Image Enhancement Method using E-spline by editorijettcs

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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(2f ))
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(2f )df
                                                                           T0                                                    (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)
              k2
                                                                                            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

								
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