Consistency proof of two denoising methods and the parameter sel...
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COL 11(12), 121201(2013) CHINESE OPTICS LETTERS December 10, 2013
Consistency proof of two denoising methods and the
parameter selection of PDE filtering method for ESPI
Zhitao Xiao (
777
)
1
, Quan Yuan (
)
1
, Fang Zhang (
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)
1∗
, Jun Wu (
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)
1
, Lei Geng (
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)
1
,
Zhenbei Xu (
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)
1
, and Jiangtao Xi (
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777
)
2
1
School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China
2
School of Electrical, Computer, and Telecommunications Engineering, University of Wollongong,
Wollongong, NSW 2522, Australia
∗
Received September 5, 2013; accepted November 11, 2013; posted online December 9, 2013
Electronic speckle pattern interferometry (ESPI) is a nondestructive, whole-field optical measurement
technique. The removal of speckle noise is fundamental to extract measurement information accurately.
In this letter, two filtering methods based on the oriented feature of ESPI fringes, i.e., the second-order
oriented partial differential equation (SOOPDE) and oriented, regularized quadratic-cost function filtering
methods, are first proven to be consistent. An important question in solving partial differential equation,
i.e., how to select suitable parameters in an adaptive manner, is then discussed. The computer-simulated
and experimentally obtained ESPI fringe patterns and phase map are processed by the SOOPDE filtering
model with adaptive selective parameters. The qualitative and quantitative analyses demonstrate that the
parameters selected by the adaptive method are effective and suitable for the SOOPDE filtering model.
OCIS codes: 120.6160, 110.6150.
doi: 10.3788/COL201311.121201.
Electronic speckle pattern interferometry (ESPI) is a
nondestructive measurement technique that has been ap-
plied in numerous areas to measure vibrations, displace-
ments, and their derivatives, as well as to reconstruct
three-climensional (3D) objects
[1]
. With ESPI, useful in-
formation is acquired by analyzing ESPI fringe patterns.
However, the original ESPI fringe patterns have strong
noises. Therefore, effectively denoising fringe patterns is
a key problem in applying the ESPI technique.
Orientation is one of the important characteristics
of ESPI fringe patterns. In past few decades, several
filtering methods based on fringe orientation have been
proposed to denoise fringe patterns. Yang et al.
[2]
pre-
sented the spin filtering method, which used a curve
window to approximate the fringe contour and fitted the
gray levels. Villa et al.
[3]
proposed the oriented, regular-
ized quadratic-cost function (ORQCF) method, which
used regularization theory in a Bayesian framework to
derive a quadratic cost function that includes informa-
tion about the fringe orientation. Tang et al.
[4]
proposed
the second-order oriented partial differential equation
(SOOPDE) model to control diffusion orientation. The
ORQCF method and the SOOPDE model can provide
good filter results to ESPI fringe patterns.
The denoising performance of partial differential equa-
tion (PDE) filter methods is related to the parameters
used when solving PDE models. These parameters in-
clude discrete time step size and iteration number, which
are typically chosen by trials. Although Szolgay et al.
[5]
presented the angle deviation error to estimate the ideal
stopping condition, only a typical iteration stopping
time was provided for the iterative image deconvolution
method. This letter discusses how to select the dis-
crete time step size and the iteration number, as well as
introduces the correlation coefficient to measure the cor-
relation between signal and noise, which is more direct
and easily understood than the angle deviation error. To
harmonize contradiction for denoising and fidelity, the
speckle index is also considered as an important standard
for selecting filter parameters.
This letter emphasizes the oriented filtering method
for ESPI fringe patterns. Firstly, two filtering methods
based on the oriented feature of fringes mentioned earlier,
i.e., SOOPDE and ORQCF, are proven to be consistent.
Secondly, an important question in solving PDE, i.e.,
how to select suitable parameters such as the discrete
time step size and the iteration number, is discussed to
obtain an adaptive solution.
ESPI fringe patterns have an obvious oriented feature.
Figure 1(a) illustrates a noise-free fringe pattern, and its
local region is shown in Fig. 1(b). In Fig. 1(b), the
adjacent fringe is approximately parallel, and the fringe
orientation varies gradually in the small window. Fringe
patterns have a stable orientation when the selected local
region is sufficiently small. Figure 1(c) shows the Fourier
spectrum of Fig. 1(b). Uniformly distributed fringe pat-
terns transform into two blobs, which are symmetrical
around the origin point. The line of the two blobs is
Fig. 1. Pictures of noise-free fringe patterns and the spectrum
diagram of its local region. (a) Noise-free fringe patterns, (b)
local region of the fringe patterns, and (c) Fourier spectrum
of (b).
1671-7694/2013/121201(6) 121201-1
c
2013 Chinese Optics Letters
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