March 10, 2007 / Vol. 5, No. 3 / CHINESE OPTICS LETTERS 149
Fast regularized image interpolation metho d
Hongchen Liu (
), Yong Feng (
úúú
), and Linjing Li (
)
Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001
Received October 11, 2006
The regularized image interpolation method is widely used based on the vector interpolation model in
which down-sampling matrix has very large dimension and needs large storage consumption and higher
computation complexity. In this paper, a fast algorithm for image interpolation based on the tensor product
of matrices is presented, which transforms the vector interpolation model to matrix form. The proposed
algorithm can extremely reduce the storage requirement and time consumption. The simulation results
verify their validity.
OCIS codes: 100.2000, 100.3190.
Image interpolation can be used in image enlargement
and local image zooming. Several common interpolation
algorithms have b een suggested, such as zero-order inter-
polation, bi-linear interpo lation
[1]
, and cubic convolution
interpolation
[2]
. However, image artifacts like blurring
or zigzag on edge may occur when these interpolation
schemes are used. In order to reduce the effect of im-
age artifacts, other new methods have been proposed,
including directional image interpolation
[3]
, convolution-
based interpolation
[4]
, and edge-directed interpolation
[5]
.
These methods take into account the edge information of
image, and the vision effect is better than the conven-
tional image interpolation methods.
Yoon et al.
[6]
presented regularized image sequence in-
terpolation by fusing low-resolution (LR) frames. The
regularized iterative image interpolation performs good
subjective quality, nevertheless, requires lots of running
time. In order to reduce the time-consuming, we present
a fast regularization image interpolation method of single
image based on matrix tensor pro duct.
Let x
c
(p, q) represent a two-dimensional (2D) spatially
continuous image, and x(m, n) is the corresponding digi-
tal image obtained by sampling x
c
(p, q), with size M ×N ,
such as
x(m, n)=x
c
(mT
v
,nT
h
),
m =0, 1, ··· ,M − 1; n =0, 1, ··· ,N − 1, (1)
where T
v
and T
h
represent the vertical and horizontal
sampling intervals respectively. In a similar way, the im-
age with four times LR in both horizontal and vertical
directions can be represented as
y(m, n)=
1
4
1
i=0
1
j=0
x(2m + i, 2n + j),
m =0, 1, ··· ,M/2 − 1; n =0, 1, ··· ,N/2 − 1.(2)
A discrete linear space-invariant degradation model for
an M/2 × N/2 LR frame obtained by sub-sampling the
original M ×N high resolution image frame, can be given
as
[7−11]
y = Hx + n, (3)
where the MN × 1 vector x represents the lexicograph-
ically ordered high resolution image frame, and the
MN/4 × 1 vectors y and n represent observed LR and
noise image frames, respectively. H is an MN/4 × MN
uniform down-sampling matrix.
The interp olation problem, therefore, can be formu-
lated as solving the least squares pr oblem for x,given
the observation y. That is, we find the estimation, ˜x,
which satisfies the following optimization problem
[6]
,
x =argminf(˜x), (4)
where
f(˜x)=n
2
= y − H˜x
2
. (5)
From the regularized image restoration theory, it is well
known that solving Eq. (3) is an ill-posed problem. In
order to make the problem better-posed, the following
functional is minimized,
f(˜x)=y − H˜x
2
+ λ L˜x
2
, (6)
where L is the regularization operator, which is prefer-
ably the three-dimensio nal (3D) Laplacian operator pro-
cess, to capture the between-channel information in the
interpolation process. The parameter λ is a global regu-
larization parameter.
In order to solve the ab ove equation given in Eq. (5),
the successive approximation equation
[8]
describing the
interpolated image ˜x,atthek + 1 iteration step, is given
by
˜x
k+1
= ˜x
k
+ βH
T
(y − H˜x
k
), (7)
where β means the function which controls the conver-
gence rate, and k represents the iteration number. The
successive approximation equation of Eq. (6) by using the
same method may be represented as
˜x
k+1
= ˜x
k
+ β(H
T
y − (H
T
H + λL
T
L)˜x
k
). (8)
In image pro cessing field, bit map is the most com-
monly used image format. Consider the bit map with
256 gray levels, that is each pixel in this image must use
8 bits (1 byte) to represent when storing in computer. If
1671-7694/2007/030149-04
c
2007 Chinese Optics Letters
下载后可阅读完整内容，剩余3页未读，立即下载
weixin_38693173
- 粉丝: 4
- 资源: 948
我的内容管理
展开
最新资源
- C++多态实现机制详解：虚函数与早期绑定
- Java多线程与异常处理详解
- 校园导游系统：无向图实现最短路径探索
- SQL2005彻底删除指南：避免重装失败
- GTD时间管理法：提升效率与组织生活的关键
- Python进制转换全攻略：从10进制到16进制
- 商丘物流业区位优势探究：发展战略与机遇
- C语言实训：简单计算器程序设计
- Oracle SQL命令大全：用户管理、权限操作与查询
- Struts2配置详解与示例
- C#编程规范与最佳实践
- C语言面试常见问题解析
- 超声波测距技术详解：电路与程序设计
- 反激开关电源设计：UC3844与TL431优化稳压
- Cisco路由器配置全攻略
- SQLServer 2005 CTE递归教程：创建员工层级结构
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
