2892 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014
A New Formula for Bivariate Hermite Interpolationon Variable Step Grids and Its Application
to Image InterpolationKonstantinos K. Delibasis and Aristides Kechriniotis
Abstract— In this paper, we present a novel formula of thebivariate Hermite interpolating (BHI) polynomial in the caseof support points arranged on a grid with variable step. Thisexpression is applicable when interpolation of a bivariate func-tion is required, given its value and the values of its partialderivatives of arbitrarily high order, at the support points. Theproposed formula is a generalization of an existing formula forthe bivariate Hermite polynomial. It is also algebraically muchsimpler, thus can be computed more efficiently. In order to applyHermite interpolation to image interpolation, we simplify theproposed (BHI) to handle support points on a regular unit-stepgrid. The values of image partial derivatives are arithmeticallyapproximated using compact finite differences. The proposedmethod is being assessed in a number of image interpolationexperiments that include a synthetic image, for which the valuesof the partial derivatives are computed analytically, as well asa collection of images from different medical modalities. Theproposed BHI with up to second-order image partial derivatives,outperforms the convolution-based interpolation methods, as wellas generalized interpolation methods with the same numberof support points that was compared with, in the majority ofimage interpolation experiments. The computational load of theproposed BHI is calculated and its behaviour with respect to itscontrolling parameters is investigated.
Index Terms— Image interpolation, bivariate Hermiteinterpolation on variable step grids.
I. INTRODUCTION
INTERPOLATION is required for every kind of geometrictransformation of images, from simple rotation and resizing
to complex elastic deformations. It is always desirable to min-imize the artifacts of these geometric operations. Therefore,image interpolation is a subject that has attracted wide atten-tion in literature, [1]–[3]. Reconstruction of medical imagesusing the radon transform [1], as well as registration (spatialalignment) of two images of the same or different modality,requires image interpolation [4]. Interpolation-induced gray
Manuscript received September 24, 2013; revised January 31, 2014 andApril 16, 2014; accepted April 26, 2014. Date of publication May 7, 2014;date of current version May 27, 2014. Portions of the research in this paper usethe CASIA-IrisV1 collected by the Chinese Academy of Sciences’ Instituteof Automation (CASIA). The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. David Frakes.
K. K. Delibasis is with the Department of Computer Science andBiomedical Informatics, University of Thessaly, Lamia 35100, Greece (e-mail:[email protected]).
A. Kechriniotis is with the Department of Electronics, TechnologicalEducational Institute of Lamia, Lamia 35100, Greece (e-mail:[email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2014.2322441
level errors of the transformed image affect the registrationerror in a number of imaging modalities [2].
A number of interpolation techniques have been proposedin the literature [1]. In this work we propose the use ofbivariate Hermite interpolation (BHI) for image interpolation.The Hermite interpolation requires the values of the derivativesof the interpolated function, up to an arbitrary order, at eachsupport point, in addition to the values of the interpolatedfunction. In [7] and [8] the bivariate Hermite interpolation isdiscussed. In [9] the use of Hermite bivariate interpolationof the first order using 2 × 2 support points is proposed,whereas in [10] its use for surface reconstruction from shadingis demonstrated.
In [5] a formula is provided for the univariate Hermiteinterpolating polynomial, expressed as sums of derivativesof rational functions. In [6] we presented a much simpleralgebraic new formula for the univariate Hermite interpolatingpolynomial that is expressed as sum of polynomials. However,the bivariate Hermite polynomial is not separable [32], [33],thus cannot be derived by applying the univariate one alongthe two dimensions.
In [11], [12], formulas for the bivariate Hermite interpolat-ing polynomial are provided, in the case of support pointsarranged on a non-equidistant grid. They are applicable inspecial conditions concerning the number of support pointsin each axis and the maximum order of partial derivativesfor each support point. These formulas are algebraicallyvery complex since they require the derivatives of rationalfunctions. In this work, we propose a generalization of theformulas provided in [11], [12], removing the constraint ofthe aforementioned special conditions. Our proposed formulais also algebraically considerably simpler, since it constructsthe Hermite polynomial as a sum of polynomials, rather thana sum of derivatives of rational functions as in [11], [12].
In addition, we make simplifying assumptions that allowthe application of Bivariate Hermite Interpolation (BHI) totwo-dimensional (2D) images. We combine the algebraicsimplicity of the propose Hermite bivariate polynomial witha computational approach based on look up tables. In thisway we achieve acceleration of the execution of the proposedBHI with selected settings, to a level that is comparable toconvolution-based interpolation with separable kernels. Thepartial image derivatives that are required by the BHI areapproximated using compact finite differences [13], rather thancentral finite differences [14]. This approach allows the use of
DELIBASIS AND KECHRINIOTIS: NEW FORMULA FOR BHI ON VARIABLE STEP GRIDS 2893
limited number of effective support points, which is a verydesirable characteristic for any interpolation method.
II. THEORY OF HERMITE INTERPOLATION
This section deals with the general case of BHI with supportpoints on variable step grids. Image interpolation is a specialcase of interpolation with support points on unit step grid andit will be explored in the next section.
A. Existing Theorems of Bivariate HermiteInterpolation on Grids
In [11] the following bivariate generalization of Hermite’sinterpolation was proven:
Theorem 1: Let m be a positive integer that denotes orderof differentiation, p(i1,i2)
j1, j2, 0 ≤ i1, i2 < m, 1 ≤ j1, j2 ≤ n
be given real numbers; let (λ j1, μ j2) be n2 distinct pointsfor 1 ≤ j1, j2 ≤ n. Then there exists a unique polynomialp (x1, x2) with degx1
p < nm, degx2p < nm such that
∂ i1+i2
∂xi11 ∂x
i22
(p(λ j1, μ j2)) = p(i1,i2)j1, j2
, 0 ≤ i1, i2 < m, 1 ≤ j1,
j2 ≤ n. This polynomial is given by:
p (x1, x2) = [L1 (x1) L2 (x2)]m
[(m − 1)!]2
×n∑
j1=1
n∑
j2=1
m−1∑
i1=0
m−1∑
i2=0
M(i1,i2)j1, j2
(x1, x2) p(i1,i2)j1, j2
, (1)
where
L1 (x1) =n∏
j1=1
(x1 − λ j1
), L2 (x2) =
n∏
j2=1
(x2 − μ j2
),
M(i1,i2)j1, j2
(x1, x2)=(
m−1i1
)(m−1
i2
)[dm−1−i1
dsm−1−i1
(1
A j1 (s)
)]
s=λ j1
·[
dm−1−i2
dtm−1−i2
(1
B j2 (t)
)]
t=μ j2
with
A j1 (s) = (s − x1)
[L1 (s)(
s − λ j1
)]m
,
B j2 (t) = (t − x2)
[L2 (t)(
t − μ j2
)]m
.
