# INTRODUCTION mage processing is a very active research area that has impact in several fields from remote sensing, Biometric authentication system, robotics, traffic Surveillance, to medicine. Automatic target recognition and Tracking, character recognition, 3-D scene analysis and reconstruction are only a few objectives to deal with. Since the real sensing systems are sometimes imperfect and also the environmental conditions are dynamic over time, the acquired images often The image are the for the most part frequent component of information representation and transmission due to the robust nature of information storage and the continuous effort to make digital image processing and presentation better. The studies have shown that the images contain information which is redundant and changing a value may cause errors in the calculation for further steps. In the space of image processing, the restoration of images is the major expanse of research Author ? ?: Department of CSE BVRIT Narsapur Hyderabad, India. e-mails: bhima.mnnit@gmail.com, jagan.amgoth@bvrit.ac.in for many decades. Many researchers have proposed various algorithms and techniques for better restoration of images for various applications. However the collection of image is strongly dependent on the imaging agent. The quality of a image possibly will suffer from a variety of impairments, Still the key bottleneck for better restoration of images are the random distortion and blurring caused to the initial images to be provided as input to the recognition system [1] [2]. The distortion and blurriness of the images are not only dependent on the capture agent, but also depends on the environmental and human errors. The causes of blurriness are studies and classified in four major kinds. Firstly, the focal length of the capture devices, Secondly, during the capture of object in a time irrelevant scale needs to be mapped with the capture speed of the agent to avoid the blurriness [3]. Thirdly, sometimes due to environmental and human causes the stabilization of the capture devices may be disturbed causing the blurriness. Fourthly, the most unavoidable situation, where the object is in higher order of colour range but the relevant background of lower order of colour range causing the blurriness. Thus to remove the effect of blurriness of the image, the most appropriate algorithms to be deployed are the momentary calculation algorithms. In the field of image processing, computer vision and allied fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation. Image moments are helpful to depict objects after segmentation. Simple properties of the image which are found via image moments include area (or total intensity), it's centric, and information about its orientation and Effects of moments in digital image processing for restoration cannot be ignored as supported by related researches. In general moments are the numeric values used to represent the nature of any functions and identify with the significant properties [3] [4]. The following are mostly used moments algorithms are Hu moment, Zernike moment and the well discussed Legendre algorithms. The moments are superior to principle component analysis for image recognition especially for image recognition [5] [6] [7]. Yet the application of moments algorithms are not been studied for digital image restoration with the comparative results for blur to 7 Year 2016 ( ) restoration algorithm efficiency mapping. Thus in this work we understand the algorithms of moments calculation proposed by Hu, Zernike and Legendre for image restoration and develop a framework for comparing the visual performance of the restoration process by applying the same algorithms. This work also demonstrates the effect of multi order Legendre for blurred image restoration. The rest of the work is organized as in Section II we understand the basic constructions of the moment algorithms and possibilities to apply for image restoration, in Section III we consider the Legendre moment in detail, in Section IV we define the components for Blurred image restoration process, in Section V, we demonstrate the proposed framework for Blurred Image recovery using multi-order Legendre moment algorithm, in Section VI we discuss the application constructed for the visual comparison for the blurred image restoration, in Section VII we discuss the results tested on multiple image datasets and in Section VIII we discuss the conclusions and future scope of this work. # II. # IMAGE MOMENTS In Image processing and computer vision processing explore the calculation of image moments or finding the image descriptors is widely accepted. The moment is a calculated on certain weighted average of any pixel taking into account the neighbourhood pixel values. Often the moment is also used to calculate to understand and extract the most significant property of a continuous function [8]. The image moments are widely accepted for image processing and used by all polynomial approaches. In