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International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 Image Enhancement Method using E-spline Ram Bichar Singh1, Anurag Jain2 and Manoj Lipton3 1 M.Tech Student, Radha Raman Institute of Technology & Science, Bhopal, India 2,3 Department of Computer Science, Radha Raman Institute of Technology & Science, Bhopal, India computerized tomography slices and X-rays may need to Abstract: This thesis introduces a new fast method for the be zoomed to search for anomalies. Reconnaissance calculation of exponential B-splines sample at regular photographs must be expanded accurately to show hidden intervals. As another approach, this paper presents an details of weapons manufacturing plants and landing exponential B-spline interpolation kernel using simple strips. With common methods of image expansion the mathematics based on Fourier approximation. A high signal distinguishing objects of such original images tend to be to noise ratio can be achieved because exponential B-spline smoothed over. This decreases the usefulness of the parameters can be set depending on the signal characteristics. expanded image in showing precise details. The analysis of these interpolated kernels shows they have better performance in high and low frequency components as The B-spline functions because of its close resemblance compared to other conventional nearest neighbor, linear, with the sinc function were being started to use spline based methods. This new method is fast and it also prominently as an interpolation function. The term considered polynomial spline as special case. This algorithm spline is used to refer to a wide class of piecewise is based on a combination of FIR and IIR filters which polynomial function jointed at certain continuity points enables a fast decomposition and reconstruction of a signal. called as knots. Until now, in the spline family, For different values of the exponential parameter the extensive research is being done for polynomial spline approximation function is obtained. In this thesis we have [Uns99a]. However, the exponential splines are more tried to get the interpolation function which uses the general representation of these polynomial splines symmetric exponential functions of 4th order. When complex [Dah87a]. In the present work, the continuous values are selected for the parameters of the exponentials, exponential function is derived at equally spaced knots complex trigonometric functions are obtained. We are using truncated power functions and for the formulation considering the real part of these functions which is used for interpolation of real signals corresponding to different of the exponential interpolated kernel this approximation exponential parameter that leads to less band limited signals function is convolved with Fourier approximation of when they are compared with polynomial B-spline the sampled exponential E-spline function [Leh99a]. The counterparts. These characteristics were verified with 1-D calculation of polynomial B-splines is a particular case, and 2-D examples. We are also going through all the when the parameters of the exponents are set to be zero. interpolation methods which are already in use. The exponential B-spline interpolation function is KEY WORDS: E-spline, Exponential B-spline, derived for symmetric case taking different Interpolation, medical imaging, X-rays. exponential parameter in consideration. A great variety of methods with confusing naming can be found in the literature of 1970’s and 1980’s. B-splines 1. INTRODUCTION sometimes are referred to as cubic splines while cubic interpolation is also known as cubic convolution, high The problem of constructing a continuously defined resolution spline interpolation and bi-cubic spline function from given discrete data is unavoidable interpolation. In 1983, parker, Kenyon and troxel whenever one wishes to manipulate the data in a way that published the first paper entitled “comparison of requires information not included explicitly in the data. interpolation methods” followed by a similar study In this age of digitization, it is not difficult to find presented by Mealand in 1988. However, previous work examples of applications where this problem occurs. the of Hou and Andrews, as well as that of keys also compare relatively easiest and in many applications often most global and local interpolation methods. In more recent desired approach to solve the problem is interpolation papers, not only hardware implementations for linear where an approximating function is constructed in such a interpolation and fast algorithm and fast algorithms for way as to agree perfectly with he usually unknown B-spline interpolation or special geometric transforms original functions at the given measurement points. In have been published. However smoothing effects are most view of its increasing relevance, it is only natural that the bothersome if large magnifications are required. In subject of interpolation is receiving more and more addition, shape based and objects based methods have attention these days. Image expansion is required in been established in medicine for slice interpolation of many facets of image processing. To generate