In [12], Theorem 1 is generalized so that the maximumorder of the partial derivative with respect to x1 at any point(λ j1, μ j2
)depends only on j1 and the maximum order of the
partial derivative with respect to x2 depends only on j2. Using[12] we conclude that the error of the interpolation formulaof Theorem 1 is the following:
R (x1, x2) = f (x1, x2) − p (x1, x2)
= [L1 (x1)]m
(nm)!∂nm
∂xnm‘1
f (ξ1, x2) + [L2 (x2)]m
(nm)!∂nm
∂xnm2
f (x1, ξ2)
− [L1 (x1) L2 (x2)]m
[(nm)!]2
∂nm+nm
∂xnm‘1 ∂xnm
2f (ξ1, ξ2) , (2)
for a point (ξ1, ξ2) ∈ [min
(x1, λ j1, . . . , λ jn
), max
(x1, λ j1,
. . . , λ jn
)]×[min
(x2, μ j1, . . . , μ jn
), max
(x2, μ j1, . . . , μ jn
) ].
An integral formula of the error R(x1, x2) is givenin [11]–[12].
B. The Proposed Bivariate Hermite Interpolation
In this subsection we derive a new formula for the bivariateHermite polynomial that: a) It is algebraically simpler than (1),since it does not contain derivatives of rational functions, thusit is calculated more efficiently and b) It can be applied to moregeneral circumstances, since the maximum order of the partialderivatives at any point
(λ j1, μ j2
)depends on both j1 and j2.
Before presenting the Theorem for the proposed expression ofthe BHI polynomial, we need the following definitions.
Let us suppose that for each support position(λ j1, μ j2
),
1 ≤ j1 ≤ n1, 1 ≤ j2 ≤ n2 two non-negativeintegers k j1, j2, m j1, j2 are given, such that the value ofall partial derivatives of a bivariate function f (x1, x2),
∂ i1+i2
∂xi11 ∂x
i22
(f(λ j1, μ j2
)), 1 ≤ i1 < k j1, j2, 1 ≤ i2 < m j1, j2 are
prescribed. The value of the aforementioned partial derivativesof f will be denoted as p(i1,i2)
j1, j2, (the two subscripts denote
the index of the support position and the two superscriptsindicate the order of partial derivative with respect to thefirst and second coordinate). The proposed BHI, constructs aunique polynomial p (x1, x2), which interpolates the value ofthe function f (x1, x2) as well as the given values of its partialderivatives of all given orders, at each support position. Wecan formalize the proposed BHI as following. Let us definethe following polynomials
K j1, j2 (x1) =∏
h1 �= j1
(x1 − λh1
)k j1 , j2
(λ j1 − λh1
)k j1 , j2,
L j1, j2 (x2) =∏
h2 �= j2
(x2 − μh2
)m j1, j2
(μ j2 − μh2
)m j1, j2. (3)
Let us also define the rows with dimensions 1 × k j1, j2 and1 × m j1, j2 respectively:
X j1, j2 (x1) = K j1, j2 (x1)
[1,
x1−λ j11! , . . . ,
(x1−λ j1
)k j1 , j2−1
(k j1 , j2−1
)!
],
(4a)
Y j1, j2 (x2) = L j1, j2 (x2)
[1,
x2−μ j21! , . . . ,
(x2−μ j2
)m j1, j2−1
(m j1, j2−1
)!
].
(4b)
Let us denote by V the vector space spanned by the polyno-
mials(x1−λ j1
)i1
i1 !(x2−μ j2
)i2
i2 ! K j1, j2 (x1) L j1, j2 (x2) , 1 ≤ j1 ≤ n1,1 ≤ j2 ≤ n2, 0 ≤ i1 < k j1, j2 , 0 ≤ i2 < m j1, j2 . Finally, let usdefine for each support position
(λ j1, μ j2
)the k j1, j2 × k j1, j2
and m j1, j2 × m j1, j2 lower unit triangular matrices � j1, j2 and� j1, j2 , as (5) and (6), shown at the top of the next page.
Theorem 2: Let(λ j1, μ j2
), 1 ≤ j1 ≤ n1, 1 ≤ j2 ≤ n2
be n1 × n2 discrete support positions that lie on a vari-able step grid with distinct λ j1 and distinct μ j2 coordinates.If k j1, j2, m j1, j2 are positive integers and p(i1,i2)
j1, j2, 0 ≤ i1 <
2894 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014
� j1, j2 =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
(00
)K j1, j2
(λ j1
)0 . . . 0
(10
) (K j1, j2
)(1) (λ j1
) (11
)K j1, j2
(λ j1
). . . 0
. . .. . . . . .
(k j1, j2 − 1
0
) (K j1, j2
)(k j1, j2 −1) (
λ j1
) (k j1, j2 − 1
1
) (K j1, j2
)(k j1 , j2−2) (
λ j1
). . .
(k j1, j2 − 1k j1, j2 − 1
)K j1, j2
(λ j1
)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, (5)
� j1, j2 =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
(00
)L j1, j2
(μ j2
)0 . . . 0
(10
) (L j1, j2
)(1) (μ j2
) (11
)L j1, j2
(μ j2
). . . 0
. . .. . . . . .
(m j1, j2 − 1
0
) (L j1, j2
)(m j1, j2−1
) (μ j2
) (m j1, j2 − 1
1
) (L j1, j2
)(m j1, j2−2
) (μ j2
). . .
(m j1, j2 − 1m j1, j2 − 1
)L j1, j2
(μ j2
)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
. (6)
k j1, j2 , 0 ≤ i2 < m j1, j2 are any real numbers, then there existsone and only one polynomial p (x1, x2) in V , such that
∂ i1+i2
∂xi11 ∂xi2
2
p(λ j1, μ j2
) = p(i1,i2)j1, j2
(7)
for all 1 ≤ j1 ≤ n1, 1 ≤ j2 ≤ n2, 0 ≤ i1 < k j1, j2 , 0 ≤ i2 < m j1, j2 .This polynomial is given explicitly by
p (x1, x2) =n1∑
j1=1
n2∑
j2=1
X j1, j2 (x1)(� j1, j2
)−1
×Pj1, j2
((� j1, j2
)−1)T (
Y j1, j2 (x2))T (8)
where Pj1, j2 is a k j1, j2 × m j1, j2 matrix given by
Pj1, j2
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
p(0,0)j1, j2
p(0,1)j1, j2
. . . p(0,m j1, j2−1
)
j1, j2
p(1,0)j1, j2
p(1,1)j1, j2
p(1,m j1, j2 −1
),
j1 j2
......
p(k j1, j2−1,0
)
j1, j2p
(k j1 , j2−1,1
)
j1, j2. . . p
(k j1 , j2−1,m j1, j2 −1
)
j1, j2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
. (9)
The complete proof of the above Theorem is given inAppendix A. If we apply Theorem 2 for n1 = n2 = n andk j1, j2 = m j1, j2 = m for any 1 ≤ j1 ≤ n, 1 ≤ j2 ≤ nthen we obtain a new algebraically simpler expression for theinterpolating polynomial p (x1, x2) that is given in Theorem 1,which will be used in the next sections. The interpolation errorof this polynomial is the same with the expression for the errorof Theorem 1, shown in Eq.(2).
