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We present the analysis of three independent and most widely used image smoothing techniques on a new fractional based convolution edge detector originally constructed by same authors for image edge analysis. The implementation was done using only Gaussian function as its smoothing function based on predefined assumptions and therefore did not scale well for some types of edges and noise. The experiments conducted on this mask using known images with realistic geometry suggested the need for image smoothing adaptation to obtain a more optimal performance. In this paper, we use the structural similarity index measure and show that the adaptation technique for choosing smoothing function has significant advantages over a single function implementation. The new adaptive fractional based convolution mask can smoothly find edges of various types in detail quite significantly. The method can now trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges.

Edge detection is an important stage in image analysis since they provide the topology and structural information of relevant object in an image [

Until recently, methods like Robert [

In this paper, we present a hybrid of median, Gaussian and cubic spline based smoothing technique on the new fractional based convolution edge detector. We show that the resulting hybrid fractional edge operator is able to detect edges very well when the smoothing function is adaptive. In section 2, the paper provides brief review of basis spline and Gaussian filters as image smoothing functions and the formulation of a fractional edge detector. In section 3, the paper discusses how the numerical experiment is setup, optimal selection of Gaussian and spline parameters and the performance analysis of smoothing functions used in the study. The last section concludes the paper.

Edges are well known to be characterized by high frequencies and so are noise as well [

A basis spline (B-spline) filter is a piecewise polynomial function of degree k in a variable t defined over a domain

Definition 1. Let

where

Definition 2. Let

where

The Gaussian filter normally written as:

has a standard deviation s which determines the width of the filter as well as the outcome of the smoothed image. The larger the value of s, the wider the frequency band of the Gaussian filter.

We begin with the following definition of Riemann Liouville fractional calculus as defined in the work Owa [

Definition 3. The Riemann Liouville fractional integral of order

where

Definition 4. The Riemann Liouville fractional derivative of order

where

Given an analytic function

Using Equation (7), it is clear that Equation (6) can be written as below:

where

Expanding Equation (8) into 2-D, we let the following:

Taking derivative in the x-direction we obtain:

while in the y-direction we have:

Finally, the fractional gradient operator is defined as:

In order to apply the mask on images, Equations ((11) and (12)) are rewritten as discrete operators as follows:

where

In this section, an experimental analysis of two most widely [

(a) Horizontal Directional Fractional Mask | ||||
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(b) Vertical Directional Fractional Mask | ||||

0 | 0 | 0 | 0 | 0 |

a performance analysis of the three selected smoothing functions are compared with justifications.

In order to carry out a more accurate experiment, the following measures and structures were considered. A performance analysis of the behaviour of some selected smoothing functions on a fractional derivative operator described in Equations ((13) and (14)) under various noise types is done. Here, three smoothing functions, Gaussian, Median and Spline functions were considered. We also considered four noise types, namely the motion blur, Gaussian, salt & pepper and speckle. Unlike the median smoothing function, the Gaussian and the spline smoothing functions are parametric and an appropriate value is required before a comparative analysis could be done. For example, the Gaussian and the spline functions require a discrete and finite length to form the filter length (size). In this study, the selection of an optimal filter length is obtained using the Structural Similarity Index Measure (SSIM). Here an optimal filter length is obtained at the maximum SSIM value (point of intersection) using Equations ((2) and (4)). The results reported in this paper is based the popular Lena image in

In the selection of an optimal filter size for the spline and Gaussian function, we assume the s parameter is constant as well as the order of the fractional gradient operator. For the purpose of demonstration, the order

In these Figures 2(a)-(d), the spline smoothing function has a decaying life cycle with varying filter length just after the optimum SSIM value is obtained. The SSIM value for the Gaussian smoothing function however remains stable after its maximum point even as the filter length grows. An indication that, unlike the spline function which obtains different SSIM values for varying filter length, changing the filter length of the Gaussian function has no significant effect on its SSIM value after its maximum point value. Apart from

In

In the last experiment, an edge operator of mask size

As stated in the previous section, you observe that, the value for the Gaussian smoothing function as provided in

In order to obtain the required optimal value for the smoothing functions, performance test was carried out on the three selected smoothing functions to ascertain which function works better and under which condition it should be considered.