this work we consider the restoration techniques using moments, thus the understanding of moments will be helpful in section IV. In case of image and vision processing calculating the image moment which is resulting in the image descriptor is performed after the image segmentation. The image properties like area, centroid, pixel values and orientation of any object in the image can be represented using the image moment. Image moments are classified into three categories such as Raw Moments, Central Moments and Scale invariant Moments [3]. In this work, we understand the moments in details: a) Raw Moment For a simple two dimensional function, denoted by ( , ) f a b , the raw moment of order ( , ) x y can be defined as . . ( , ). . In order to simply the calculations by considering the probabilistic measure for image analysis, the Eq. 2 needs to be normalized by the The central moments for order k can be represented as 1 2 1 21 2 ( ) ( ) 1 2 . # .( ) .( ) . y x x k y k xy k k k k x y x y M k k µ ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ?? ?Eq 6 The central moments are considered as translation invariant. # c) Scale invariant Moment The moment of order (x + y) where x + y ? 2 can be obtained by dividing the central moment with 0 th moment as following: # APPLICABILITY OF LEGENDRE MOMENT The most adopted method for image pattern or image restoration is the use of moments. The recent advancements demonstrate the use of moment calculation methodologies as geometric and orthogonal moments. Further studies have demonstrated that the orthogonal moments are better than the geometric moments. Among the orthogonal moments the most widely accepted method is to deploy the Legendre moment. But the application of Legendre moment is also restricted for the blurred or distorted images. Here we understand Legendre Moments in detail [3] [13]: Legendre Moment for of order (a + b) is defined as: 1 1 1 1 (2 1)(2 1) ( ). ( , ). 4 ab a b a b P i P i j didj ? + + ? ? + + = ? ? ?Eq 8 Where a, b is ranging from 1 to?. Hence the k th order Legendre polynomial is written as: 2 2 (2 )! (2 )! ( ) ... 2 ( !) 2 !( 1)!( 2)! k k k k k k k k P i i i k k k ? ? = ? + ? ? K th Term ?Eq9 Where, D(k) = k/2 or (k-1)/2, is an positive integer. IV. # CHARACTERISTICS OF BLURRED IMAGE In the Blurred or noisy image, the objects vary in terms of contrast and size. The objects in the image can represent large to small item or the items with detailed visibility. The primary effect of the blurriness on the imageis to reduce the contrast and visibility of the images. The reduced visibility images causes less detailed information in the images [10] [11]. The objects in the images are generally differentiated by the pixel difference between the object and the background at the object edges. The blurriness of the image actually reduces the pixel difference at the object edges [12]. The blurriness of the image can be measured in terms of units of lengths. The length of the images denotes the blurriness of the image [Table - # FRAMEWORK FOR BLURRED IMAGE RESTORATION PROCESS The two dimensional Legendre Moment for the blurred image of g (a, b) can be defined as [3] [13]: 1 1 , 11 # ( ) ( ). ( ). ( , ). x y x y L g P a P b g a b dadb + + ? ? = ? ? ?Eq10 With the understanding of blurriness effect on the image, the image pixel will be multiplied by random value generated by the noise function. 1 1 , 11 ( ) ( ). ( ).( * ). x y x y L g P a P b f h dadb + + ? ? = ? ? ?Eq 11 Legendre moment of the blurred image can be represented as 1 1 , 11 # ( ) ( , ).( ( ). ( ) ( , ) ). x y x y L g h i j P a i P b j f a b dadb didj + + +? +? ? ? ?? ?? = + + ? ? ? ? ?Eq 12 Image restoration procedure using moments:- ? Capturing image using capture device. ? Captured image is stored and referred for pre processing ? Blur function is applied on Image and also calculates image moment using Legendre polynomials. ? Comparison of original blurred and restored image. Thus the process of restoring the blurred image using Legendre Moment is presented in this work [Figure -1]. visual advantages of Legendre Moments over other available moments. MATLAB is a highly popular multipurpose numeric programming language for the wide variety of build in library functions ranging from image processing to higher order numeric calculation. The built in library is capable of generating matrix based calculation and graph plotting in multi-dimensional space. The MATLAB is considered as the fourth generation programming language. In the implementation we also propose the multi order Legendre Moments to restore the blurred and noisy image. # VII. # RESULTS AND DISCUSSIONS In this section, we have considered three different image dataset of fingerprint, bird and human face for restoration using various methods such as Hu, Zernike and Legendre moments. Henceforth we compare the initial image and restored image generated by the Hu, Zernike and Legendre moments using the following formulation: The difference between the original