precise three dimensional (3-D) data sets. In 1996, Apperdorn maps of the earth’s surface, cartographers, must expand presented a new approach to the interpolation of sampled small regions of satellite image data. In medical imaging, data. His interpolated functions are generated from a Volume 1, Issue 4 November - December 2012 Page 35 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 linear sum of a Gaussian function and their even possible setup, which goes back to the pioneering work of derivatives. Our work presents a comprehensive survey of Schoenberg on polynomial splines in 1946. Since then, existing expansion methods. there have been many theoretical advances and the Here, we propose E-spline method for image expansion methods of spline construction have been extended for and compare to other methods such as Linear and cubic non uniform grids and many other types of non spline. Many techniques currently exist for interpolation polynomial basis functions. The good news is that the and expansion. Commonly used methods are linear and choice of a uniform grid leads to important cubic spline expansion smooths the image data in simplifications. discontinuous regions, producing a large image which In the present study, we concern ourselves with matters appears rather blurry. Image interpolation has many related to the computation of exponential splines. Our applications in computer vision. It is the first two basic primary goal is to develop new tension parameters re-sampling steps and transforms a discrete matrix into a selection algorithms that expanded the image. It must be continuous image. Subsequent sampling of this emphasized that the lack heretofore of viable tension intermediate result produces the resample discrete image. parameter selection schemes has greatly diminished the Image expansion methods have occupied a peculiar practical utility of exponential splines. position in medical image processing. They are required for image generation as well as image post processing. In computer tomography or magnetic resonance imaging, 2. B-SPLINE APPROXIMATION Image reconstruction requires interpolation to Basis splines (B-splines) are one of the most commonly approximate the discrete functions to be back projected used family of spline functions. It can be derived by for inverse. The goal of this study was not to determine several self convolution of a so called basis function. overall best method but to present a comprehensive Actually, the linear interpolant kernel can be considered catalogue of interpolation methods using E-spline, to as the result of convolving the rectangular nearest define general properties and requirements of E-spline neighbor kernel so it is given as: techniques. Exponential spline plays a fundamental role in classical system theory. During the past decade there has been (1) number of articles devoted to the use of polynomial Therefore, Uniform B-splines can be obtained by splines in image expansion. E-splines are a natural multifold convolution of rect functions: extension of B-splines and have very similar properties. B-splines are just a special case of E-splines (with parameter alpha=0) these spline based algorithms have (2) been found to be quite advantageous for image processing We obtained the quadratic B-spline for N=3 which in and medical imaging, especially in the context of high fact, equal the previously mentioned quadratic quality interpolation where it has been demonstrated that approximation. Now, we obtain cubic B-spline for N=4 they yield the best cost quality tradeoff among all linear and is given as: techniques. The interest in these techniques grew after it was shown that most classical spline fitting problems on a uniform grid could be solved efficiently using recursive digital filtering techniques. In continuous time signal and (3) system theory are the exponentials which plays a pivotal The figure is shown below: role having made this observation and motivated by the search for a unification between the continuous and discrete time approaches to signal processing we decided to undertake the task to find the parameter for image expansion using Exponential splines. These splines, as their name suggests are made up of exponential segments that are connected together in smooth fashion. They form a natural extension of the polynomial splines and have been characterized mathematically in relatively general terms. Even though there have not been many computational applications of E-splines, we believe that image expansion is one of the attractive and decent applications of E-splines. The kinds of splines that are most appropriate for signal processing are the cardinal Fig 2.0.1 Kernel of cubic B-spline approximation ones which are defined on uniform grid. It is proposed by Michael Unser and Thierry Blu in their latest paper on E- Note that the B-spline kernel fails to occupy the null splines. Mathematically, this corresponds to the simplest positions of the sinc function h4 (-1) =h4 (1) =1/6. And Volume 1, Issue 4 November - December 2012 Page 36 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 therefore, the B-spline Kernel is actually not an interpolation but rather than an approximation kernel. 