III. APPLICATION OF BHI TO IMAGE PROCESSING
In this section, we apply a simplified version of the proposedBHI to interpolation of discrete images. A digital image is atwo dimensional function that has been sampled on a regulargrid with integer coordinates. Image interpolation deals withthe problem of determining the value of an image at a non-integer coordinate, which is a issue that arises when the spatial
resolution of the image changes or when a geometric transformis applied [15], [16]. The values of the image derivativesthat are required by the proposed BHI can be arithmeticallyapproximated.
The number of arithmetic operations when using (8) inits generic formulation may become prohibitive for imageprocessing applications. However, a number of assumptionscan be made, which allow us to accelerate the execution ofthe BHI to a level that is comparable to other convolutionbased interpolation methods: a) The support points are on aregular unit-step grid, b) instead of a single polynomial, weconstruct a number of small-support polynomial patches withappropriate boundary conditions and c) the maximum numberof differentiation is the same for all support points. Below weelaborate on the aforementioned simplifications.
Since any image consists of hundreds of pixels in eachdimension and in order to avoid excessively high polynomialdegree, we restrict the number of support positions n1, n2for each required image location to small values. We use theterm “nominal support points” when referring to the n1 × n2support points described in Theorem 2, in order to discriminateagainst the “effective support points” that are introduced by thearithmetic approximation of image partial derivatives (see nextparagraph). In this work we used n1 = n2 = n. The numberof nominal support points per image axis n is an importantcontrolling parameter of the proposed BHI.
In the case of image interpolation the n×n support positionsare distributed on a regular grid of unit step. Assumingthat we require the image value at the non-integer position(λreq, μreq), the n×n support points of Theorem 2 are selectedas following. For n even integer:
(λ j1, μ j2
) =(⌊
λreq⌋ − n
2+ j1,
⌊μreq⌋ − n
2+ j2
),
j1, j2 ∈ {1, 2, . . . , n}
where �z� is the integer part of z. These supportpoints may be used for any point (λ, μ) ∈ (�λreq� ,�λreq� + 1) × (�μreq� , �μreq� + 1). For n odd integer the
DELIBASIS AND KECHRINIOTIS: NEW FORMULA FOR BHI ON VARIABLE STEP GRIDS 2895
Fig. 1. Graphic illustration of support point determination for (a) n×n = 4×4(even number). Circles indicate the image pixels and the star is used for thearbitrary position for which interpolation is required.
support points are defined as:
(λ j1, μ j2
) =([
λreq ] − n + 1
2+ j1,
[μreq] − n + 1
2+ j2
),
j1, j2 ∈ {1, 2, . . . , n}where [z] is the nearest integer to z. These support points maybe used for any point:
(λ, μ) ∈ [[λreq ] − 0.5,
[λreq ] + 0.5
)
× [[μreq] − 0.5,
[μreq] + 0.5
).
Fig. 1 gives a graphic illustration of the selection of supportpoints for n = 4 (4 × 4) support points.
The maximum order of partial differentiations M is consid-ered constant for all support positions: k j1, j2 = m j1, j2 = M+1for all ( j1, j2). M is also an important controlling parameterof the proposed BHI.
According to the aforementioned assumptions the proposedBHI operates in a manner of bivariate polynomial patches withspecial continuity at its border. If we set n = 2 (2×2 nominalsupport points), then for each group of 2 × 2 source imagepixels the Hermite polynomial of Eq.(8) is defined, whosecoefficients are calculated by the values of the image, as wellas the values of the image partial derivatives at these 4 pixels.This polynomial patch has 8 neighboring polynomial patches.The values of every partial derivative of the Hermite surface ateach support point are equal to the values of the image partialderivative, as required by Eq.(7) in Theorem 2.
From the previous assumption it follows that the polynomi-als K j1, j2 (x1) in (3) are independent from j2 and L j1, j2 (x2)are independent from j1, therefore K j1, j2 (x1) = K j1 (x1),L j1, j2 (x2) = L j2 (x2). Consequently, the square lower triangu-lar matrices � j1, j2,� j1, j2 in (5) and (6) are also independentfrom j2 and j1 respectively, therefore we can simplify thesymbols as: � j1, j2 = � j1 for all j2 and � j1, j2 = � j2 for allj1. Since X j1 is function of x1 and Y j2 is function of x2, we areable to precalculate and store in a look up table the productsX j1
(� j1
)−1 and(� j2
)−1 (Y j2
)T that are required in (8). Thex1 and x2 coordinates are discretized by a step of 0.01. Foreach discrete value of the x1 (x2) coordinate, the look up tableholds one row of 1 × (M+1) for each of the n1 (n2) nominalsupport points. Assuming n1 = n2 = n = 2 and M = 2 which
are values that produce very competitive interpolation results,the size of the look up table becomes n1(M + 1) = 6 floatingpoint numbers for each one of the 101 possible values of x1and x2 coordinate. The look-up tables are precalculated andthey are data / application independent, so they impose nooverhead to the implementation of the BHI.
A. Approximation of Image Partial Derivatives
This subsection discusses the arithmetic approximation ofimage partial derivatives. It has to be mentioned however, thatif the partial derivatives are not approximated, but measuredas part of data acquisition, this subsection would not beapplicable.
The partial derivatives of any given image may be approx-imated by convolution of one-dimensional masks using cen-tered finite differences [14]. However, in this work we usethe compact finite differences, as described in the work ofLele [13]. Compact (or implicit) derivatives have been recentlyapplied to image processing [17]–[19] and require less supportpoints than the centered differences. Let us consider the 1Dcase and let us denote as fi and f ′
i the sampled signal valueand its 1st derivative approximation at sample i . According tothe work of Lele [13, (2.1)], the following holds:
q2 f′i−2 + q1 f
′i−1 + f
′i + q1 f
′i+1 + q2 f
′i+2
=(c3
6,
c2
4,
c1
2, 0,−c1
2,−c2
4,−c3
6
)∗ f (10)
where q1, q2 and c1, c2, c3 are coefficients whose values definethe order of accuracy d1 of the 1st derivative approximationand the symbol ∗ stands for linear convolution. (The order ofaccuracy for derivative approximation is not to be confusedwith the order of the derivative itself).