image and the restored image using movements' algorithms considered as K 1 and the difference between the original image and blurred image is considered as K 2 . Hence the comparative difference between the K 1 and K 2 is considered K, demonstrating the amount of successful restoration for any given image using any given moment algorithm. The testing results clearly demonstrate the comparative study on different data sets such as fingerprint, bird and human face for restoration using Hu, Zernike and Legendre moment. For fingerprint Hue method exhibit better results, Zernike and Legendre shows better results for bird. In the case of human face Legendre moments demonstrates better results. # VIII. # Conclusion In this paper focus on the analysis of three categories of moments such as Raw Moments, Central Moments and Scale invariant Moments and the basic mathematics functions behind those moments. In order to achieve better understanding of image restoration process, we have also understood the nature of blurred images. The understanding of the difference of lengths for normal and blurred image based on the length for various capture device types. Henceforth, this work proposes a theoretical framework using Hu, Zernike and Legendre moment to restore blurred images. The theoretical model is also validated using the image dataset and the results are also been tested. The result of image dataset is satisfactory for restoring the blurred images. The application is been tested on three types of image such as Fingerprint, bird and human face. For majority of the image restoration Legendre moments demonstrate good results. 1![all positive integers of x and y The function ( , ) f a b denoting any greyscale image with pixel intensity of I(a, b) will be denoting the moment as . . ( , )](image-2.png "1 For") ![MomentFor a simple two dimensional function, denoted by( , ) f a b , the central moment of order ( , )x y can be defined as ( ) .( ) . ( , ). .](image-3.png "") ![and b are the generic components of the centroid of the image and can be defined as case of a digital image, the Eq. 4 can be represented as the following:](image-4.png "a") 7![The scale invariant is neutral for scale change.Global Journal of Computer Science and TechnologyVolume XVI Issue I Version I](image-5.png "?Eq 7") 1![Figure 1 : Framework for Blurred Image RestorationVI. REAL TIME BLURRED IMAGE RESTORATIONIn order to prove the findings and theoretical framework proposed in this work, we provide the MATLAB implementation of this framework to test the](image-6.png "Figure 1 :") 21![Figure 2. 1 : Restoration of fingerprint Image using moments The input fingerprint image is blurred with length of 10mm and been tested for restoration with Hu, Zernike and Legendre moments of 50 order [Figure -2.1].](image-7.png "Figure 2 . 1 :") 22![Figure 2. 2 : Restoration of fingerprint Image using moments The input fingerprint image is blurred with length of 20mm and been tested for restoration with Hu, Zernike and Legendre moments of 50 order [Figure -2.2].](image-8.png "Figure 2 . 2 :") 23![Figure 2. 3 : Restoration of fingerprint Image using moments The input fingerprint image is blurred with length of 30mm and been tested for restoration with Hu, Zernike and Legendre moments of 50 order [Figure -2.3].](image-9.png "Figure 2 . 3 :") 24![Figure 2. 4 : Restoration of Bird Image using moments The input bird image is blurred with length of 10mm and been tested for restoration with Hu, Zernike and Legendre moments of 50 order [Figure -2.4].](image-10.png "Figure 2 . 4 :") 2526![Figure 2. 5 : Restoration of Bird Image using moments The input bird image is blurred with length of 20mm and been tested for restoration with Hu, Zernike and Legendre moments of 50 order [Figure -2.5].](image-11.png "Figure 2 . 5 :Figure 2 . 6 :") ICapture Agent TypeRange ofBlur Value(In MM)Gamma Ray Camera10 to 2Ultrasonic Camera5 to 2.1Magnetic Resonance Camera3.4 to 1Computed Thermography Camera2 to 1.3Motion Capture Camera2.8 to 0.3Radio Active Camera0.5 to 0.1V. IIYear 2016)f(Input ImageBlur LengthHu Moment (In %)Zernike Moment (In %)Legendre Moment (In %)10 mm786553Fingerprint20 mm80685830 mm83716810 mm236371Bird20 mm24687530 mm24717910 mm375381Human Face20 mm41578330 mm536187Study of Hu, Zernike and Legendre moment based on K value in Eq. 15.For dataset of fingerprint, bird and human face © 2016 Global Journals Inc. (US) 1 © 2016 Global Journals Inc. (US) * Combined invariants to similarity transformation and to blur using orthogonal Zernike moments BChen HShu HZhang GCoatrieux LLuo JLCoatrieux IEEE Trans. Image Process 20 2 2011 * Zitová Moments and Moment Invariants in Pattern Recognition JFlusser TSuk B 2009 Wiley * Blurred image recognition by Legendre moment invariants HZhang HShu G.-NHan GCoatrieux LLuo JLCoatrieux IEEE Trans. Image Process 19 3 2010 * Combined invariants to blur and rotation using Zernike moment descriptors HZhu MLiu HJi YLi Pattern Anal. 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