2.1 Exponential B-spline Interpolation To create an Interpolating B-spline kernel, the B-spline approximation is applied to a different set of samples t (9) (k). Since the B-spline kernel is symmetrical and separable, the reconstruction (1) yields s ( x ) t ( k ) h( x k ) (4) k (10) With h=h4, as defined in (8). Note that the general case Hence (5) can be written as u=t*h4=v* Splineh(x) and (8) reduces to (1) if the samples are taken directly from with (8) we finally obtain the image data: t (k) =s (k). Here, the t (k) must be derived from the image’s sample point’s s (k) in such a way that the resulting curve interpolates the discrete (11) image. From (8) and (7) we obtain The figure is shown below: (5) Which, ignoring edge effects, results in a set of equations to solve Fig 2.1.2(a) B-spline Fig 2.1.2(b) B-spline interpolated image interpolation kernel The interpolation kernel is symmetric, passes through the integer points. Fig. 4 shows the interpolation kernel for only [0,3], moreover the kernel is symmetric around x=0. The Fourier domain response of the interpolation (6) kernel is shown in Fig. 4 for different values of α Labeling, the three Matrices above as S, C, and T exponential parameters. As the value of alpha is changed respectively. The coefficients in T may be evaluated by the filter response deviates from the ideal low pass multiplying the known data points S with the inverse of filter. However, the interpolation kernel is band limited passing the high frequency components near the cut-off the tri-diagonal matrix C. frequency, which can be used to preserve the edge information in the images. With the increase in value of the transaction part of the filter decreases the (7) magnitude of the low frequency and increases the In all other methods, the coefficients used for convolution number and magnitude of the high frequency components. with the interpolation kernel are taken to be the data Taking Fourier series approximation samples themselves. Because the coefficients for B-spline interpolation are determined by solving a tri-diagonal T ( f ) V ( f ) (a0 an cos(2f )) matrix system, the resulting kernel Splineh(x). (12) n 1 For its simplification the interpolated image s(x) and the data samples s (k) now are called u and v respectively. T 1 T From (10) we obtain v=t*c and in the frequency domain a0 G ( f )df (13) 0 T 1 an G ( f ) cos(2f )df T0 (14) Taking back in the spatial domain Hence final equation: (8) a Inversion of (8) yields h Espline ( x) ( x ) * ( 0 ( x ) a n ( ( x n) ( x n))) (15) 2 n 1 a0 t ( x) v ( x ) * ( ( x) an ( ( x n) ( x n))) 2 n 1 Volume 1, Issue 4 November - December 2012 Page 37 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 Cubic polynomials are frequently used because of theory (16) ability to fit C2-continuous. Also, the B-spline approximate h4 as defined in (8) are constructed piecewise from cubic polynomials. Of course, cubic polynomials also can be used to approximate the sinc function. Two point Interpolation: in the case of cubic interpolation with two points, a symmetric kernel can be defined with (a) (b) Fig 2.1.3 (a) Fourier domain magnitude plot and, (b) log (17) plot of Exponential B-spline for (α, α,-α, - α) with The parameters A to D can be determined by applying the different values of α. following boundary conditions: h ( k ) h ( k ), c 0 continuity ; Table2.1.1The percentage energy distribution in h ' ( k ) h ' ( k ), c1 continuity ; exponential B-spline interpolation kernel for different α (18) after truncation. h ( k ) 1, k 0 Expone Energy Energy ntial distributed distributed h ( k ) 0, k 0 parame between -3 to 3 between -2 For N=2, those boundary conditions yield four equations 0 0.999800 0.998423 for the four parameters resulting in; 1 0.999941 0.998964 2 0.999991 0.999711 3 1.000000 0.999961 (19) 4 1.000000 0.999997 It should be pointed out that, by definition, the above The energy distribution in exponential B-spline cubic function is a DC-constant interpolator. The interpolation kernel is minimum for lower value of resulting curves are similar to those obtained by linear interpolation, but the pieces fit C1-continuously in the exponential parameter. As the value of these spatial domain. Here, only DC-constant interpolators parameter increases, energy decreases in both have been derived in this subsection. The figure is shown between -2 to 2 and -3 to 3 which is shown in below: Table1. Fig. 2.4 plots the sum of sampled interpolation kernel from equation (9) as a function of displacement d. The summation is done after truncating the kernel from -3 to 3. It is clear that for alpha closer to 1.2, the sum of sampled interpolated kernels is closer to 1, hence the value close to 1.2 give better interpolation. This can be verified with PSNR given in table 2. Fig 2.2.5 Cubic interpolation=4 2.3 Gaussian Interpolation: Appledorn has recently introduced a new approach to the generation of interpolation kernels. The objective was to exploit the characteristics of the Gaussian function in both the spatial and the frequency domain. In particular, the Gaussian function is recurrent with respect to operations such as derivation and Fourier transform. Hence, Appledorn published a scheme to develop simple interpolation kernels that are both locally