the general performance of our proposed mask using different fractional orders of the differential operator. Here, the peak signal-to-noise ratio (PSNR) and the mean square error (MSE) of the edge information are extracted during the implementation of the fractional mask and Canny. These algorithms are applied to the images 2 tagged as Linear, Non-linear, and Medical image-1. It is interesting to note that these algorithms behave differently with varying mask size. In this work, a mask of size \(3\times3\), \(5\times5\), \(7\times7\), and \(9\times9\) is tested on each image for a fair comparison. It is expected that the higher the value of the PSNR, the better the edge information extracted. In contrast, the lower the MSE value, the closer the edge information extracted is, to the true and actual edge map.
Observation on a \(3\times3\) mask
Proposed method with Canny using a \(\pmb{3\times3}\) mask size
Method | Linear image | Non-linear image | Medical image | |||
---|---|---|---|---|---|---|
MSE | PSNR | MSE | PSNR | MSE | PSNR | |
CANNY | 0.2571 | 5.8995 | 0.3012 | 5.2111 | 0.3178 | 4.9782 |
PRO α = 0.1 | 0.2476 | 6.0633 | 0.2945 | 5.3086 | 0.2888 | 5.3935 |
PRO α = 0.2 | 0.2479 | 6.0564 | 0.2943 | 5.3118 | 0.2892 | 5.3877 |
PRO α = 0.3 | 0.2471 | 6.0718 | 0.2945 | 5.3087 | 0.2907 | 5.3650 |
PRO α = 0.4 | 0.2476 | 6.0632 | 0.2947 | 5.3059 | 0.2911 | 5.3597 |
PRO α = 0.5 | 0.2465 | 6.0817 | 0.2953 | 5.2973 | 0.2916 | 5.3516 |
PRO α = 0.6 | 0.2476 | 6.0623 | 0.2957 | 5.2915 | 0.2922 | 5.3432 |
PRO α = 0.7 | 0.2480 | 6.0558 | 0.2960 | 5.2864 | 0.2940 | 5.3165 |
PRO α = 0.8 | 0.2475 | 6.0637 | 0.2960 | 5.2874 | 0.2942 | 5.3132 |
PRO α = 0.9 | 0.2478 | 6.0588 | 0.2962 | 5.2845 | 0.2957 | 5.2919 |
Observation on a \(7\times7\) mask
Proposed method with Canny using a \(\pmb{7\times7}\) mask size
Method | Linear image | Non-linear image | Medical image | |||
---|---|---|---|---|---|---|
MSE | PSNR | MSE | PSNR | MSE | PSNR | |
CANNY | 0.1296 | 8.8727 | 0.1025 | 9.8911 | 0.1462 | 8.3508 |
PRO α = 0.1 | 0.0308 | 15.1136 | 0.0509 | 12.9351 | 0.0761 | 11.1853 |
PRO α = 0.2 | 0.0324 | 14.8995 | 0.0512 | 12.9059 | 0.0826 | 10.8284 |
PRO α = 0.3 | 0.0349 | 14.5731 | 0.0525 | 12.8024 | 0.0857 | 10.6699 |
PRO α = 0.4 | 0.0386 | 14.1369 | 0.0540 | 12.6788 | 0.0878 | 10.5664 |
PRO α = 0.5 | 0.0412 | 13.8542 | 0.0559 | 12.5255 | 0.0906 | 10.4278 |
PRO α = 0.6 | 0.0441 | 13.5601 | 0.0581 | 12.3555 | 0.0929 | 10.3218 |
PRO α = 0.7 | 0.0459 | 13.3825 | 0.0604 | 12.1876 | 0.0966 | 10.1494 |
PRO α = 0.8 | 0.0529 | 12.7643 | 0.0635 | 11.9705 | 0.0999 | 10.0030 |
PRO α = 0.9 | 0.0571 | 12.4334 | 0.0647 | 11.8881 | 0.1039 | 9.8346 |
Observation on a \(5\times5\) mask
Proposed method and Tiansi with Canny using a \(\pmb{5\times5}\) mask size
Method | Linear image | Non-linear image | Medical image | |||
---|---|---|---|---|---|---|
