MATLAB

Functionality:

1) Convert the matlab code Fig.20 (Page 12 of attached file) into java function.

2) convert matlab function(imread) into java function.

3) Takes in a fringe pattern (noisy) as a jpg, convert it into matrix and process it using the java code in (1)

4) output the product into a jpg file (clean)

It is a web-based processing GUI that give the user the option to use one of the two method that is specified; WFF or WFR. Once user input the image file, using the java functionality change the input image into matrix (matlab:imread). the result is then processed using the wft2 function and converted back into jpg

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Optics and Lasers in Engineering 45 (2007) 304–317

Two-dimensional windowed Fourier transform for fringe pattern
analysis: Principles, applications and implementations

Qian Kemao

School of Computer Engineering, Nanyang Technological University, Singapore 639798, Singapore

Abstract

Fringe patterns from optical metrology systems need to be demodulated to get the desired parameters. Two-dimensional windowed

Fourier transform is chosen for the determination of phase and phase derivatives. Two algorithms, one based on filtering and the other

based on similarity measure, are developed. Some applications based on these two algorithms are explored, including strain

determination, phase unwrapping, phase-shifter calibration, fault detection, edge detection and fringe segmentation. Various examples

are given to demonstrate the ideas. Finally implementations of these algorithms are addressed. Most of the work has appeared in various

papers and its originality is not claimed. Instead, this paper gives an overview and more insights of our work on windowed Fourier

transform.

Keywords: Windowed Fourier transform; Fringe demodulation; Optical metrology; Noise reduction; Strain; Phase unwrapping; Phase-shifter calibration;

Fault detection; Edge detection; Fringe segmentation

1. Introduction

In optical metrology, the output is usually in the form of
a fringe pattern, which should be further analyzed [1–4].
For example, phase retrieval from fringe patterns is often
required. Two traditional techniques for phase retrieval are
phase-shifting technique [1,5] and carrier technique

with

Fourier transform [1,6]. Phase-shifting technique processes
the fringe patterns pixel by pixel. Each pixel is processed
separately and does not influence the others. However, this
technique is sensitive to noise. As an example, Fig. 1(a)
shows one of four phase-shifted fringe patterns. Phase
extracted using phase-shifting algorithm is shown in
Fig. 1(b), which is obviously very noisy. On the contrary,
carrier technique with Fourier transform processes the
whole frame of a fringe pattern at the same time. It is more
tolerant to noise, but pixels will influence each other. As an
example, A carrier fringe pattern and its phase extracted
using Fourier transform are shown in Fig. 2(a) and (b),
respectively. A better result, if possible, is expected.

Thus a compromise between the pixel-wise processing
and global processing is necessary. A natural solution is to

tlaseng.2005.10.012

ess: mkmqian@ntu.edu.sg.

process the fringe patterns locally, or block by block. A
smoothing filter is a typical local processor [7]. It assumes
that the intensity values in a small block around each pixel
ðu; vÞ are the same and hence the average value of that
block is taken as the value of pixel ðu; vÞ. Obviously it is not
reasonable for a fringe pattern since its intensity undulates
as a cosine function (see next paragraph). Because of this,
more advanced and effective techniques, such as regular-
ized phase tracking (RPT) [4,8–12], wavelet transform
[13–23], Wigner–Ville distribution [19,24] and windowed
Fourier transform (WFT) [19,25,26], were proposed. In
this paper, principle of WFT will be emphasized and
compared with other techniques. Then various applications
of WFT and the implementation issues will be introduced.
Figs. 1(c) and 2(c) show the effectiveness of WFT at a first
glance. Again it is emphasized that most of the work has
appeared in various papers and its originality is not
claimed. Instead, this paper gives an overview and more
insights of our work on WFT.
Before further discussion, a brief definition and

analysis

of fringe patterns are given, which will be used throughout
the paper. A fringe pattern can be generally expressed as

f ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos½jðx; yÞ

�

(1)

dx.doi.org/10.1016/j.optlaseng.2005.10.012

mailto:mkmqian@ntu.edu.sg

ARTICLE IN PRESS

Fig. 2. Phase retrieval from a carrier fringe: (a) a carrier fringe pattern;

(b) phase by Fourier transform and (c) phase by windowed Fourier

transform.