compact in the signal space and almost band limited in the frequency domain and in, addition are easy to manipulate Fig 2.1.4 Sum of sampled interpolation kernels as a analytically. function of the displacement for different α. Consequently, we will denote the Mth partial derivative of the unit area Gaussian function 2.2 Cubic Interpolation: Volume 1, Issue 4 November - December 2012 Page 38 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 B-splines are just a special case of E-splines (with parameter alpha=0) these spline based algorithms have been found to be quite advantageous for image processing ; with zero mean and and medical imaging, especially in the context of high variance as : quality interpolation where it has been demonstrated that they yield the best cost quality tradeoff among all linear techniques. The interest in these techniques grew after it (20) was shown that most classical spline fitting problems on a Hence, we obtain uniform grid could be solved efficiently using recursive digital filtering techniques. In continuous time signal and system theory are the exponentials which plays a pivotal role having made this observation and motivated by the search for a unification between the continuous and discrete time approaches to signal processing we decided to undertake the task to find the parameter for image expansion using Exponential splines. These splines, as (21) their name suggests are made up of exponential segments that are connected together in smooth fashion. They form Then, the Mth–order Gaussian interpolation kernel is a natural extension of the polynomial splines and have given by been characterized mathematically in relatively general terms. Even though there have not been many (22) computational applications of E-splines, we believe that The weighting factors alpha and the variance are image expansion is one of the attractive and decent determined from the following constraints: applications of E-splines. The kinds of splines that are most appropriate for signal processing are the cardinal The Gaussian kernels should equal the ideal ones which are defined on uniform grid. It is proposed by interpolator, at least for x=0 Michael Unser and Thierry Blu in their latest paper on E- splines. Mathematically, this corresponds to the simplest possible setup, which goes back to the pioneering work of Schoenberg on polynomial splines in 1946. Since then, there have been many theoretical advances and the The Fourier transforms of the methods of spline construction have been extended for Gaussian kernels should equal those of the ideal non uniform grids and many other types of non polynomial basis functions. The good news is that the interpolator, at least for choice of a uniform grid leads to important simplifications. In the present study, we concern ourselves with matters Furthermore, should be flat as possible related to the computation of exponential splines. Our without any slope or curvature for primary goal is to develop new tension parameters selection algorithms that expanded the image. It must be emphasized that the lack heretofore of viable tension (23) parameter selection schemes has greatly diminished the Note that the first and the second constraints cover only practical utility of exponential splines. one part of the interpolation condition (4) and the DC- constant respectively. The latter constraint is imposed to 3.2 Review of Theory: approximate the pass band characteristics of the ideal low pass filter and therefore to minimize the corruption of the A family of continuous piecewise basic spline functions image’s Fourier spectrum by the interpolation. can be obtained by multifold convolution of functions 3. E -SPLINES 3.1 E-splines and its computations: (24) Exponential spline plays a fundamental role in classical Where denotes the convolution integral operation. system theory. During the past decade there has been These weights are defined in the number of articles devoted to the use of polynomial domain x [1/ 2,1/ 2] , and being 0 otherwise for the splines in image expansion. E-splines are a natural case of the centered basic splines and the extension of B-splines and have very similar properties. domain x [0,1] for the shifted ones. But in the case of Volume 1, Issue 4 November - December 2012 Page 39 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 polynomial B-splines, w i (.) is a rectangular function of height 1. In order to obtain the exponential B-splines, the weights will be exponential functions (33) i i w ( x) w1 exp(i x) with i C being the parameter of the exponential function and w1 i C a normalization factor. So, by defining the vector ( 1, .......... .., r )T the exponential B-spline can (34) also be expressed as: 3.3 Calculation of E-spline Basis Function: A general solution of exponential spline function of order (25) n=4 is given below where the factors are defined as: Where the single and multiple discrete difference functions are given by (26) Respectively, with [k ] denoting the Kronenecker’s delta function. Also, n (.) is the continuous exponential truncated power or exponential cone spline. The later can be defined by (27) (35) And then recursively calculated by (36) Here, we are considering for the