图像不变矩的发展简史与改进策略
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"图像不变矩的发展概述"
在计算机视觉和图像处理领域，不变矩是一种重要的图像描述符，用于识别和分类图像。"Development of image invariant moments—a short overview"这篇文章由Kejia Wang、Ziliang Ping和Yunlong Sheng等人撰写，主要探讨了基于矩的图像描述方法——即图像不变矩和正交矩——超过50年的发展历程。
图像不变矩的核心在于它们对图像的旋转、缩放和翻转等几何变换具有不变性，这使得它们在图像分析和模式识别中十分有用。自20世纪中期以来，研究人员一直在不断优化这些方法，以提高其性能和鲁棒性。
文章提出了一些改进图像矩基方法的基本思想。首先，低阶径向矩的使用有助于减少信息抑制的缺点。低阶矩通常能捕获图像的主要特征，同时减少了计算复杂性和存储需求，这对于实时系统尤其重要。通过聚焦于这些基本特征，可以更有效地保持图像的关键信息，即使在有噪声或不完全对齐的情况下也是如此。
其次，将径向基与圆形谐波基分离，允许自由选择正交径向多项式。这种方法为设计更灵活的矩表示提供了可能，可以根据具体应用定制最佳的矩基，以增强对特定图像变换的不变性。这种分离方法可能包括使用Hermite、Zernike或其他类型的正交多项式，它们在保持不变性的同时，可以提高匹配精度和稳定性。
此外，文章还可能讨论了不同类型的不变矩，如Hu不变矩、Chen不变矩等，这些都为图像识别和分析提供了不同的理论框架和计算策略。不变矩的计算效率、稳定性和抗噪声能力是衡量其性能的关键指标，而文中提到的这些改进方法旨在优化这些指标。
"Development of image invariant moments—a short overview"这篇文章是对图像处理领域中一个关键主题的总结，它关注了如何通过改进图像不变矩和正交矩的方法来提升图像分析系统的性能。通过深入理解这些技术，可以更好地设计和实现图像识别算法，从而应用于自动驾驶、机器人导航、医学成像和安全监控等多个领域。
ψðrÞ¼
4α
nþ1
2πðn þ1Þ
p
σ cosð2πf
0
ð2r − 1ÞÞexp
−
ð2r − 1Þ
2
2σ
2
ðn þ1Þ
;
(6)
where n ¼ 3, α ¼ 0.697, f
0
¼ 0.409, and σ
2
¼ 0.561. The
wavelet transform is a convolution of the image’s circular
harmonic function
f
m
ðrÞ¼
Z
2π
0
e
−jmθ
f ðr; θÞdθ; (7)
with the wavelets, which are scaled, and translated
mother wavelets that are shown as
ψ
a;b
ðrÞ¼
1
a
p
ψ
r − b
a
; (8)
where the discrete scaling factor is a ¼ð1∕2Þ
p
, the discrete
translation factor is b ¼ðq∕2Þð1∕2Þ
p
, and p and q are pos-
itive integers. The continuous wavelets such as the cubic
B-spline functions are bandpass filters
[19]
. By adaptively
choosing the appropriate scale a, the wavelet transform
can highlight the local features such as edges in the image.
However, the wavelet moments as defined in Eq. (
5) are
not orthogonal and, therefore, can be used as image fea-
tures for the image description but not for the image
reconstruction. Moreover, the wavelet moments are not
scale invariant. The image must be resized to a fixed size
before computing the wavelet moments
[17,18]
.
3. Orthogonal moments
The orthogonal moments are an expansion of the image
into the orthogonal polynomial bases. The orthogonal
moments are information independent from each other,
allowing the image reconstruction for assessing the quality
of the image description.
A. Rotation-invariant orthogonal moments
The orthogonal polynomials result from the power series
solutions of the ordinary differential equations with the
orthogonality of the polynomials related to the Sturm-
Liouville forms of the differential equations plus the
boundary conditions. They are referred to as the special
functions and are given special names because of their
frequent uses. On the other hand, the orthogonal polyno-
mials can be easily generated using the Gram-Schmidt
orthonormalization, so that there can be an infinite num-
ber of orthogonal polynomials. The well-known orthogo-
nal polynomials include the Jacobi, Hermite, Laguerre,
and Bessel polynomials. The interrelations between
the polynomials exist. For instance, the Gegenbauer,
Legendre, Chebyshev, and Zernike polynomials are special
cases of the Jacobi polynomials, which are the solutions of
the hypergeometric differential equations.
Many orthogonal polynomials are proposed as the bases
for the image analysis. For the 2D expansion of an image
in the polar coordinate system, the radial orthogonal poly-
nomials are the radial basis, and the Fourier kernels
expðjmθÞ are used for the circular harmonic analysis.
As pointed out by Bhatia and Wolf
[20,21]
, for the rotation
invariance in the form, the circular Fourier kernel is a
unique solution to be included as the basis function.
The scale invariance is achieved by using the normalized
radial coordinate.
1. Zernike and Pseudo-Zernike moments
The Zernike moments are orthogonal on the unit circle.
The Zernike radial polynomials play a dominant role
for the aberration characterization in the optical design.
The Zernike moments were proposed in 1980 for image
analysis
[22]
as
A
nm
¼
n þ 1
π
Z
2π
0
Z
1
0
R
nm
ðrÞ expð−jmθÞf ðr; θÞrdrdθ;
(9)
where the Zerni ke radial polynomials are
R
nm
ðrÞ
¼
X
ðn−jmjÞ∕2
s¼0
ð−1Þ
s
ðn − sÞ!
s!ððn þjmjÞ∕2 − sÞ!ððn − jmjÞ∕2 − sÞ!
r
n−2s
;
(10)
where m is the circular harmonic order, and n is the degree
of the Zernike polynomial. The Zernike moments are de-
signed to be presented in the Cartesian coordinate system,
so that the Zernike radial polynomials must not contain
powers of r smaller than the circular harmonic order,
n ≥ jmj and n − jmj are even
[20]
. In the Zernike moments,
each term of the radial power in R
nm
ðrÞ corresponds to a
radial moment of a specific order, so that the Zernike
moments can be computed as the algebra ic combinations
of the Fourier-Mellin moments.
Bhatia and Wolf derived
[20]
another orthogonal set of
polynomials in x and y, which contains only the powers
of r higher than circular harmonic order n ≥ jmj, but with
the condition for even n − jmj removed
[20]
. This moment
set is now referred to as the pseudo-Zernike moments
[23]
.
When high circular harmonics of the order m are used
to represent the image of the angular variation in the
range ½0; 2π
[15]
, the Zernike and pseudo-Zernike moments
with the radial polynomial degrees n ≥ jmj must suffer
from severe information suppression drawback by the high
radial moment orders.
2. Orthogonal Fourier-Mellin moments
The orthogonal Fourier-Mellin moments
[24]
are design ed to
avoid the information suppression issue in the Zernike
moments. The orthogonal Fourier-Mellin moments are
defined as
Φ
nm
¼
1
2πa
n
Z
2π
0
Z
1
0
Q
n
ðrÞ expð−jmθÞf ðr; θÞrdrdθ;
(11)
COL 14(9), 091001(2016) CHINESE OPTICS LETTERS September 10, 2016
091001-3
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