The second order derivative f ′′i of fi can be described in a
manner similar to (12), in [13, Eq.(2.2)]:
q2 f ′′i−2 + q1 f ′′
i−1 + f ′′i + q1 f ′′
i+1 + q2 f ′′i+2
=(c3
9,
c2
4,
c1
2,−2
(c3
9+ c2
4+ c1
2
),
c1
2,
c2
4,
c3
9
)∗ f (11)
The parameters q1, q2 and c1, c2, c3 define the accuracy orderd1, d2 of the 1st and 2nd derivative approximation respectivelyand can be obtained for d1 = 6, 8, 10 and d2 = 4, 6, 8 and10, according to (2.1.12) and (2.1.14) and (2.2.7) - (2.1.11)in [13], or according to Table II in [19].
The left hand sides of (12) and (13) involve inverse con-volution with a symmetric kernel of up to 5 points. Therequired values f ′, f ′′ can be obtained by solving a linearsystem of simultaneous equations, or by using the prefilteringalgorithm described in the work of Unser et al [20, section IIA,Eq.(2.5),(2.6)]. The details of this implementation are givenin Section 2.2 in [19]. The calculation of image derivativestakes place by applying (10) and (11) along image lines andsubsequently along columns.
B. Parameters of the Proposed BHI and the Conceptof Effective Support Points
The number of nominal support points per image axis n andthe maximum order of partial differentiations M are important
2896 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014
TABLE I
NEIGHBORHOOD SIZE OF THE EFFECTIVE SUPPORT POINTS
OF THE PROPOSED BHI FOR 2 × 2 NOMINAL SUPPORT
POINTS, AS FUNCTION OF d1 AND d2
Fig. 2. Graphic representation of the case of 2 × 2 nominal support pointsand the effective support points occupying the 2 central lines and columns ofan 8 × 8 neighborhood.
controlling parameter of the proposed BHI. The order ofaccuracy for the 1st and 2nd derivative approximation, d1, d2respectively, are also parameters of the proposed BHI. Theforward convolution of the right hand side of (10) and (11)involves more support positions per image axis than the nomi-nal support positions n1 ×n2. Thus, the effective support posi-tions of the proposed BHI are the central lines and columnsof a rectangular neighborhood, whose size is a function of d1,d2. Table I is constructed for the selected values of d1 and d2,using n1 = n2 = 2 nominal support points and M = 2.
A graphic explanation of the concept of effective supportpoints is given in Fig. 2 for 2×2 nominal support points. Theapproximation of the 1st and 2nd order image derivatives ateach one of the 4 nominal support points, requires the use of upto 7 pixels along the current image line and column, shown forthe lower left and upper right nominal support position. Thus,although 2×2 nominal support points are required for the BHI,the calculation of image derivatives at each of these pointsinvolves effectively more points that lie inside a neighborhoodof 8×8 image pixels. These points are called effective supportpoints.
On the other hand, if the values of the image derivativesare given, or can be analytically computed, then the numberof nominal support points is identical to the effective ones.This is the case of the synthetic image used in par. 4.4.1.
1) The Issue of Non-Separability: The expression for thebivariate Hermite interpolating polynomial cannot be writtenin separable form (i.e. there are no univariate polynomialsp1(x1), p2(x2), such that p(x1, x2) = p1(x1)p2(x2) for anyvalue of m1
j1, j2, m2
j1, j2> 0). Non-separability does increase
TABLE II
COMPARISON OF THE PROPOSED BHI UNDER 3 DIFFERENT SETTINGS
WITH STATE OF THE ART METHODS, IN TERMS OF RMSE AND
NSD FOR THE SYNTHETIC IMAGE INTERPOLATION IN THE
CASE OF 4 AND 9 ROTATIONS
significantly the computational work load in the case where thefractional parts of the required image coordinates λreq and μreq
are constant for any pixel location (e.g. image upsampling).Such an example is provided in [21], where it is stated thatin the case of n point kernel applied in two dimensions,2n evaluations are needed, compared to n2 evaluations for anon-separable kernel.
However more elaboration is needed for a more generalgeometric transformation, like the affine one, in which thex1 and x2 depend on the pixel position. In this case, in order toexploit the separability property of a kernel, the transformationneeds to be decomposed into a series of one-dimensional trans-formations [22], (a relevant example of rotation decompositioninto three one-dimensional translation or skew transformationsis given in Gotchev et al. [23, page 321]). It has been shownthat any 2D affine transformation can be decomposed into3 one-dimensional transformations [22], [24]. Therefore, ann point separable kernel has to be applied 3 times in thecase of a 2D affine geometric transformation. Equivalently,no computational gain is achieved for n < 4, (n the numberof nominal support points per image axis). In this work, theproposed BHI method is employed for n = 2 (2 × 2 nominalsupport points), since as it will be shown in the Results section,in most of the experiments it outperforms the state of theart methods in comparison with same number of effectivesupport points, including to O-MOMS and B-spline of 7th
degree. The calculation of discrete derivatives using compactdifferences possesses the separability property, [17], [18]. Fur-thermore, if the 2D geometric transformation is not separableinto a series of one-dimensional transformations (such as anon-linear or elastic transformation), a separable kernel offersno computational advantage.
DELIBASIS AND KECHRINIOTIS: NEW FORMULA FOR BHI ON VARIABLE STEP GRIDS 2897
Fig. 3. The synthetic image, one mammographic and one iris image used inthe interpolation experiments with their portions used in Fig. 4 and 5 outlined.
It would seem convenient to express the proposed BHIas a two-dimensional kernel. For instance, in the case ofthe univariate Hermite interpolating polynomial, describedin [5] and more recently in [6], the corresponding kernelshave been produced for different number of support points[26], approximating discrete derivatives by central differences.In the current work however, the non-separable bivariateHermite interpolating polynomial causes the correspondingHermite kernels to be two-dimensional, thus they would havea large number of branches. In addition, the use of compactfinite differences necessitates one such kernel for each imagepartial derivative, thus the kernel option becomes less practical.A single kernel could be generated using central differencesinstead of compact differences for derivative approximation,however the performance of the BHI would be inferior and thenumber of the required effective support points would increase.Therefore, the direct implementation of (8) is preferred.