MSE | PSNR | MSE | PSNR | MSE | PSNR | |
CANNY | 0.2266 | 6.4483 | 0.2670 | 5.7342 | 0.2393 | 6.2102 |
PRO α = 0.1 | 0.1580 | 8.0138 | 0.1607 | 7.9402 | 0.1538 | 8.1298 |
PRO α = 0.2 | 0.1594 | 7.9754 | 0.1659 | 7.8009 | 0.1594 | 7.9746 |
PRO α = 0.3 | 0.1651 | 7.8232 | 0.1730 | 7.6206 | 0.1669 | 7.7753 |
PRO α = 0.4 | 0.1718 | 7.6488 | 0.1784 | 7.4856 | 0.1732 | 7.6155 |
PRO α = 0.5 | 0.1759 | 7.5465 | 0.1858 | 7.3088 | 0.1786 | 7.4805 |
PRO α = 0.6 | 0.1806 | 7.4329 | 0.1945 | 7.1100 | 0.1419 | 8.4810 |
PRO α = 0.7 | 0.1849 | 7.3301 | 0.2003 | 6.9842 | 0.1462 | 8.3519 |
PRO α = 0.8 | 0.1889 | 7.2382 | 0.1591 | 7.9826 | 0.1502 | 8.2345 |
PRO α = 0.9 | 0.1930 | 7.1440 | 0.1655 | 7.8130 | 0.1573 | 8.0340 |
TIA α = 0.1 | 0.1702 | 7.6899 | 0.1765 | 7.5337 | 0.1686 | 7.7307 |
TIA α = 0.2 | 0.1764 | 7.5344 | 0.1825 | 7.3883 | 0.1762 | 7.5388 |
TIA α = 0.3 | 0.1835 | 7.3626 | 0.1901 | 7.2093 | 0.1842 | 7.3468 |
TIA α = 0.4 | 0.1896 | 7.2214 | 0.1981 | 7.0312 | 0.1484 | 8.2843 |
TIA α = 0.5 | 0.1957 | 7.0833 | 0.2081 | 6.8166 | 0.1542 | 8.1201 |
TIA α = 0.6 | 0.2035 | 6.9149 | 0.2171 | 6.6326 | 0.1640 | 7.8520 |
TIA α = 0.7 | 0.2140 | 6.6951 | 0.1865 | 7.2938 | 0.1785 | 7.4829 |
TIA α = 0.8 | 0.1681 | 7.7450 | 0.2008 | 6.9718 | 0.1922 | 7.1620 |
TIA α = 0.9 | 0.1830 | 7.3749 | 0.2159 | 6.6569 | 0.2077 | 6.8246 |
Observation on a \(9\times9\) mask
Proposed method with Canny using a \(\pmb{9\times9}\) mask size
Method | Linear image | Non-linear image | Medical image | |||
---|---|---|---|---|---|---|
MSE | PSNR | MSE | PSNR | MSE | PSNR | |
CANNY | 0.0810 | 10.9149 | 0.0762 | 11.1777 | 0.1148 | 9.4000 |
PRO α = 0.1 | 0.0142 | 18.4743 | 0.0347 | 14.5988 | 0.0532 | 12.7443 |
PRO α = 0.2 | 0.0140 | 18.5449 | 0.0360 | 14.4375 | 0.0543 | 12.6515 |
PRO α = 0.3 | 0.0143 | 18.4441 | 0.0368 | 14.3451 | 0.0558 | 12.5306 |
PRO α = 0.4 | 0.0146 | 18.3638 | 0.0381 | 14.1945 | 0.0568 | 12.4597 |
PRO α = 0.5 | 0.0146 | 18.3684 | 0.0400 | 13.9797 | 0.0588 | 12.3071 |
PRO α = 0.6 | 0.0149 | 18.2827 | 0.0417 | 13.7940 | 0.0605 | 12.1796 |
PRO α = 0.7 | 0.0193 | 17.1344 | 0.0436 | 13.6057 | 0.0702 | 11.5370 |
PRO α = 0.8 | 0.0210 | 16.7856 | 0.0448 | 13.4854 | 0.0724 | 11.4034 |
PRO α = 0.9 | 0.0227 | 16.4349 | 0.0462 | 13.3500 | 0.0754 | 11.2258 |
General remarks
From all the observations made on \(3\times3\), \(5\times5\), \(7\times7\), and \(9\times9\) it was clear that increasing the mask size irrespective of the image type or the derivative operator increases the performance metric value. This is due to the choice of the standard deviation, σ in the Gaussian filter which has a direct influence on the size of the mask. Using the same σ value in the Gaussian filter on all three methods also meant that smoothing techniques were the same and therefore should not affect the results. The only part which was varied was the choice of the mask.