Fig. 1. Phase retrieval from phase-shifted fringes: (a) one of four phase-

shifted fringe patterns; (b) phase by phase-shifting technique and (c) phase

by windowed Fourier transform.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317 305

where f ðx; yÞ, aðx; yÞ, bðx; yÞ and jðx; yÞ are the recorded
intensity, background intensity, fringe amplitude and phase
distribution, respectively. Fringe patterns are classified into
four types: (I) exponential phase fringe patterns, (II)
wrapped phase fringe patterns, (III) carrier fringe patterns
and (IV) closed fringe patterns. Exponential phase fringe
patterns, which are analytic signals, are fundamental for
fringe processing [27] and are basic patterns considered in
this paper. They can be obtained from, say, phase shifting
technique. For example, given four phase-shifted fringe
patterns as

f iðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos½jðx; yÞ þ ði � 1Þp=2�,
i ¼ 1; 2; 3; 4,

the combination of 1
2
½f 1ðx; yÞ � f 3ðx; yÞ þ j f 4ðx; yÞ �

j f 2ðx; yÞ� gives
f Iðx; yÞ ¼ bðx; yÞ exp½jjðx; yÞ�,

(2)

where j ¼

ffiffiffiffiffiffiffi
�1
p

. The phase can be obtained by taking the
angle of f Iðx; yÞ, which is usually very noisy. Sometimes
wrapped phase maps are given. A typical example is phase
unwrapping. This type of fringe patterns can be written as

f IIðx; yÞ ¼ jwðx; yÞ,

(3)

where jwðx; yÞ denotes a wrapped phase map. It can be
easily converted to fI by simply multiplying it with j and
taking its exponential value. In this paper, fII is always
converted to fI before any processing. The third type,
carrier fringe patterns, can be written as

f IIIðx; yÞ ¼ aðx; yÞ þ bðx; yÞ
� cos½ocxxþ ocyyþ jðx; yÞ�, ð4Þ

where ocx and ocy are carrier frequencies along x and

directions, respectively. They are usually constants. Since

cosðtÞ ¼ expðjtÞ=2þ expð�jtÞ=2, fIII consists of a back-
ground field and two conjugated fringe patterns of fI.
These three items are separable in Fourier domain
provided that the carrier frequencies are high enough.
Thus, it can also be converted to fI. However, this
conversion is unnecessary as the separation is realized
automatically in the WFT algorithms. The fourth type,
closed fringe patterns, can be written as

f IVðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos½jðx; yÞ�.

(5)

It also consists of a background field and two conjugated

fringe patterns of fI, but they are not separable. Thus
converting fIV to fI is generally not easy. Other cues, such
as a fringe follower [12] or fringe orientation [28], are
required. Note that though the expressions of Eqs. (1) and
(5) are the same, the former refers to general fringe patterns
while the latter refers to fIV.
Local frequencies (or instantaneous frequencies) are used

to express phase derivatives as

oxðx; yÞ ¼
qjðx; yÞ

(6)

oyðx; yÞ ¼
qjðx; yÞ

qy
. (7)

Phase distribution in a block around a pixel ðu; vÞ can
thus be approximated as a small plane,

jðx; yÞ � oxðu; vÞðx� uÞ þ oyðu; vÞðy� vÞ þ jðu; vÞ. (8)

For fIII, the carrier frequencies are usually included into
the local frequencies. Note that there is slight difference in
the definition of instantaneous frequencies [29]. Also note
that in this paper, ðx; yÞ and ðu; vÞ are sometimes used
interchangeably if no confusion is raised.

2. Principles of windowed Fourier transform

In this section, the WFT is first introduced, based on
which, two algorithms, windowed Fourier filtering (WFF)
and windowed Fourier ridges (WFR) are developed and
discussed. To process the fringe patterns block by block,
the WFT is by default two-dimensional (2-D) throughout
this paper.