symmetric case. By (28) putting the value of alpha1= , alpha2= , alpha3=- , alpha4=- . The basis function which we get after **Convention of sign: here varying the value of alpha is given Table 1. Table 3.2 Piecewise function for the E-spline In linear case for n=2 we have 2 (1, 2 )T and from approximation of order n=4 above equation (29) Table 3.3.2 We see the asymmetrical behavior of spline approximat- (30) ion function from the above table and figure for different order and are shown below: And the truncated exponential can be simplified as (31) And therefore, Fig 3.3..6 (a) Nth order E-spline Fig 3.3.6 (b): Approximation (32) with varying with from function graph varying value Hence, 2 to 4 alpha Volume 1, Issue 4 November - December 2012 Page 40 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 v( x) t ( x) * ( f (1) ( x 1) f (0) ( x) f (1) ( x 1)) (42) The graph for varying alpha in symmetric case is shown V ( f ) T ( f ) ( f (0) 2 f (1) cos(2 f )) above in fig 3.3(b) Inversion of above equation yields 3.4 E-spline Interpolation: T ( f ) V ( f ) /( f (0) 2 f (1) cos(2 f )) To create an interpolating E-spline kernel, the E-spline kernel approximation is applied to a different set of examples. Since we are dealing with symmetric E-spline v ( f ) (a0 an cos 2 nf bn sin 2 nf ) (43) kernel the reconstruction yields: n 1 n 1 Here bn =0 and taking the inverse Fourier transform of s ( x ) t ( k ) h( x k ) (37) the above equation, the following equation will come: k Here, h is approximation kernel which is discussed a t ( x) v( x) a0 ( x) n [ ( x n) ( x n)] (44) above. The general case of given equation is reduces to n 1 2 following equation Here bn =0 and taking the inverse Fourier transform of the above equation, the following equation will come: s( x, y ) s ( k , l ) 2D h ( x k , y l ) (38) a (45) k l t ( x) v( x ) a0 ( x) n [ ( x n ) ( x n)] If the samples are taken directly from the image data: n 1 2 t (k ) s(k ) .here the t (k ) must be derived from the Hence, equation 1 can be written as image’s sample points s (k ) in such a way that the u t * h4 v * Esplineh and with (13) we finally resulting curve interpolates the discrete image. From obtain equation 1 and 2 w e obtain Esplineh ( x ) = h4 * t ( x ) (46) k2 The graph is show below within the interval s(k ) t(m ) h m k 2 4 (k m ) x 3 .Although the kernel is infinite; the amplitudes of = f (1)t ( k 1) f (0)t ( k ) f (1)t ( k 1) (39) the half waves are reduced significantly when compared with that of ideal IIR-interpolation. That results in a set of questions to solve when we ignore edge effects: s (0) f (0) f ( 1) 0 t (0) s (1) f (1) f (0) f ( 1) t (1) (40) s (2) f (1) f (0) t (2) s ( K 2) f t ( K 2) s ( K 1) 0 f (1) f (0) t ( K 1) Labeling the three matrices above as S, C and T respectively, the coefficients in T may be evaluated by multiplying the known data points S with the inverse of the tri diagonal matrix C . T C 1 S (41) In all other methods included in this, the coefficients Fig 3.4.7 The interpolation Exponential B-spline kernel used for convolution with the interpolation kernel are with varying . taken to be the data samples themselves. Because the coefficients for E-spline interpolation are determined by The Fourier response of the Esplineh ( x ) is given below solving a tri diagonal matrix system Esplineh ( x ) is for varying the value of . Here solid line shows the infinite. frequency response of =1. Subsequently, the dotted is for =2. The following figure illustrated the side lobes To simplify its analytical derivation, the interpolated behavior of E-spline interpolation. image s ( x) and the data samples s (k ) now are called u and v respectively. From we obtain v t * c and in the frequency domain Volume 1, Issue 4 November - December 2012 Page 41 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 Fig 3.4.9 (a) Original Image Fig 3.4.9 (b) =0.2 Fig 3.4.8 4th order Exponential B-spline approximation and normalized Exponential B-spline for ( , ,- ,- ) where =.2,.5,1,2,3 Fig 3.4.9 (c) =0.5 Fig 3.4.9 (d) =1 Here we are considering the case for n=4( , , - ,- ). The following figure shows the impulse response in spatial domain of 4th order , , - , . Cardinal exponential B-spline decays faster than polynomial spline and also we see that increase the value of ranging from .2 to 3. The ringing behavior is reduced. The second figure shows the logarithmic plot of Fourier magnitude which shows energy distribution for different frequencies in the main lobe and side lobes. The energy distribution Fig 3.4.9 (e) =1.5 Fig 3.4.9 (f) =2 in the E-spline is minimal for lower value of alpha and as the values of alpha increases the energy decreases in both Fig 3.4.5 Expansion of Lena image by Exponential spline between from -2 to 2 and -3 to 3 which is shown in table with varying 3.4.3. 