5.2.2 Experimental result 2: noise immunity
- (1)
Gauss: Gaussian white noise with constant mean and variance.
- (2)
S & P: salt & pepper noise.
- (3)
localvar: zero mean Gaussian white noise with an intensity dependent variance
- (4)
speckle: speckle or Multiplicative noise.
- (5)
Poisson: shot noise.
- (6)
motion: Motion Blur (blurry pixels).
- (7)
erosion: morphological erosion.
- (8)
dilation: morphological dilation.
- (9)
jpg compression blocking effect: compression artifact.
Noise immunity with \(3\times3\) mask
Noise immunity with \(\pmb{3\times3}\) Canny mask at varying standard deviations (Noise SD)
Noise type | Noise SD | 20 | 25 | 30 | 35 | 40 | 45 |
---|---|---|---|---|---|---|---|
Motion | Linear image | 0.7536 | 0.7094 | 0.6124 | 0.6013 | 0.5848 | 0.5910 |
Non-linear image | 0.5785 | 0.4246 | 0.3709 | 0.3618 | 0.3529 | 0.3386 | |
Medical image | 0.6707 | 0.4566 | 0.3815 | 0.3785 | 0.3774 | 0.3681 | |
Gauss | Linear image | 0.1081 | 0.0892 | 0.0777 | 0.0655 | 0.0563 | 0.0555 |
Non-linear image | 0.0861 | 0.0650 | 0.0535 | 0.0431 | 0.0375 | 0.0331 | |
Medical image | 0.0426 | 0.0361 | 0.0235 | 0.0208 | 0.0155 | 0.0129 | |
S & P | Linear image | 0.7103 | 0.5708 | 0.4780 | 0.4003 | 0.3392 | 0.2523 |
Non-linear image | 0.6731 | 0.5846 | 0.4842 | 0.4230 | 0.3655 | 0.2897 | |
Medical image | 0.7241 | 0.6630 | 0.5875 | 0.4919 | 0.3687 | 0.3162 | |
Speckle | Linear image | 0.2618 | 0.2416 | 0.2266 | 0.2132 | 0.2021 | 0.1947 |
Non-linear image | 0.2303 | 0.1966 | 0.1641 | 0.1513 | 0.1338 | 0.1147 | |
Medical image | 0.5528 | 0.5280 | 0.4972 | 0.4614 | 0.4571 | 0.4373 |
Noise immunity with a \(\pmb{3\times3}\) proposed mask at \(\pmb{\alpha= 0.5}\) and varying standard deviations (Noise SD)
Noise type | Noise SD | 20 | 25 | 30 | 35 | 40 | 45 |
---|---|---|---|---|---|---|---|
Motion | Linear image | 0.7788 | 0.7137 | 0.6056 | 0.5839 | 0.5704 | 0.5569 |
Non-linear image | 0.6364 | 0.4761 | 0.4056 | 0.3939 | 0.3847 | 0.3647 | |
Medical image | 0.7040 | 0.5028 | 0.4251 | 0.4174 | 0.4127 | 0.4025 | |
Gauss | Linear image | 0.1249 | 0.1007 | 0.0893 | 0.0785 | 0.0686 | 0.0604 |
Non-linear image | 0.1033 | 0.0793 | 0.0665 | 0.0559 | 0.0480 | 0.0430 | |
Medical | 0.0593 | 0.0451 | 0.0336 | 0.0274 | 0.0199 | 0.0178 | |
S & P | Linear image | 0.6006 | 0.5064 | 0.4258 | 0.3549 | 0.3100 | 0.2409 |
Non-linear image | 0.6909 | 0.5616 | 0.4560 | 0.3974 | 0.3267 | 0.2580 | |
Medical image | 0.7545 | 0.6764 | 0.5963 | 0.4477 | 0.3870 | 0.2940 | |
Speckle | Linear image | 0.2526 | 0.2276 | 0.2298 | 0.2117 | 0.2060 | 0.1967 |
Non-linear image | 0.2887 | 0.2305 | 0.2005 | 0.1915 | 0.1629 | 0.1436 | |
Medical image | 0.6337 | 0.5858 | 0.5691 | 0.5267 | 0.5125 | 0.4960 |
Noise immunity with \(7\times7\) mask
Noise immunity with a \(\pmb{7\times7}\) Canny mask at varying standard deviations (Noise SD)
Noise type | Noise SD | 20 | 25 | 30 | 35 | 40 | 45 |
---|---|---|---|---|---|---|---|