IV. RESULTS
A. Data
The proposed evaluation scheme has been applied to onesynthetic image, as well as to a number of images fromdifferent medical modalities. More specifically, the syntheticimage with dimensions 512 × 512 pixels was constructed asa radial sinusoidal chirp with higher frequency at the imagecenter (see Fig. 3), similar to the one used in the work ofThevenaz et al [25]. The synthetic image was selectedsince its partial derivatives can be calculated analytically, aswell as approximated arithmetically, thus allowing to studythe behavior of the proposed BHI in more detail. Fromthe medical modalities, the following were used: 8 images1024 × 1024 from the Mini-MIAS mammography data-base [27] and 8 images from the Iris database v1 withdimensions 320 × 200 [28]. One iris and one mammographicimage are shown in Fig. 3, each with a portion indicated tobe used for error visualization in Figs. 4 and 5. Furthermore,10 transverse slices at the level of lower abdomen from a Com-puter tomography study, denoted as ‘CT’, with dimensions of512×512 and 10 sagittal slices, 256×256 from an MRI brainstudy (‘MRI’).
B. Scheme for the Evaluation of BHI
The BHI is quantitatively evaluated using a series of affinegeometric transformations that are applied sequentially on anygiven image I0 and result in a final image I f that ideally
Fig. 4. The absolute difference between the ideal image and the result of9 rotations of the synthetic image, for the proposed BHI using n = 2, M = 2,with analytic derivatives, as well as with arithmetic derivative approximation(d1 = 10, d2 = 6). Other state of the art interpolation methods are alsoincluded.
should be identical to the original image. In this work, weconsider the following:
A) Series of 2, 4 and 8 pairs of random forward – inverseaffine transformation: a random forward affine transformationis applied to a given image I0 to produce the image I1,followed by the application of its inverse applied to theresulting I1. A series of pairs of forward – inverse transfor-mations are applied successively (see [2]–[29]). Each affinetransformation consists of image rotation and scaling withrespect to its centre. The angle and scaling factors are selectedrandomly in the range [0, π/3] and [0.9, 1.1].
B) Series of 2, 4 and 9 randomly selected successiverotations, which add to a total of 2π are applied to a givenimage to produce the final image I f , in an approach similarto [1], [2], [25, subsection III-A], [37 subsection 3.4].
The geometric transformations are implemented using theproposed BHI as well as the other interpolation methods undercomparison. The root mean square error (RMSE) betweenI f and I0 was used as an objective measure of the qual-ity of the investigated interpolation method [2]. We havealso calculated the number of sites of disagreement (NSD)(see [30], [31]) between I f and I0, defined as the number
2898 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014
Fig. 5. The resulting absolute difference between the ideal and the interpolated portion of an iris image (right) and a mammography image (left), forthe proposed BHI and other state of the art interpolation methods, for 8 pairs of forward-inverse affine transforms. Black indicates zero difference. Theinterpolation methods are arranged in 2 × 4 as shown in the central part of the Figure. The image portions are shown on the original images in Fig. 3.
of pixels with values that differ from the correct onesby more than a predefined threshold. This threshold wasset to 10 for the image interpolation experiments inthis work. In order to avoid border effects, the imageswere mirrored in both dimensions ([2, subsection 4.1],[32, subsection 2.1]).
C. Other Interpolation Techniques in Comparison
A number of different interpolation techniques were selectedto compare against the proposed BHI interpolation, includinglinear interpolation and Lagrange interpolation [33] for 4 and 6support positions, implemented as in [1]. From the family ofcubic interpolation kernels with 4 support positions, definedin [1,(23)], we used the ones generated by setting the kernelfree parameter a to −1/2 which corresponds to Catmull-Rominterpolation [34], −3/4 and −1. We also included comparisonwith the Mitchell-Netravali (MN) kernel [35] (as defined inEq.(26) from [1] using b = c = 1/3), the cubic polynomialinterpolation using 6 and 8 support positions using the kernel,provided in [1 (24)] and the 4-point Lanczos interpolation[15 (16.61)].
From the methods of generalized interpolation, which areconsidered state of the art, we included B-spline interpolation([20], [36]), as well as the optimal smallest-support supportfunctions (O-MOMS) [37] of up to the 7th degree. This degreewas selected since the kernels use the same number of supportpoints per image dimension (8 × 8) with the effective support
points of the proposed BHI for n1 = n2 = 2, d1 = 10,d2 = 6, 8. The BHI for n1 = n2 = 2, with d1 ≤ 8, d2 ≤ 6,involves effective support points in a 6 × 6 neighborhood,thus it may be compared to the B-spline and O-MOMS ofthe 5th degree. Finally, we included the FT-based two stageresampling with magnification factor of 2, proposed in [32],in conjunction with the cubic polynomial 6- and 8-pointinterpolation, as well as in conjunction with the B-splines andthe O-MOMS. All interpolation methods in comparison wereimplemented using Matlab.
D. Comparative Results
1) Synthetic Image: Table II shows the interpolation errorin terms of RMSE and NSD for 4 and 9 successive rotations ofthe synthetic image. The time required for one rotation usingthe aforementioned Matlab implementations is also given. Tofacilitate comparison, the interpolation methods are groupedaccording to their number of support points (see [2, p119]).Three entries for the proposed BHI are shown, usingn1 = n2 = 2, M = 2.
In the first BHI entry, the image partial derivatives arecomputed from the analytic expression of the image data, thusthe arithmetic approximation is not applied. Therefore, onlyimage information on the 2 × 2 nominal support points isused in the interpolation process. The other two BHI entriesare using the arithmetic image derivative approximation withd1 = 8, d2 = 6 (thus using 6 × 6 effective support points)
DELIBASIS AND KECHRINIOTIS: NEW FORMULA FOR BHI ON VARIABLE STEP GRIDS 2899
TABLE III
THE RMSE OF THE PROPOSED BHI AND THE OTHER INTERPOLATION METHODS, AVERAGED OVER THE AVAILABLE IMAGES OF EACH MODALITY FOR
THE SERIES OF FORWARD – INVERSE AFFINE TRANSFORMATION EXPERIMENTS, GROUPED BY THE NUMBER OF SUPPORT POINTS. THE PROPOSED BHI
WITH 6 × 6 EFFECTIVE SUPPORT POINTS HAS PARAMETERS n1 = n2 = 2, M = 2, d1 = 8, d2 = 6, WHEREAS FOR 8 × 8
EFFECTIVE SUPPORT POINTS BHI HAS PARAMETERS n1 = n2 = 2, M = 2, d1 = 10, d2 = 6
and for d1 = 10, d2 = 6 (8 × 8 effective support points). Theentries referring to the two-stage resampling method [32] areindicated by “FT”.
