Image Enhancement using E-spline Functions M.F. Fahmy 1 , G.Fahmy 2 and O. M. Fahmy 3 mamdouhffahmy@gmail.com 1 , g.fahmy@mu.edu.sa 2 , omar.fahmy@hotmail.com 3 1 Electrical Engineering Dept., Assiut University, Egypt 2 Electrical Engineering Dept., Majmaah University, KSA 3 Future University in Egypt, Postdoctoral researcher at university of Vigo, Spain Abstract- Exponential spline polynomials (E-splines) represent the best smooth transition between continuous and discrete domains. As they are constructed from convolution of exponential segments, there are many degrees of freedom to optimally choose the most convenient E-spline, suitable for a specific application. In this paper, the parameters of these E- splines were optimally chosen, to enhance the performance of image de-noising as well as image zooming schemes. The proposed technique is based on minimizing the total variation function of the detail coefficients of the E-spline based wavelet decomposition. In image de-noising schemes, apart from E- spline parameter estimations, the thresholding levels of their detail coefficients, are also optimally chosen. In zooming applications, the quality of interpolated images are further improved and sharpened by applying ICA technique to them, in order to remove any dependency. Illustrative examples are given to verify image enhancement of the proposed e-spline scheme, when compared with the existing approaches. Keywords-Image de-noising, interpolators, E-spline functions 1. INTRODUCTION During the past decade, there have been an increasing number of papers devoted to the use of polynomial splines in different signal processing applications, [1-3]. B-spline polynomials, is a class of these polynomial splines that find extensive applications in many engineering applications. In [4], a complete analysis for a B-spline Perfect Reconstruction (PR) frame work with a derivation for the scaling and wavelet functions was presented. However, as they are constructed using Haar functions, there is no much degree of freedom to use in optimizing the performance of some signal processing applications like the design of digital interpolators. On the other hand, Exponential splines enjoy a unique feature of being able to convert from analog to digital applications. This is crucial in several signal processing applications such as differential operators, fractional delays, interpolators and sampling rate converters, [5-6]. Moreover, E-splines has many degrees of freedom if they are optimized in a specific application, as they are constructed from the convolution of exponential segments with different rates. In [7], a preliminary application for the usage of E- splines in image zooming and interpolation was presented. In this paper, it is proposed to use E-splines in enhancing the performance of image de-noising as well as image zooming schemes. In denoising applications, the proposed denoising technique is based on total variation function minimization, [8-9]. Using a recently developed E-spline wavelet decomposition, [10-14], the E-spline parameters as well as the thresholding levels of the E- spline detail coefficients are optimally chosen to minimize the total variation of the E-spline detail wavelet coefficients. In image zooming applications, E-spline based interpolators are used in image interpolation. In this case, the parameters of E-spline polynomials are chosen to boost the high frequency detail energy of the interpolated image. Further improvement is possible by estimating the missing high frequency details from the given low resolution image using the information contained in the whole low frequency image as described in sec. 4,[15-18]. Further image enhancement is also possible, by applying ICA techniques [19]; to the interpolated images in order to boost high frequency details and reduce any dependency between them. Illustrative examples are given to verify the ability of E- spline polynomials to significantly enhance image quality. The paper is organized as follows: in sec. 2, a brief description of E-spline polynomials and wavelet Perfect Reconstruction PR systems is given. In sec. 3, the proposed denoising technique of E-spline polynomials is described. Sec. 4 describes the design of digital interpolator using E-spline polynomials. Sec. 5 concludes the paper. 2. MATHEMATICAL BACKGROUND The Exponential m th order spline polynomial ) t ( B m   , is constructed as m successive convolution of lower ones, i.e: ) ( * ... * ) ( * ) ( ) ( 1 1 1 2 1 t B t B t B t B m m       (1) where 1 0 , ) ( 1    t e t B t   .The vector   can assume any positive, negative or even complex conjugate values. This means a considerable flexibility over Cardinal B- spline polynomials that only use Haar functions. ) (t B m   is of finite support and equals zeros at 0  t and m t  . Between the knots t =1,2,…,m-1, it is represented by polynomials of order (m-1) in t, [5]. Due to its continuity and smoothness, it is used to expand continuous signals s(t). In the discrete case, s(n) can be expressed using the convolutional relation.    k m k n B k c n s ) ( ) ( ) (   (2) The c k coefficients are obtained using the concept of