Motion | Linear image | 0.9770 | 0.9352 | 0.9112 | 0.8528 | 0.7886 | 0.7685 |
Non-linear image | 0.8399 | 0.7029 | 0.5816 | 0.5188 | 0.4836 | 0.4618 | |
Medical image | 0.8096 | 0.6450 | 0.5414 | 0.5034 | 0.4873 | 0.4756 | |
Gauss | Linear image | 0.3329 | 0.2984 | 0.1984 | 0.2002 | 0.1816 | 0.1355 |
Non-linear image | 0.2168 | 0.1610 | 0.1409 | 0.1130 | 0.0946 | 0.0866 | |
Medical image | 0.3416 | 0.2301 | 0.1167 | 0.1014 | 0.0668 | 0.0595 | |
S & P | Linear image | 0.7577 | 0.6590 | 0.5930 | 0.5193 | 0.4242 | 0.3738 |
Non-linear image | 0.7300 | 0.6223 | 0.5186 | 0.4388 | 0.3857 | 0.3127 | |
Medical image | 0.6902 | 0.5847 | 0.5017 | 0.3985 | 0.3274 | 0.2593 | |
Speckle | Linear image | 0.5638 | 0.4109 | 0.4196 | 0.3537 | 0.3592 | 0.3213 |
Non-linear image | 0.6205 | 0.4995 | 0.4910 | 0.4148 | 0.3456 | 0.3378 | |
Medical image | 0.7696 | 0.7213 | 0.7005 | 0.6748 | 0.6673 | 0.6574 |
Noise immunity with a \(\pmb{7\times7}\) proposed mask at \(\pmb{\alpha= 0.5}\) at varying standard deviations (Noise SD)
Noise type | Noise SD | 20 | 25 | 30 | 35 | 40 | 45 |
---|---|---|---|---|---|---|---|
Motion | Linear image | 0.9820 | 0.9540 | 0.9259 | 0.8792 | 0.8204 | 0.7992 |
Non-linear image | 0.8665 | 0.7562 | 0.6426 | 0.5662 | 0.5219 | 0.4961 | |
Medical image | 0.8466 | 0.7085 | 0.5900 | 0.5357 | 0.5082 | 0.4939 | |
Gauss | Linear image | 0.6136 | 0.3837 | 0.3749 | 0.2832 | 0.2750 | 0.2799 |
Non-linear image | 0.5000 | 0.3616 | 0.2218 | 0.1723 | 0.1367 | 0.1275 | |
Medical image | 0.5370 | 0.3380 | 0.3544 | 0.1866 | 0.1445 | 0.1009 | |
S & P | Linear image | 0.8089 | 0.7549 | 0.6689 | 0.6732 | 0.5958 | 0.4860 |
Non-linear image | 0.8150 | 0.7181 | 0.6169 | 0.5534 | 0.4522 | 0.4559 | |
Medical image | 0.6529 | 0.5854 | 0.4799 | 0.3947 | 0.3259 | 0.2649 | |
Speckle | Linear image | 0.7800 | 0.6257 | 0.6336 | 0.4921 | 0.4228 | 0.4460 |
Non-linear image | 0.7456 | 0.7243 | 0.6520 | 0.5742 | 0.5087 | 0.4310 | |
Medical image | 0.7720 | 0.7466 | 0.7289 | 0.7109 | 0.6961 | 0.6829 |
Noise immunity with \(9\times9\) mask
Noise immunity with a \(\pmb{9\times9}\) Canny mask at varying standard deviations (Noise SD)
Noise type | Noise SD | 20 | 25 | 30 | 35 | 40 | 45 |
---|---|---|---|---|---|---|---|
Motion | Linear image | 0.9714 | 0.9355 | 0.9083 | 0.8639 | 0.7971 | 0.7770 |
Non-linear image | 0.8716 | 0.7344 | 0.6140 | 0.5439 | 0.5036 | 0.4792 | |
Medical image | 0.8199 | 0.6667 | 0.5618 | 0.5152 | 0.4919 | 0.4815 | |
Gauss | Linear image | 0.4364 | 0.2718 | 0.2716 | 0.2771 | 0.2089 | 0.1944 |
Non-linear image | 0.3338 | 0.2347 | 0.1542 | 0.1469 | 0.1256 | 0.1021 | |
Medical image | 0.4525 | 0.2374 | 0.1634 | 0.1323 | 0.1140 | 0.0881 | |
S & P | Linear image | 0.7592 | 0.6748 | 0.6125 | 0.5529 | 0.4473 | 0.4384 |
Non-linear image | 0.7285 | 0.6679 | 0.5631 | 0.4614 | 0.4139 | 0.3344 | |
Medical image | 0.6997 | 0.5756 | 0.4879 | 0.4012 | 0.3158 | 0.2577 | |
Speckle | Linear image | 0.6689 | 0.4865 | 0.5035 | 0.4113 | 0.4316 | 0.3536 |