The proposed BHI with analytically calculated image deriv-atives, although it uses only 2 × 2 support points, achievesthe smallest RMSE and NSD of all the interpolation methodsin comparison, including the two implementations of theproposed BHI that use arithmetic image derivative approxi-mations. The BHI with 6 × 6 and 8 × 8 effective supportpoints outperform the state of the art methods in compar-ison with the same number of support points (includingthe B-spline and the O-MOMS of the 5th and 7th degreerespectively). It can also be observed that the BHI with6×6 effective support points outperforms all the interpolationmethods, including those with 8 × 8 support points aswell, including the methods combined with the FT-basedtwo stage resembling. The FT resampling causes a markedimprovement in the case of cubic polynomial interpolation,or 3rd degree B-spline and more modest improvement whencombined with the 7th degree O-MOMS. Furthermore, itis verified that the generalized interpolation methods (theB-spline and the O-MOMS of various degrees) clearly out-perform the classic interpolation methods (e.g. the 3rd degreeB-splines with 4 × 4 support points, achieves smaller RMSEthan the cubic polynomial interpolator with 8 × 8 supportpoints).
Fig. 4 visualizes the resulting interpolated image after9 rotations of the synthetic image for selected interpolationtechniques, including BHI with analytic and numeric deriva-tive calculation and the state of the art B-spline and O-MOMS
interpolators. The images appear to correlate well with thenumerical results in Table II.
2) Real Images: Tables III and IV show the RMSE and theNSD of the interpolation methods, averaged over the availableimages, from mammography, iris, CT and MRI studies, for2, 4 and 8 pairs of forward – inverse affine transformations.Table V shows the RMSE for 2, 4 and 9 successive rotations.The proposed BHI for n1 = n2 = 2, M = 2 is evaluated fortwo different settings of the order of accuracy of the first andsecond partial derivatives: a) d1 = 8, d2 = 6 that correspondsto 6 × 6 effective support points and b) d1 = 10, d2 = 6that corresponds to 8×8 effective support points. To facilitatecomparison, the interpolation methods are grouped accordingto their number of support points [2, p119]. It can be observedthat in 21 out of 24 interpolation experiments in Tables IIIand V, the proposed BHI with d1 = 8, d2 = 6 outperformsin terms of RMSE the state of the art interpolation with thesame number of support points (6 × 6) or less (including theB-spline and the O-MOMS of the 5th degree and the cubicinterpolating polynomial, even combined with the FT-basedtwo-stage resampling). In terms of NSD, the proposed BHIwith d1 = 8, d2 = 6 outperforms the other interpolationmethods with same or less support points (6 × 6) in 8 outof the 12 experiments of Table IV. Furthermore, in 19 outof the 24 interpolation experiments reported in Tables III andIV, the proposed BHI with d1 = 10, d2 = 6 achieves lowerRMSE than any of the state of the art interpolation methodswith the same number of support points (8 × 8) or less. Interms of NSD, the proposed BHI with d1 = 10, d2 = 6outperforms the other interpolation methods with same or
2900 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014
TABLE IV
THE NUMBER OF SITES OF DISAGREEMENT (NSD) FOR THE INTERPOLATION METHODS AND THE EXPERIMENTS OF TABLE III
TABLE V
THE RMSE FOR THE ROTATION EXPERIMENTS (SEE CAPTION OF TABLE III)
less support points (8 × 8) in 9 out of the 12 experimentsof Table IV.
As expected, the generalized convolution methods(O-MOMS and B-spline) consistently outperform thetraditional convolution methods (the Lanczos, MN, Lagrangeand cubic polynomial interpolator with various settings).The FT-based two-stage resampling improves significantlythe performance of the 6-point and 8-point cubic polynomialinterpolation, both in terms of RMSE and NSD. Significant
improvement is also achieved when the FT-based two-stageis combined with B-spline of 3rd degree, whereas marginalimprovement is observed with the combination of the FTtwo-stage with the 7th degree O-MOMS. In order to assessthe performance of the proposed BHI, we present in Fig. 5the resulting difference images for 8 affine transformationpairs for a portion of an iris image and for a portion of oneof the mammography images (image portions are shown inFig. 3). The BHI with 8 × 8 support points and 7 state of
DELIBASIS AND KECHRINIOTIS: NEW FORMULA FOR BHI ON VARIABLE STEP GRIDS 2901
Fig. 6. The achieved RMSE of various state of the art interpolation methodsgrouped by the number of support points, for the 4-rotation experiment of theMammography images.
the art interpolation methods are arranged in 2 columns and4 rows, as shown in the central part of Fig. 5. The visualappearance of the difference images correlates well with thenumerical results in Table III.
Fig. 6 summarizes the RMSE for various state of theart interpolation methods, grouped by the number of theirsupport points (the effective number of support points wasused for the BHI). The 4-rotation experiment using theMammography images was selected, as a typical exam-ple of the majority of experiments in which the proposedBHI using 8 × 8 support points outperforms the othermethods.
V. DISCUSSION
A. Computational Complexity of the Proposed BHI forImage Interpolation
Considering the computational implementation of BHIusing look up tables, as described previously, the calculationsof (7) involve one multiplication of the 1 × (M + 1) row,times the (M + 1) × (M + 1) matrix, times one column(M +1)×1, which add up to (2M +1)(M +1)+ M +1+ M =(2M +1)(M +2) arithmetic operations for each support point.Thus, the complexity for n×n nominal support points becomesn2(2M + 1)(M + 2).
The approximation of the image partial derivatives imposesa computational cost, which depends on the order of accuracyd1, d2. Setting d1 equal 6, 8 and 10, the size of the rowconvolution kernel at the right hand side of Eq.(12) is 5, 5 and7 respectively. The number of arithmetic operations introducedby the left hand side of (10) and (11) is O(4N), O(8N) andO(8N) respectively, as described in detail in [19], [20].
Concerning the generalized interpolation (B-splines andO-MOMS), the cost of IIR prefiltering for each image pixelis obtained as described in detail in [20]. The cost of thesubsequent convolution per image pixel with an n×n separablekernel is 3(2n − 1) arithmetic operations in the case of anaffine geometric transformation that can be decomposed into3 one-dimensional transformations (see par. 2.2.3), or 2n2 − noperations in the case of a non-decomposable geometric
Fig. 7. The RMSE of the BHI for the synthetic image with 4 rotations, asfunction of n and M. The RMSE of the 7th degree O-MOMS is also shown.
transformation, in which case the convolution step needs tobe performed in two dimensions.
The proposed BHI can be computationally more demandingthan the generalized interpolation methods, for separable affinegeometric transformations, but it presents the following advan-tages: a) it achieves lower interpolation RMSE, as well as NSDin the majority of the interpolation experiments, compared tokernels with the same number of support points and b) itis the method of choice when the values of the derivativesare not arithmetically approximated from the image data,but originally acquired (analytically or by measurements), asit has been shown in the results with the synthetic image.For geometric transformations that cannot be decomposed inseries of 1D transformations, the difference in computationalcomplexity becomes less significant.