4. EXPERIMENTAL RESULTS Here we perform on lena image (64x64). The given image Exponential Energy between Energy in other is the expansion by factor of 8 from a section of the image parameter α -3 to 3 part of Lena. After performing, E-spline interpolation with 0.2 99.9800 0.0112 varying . We get the smooth area corresponding to the 1 99.9941 0.0059 2 99.9993 0.00085739 face of Lena and higher frequency parts which are the 3 100 0 back ground. In above figure it can be observed that the 4 100 0 resulting interpolated images are almost the same at a glance, but more jaggedness appears in the high frequency areas in the exponential B-spline case due to its Exponential Energy between Energy in other fewer bands–limited characteristics. The Fourier analysis parameter α -2 to 2 part of pass band, stop band, and cutoff frequency the nearest 0.2 99.8431 0.1578 neighbor and linear interpolation should be avoided while 1 99.8970 0.1036 the preferred method is the Gaussian kernel with large 2 99.9973 0.0289 sizes. To judges the interpolation quality, it solely 3 99.9965 0.0038 4 99.9997 0.00030095 depends on the application. Interpolation versus Table 3.4.3 Approximation: in this E-spline methods are well suited for the images containing high frequency components. The given image is the expansion by a factor 8 by using Gaussian kernels are not suited for this method. These the above E-spline technique in 2-D. Here, we are kernels have been compared on various images of Lena. performing E-spline interpolation on Lena image In each case, the efficiency and accuracy of a particular (64x64). Lena image is expanded by factor 8. Here are the interpolation technique was evaluated by analyzing its given results of varying the value of . Fourier properties, visual quality and run time measurement. 4.1 Runtime measurements: The runtime of the various interpolation schemes were measured on the standard machine. Sources have been compiled using MATLAB 6.5. The rotation is quite time consuming in MATLAB environment. It shows that simple interpolation methods such as nearest neighbor ,linear and 2x2 cubic interpolation are fairly fast and requires less time than the rotations of pixel coordinates. Volume 1, Issue 4 November - December 2012 Page 42 International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email: editor@ijettcs.org, editorijettcs@gmail.com Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 Gaussian interpolation required more time due to the get better result in interpolation methods by finding the evaluation of the exponential function necessary to adaptive method for the appropriate image. determine weights. Here are the results of all interpolated image and we compare all interpolated image to original one. The resultant image is show below: REFERENCES [1] M.Unser, ”Splines: A perfect fit for signal and image processing,” IEEE Signal Processing Magazine, Vol.16, No.6, pp.22-38,November 1999. [2] Raghubansh B. gupta, Byung Gook Lee, Joon Jae Lee: A new Image Interpolation Technique using Exponenial B-Spline. [3] Thomas M.Lehmann, Claudia Gonner, and Klaus Spitzer,” Survey: Interpolation Methods in Medical Fig 4.1.10 (a) Original image Fig 4.1.10(b) Nearest Image Processing,” IEEE Transactions on Medical neighbor Imaging, Vol. 18, No. 11, November 1999. [4] Takeshi Asahi, Koichi Ichige,and Rokuya Ishlii, ”Fast Computation of Exponential Splines” Proceedings of the IEEE. [5] M.Unser,A.Aldroubi and M.Eden, ”Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. On Pattern Anal. & Machine Intell. Vol.13, No.3, pp.277-285, March 1991. [6] Erik Meijering,”A Chronology of Interpolation: Fig 4.1.10(c) Linear image Fig 4.1.10(d) B-spline From Ancient Astronomy to Modern Signal and image Image Processing,” Proceedings of the IEEE, vol.90, Fig 4.1.10 Interpolated image of Lena (64x64) using No.3, March 2002. various types of interpolation methods. [7] Michael Unser and Thierry Blu, ”Cardinal Exponential Splines: Part I- Theory and Filtering Algorithms,” IEEE Transactions on Signal 5. CONCLUSIONS Processing, Vol. 53, No.4, April 2005. [8] Michael Unser, ”Cardinal Exponential Splines: Part This thesis is focused on interpolation methods using E- II- Think Analog, Act Digital,” IEEE Transactions spline. The method is fast for calculation of E-spline on Signal Processing, Vol. 53, No.4, April 2005. (where the calculation of B-spline is a particular case). [9] Brian J.Mcartin,”Compuatation of Exponential When complex parameters are used in exponential Splines” SIAM J. Sci STAT. Comput. Vol. 11, No. 2, functions, trigonometric spline are obtained and here we pp.242-262, March 1990. are dealing with only real parts of the signal. Less band [10] Poth Miklos,”Image Interpolation techniques”. limited functions can be achieved by using the real par of this later functions and comparing them with the polynomial splines counterparts. Although more amount of computation is required, it is specially used for images containing high frequency components. As demonstrated by codes in this thesis, the interpolation methods using E- spline is easy to implement. The E-spline is general case of B-spline polynomials and performs better interpolation for images containing high frequency parts. 6. FUTURE WORK Since I have a good result for the interpolation using E- spline techniques, the further study on interpolation should be carry on in the future. Interpolation of images is popular problem until yet but now we have to find kernel that enhances high frequency parts. If we get the required result then we go for L-spline that is general case of E-spline. Future work will address techniques to Volume 1, Issue 4 November - December 2012 Page 43