Non-linear image | 0.6958 | 0.5763 | 0.5672 | 0.4927 | 0.3948 | 0.3511 | |
Medical image | 0.7557 | 0.7328 | 0.7107 | 0.6896 | 0.6820 | 0.6588 |
Noise immunity with \(\pmb{9\times9}\) proposed mask at \(\pmb{\alpha= 0.5}\) at varying standard deviations (Noise SD)
Noise type | Noise SD | 20 | 25 | 30 | 35 | 40 | 45 |
---|---|---|---|---|---|---|---|
Motion | Linear image | 0.9818 | 0.9493 | 0.9270 | 0.8973 | 0.8447 | 0.8151 |
Non-linear image | 0.9098 | 0.8058 | 0.7038 | 0.6222 | 0.5671 | 0.5380 | |
Medical | 0.8805 | 0.7732 | 0.6670 | 0.6031 | 0.5667 | 0.5427 | |
Gauss | Linear image | 0.8422 | 0.6467 | 0.6009 | 0.4269 | 0.4777 | 0.3587 |
Non-linear image | 0.6410 | 0.5999 | 0.4123 | 0.3266 | 0.2665 | 0.1951 | |
Medical image | 0.6285 | 0.5280 | 0.4705 | 0.3381 | 0.2510 | 0.2579 | |
S & P | Linear image | 0.9254 | 0.8855 | 0.8496 | 0.7843 | 0.7707 | 0.7043 |
Non-linear image | 0.8621 | 0.8072 | 0.7358 | 0.6759 | 0.5891 | 0.6016 | |
Medical | 0.6925 | 0.5735 | 0.4810 | 0.3943 | 0.4067 | 0.3767 | |
Speckle | Linear image | 0.9186 | 0.8753 | 0.8085 | 0.6825 | 0.5964 | 0.6200 |
Non-linear image | 0.8521 | 0.8050 | 0.7481 | 0.6977 | 0.6382 | 0.5735 | |
Medical image | 0.8218 | 0.7981 | 0.7731 | 0.7530 | 0.7426 | 0.7194 |
Noise immunity with \(5\times5\) mask
5.2.3 Experimental result 3: segmentation
One of the most important tasks in Medical image analysis is segmentation, which is the process of partitioning an image into a set of distinct regions, which are different in some important qualitative or quantitative way. It therefore becomes a critical intermediate step in all high-level object recognition tasks, especially in computer assisted imagery. To test the proposed method in this context, three standard medical test images were selected and their results compared to that of Canny. The Single Seed Region Growing algorithm was employed at this stage for the segmentation based on the output of the edge maps generated by Canny and the proposed method. A mask size of \(9\times 9\) was used for this purpose.
Segmentation on Medical image-1
Segmentation on Medical image-2
Segmentation on Medical image-3
6 Conclusion
We have presented another way of constructing a fractional-based convolution mask for image edge analysis using the Riemann-Liouville fractional derivative formulation. Unlike other constructions, we extracted the mask, maintaining enough memory without the need for complicated optimization criteria. We performed both quantitative and qualitative comparative analysis with existing edge detectors and have demonstrated the effectiveness and efficiency of the proposed construction in detecting several edge types including step, Dirac edges and hidden edges found in the images used to perform the experiments. In addition, we have shown that the resulting mask is robust to noise. We also performed object identification using the resulting mask and generated mostly significant improvement over the methods studied.
