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ISBN 978-952-5726-09-1 (Print)
Proceedings of the Second International Symposium on Networking and Network Security (ISNNS ’10)
Jinggangshan, P. R. China, 2-4, April. 2010, pp. 267-270
Parabola Interpolation With Adaptive Error
Compensation
Guangming Yang 1 , and Fengqi Yu 2
1,2 Department of Integrated Electronics,Shenzhen Institute of Advanced Technology, CAS
Shenzhen, China, 518067
1 Email: mark111yang@hotmail.com
2 Email: fq.yu@siat.ac.cn
Abstract—This paper proposes a novel scheme to interpolate
images based on Lagrange Interpolation Theory. Parabola
polynomial with an adaptive Lagrange error compensation
is adopted to determine 1-D pixels. In order to predict 2-D
pixels more accurately, the direction perpendicular to
gradient vector is determined. To simplify the interpolation
process, we give a range of the direction. Simulation results
show that the proposed method can get clearer and sharper
image than traditional methods. From objective point of
view, average gradient and mean structural similarity
(MSSIM) are calculated to demonstrate its superior to other
methods.
Index Terms -parabola interpolation, Lagrange
interpolation error, Sobel operator, gradient direction ,
MSSIM.
I. INTRODUCTION
Image interpolation is a method to improve resolution
of an image. It is widely applied in outer space image,
medical image, and images in consumer electronics, etc.
There are many image interpolation techniques, e.g.
bilinear, cubic convolution [1], and bicubic B-spline
interpolators. However, These techniques suffer from
artifacts such as zigzag, ringing and blurriness. Especially
the blurriness artifact blurs image detail so badly that we
can’t do any further image analysis.
Efforts have been made by many people to improve
the image quality, Lei Zhang [6] proposed an interpolation
algorithm based on directional filtering and data fusion,
which can preserve the edge information to a certain
extent. Schultz proposed MAP (maximum a posteriori)
algorithm to expand image [4], which places the
interpolation problem into a statistical framework. For the
purpose of reducing the interpolation error, EASE method
was proposed by Cha [2]. In his scheme, based on bilinear
method, Lagrange interpolation error theory is used to
compensate the pixel error in 1-D signal, which makes the
pixel values more accurate. However, Zhang’s algorithm
and MAP require more computations. Although EASE
applies error compensation to interpolation successfully,
the quality of the interpolated image cannot meet some
requirements because it is based on biline method in 1-D.
To further improve image resolution, we propose a
parabola interpolation method which interpolates the 1-D
direction pixels with a parabola interpolation polynomial,
where an error compensation technique is considered.
Then based on bilinear method, 2-D pixels along the
direction perpendicular to gradient direction are
interpolated. The gradient direction is calculated by
vertical and horizontal Sobel operators.
This paper is organized as follows: Section Ⅱ briefly
describes the parabola interpolation and the error
estimation theory. Section Ⅲ presents the proposed
algorithm of 1-D and 2-D interpolation. Experiment
results and performance analysis are presented in section
Ⅳ , where average gradient and MSSIM quality
assessment methods are adopted from objective point of
view. A brief conclusion is presented in Section Ⅴ.
II.
REVIEW OF PARABOLA INTERPOLATION AND ERROR
ESTIMATION
As a special part of Lagrange interpolation, parabola
interpolation polynomial L 2 (x) can be formulated as [5]:
( x−xk)( x−xk+ 1) ( x−xk− 1)( x−xk+
1)
L2()
x = yk−
1
+ yk
( xk− 1−xk)( xk− 1−xk+ 1) ( xk−xk− 1)( xk−xk+
1)
( x−xk−
1)( x−xk)
+ yk
+ 1
( xk+ 1−xk− 1)( xk+
1−xk)
(1)
where each variable is interpreted in figure 1.
x k −2
x k −1
f ( x)
x
x k x
k + 1
x
k + 2
Figure 1. Interpolation sketch map
The value of x is estimated by the surrounding
elements with corresponding weights. The interpolation
error is discussed in ref. [5].
Assume f (n) (x) is continuous in the range x∈ [x 0 , x 1 ] ,
and f (n+1) (x) exists when x∈ (a, b), L n (x)is the interpolation
polynomial. The interpolation error is:
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