2.1. Windowed Fourier transform

The WFT and inverse WFT (IWFT) are a pair of
transforms [25,29] as

Sf ðu; v; x; ZÞ ¼

Z 1
�1

f ðx; yÞg�u;v;x;Zðx; yÞdxdy, (9)

f ðx; yÞ ¼ 1
4p2

Z 1
�1
Z 1
�1
Z 1
�1
Z 1
�1

Sf ðu; v; x; ZÞ

�gu;v;x;Zðx; yÞdxdZdudv, ð10Þ

where the symbol � denotes the complex conjugate
operation. Eq. (9) deconstructs a 2-D image f ðx; yÞ onto

ARTICLE IN PRESS

Fig. 3. Scheme of WFF. The input is a noisy fringe pattern and the output

is a filtered fringe pattern.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317306

WFT basis gu;v;x;Zðx; yÞ, resulting in 4-D coefficients (or
WFT spectrum) Sf ðu; v; x; ZÞ, while Eq. (10) reconstructs
the image. The WFT basis consists of a series of windowed
Fourier elements (or building blocks, atomic functions) as

gu;v;x;Zðx; yÞ ¼ gðx� u; y� vÞ expðjxxþ jZyÞ. (11)

Compared with the Fourier basis expðjxxþ jZyÞ, which
has an infinite spatial extension [30,31], WFT element has a
limited spatial extension due to the window function
gðx� u; y� vÞ. Consequently, the WFT spectrum,
Sf ðu; v; x; ZÞ, gives the frequency information at each pixel
in the image, which is impossible for Fourier transform.
The cost is that WFT computation is heavier because its
basis is redundant and not orthogonal [29]. WFT is also
known as short-time Fourier transform [29]. It is also
called Gabor transform if gðx; yÞ is a Gaussian function
given as [32]

gðx; yÞ ¼ exp½�x2=2s2x � y
2=2s2y�, (12)

where sx and sy are the standard deviations of the
Gaussian function in x and y directions, respectively,
which control the spatial extension of gðx; yÞ. A Gaussian
window is often chosen as it provides the smallest
Heisenberg box [29,32], which is useful in, for example,
multispectral estimation [29]. The Gaussian window is
selected throughout this paper, although a simple square
window also works quite well. The Gaussian window
function is divided by

ffiffiffiffiffiffiffiffiffiffiffiffipsxsyp for normalization such that
jjgðx; yÞjj2 ¼ 1.

2.2. Windowed Fourier filtering

The similarity of WFT and Fourier transform reminds
us to filter a fringe pattern by processing its WFT
spectrum [31]. The scheme is shown in Fig. 3. A fringe
pattern is transformed into its spectrum. Generally the
noise permeates the whole spectrum domain with very
small coefficients due to its randomness and incoherence
with the WFT basis. Thus it can be suppressed
by discarding the spectrum coefficients if their
amplitudes are smaller than a preset threshold. A smooth
image is produced after an IWFT. The scheme can be
expressed as

f̄ ðx; yÞ ¼ 1
4p2

Z 1
�1
Z 1
�1

Z Zh
Zl

Z xh
xl

Sf ðu; v; x; ZÞ

�gu;v;x;Zðx; yÞdxdZdudv ð13Þ

with
Sf ðu; v; x; ZÞ ¼

Sf ðu; v; x; ZÞ
if jSf ðu; v; x; ZÞjXthr;

if jSf ðu; v; x; ZÞjothr;

8>>>>< >>>>:

(14)

where Sf ðu; v; x; ZÞ denotes the thresholded spectrum
and f̄ ðx; yÞ denotes the filtered fringe pattern; ‘thr’ denotes

the threshold. The WFF can be applied to all the four
types of fringe patterns. For fIII, if only the spectrum of one
side-lobe around the carrier frequencies is selected,