Declarations
Acknowledgements
We would like to acknowledge the support received from the National Institute for Mathematical Sciences, Ghana for this study
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Ghimpeteanu, G, Batard, T, Bertalmio, M, Levine, S: A decomposition framework for image denoising algorithms. IEEE Trans. Image Process. 25, 388-399 (2016) MathSciNetView ArticleGoogle Scholar
- Yan, R, Shao, L: Blind image blur estimation via deep learning. IEEE Trans. Image Process. 25, 1910-1921 (2016) MathSciNetGoogle Scholar
- Jalab, HA, Ibrahim, RW: Texture enhancement based on the Savitzky-Golay fractional differential operator. Math. Probl. Eng. 2013, Article ID 149289 (2013) MathSciNetMATHGoogle Scholar
- Oram, JJ, McWilliams, JC, Stolzenbach, KD: Gradient-based edge detection and feature classification of sea-surface images of the Southern California Bight. Remote Sens. Environ. 112(5), 2397-2415 (2008) View ArticleGoogle Scholar
- Shrivakshan, GT, Chandrasekar, C: A comparison of various edge detection techniques used in image processing. Int. J. Comput. Sci. Issues 9(1), 269-276 (2012) Google Scholar
- Yasri, I, Hamid, NH, Yap, VV: An FPGA implementation of gradient based edge detection algorithm design. In: International Conference on Computer Technology and Development. IEEE Press, New York (2009) Google Scholar
- El-Sayed, MA: A new algorithm based entropic threshold for edge detection in images. Int. J. Comput. Sci. Issues 8(1), 71-78 (2011) Google Scholar
- Guo, W, Lai, M-J: Box spline wavelet frames for image edge analysis. SIAM J. Imaging Sci. 6(3), 1553-1578 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Yang, Z, Lang, F, Yu, X, Zhang, Y: The construction of fractional differential gradient operator. J. Comput. Inf. Syst. 7, 4328-4342 (2011) Google Scholar
- Oustaloup, A, Mathieu, B, Melchior, P: Edge detection using non integer derivation. In: IEEE European Conference on Circuit Theory and Design (ECCTD ’91), Copenhagen, Denmark, 3-6 September (1991) Google Scholar
- Garg, V, Singh, K: An improved Grunwald-Letnikov fractional differential mask for image texture enhancement. Int. J. Adv. Comput. Sci. Appl. 3(3), 130-135 (2012) Google Scholar
- Gao, C, Zhou, J, Zhang, W: Edge detection based on the Newton interpolation’s fractional differentiation. Int. Arab J. Inf. Technol. 11(3), 223-228 (2014) Google Scholar
- Pu, Y, Wang, W, Zhou, J, Wang, Y, Jia, H: Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation. Sci. China, Ser. F 51(9), 1319-1339 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Jalab, HA, Ibrahim, RW: Texture feature extraction based on fractional mask convolution with Cesaro means for content-based image retrieval. In: 12th Pacific Rim International Conference on Trends in Artificial Intelligence (PRICAI ’12), pp. 170-179 (2012) Google Scholar
- Dalir, M, Bashour, M: Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021-1032 (2010) MathSciNetMATHGoogle Scholar
- Jalab, HA, Ibrahim, RW: Denoising algorithm based on generalized fractional integral operator with two parameters. Discrete Dyn. Nat. Soc. 2012, Article ID 529849 (2012) MathSciNetView ArticleMATHGoogle Scholar
- McAndrew, A: An introduction to digital image processing with Matlab notes for SCM2511 image processing. School of Computer Science and Mathematics, Victoria University of Technology (2004) Google Scholar
- El-Zaart, A: A novel method for edge detection using 2 dimensional gamma distribution. J. Comput. Sci. 6(2), 199-204 (2010) View ArticleGoogle Scholar
- Roushdy, M: Comparative study of edge detection algorithms applying on the grayscale noisy image using morphological filter. GVIP, special issue on edge detection, 51-59 (2007) Google Scholar
- Wang, M, Yuan, S: A hybrid genetic algorithm based edge detection method for SAR image. In: IEEE Proceedings of the Radar Conference, 9-12 May, pp. 503-506 (2005) Google Scholar
- Chan, TF, Vese, LA: Active contours without edges. IEEE Trans. Image Process. 10, 266-277 (2001) View ArticleMATHGoogle Scholar
- Mumford, D, Shah, J: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 557-685 (1989) MathSciNetView ArticleMATHGoogle Scholar
