基于密度的同态偏微分方程滤波方法用于电子散斑干涉条纹噪声抑制
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"基于fringe密度的同态偏微分方程滤波方法用于电子散斑干涉条纹噪声去除"
在电子散斑干涉术（Electronic Speckle Pattern Interferometry, ESPI）中，噪声的降低是一项至关重要的任务。研究者提出了一种新的去噪方法，即基于条纹密度的同态偏微分方程滤波法，专门针对干涉条纹图案。这种方法的核心是根据条纹的密度来确定方程的扩散速度，以此来优化滤波效果。
同态滤波是一种处理图像亮度函数的非线性滤波技术，它能够分别对图像的对数域和线性域进行操作，有效保留图像的边缘信息，同时减少噪声。在本研究中，这一特性被应用于ESPI的干涉条纹，通过对条纹密度的分析，动态调整滤波过程，使得在消除噪声的同时，尽可能保持条纹的细节和结构完整性。
为了验证该方法的有效性，研究者首先在计算机模拟的条纹图案上进行了测试。通过模拟实验，他们能够控制不同类型的噪声并评估新方法的滤波性能。接着，研究人员在实际实验中获取了干涉条纹图案，并应用同态偏微分方程滤波器进行处理。经过对比分析，无论是定性还是定量的评估结果都表明，这种技术在去除噪声、提高图像质量方面表现出色。
此外，该研究还可能对070.6110分类的光学工程和120.6160分类的电子技术领域产生积极影响。具体来说，它为ESPI数据处理提供了一种创新工具，有助于提高光学测量的精度和可靠性。对于那些需要高精度表面形貌和微小变形测量的应用，如材料力学性质评估、结构健康监测等，这项技术将带来显著的改进。
基于条纹密度的同态偏微分方程滤波方法为电子散斑干涉术中的噪声去除提供了一种高效且具有潜力的新途径。其在理论与实践上的成功应用，不仅强化了ESPI的图像处理能力，也为未来相关领域的研究和应用提供了新的思路。
210 CHINESE OPTICS LETTERS / Vol. 7, No. 3 / March 10, 2009
Homomorphic partial differential equation filtering method
for electronic speckle pattern interferometry fringes
based on fringe density
Fang Zhang (
ÜÜÜ
ǑǑǑ
)
1,2∗
, Wenyao Liu (
444
©©©
)
1,2
, Lin Xia (
ggg
)
1,2
,
Jinjiang Wang (
AAA
õõõ
)
1,2
, and Yue Zhu (
ÁÁÁ
)
1,2
1
College of Precise Instrument and Optical Electronic Engineering, Tianjin University, Tianjin 300072
2
Key Laboratory of Opto-Electronics Information Technical Science, Ministry of Education, Tianjin 300072
∗
E-mail: hhzhangfang@126.com
Received July 1, 2008
Noise reduction is one of the most exciting problems in electronic speckle pattern interferometry. We
present a homomorphic partial differential equation filtering method for interferometry fringe patterns.
The diffusion speed of the equation is determined b ased on the fringe density. We test the new method
on the computer-simulated fringe pattern and ex perimentally obtain the fringe pattern, and evaluate its
filtering performance. The qualitative and quantitative analysis shows that this technique can filter off the
additive and multiplicative noise of the fringe patterns effectively, and avoid blurring high-density fringe.
It is more capable of improving the quality of fringe patterns than the classical filtering methods.
OCIS codes: 120.6160, 070.6110.
doi: 10.3788/COL20090703.0210.
Electronic sp e ckle pattern interfero metry (ESPI) is a
well-known, nondestructive, and full-field technique for
displacement measurements
[1−3]
. Accurate extraction of
phase from fringe patterns is of fundamental importa nce
for the successful application of ESPI in obtaining the
displacement. However, the strong grain-shape random
noise in fringe patterns leads to heavy restr aint to phase
extraction. Therefore, it is important to filter off the
noise from fringe patterns to make phas e extraction eas-
ier, more robust, and more a c c urate
[2]
. In general, the
traditional filtering methods including spatial filtering
and frequency filtering, usually result in blurring effect.
Therefore, some algorithms have been proposed to reduce
the noise for ESPI fringe patterns, including Lee filtering,
which is a noise filtering by the use of local statistics
[4]
,
span filtering, which filters off the fr inge noise in a curva-
ture window
[2]
, and recursive a lgorithm, which has high
computation speed but the result has low contrast
[5]
.
Since the initial evolving equation, heat conduction
equation, was used in image filtering in 1983
[6]
, various
forms of partial differe ntial equations (PDEs) have been
proposed for the filtering of noisy images
[7−12]
. How-
ever, the existing PDE filtering methods usually lead to
insufficient denoising in the interior ar e a of fringe pat-
terns, or blurring effect on the fringe patterns with high
fringe density. This is because these general PDE meth-
ods do not consider the fr inge features, i.e., the multi-
plicative noise and the fringe density.
Perona et al. proposed a nonlinear anisotropic
diffusion equation (called PM equation)
[7]
, which has
been widely used in image denoising and enhancement.
The gray levels of an image u(x, y, t) : Ω × [0, +∞) → R
are diffused according to
∂
t
u = div (c (|∇u|) ∇u) , u (x, y, 0) = I (x, y) , (1)
where I (x, y) is the initial image, ∇u is the g radient of
u, and diffusion ge ne c is a nonincreasing function of the
gradient such as 1/(1 + |∇u|
2
/k
2
). It implies a direct
relation between the image smoothness at a point and
the image gradient
[8]
. Unfortunately, this model is very
sensitive to noise.
By formally developing the divergence term, the PM
equation c an be desc rib e d in the inner orthogonal co-
ordinates based on the image featur e
[9]
. We define the
inner orthogonal coordinates, i.e., the normal direction
~
N and the tangent direction
~
T as
~
N = (u
x
, u
y
)/
q
u
2
x
+ u
2
y
,
~
T = (− u
y
, u
x
)/
q
u
2
x
+ u
2
y
. (2)
Then Eq. (1) can be put in terms of the second der iva-
tives of
~
N and
~
T :
∂
t
u =
1
1 + |∇u|
2
.
k
2
1 − |∇u|
2
.
k
2
1 + |∇u|
2
.
k
2
u
NN
+ u
T T
, (3)
where
u
NN
=
u
2
x
u
xx
+ u
2
y
u
yy
+ 2u
x
u
y
u
xy
u
2
x
+ u
2
y
,
u
T T
=
u
2
x
u
yy
+ u
2
y
u
xx
− 2u
x
u
y
u
xy
u
2
x
+ u
2
y
. (4)
From Eq. (3), one can find that if |∇u|/k ≤ 1, it is a for-
ward diffusion model and the image becomes denoised;
but if |∇u|/k > 1, it includes a backward diffusion along
the normal directio n
~
N and the equation is ill-posed in
mathematics. Qia n et al. modified the diffusion gene
and pr oposed a new model
[9]
∂
t
u =
1
r
1 + |∇u|
2
.
k
2
1
1 + |∇u|
2
.
k
2
u
NN
+ u
T T
,
(5)
1671-7694/2009/030210-04
c
2009 Chinese Optics Letters
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