03-05-2012, 12:42 PM
Curvature Interpolation Method for Image Zooming
Curvature Interpolation Method for Image Zooming.pdf (Size: 2.29 MB / Downloads: 77)
Abstract
We introduce a novel image zooming algorithm,
called the curvature interpolation method (CIM), which is partial-
differential-equation (PDE)-based and easy to implement. In
order to minimize artifacts arising in image interpolation such as
image blur and the checkerboard effect, the CIM first evaluates
the curvature of the low-resolution image. After interpolating
the curvature to the high-resolution image domain, the CIM
constructs the high-resolution image by solving a linearized curvature
equation, incorporating the interpolated curvature as an
explicit driving force. It has been numerically verified that the new
zooming method can produce clear images of sharp edges which
are already denoised and superior to those obtained from linear
methods and PDE-based methods of no curvature information.
Various results are given to prove effectiveness and reliability of
the new method.
Index Terms—Curvature interpolation method (CIM), image
zooming, interpolation, partial differential equation (PDE),
super-resolution (SR).
I. INTRODUCTION
I MAGE resampling is necessary for every geometric transform
of discrete images except shifts over integer distances
or rotations about multiples of 90 . When an image is to be
resampled, an interpolation method should be applied, as the
first of two basic resampling steps, to transform the discrete
data into a continuous function. (The second step of image resampling
is the sampling evaluation, which produces a discrete
image.) Image interpolation techniques are required in various
tasks in image processing and computer vision such as image
generation, compression, and zooming. Thus, image interpolation
methods have occupied an especial position in image processing
and computer graphics [2]–[4], [6], [7], [12], [13], [18].
Interpolation techniques can be characterized into three
methods: linear, nonlinear, and variational ones. For linear
methods, various interpolation kernels of finite size have been
introduced, in the literature, as approximations of the ideal
interpolation kernel (sinc function) which is spatially unlimited;
Manuscript received February 10, 2010; revised November 05, 2010; accepted
January 04, 2011. Date of publication January 20, 2011; date of current
version June 17, 2011. The work of Y. Cha was supported by the National Research
Foundation of Korea T funded by the Korea Government (MEST) under
Grant 2010-0016427. The work of S. Kim was supported in part by the National
Science Foundation under Grant DMS-0609815. The associate editor coordinating
the review of this manuscript and approving it for publication was Prof.
Kai-Kuang Ma.
H. Kim and S. Kim are with the Department of Mathematics and Statistics,
Mississippi State University, Mississippi State, MS 39762-5921 USA (e-mail:
hk246[at]msstate.edu; skim[at]math.msstate.edu).
Y. Cha is with the Department of Applied Mathematics, Sejong University,
143-747 Seoul, Korea (e-mail: yjcha[at]sejong.ac.kr).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2011.2107523
see [13] and references therein. The simplest linear approximations
are the nearest-neighbor interpolation and the bilinear
interpolation. Higher order interpolation methods involving
larger numbers of pixel values have shown superior properties
for some classes of images. However, the linear interpolation
methods have been introduced with no count on the image local
profile or edges. Thus, they may bring up either image blur or
the checkerboard effect (looking blocky) to the resulting image.
Various nonlinear interpolation methods have been studied to
overcome the artifacts of linear methods [1], [4], [10], [15]. The
major step in the nonlinear methods is to either fit the edges
with some templates or predict edge information for the highresolution
(HR) image from the low-resolution (LR) one. Recently,
variational methods have been suggested to form reliable
edges by integrating partial differential equation (PDE) models;
see [8] and [16] for a total variation (TV)-based interpolation
method and [2] and [3] for edge-forming anisotropic diffusion
models.
In this paper, we are interested in a novel PDE-based interpolation
method which is able to produce zoomed images having
the same curvature profile as in the LR one, either in color or
grayscaled. The new method utilizes the gradient-weighted curvature
measured from the LR image, as a driving force, in order
to construct an HR image having an accurate curvature profile.
In this paper, we call it the curvature-incorporating method
(CIM) for image zooming. The provided work is a mathematical
framework of CIM which can appropriately quantify the
curvature profile for the HR image domain and minimize interpolation
artifacts, for image zooming of arbitrary magnification
factors.
An outline of this paper is as follows. In Section II, we review
briefly the linear interpolation methods and the PDE-based
edge-forming method. Section III introduces the CIM for image
zooming, having three steps: the evaluation of the curvature for
the given image, the interpolation of the curvature, and the construction
of the zoomed image. Numerical schemes are presented
in detail for each step. In Section IV, various numerical
examples are given to show effectiveness and reliability
of the new method. Section V concludes our development and
experiments.