矩阵张量乘积快速图像插值算法
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"Fast regularized image interpolation method"
在图像处理领域，图像插值是一种常见的技术，用于扩大图像尺寸或实现局部图像缩放。传统的插值算法，如零阶插值、双线性插值和三次卷积插值，尽管在一些场景下效果良好，但它们可能会导致图像失真，如模糊或锯齿状边缘。为了克服这些问题，正则化的图像插值方法被广泛采用。然而，基于向量插值模型的正则化方法通常涉及大规模的降采样矩阵，这需要大量的存储空间并增加了计算复杂度。
本文提出了一种基于张量积的快速图像插值算法，该算法将向量插值模型转换为矩阵形式。张量积，也称为外积，是将多个向量空间的元素组合起来形成一个新的向量空间的方法。在本研究中，这种数学工具被巧妙地应用于减少存储需求和计算时间。通过这种方式，算法的效率得到了极大的提升，使得处理高维数据变得更加高效。
具体来说，论文中提到的快速算法首先将原始的向量插值问题转化为矩阵运算，这样可以利用矩阵运算的优化策略来减小计算负担。例如，通过矩阵分解或使用高效的矩阵乘法算法，可以显著降低计算复杂度。同时，由于矩阵形式的数据结构，存储要求也得到了大幅降低，这对于处理大数据量的图像至关重要。
实验结果证明了该快速算法的有效性。模拟研究表明，该算法不仅在保持图像质量的同时减少了计算时间，而且在存储需求上也优于传统方法。这些优势使得该算法在实时图像处理和高分辨率图像渲染等应用中具有很大的潜力。
此外，文章引用的OCIS代码"100.2000"和"100.3190"分别对应于光学领域的图像处理和信号处理，进一步表明该工作在这些专业领域内具有重要价值。总体来说，这项研究提供了一个创新的解决方案，以解决高维度图像插值中的计算和存储挑战，对于图像处理和计算机视觉的研究者和工程师来说，这是一个非常有价值的贡献。
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
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