B. The Behaviour of BHI With Respect to ItsControlling Parameters
As it has been mentioned previously, the controlling para-meters of the BHI are the nominal number of support pointsper image axis n(n × n points in two dimensions), themaximum order of partial differentiation M and the orderof accuracy of the 1st and 2nd image derivatives d1, d2respectively, which define the number of effective supportpoints. We can rewrite the error expression (2) for Theorem 1for any point (ξ1, ξ2) ∈ [ j1, j1 + n − 1] × [ j2, j2 + n − 1],under the assumptions presented in section 3:
f (x1, x2) − p (x1, x2)
= (x1 − j1)M+1 K j1 (x1)
(n (M + 1))!∂n(M+1)
∂xn(M+1)1
f (ξ1, x2)
+ (x2 − j2)M+1 L j2 (x2)
(n (M + 1))!∂nm
∂xn(M+1)2
f (x1, ξ2)
− ((x1 − j1) (x2 − j2))M+1 K j1 (x1) L j2 (x2)
[(n (M + 1))!]2
× ∂2n(M+1)
∂xn(M+1)‘1 ∂xn(M+1)
2
f (ξ1, ξ2) , (12)
Fig. 7 investigates the behaviour of the proposed BHI withrespect to its most important controlling parameters: n, M ,whereas d1 and d2 were set equal to 10 and 6 respectively
2902 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014
Fig. 8. The RMSE of the proposed BHI for an image of the mammographydatabase and 1 pair of forward - inverse affine transformations, as a functionof 1st and 2nd derivative accuracy order.
(d2 is not applicable for M = 1). The synthetic imageunder 4 rotations with arithmetic approximation of the partialderivatives was used. It can be observed that increasing nwhile keeping M = 1, results in significant error improvementfor M = 1 (Fig. 7). This behavior is expected from theexpression of the interpolation error (12), since all the treeerror terms contain n! in the denominator. On the other hand,each error term also contains the value of the (n(M +1)) orderof the function partial derivatives, which cannot be predicted,which may cause the behavior for M = 2. Furthermore, allBHI with M = 2 outperform the BHI with M = 1, at theexpense of slight increment to its computational complexity.This behavior is also expected from (12). As M increases,the denominator of each error term increases in a factorialmanner. If n = 2 the nominator of each fraction decreasesexponentially. If n > 2 the nominator increases exponentially.Therefore, the overall value of the fraction of each error termdecreases.
In Fig. 8 the RMS error of the BHI is calculated for animage of the mammography database and 1 pair of forward- inverse affine transformations, using all combinations of
d1 ∈{6,8,10}, d2 ∈{ 4,6,8} . The rest of the BHI parameterswere set to n = 2 (2 × 2 nominal support points) andM = 2, as discussed before. The RMSE of O-MOMS of7th degree (8 × 8) is also included. It can be observer thata significant RMSE improvement occurs when increasing theorder of accuracy of the 1st image derivative d1 from 6 to 8and a more marginal improvement by setting d1 = 10. This isexpected, since increasing d1 results in increasing the numberof effective support points. Similar behaviour is observed inthe majority of the interpolation experiments.
Increasing d2 from the value of 4 up to 8 also causes areduction of the RMSE, when d1 = 8 or 10, although lessnoticeable. The RMSE of O-MOMS of 7th degree (one of thebest performing methods in comparison), is also included inthe Figure. Similar behaviour has been observed in most ofthe experiments with the other image modalities.
VI. CONCLUSIONS
A new formula for the bivariate Hermite interpolation forgrids (BHI) for support points arranged on a non-uniformlyspaced grid is derived in this work. The formula is moregeneral and algebraically simpler than the previously pro-posed ones. A number of simplifying assumptions are madeto achieve an efficient computational implementation forimage interpolation. The required image partial derivativesare calculated using compact differences. The controllingparameters of the BHI are identified and their effect on theproposed method is studied, both theoretically and experimen-tally. The computational complexity of the proposed method isalso studied in detail. The proposed BHI is compared againsta number of state of the art interpolation techniques, usingextensive experiments for a synthetic image with known partialderivatives, as well as a number of images from differentmedical imaging modalities. Results show that the proposedBHI outperforms the state of the art interpolation techniquesthat it was compared with, in terms of RMS error and NSDin the majority of the interpolation experiments, while usingthe same number of support points.
∂ i1+i2
∂xi11 ∂xi2
2
((x1 − λt1
)s1(x2 − μt2
)s2
s1!s2! K j1, j2 (x1) L j1, j2 (x2)
)
(x1,x2)=(λt1 ,μt2
)
=
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
0 i f (t1, t2) �= ( j1, j2)
0 i f (t1, t2) = ( j1, j2) and (i1 < s1ori2 < s2)(i1
s1
)(i2
s2
)(K j1, j2
)(i1−s1)(λ j1
) (L j1, j2
)(i2−s2)(μ j2
), i f (t1, t2) = ( j1, j2) , s1 ≤ i1, s2 ≤ i2
(A1)
⎡⎢⎢⎢⎢⎢⎢⎣
c(0,0)j1, j2
c(0,1)j1, j2
. . . c(0,m j1, j2 −1
)
j1, j2
c(1,0)j1, j2
c(1,1)j1, j2
c(1,m j1, j2−1
)
j1, j2...
...
c(k j1 , j2 −1,0
)
j1, j2c(k j1 , j2−1,1
)
j1, j2. . . c
(kj1, j2−1,m j1, j2 −1
)
j1, j2
⎤⎥⎥⎥⎥⎥⎥⎦
=(� j1, j2
)−1
⎡⎢⎢⎢⎢⎢⎢⎣
p(0,0)j1, j2
p(0,1)j1, j2
. . . p(0,m j1, j2−1
)
j1, j2
p(1,0)j1, j2
p(1,1)j1, j2
p(1,m j1, j2−1
)
j1, j2...
...
p(k j1 , j2−1,0
)
j1, j2p
(k j1 , j2 −1,1
)
j1, j2. . . p
(k j1 , j2−1,m j1, j2−1
)
j1, j2
⎤⎥⎥⎥⎥⎥⎥⎦
((� j1. j2
)−1)T
(A2)
DELIBASIS AND KECHRINIOTIS: NEW FORMULA FOR BHI ON VARIABLE STEP GRIDS 2903
� j1, j2
⎡⎢⎢⎢⎢⎢⎢⎣
c(0,0)j1, j2
c(0,1)j1, j2
. . . c(0,m j1, j2−1
)
j1, j2
c(1,0)j1, j2
c(1,1)j1, j2
c(1,m j1, j2−1
)
j1, j2...