- He, W, Lai, MJ: Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities. Appl. Comput. Harmon. Anal. 6, 53-74 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Lai, MJ: Construction of multivariate compactly supported prewavelets in \(L_{2}\) spaces and pre-Riesz basis in Sobolev spaces. J. Approx. Theory 142, 83-115 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Aurich, V, Weule, J: Nonlinear Gaussian filters performing edge preserving diffusion. In: Proceedings of the 17th Deutsche Arbeitsgemeinschaft für Mustererkennung (DAGM) Symposium, Bielefeld, Germany, 13-15 September, pp. 538-545. Springer, Berlin (1995) Google Scholar
- Basu, M: A Gaussian derivative model for edge enhancement. Pattern Recognit. 27, 1451-1461 (1994) View ArticleGoogle Scholar
- Deng, G, Cahill, LW: An adaptive Gaussian filter for noise reduction and edge detection. In: Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference, 31 October-6 November, pp. 1615-1619. IEEE Xplore Press, San Francisco (1993) Google Scholar
- Kang, C, Wang, W: A novel edge detection method based on the maximizing objective function. Pattern Recognit. 40, 609-618 (2007) View ArticleMATHGoogle Scholar
- Siuzdak, J: A single filter for edge detection. Pattern Recognit. 31, 1681-1686 (1998) View ArticleGoogle Scholar
- Zhu, Q: Efficient evaluations of edge connectivity and width uniformity. Image Vis. Comput. 14, 21-34 (1996) View ArticleGoogle Scholar
- Ambrosio, L, Tortorelli, VM: On the approximation of free discontinuity problems. Boll. Unione Mat. Ital., B 7(6), 105-123 (1992) MathSciNetMATHGoogle Scholar
- Osher, S, Sethian, JA: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12-49 (1988) MathSciNetView ArticleMATHGoogle Scholar
- Lai, MJ, Stockler, J: Construction of multivariate compactly supported tight wavelet frames. Appl. Comput. Harmon. Anal. 21, 324-348 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Qi, D, Guo, F, Yu, L: Medical image edge detection based on omnidirectional multi-scale structure element of mathematical morphology. In: Proceedings of the IEEE International Conference on Automation and Logistics, pp. 2281-2286. IEEE, Washington (2007) Google Scholar
- Stefan, W, Renaut, RA, Gelb, A: Improved total variation-type regularization using higher-order edge detectors. SIAM J. Imaging Sci. 3, 232-251 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Mahmoodi, S: Edge detection filter based on Mumford-Shah Green function. SIAM J. Imaging Sci. 5, 343-365 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, L, Butler, A, Sun, C: Fractal dimension assessment of brain white matter structural complexity post stroke in relation to upper-extremity motor function. Brain Res. 1228, 229-240 (2008) View ArticleGoogle Scholar
- Hristov, J: Transient heat diffusion with a non-singular fading memory Therm. Sci. 20(2), 757-762 (2016) MathSciNetView ArticleGoogle Scholar
- Caputo, M, Fabrizio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2, 73-85 (2015) Google Scholar
- Atangana, A, Baleanu, D: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763-769 (2016) View ArticleGoogle Scholar
- Ibrahim, RW: On generalized Srivastava-Owa fractional operators in the unit disk. Adv. Differ. Equ. 2011, 55 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, Y, Pu, Y, Zhou, J: Construction of fractional differential masks based on Riemann-Liouville definition. J. Comput. Inf. Syst. 6(10), 3191-3199 (2010) Google Scholar
- Chen, X, Fei, X: Improving edge-detection algorithm based on fractional differential approach. In: International Conference on Image, Vision and Computing, pp. 1-6 (2012) Google Scholar
- Loverro, A: Fractional calculus: history, definitions and applications for the engineer. In: Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, pp. 1-28 (2004) Google Scholar
- Sheikh, HR, Wang, Z, Bovik, AC: LIVE Image Quality Assessment Database, Release 2. http://live.ece.utexas.edu/research/quality (2005). Accessed 25 Jan 2015
- Ma, X, Li, B, Zhang, Y, Yan, M: The Canny edge detection and its improvement. In: Artificial Intelligence and Computational Intelligence. Lecture Notes in Computer Science, vol. 7530, pp. 50-58 (2012) Google Scholar
- Wang, Z, Bovik, AC, Sheikh, HR, Simoncelli, EP: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600-612 (2004) View ArticleGoogle Scholar