WFF

gives an exponential field, from which phase can be
extracted [33]. This is similar to the traditional Fourier
transform technique for demodulation of carrier fringe
patterns [6]. For fIV, if the spectrum of all the possible
frequencies is selected, WFF gives a filtered fringe pattern.
The output is usually a complex field and its real part
should be used. If either x or Z is enforced to be positive,
then WFF gives an exponential phase field. Its angle
gives an ambiguous phase distribution [26]. This is similar
to Kreis’ approach [34]. When the phase is obtained,
local frequencies can be obtained according to Eqs. (6)
and (7) [35].
Note that instead of from �N to N, the integration

limits in Eq. (13) are set to be from xl to xh and from Zl to
Zh for x and Z, respectively. This means that only the
spectrum of ½x; Z� 2 ½xl; xh� � ½Zl; Zh� should be computed if
the frequencies of the fringe pattern are known to be in this
range. It is unnecessary to compute other coefficients and
thus the computational cost is saved. The range can be
estimated by analyzing the density or Fourier spectrum of a
fringe pattern. If the useful frequency components are
estimated as ½x; Z� 2 ½a; b� � ½c; d�, then the range for WFF
is expanded as ½x; Z� 2 ½a� 3=sx; bþ 3=sx� � ½c� 3=sy; dþ
3=sy�. The reason for the expansion is as follows. For a
signal with frequencies of ½x0; Z0� around ðu; vÞ, due to the
Gaussian window function, its WFT spectrum Sf ðu; v; x; ZÞ
extends infinitely along x and Z axes. However, 99.5%
of energy is concentrated in ½x0 � 3=sx; x0 þ 3=sx��
½Z0 � 3=sy; Z0 þ 3=sy�. An expansion of �2=sx and �2=sy
can be practically selected. This expansion is necessary for
recovering the image accurately through IWFT.

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Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317 307

2.3. Windowed Fourier ridges

WFR is another algorithm to process fringe patterns.
Consider only a small block of a fringe pattern around a
pixel (u,v) and compare it with a WFT element gu;v;x;Zðx; yÞ.
The values of x and Z are continuously changed. There
exists one WFT element that gives the highest similarity,
which is usually referred to as a ridge [29,36,37]. The values
of x and Z that maximize the similarity are taken as the
local frequencies at pixel (u,v). Local frequencies for all the
pixels can be estimated by sliding the block. The

WFR

scheme is illustrated in Fig. 4. Since the similarity can be
represented by spectrum amplitude, it can thus be
expressed as

½oxðu; vÞ;oyðu; vÞ� ¼ arg max
x;Z
jSf ðu; v; x; ZÞj (15)

which means that oxðu; vÞ and oyðu; vÞ take the value of x
and Z, respectively, when these values maximize the
amplitude spectrum jSf ðu; v; x; ZÞj. The ridge and phase
values are consequently determined as follows [25]:

rðu; vÞ ¼ jSf ½u; v;oxðu; vÞ;oyðu; vÞ�j, (16)

jðu; vÞ ¼ anglefSf ½u; v;oxðu; vÞ;oyðu; vÞ�g
þ oxðu; vÞuþ oyðu; vÞv. ð17Þ

The phase is wrapped and phase unwrapping is
necessary. This is called phase from ridges [25]. The phase
can be obtained in another way. Since oxðu; vÞ and oyðu; vÞ
are obtained, the phase can be computed by integrating
them. This is called phase by integration [25]. Both of them
have limitations. For the phase from ridges, when the
window size is not very small, say sx ¼ sy ¼ 10, the phase
should be corrected by

�1
2
arctan½s2xjxxðu; vÞ� � 12 arctan½s

2
yjyyðu; vÞ�,

where jxxðu; vÞ ¼ q2j=qx2 and jyyðu; vÞ ¼ q2j=qy2 are
second-order derivatives. On the other hand, for the phase

Fig. 4. Scheme of WFR. The input is a noisy fringe pattern, the output (fro

directions, respectively. In this illustration, as an example, a closed fringe patte

output.

by integration, when oxðu; vÞ and oyðu; vÞ are integrated,
their errors are also accumulated, which might lead to a
large phase error [25]. Finally, in order to save time, the
possible frequencies could be tested within an estimated
range of ½x; Z� 2 ½xl; xh� � ½Zl; Zh�. Unlike the WFF, no
expansion is needed since IWFT is not involved. The
WFR can be applied to all the four types of fringe patterns.
For fIV, the local frequencies and phase distribution are
ambiguous, due to the fact that if oxðu; vÞ, oyðu; vÞ and
jðu; vÞ is a solution, so is �oxðu; vÞ, �oyðu; vÞ and �jðu; vÞ.
An example is shown in Fig. 4.