...
c(k j1 , j2−1,0
)
j1, j2c(k j1, j2 −1,1
)
j1, j2. . . c
(k j1 , j2 −1,m j1, j2−1
)
j1, j2
⎤⎥⎥⎥⎥⎥⎥⎦
(� j1, j2
)T
=
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
p(λ j1, μ j2
)∂
∂x2p
(λ j1, μ j2
). . . ∂
m j1, j2−1
∂xm j1, j2
−1
2
p(λ j1, μ j2
)
∂∂x1
p(λ j1, μ j2
)∂2
∂x1∂x2p
(λ j1, μ j2
) · · · ∂m j1, j2
∂x1∂xm j1, j2
−1
2
p(λ j1, μ j2
)
......
∂k j1, j2
−1
∂xk j1, j2
−1
1
p(λ j1, μ j2
)∂
k j1, j2
∂xk j1, j2
−1
1 ∂x2
p(λ j1, μ j2
). . . ∂
k j1, j2+m j1, j2
−2
∂xk j1, j2
−1
1 ∂xm j1, j2
−1
2
p(λ j1, μ j2
)
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
(A4)
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
p(λ j1, μ j2
)∂
∂x2p
(λ j1, μ j2
). . . ∂
m j1, j2−1
∂xm j1, j2
−1
2
p(λ j1, μ j2
)
∂∂x1
p(λ j1, μ j2
)∂2
∂x1∂x2p
(λ j1, μ j2
) · · · ∂m j1, j2
∂x1∂xm j1, j2
−1
2
p(λ j1, μ j2
)
......
∂k j1 , j2
−1
∂xk j1, j2
−1
1
p(λ j1, μ j2
)∂
k j1, j2
∂xk j1, j2
−1
1 ∂x2
p(λ j1, μ j2
). . . ∂
k j1 , j2+m j1, j2
−2
∂xk j1, j2
−1
1 ∂xm j1, j2
−1
2
p(λ j1, μ j2
)
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
=
⎡
⎢⎢⎢⎢⎢⎢⎣
p(0,0)j1, j2
p(0,1)j1, j2
. . . p(0,m j1, j2 −1
)
j1, j2
p(1,0)j1, j2
p(1,1)j1, j2
p(1,m j1, j2−1
),
j1 j2...
...
p(k j1 , j2−1,0
)
j1, j2p
(k j1 , j2−1,1
)
j1, j2. . . p
(k j1, j2 −1,m j1, j2 −1
)
j1, j2
⎤
⎥⎥⎥⎥⎥⎥⎦
VII. APPENDIX A
Using the Leibniz’s rule for derivatives, we easily get thefollowing Lemma:
Lemma 1: For 1 ≤ j1, t1 ≤ n1, 1 ≤ j2, t2 ≤ n2,0 ≤ i1, s1 < k j1, j2, 0 ≤ i2, s2 < m j1, j2 as (A1), shown atthe bottom of the previous page.
Proof of Theorem 1: It can be observed that (8)can be equivalently written in the following form:
p (x1, x2) =n1∑
j1=1
n2∑j2=1
k j1 , j2−1∑
i1=0
m j1, j2−1∑
i2=0c(i1,i2)
j1, j2
(x1−λ j1
)i1
i1 !(x2−μ j2
)i2
i2 !K j1, j2 (x1) L j1, j2 (x2), where as (A2), shown at the bottom ofthe previous page.
Now by calculating the derivative of order (i1, i2) of thepolynomial p at
(λ j1, μ j2
)and using (A1) of Lemma 1 we
obtain:
∂ i1+i2
∂xi1i ∂xi2
2
p(λ j1, μ j2
) =i1∑
k1=0
i2∑
k2=0
c(k1,k2)j1, j2
(i1k1
) (i2k2
)
(K j1, j2
)(i1−k1) (λ j1
) (L j1, j2
)(i2−k2) (μ j2
)(A3)
for all 1 ≤ j1 ≤ n1 , 1 ≤ j2 ≤ n2, 0 ≤ i1 < k j1, j2 ,0 ≤i2 < m j1, j2 . The system of equations (A4) can be rewritten as(A4), shown at the top of the page.
Substituting (A2) in (A4), we get the equation shown at thetop of the page below eq.(A4)
Thus, we derive that ∂ i1+i2
∂xi1i ∂x
i22
p(λ j1, μ j2
) = p(i1,i2)j1, j2
for
all 1 ≤ j1 ≤ n1, 1 ≤ j2 ≤ n2, 0 ≤ i1 < k j1, j2 ,0 ≤ i2 < m j1, j2 . Finally, for the uniqueness of thepolynomial p it is sufficient to prove that the polynomials(
x1−λ j1
)i1
i1!(x2−μ j2
)i2
i2 ! K j1, j2 (x1) L j1, j2 (x2) , for all i1, i2, j1, j2are linearly independent: We start from p (x1, x2) =
n1∑j1=1
n2∑j2=1
k j1, j2−1∑
i1=0
m j1, j2−1∑
i2=0c(i1,i2)
j1, j2
(x1−λ j1
)i1
i1!(x2−μ j2
)i2
i2! K j1, j2 (x1)
L j1, j2 (x2) = 0. Then since the polynomial p (x1, x2) isidentically equal to zero, we have that (A2) states byp(i1,i2)
j1, j2= 0 for all 1 ≤ j1 ≤ n1, 1 ≤ j2 ≤ n2, 0 ≤ i1 < k j1, j2 ,
0 ≤ i2 < m j1, j2 . Therefore from (A2) we get c(i1,i2)j1, j2
= 0.
ACKNOWLEDGMENT
Portions of the research in this paper use the CASIA-IrisV1collected by the Chinese Academy of Sciences’ Institute ofAutomation (CASIA).
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Konstantinos K. Delibasis received the degreefrom the Department of Physics, University ofAthens, Athens, Greece, in 1990, and the M.Sc.and Ph.D. degrees from the Department of Bio-medical Physics and Bioengineering, University ofAberdeen, Aberdeen, U.K., in 1991 and 1995,respectively. He is currently an Assistant Professorwith the Department of Computer Science and Bio-medical Informatics, University of Thessaly, Volos,Greece. His research interests include processing andanalysis of biomedical signal, image, and video.
Aristides Kechriniotis received the degree from theDepartment of Mathematics, University of Patras,Patras, Greece, in 1975, and the Ph.D. degree fromthe Department of Engineering Sciences, Polytech-nic School, University of Patras, in 2002. He iscurrently a Visiting Lecturer with the Technologi-cal Educational Institute of Sterea Ellada, Lamia,Greece. His research interests include mathemati-cal analysis, applied linear algebra, mathematicalphysics, and applications to signal processing.
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