Mask Size | Motion Blur | Gaussian | Salt & Pepper | Speckle | |
---|---|---|---|---|---|

Spline | 3 | 9 | 9 | 9 | |

3 | 9 | 9 | 9 | ||

3 | 7 | 7 | 7 | ||

3 | 7 | 5 | 7 | ||

Gaussian | 3 | 5 | 17 | 7 | |

3 | 9 | 13 | 9 | ||

7 | 3 | 5 | 7 | ||

3 | 7 | 9 | 5 |

Mask Size | Motion Blur | Gaussian | Salt & Pepper | Speckle | ||||
---|---|---|---|---|---|---|---|---|

sigma | filter | sigma | filter | sigma | filter | sigma | filter | |

1 | 3^{*} | 6 | 9 | 4 | 9 | 4 | 15 | |

9 | 3 | 4 | 9^{**} | 3 | 11 | 5 | 11 | |

1 | 7^{*} | 4 | 11 | 3 | 19 | 3 | 11 | |

1 | 3^{*} | 4 | 9 | 3 | 13 | 2 | 7 |

Mask Size | Motion Blur | Gaussian | Salt & Pepper | Speckle | |
---|---|---|---|---|---|

Median | 0.6654 | 0.1335 | 0.6701 | 0.2526 | |

Spline | 0.7143 | 0.5995 | 0.6374 | 0.6127 | |

Gaussian | 0.6992 | 0.5988 | 0.6464 | 0.6208 | |

Median | 0.8511 | 0.2032 | 0.8654 | 0.3852 | |

Spline | 0.8808 | 0.7573 | 0.8008 | 0.7722 | |

Gaussian | 0.8763 | 0.7681 | 0.8012 | 0.7790 | |

Median | 0.9099 | 0.2962 | 0.9392 | 0.5165 | |

Spline | 0.9441 | 0.8066 | 0.8720 | 0.8441 | |

Gaussian | 0.9385 | 0.8225 | 0.8773 | 0.8560 | |

Median | 0.9317 | 0.3949 | 0.9416 | 0.6056 | |

Spline | 0.9527 | 0.8472 | 0.8880 | 0.8635 | |

Gaussian | 0.9543 | 0.8485 | 0.9002 | 0.8742 |

Salt & Pepper) to the image. The smoothing functions (spline, median and Gaussian) iare then applied to the noisy image. The fractional mask is subsequently applied to the smoothened image. Finally the SSIM is computed for each mask type against a selected smoothing function using the optimal filter length obtained in

From

In this paper, we have presented the analysis and optimal technique for adapting an image smoothing function on a new fractional based convolution mask for image edge detection. Experimental results from the study tabulate the structural similarity index measure on three image smoothing functions and different noise types. In particular, the effect of the Gaussian, median filter and spline function on the mask is discussed. The paper also discussed the selection of an optimal filter length required to attain a higher structural similarity index as well as optimal selection of s for the Gaussian function. Finally, a performance analysis of the three selected smoothing functions was compared. The results show that the new adaptive fractional based convolution mask can smoothly find edges of various types in detail quite significantly. The method can now trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges as opposed to using single Gaussian smoothing function.

We will like to acknowledge the support received from the National Institute for Mathematical Sciences, Ghana for this study.

Peter Amoako-Yirenkyi,Justice Kwame Appati,Isaac Kwame Dontwi, (2016) Performance Analysis of Image Smoothing Techniques on a New Fractional Convolution Mask for Image Edge Detection. Open Journal of Applied Sciences,06,478-488. doi: 10.4236/ojapps.2016.67048