2.4. Comparison with other techniques

WFF and WFR are compared with other techniques to
clarify their similarities and differences, which would be
helpful for proper use of these techniques.

2.4.1. WFF vs. Fourier and orthogonal wavelet filtering

The WFF scheme is very similar to Fourier transform
denoising [31] and discrete wavelet transform denoising
[38,39]. The main difference is the transform bases. WFT
usually gives better noise reduction effects due to the
following reasons. Firstly, WFT basis has higher coherence
to the fringe patterns. As can be seen from Eqs. (2)–(5), a
fringe pattern basically consists of one (fI and fII) or two
(fIII and fIV) exponential phase fields. Since the phase in
each local area can be approximated by a plane, the fringe
pattern can be well represented by a WFT element,
resulting in sparse and large spectrum coefficients. This
representation is easier for further processing. For exam-
ple, a simple thresholding proposed in this paper usually
gives good results. Secondly, when the spectrum is
thresholded, the useful information might also be slightly
destroyed. As a redundant basis, WFT is more tolerant to
the disturbance of spectrum [29,40,41].

m left to right) are phase distribution and local frequencies in x and y

rn is taken as the input, consequently sign ambiguities are observed in the

ARTICLE IN PRESS

Fig. 5. Technique of 1-D wavelet ridges: (a) The central row of Fig. 2(a).

The structure of the fringe pattern can hardly be observed; (b) retrieved

phase distribution when the fringe pattern is noiseless and (c) Retrieved

phase distribution when fringe pattern is noisy. The noisy fringe pattern is

same as Fig. 2(a).

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317308

2.4.2. WFR vs. RPT

The WFR is similar to the RPT in their basic principles.
The RPT constructs different cosine elements as

gRPTu;v;x;Z;jðx; yÞ ¼ gðx� u; y� vÞ cos½xxþ Zyþ jðu; vÞ� (18)

and looks for the most similar one to the fringe pattern in
each local area. It is basically used for fIV [8], but can be
easily extended to process fI [9], fII [10] and fIII [11]. The
cosine elements of RPT can be deconstructed into two
conjugated exponential elements, i.e., two WFT elements.
Hence WFT elements are more fundamental than RPT
elements. For WFR, all the possible combinations of x and
Z can be tested and the best WFT element for each local
area can be found; but for RPT, three variables, x, Z and j
need to be estimated simultaneously. Testing all the
possible combinations is impractical and hence an optimi-
zation strategy is necessary. Due to the success of RPT in
fIV demodulation, it would be interesting to explore the
possibility of using WFR to demodulate fIV automatically
without phase ambiguity.

2.4.3. WFR vs. wavelet ridges

Wavelet transform is another popular approach for
phase retrieval [13–23]. The Gabor wavelet (or Morlet
wavelet) is most often used. It can be written as

gWavelet2Du;v;x;Z ðx; yÞ ¼ g
ðx� uÞx

;
ðy� vÞZ

� �

� expðjxxþ jZyÞ, ð19Þ

where x0 and Z0 are the preset constants. Compared with
Eq. (11), it can be seen that the only difference of wavelet
basis and WFT basis is the window size. For WFT basis,
the window size is fixed, while for wavelet basis, it is
variable. The window size increases when frequency x or Z
decreases. Thus, it is troublesome in processing fringe
patterns with very low frequencies. For example, for the
carrier fringe pattern shown in Fig. 2(a), the frequency of
the fringe pattern along y direction is extremely low, which
requires the window size of wavelet along y direction to be
nearly infinite. Thus it is not practical to construct a 2-D
wavelet and perform a 2-D continuous wavelet transform.
Instead, the fringe patterns are often processed row by row
through 1-D wavelet transform with following wavelet
basis:

gWavelet1Du;x ðxÞ ¼ g
ðx� uÞx

� �
expðjxxÞ. (20)

But this introduces another problem. Fig. 5(a) shows the
central row of Fig. 2(a). Its structure can hardly be
discriminated. Consequently, the result by 1-D wavelet
transform is not as good as that by WFT [25]. As an
example, a noiseless version of carrier fringe pattern
corresponding to Fig. 2(a) is simulated. Both noiseless
and noisy carrier fringe pattern are processed using the
technique of 1-D wavelet ridges and the phase distributions
are shown in Fig. 5(b) and (c), respectively. It is obvious

that 1-D wavelet transform is quite sensitive to noise. For
this reason, wavelet is often used in dynamic problems
where the signals are 1-D [22,23], or in fringe projection
profilometry [20,21] and moiré interferometry [15,16] where
the fringe patterns are less noisy.
As wavelet automatically changes the window size, WFT

is often criticized that it has to fix the window size and is
not adaptive to the fringe pattern. However this is rather
misleading. The following comments by Laine are more
reasonable. ‘If the signal consists mostly of time-harmonic
components, which, even at high frequencies, have a long
correlation time, then a windowed Fourier transform, with
building blocks that share these characteristics, is best. If
the signal consists of a wide range of frequencies, with
much shorter correlation times for the high frequencies
than for the low frequencies (which is typically the case
with transients superposed on more slowly changing
components or short-lived transients between smoother
parts of the signal), then the ‘‘zoom-in’’ quality of the
wavelet transform is more useful, because it has a very
small field of vision for high frequencies but can be used to
view low frequencies at a larger scale.’ [42]. It is assumed
that for a natural scene, low-frequency components usually

ARTICLE IN PRESS

fI, fII, fIII

WFR

Integration

D
if

fe
re

nt
ia

tio
n

Phase derivatives

Wrapped phase

WFF WFR

Phase unwrapping

Unwrapped phase

Fig. 6. WFF and WFR for fI, fII and fIII.

fIV

WFR

Ambiguous phase and
phase derivatives (the

ambiguity can be removed in
high frequency parts)

WFF

Filtered fringe

Fig. 7. WFF and WFR for fIV.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317 309

last for long durations while high-frequency components
usually last for short durations, which is suitable to be
analyzed using wavelet. However, this assumption is
obviously not reasonable for the fringe patterns. For
example, a typical fringe pattern shown as the input of
Figs. 3 and 4 has same duration for all the frequency
components. It is fair to have a fixed window size. Further,
recall that a fringe pattern is either an exponential field or a
cosine function with its phase locally approximated by
Eq. (8). For each pixel ðu; vÞ, only three unknowns, oxðu; vÞ,
oyðu; vÞ and jðu; vÞ, need to be determined. For noiseless
fringe patterns, they can be accurately recovered in either a
small window or a large window, regardless of the value of
frequencies. A small window reduces the approximation
error while a large window is more robust against noises.
According to our experience, sx ¼ sy ¼ 10 gives good
results. It is emphasized that the choice of window size is
based on a trade-off between accuracy of phase approx-
imation and immunity to noise, instead of on the duration
of frequency components of a fringe pattern.

2.4.4. WFR vs. Gabor filters and wavelets for texture

analysis

Gabor filters were proposed by Bovik et al. [43] for
texture analysis, where the texture is assumed to be a cosine
function. Though this assumption is somewhat idealized,
they argued that ‘‘the global is to segment images based on
texture, rather than to form a complete description of the
textures’’ [43]. To deal with emergent image frequencies in
nonstationary signals, they turned to Gabor wavelets [44].
As mentioned in Section 2.4.3, a fringe pattern is either an
exponential field or a cosine function, which can be
suitably processed by WFR. Further, accurate local
frequency and phase retrieval, as a complete description
of a fringe pattern, is required and always emphasized in
WFR due to its high importance in metrology, which is
different from texture analysis. Though WFR can be seen
as a simple and special case of Gabor filters, different
motivations lead to different algorithms.

3. Applications of WFT

3.1. Phase and frequency retrieval by WFF and WFR

It can be seen from Section 2 that the phase distribution
and local frequencies can be extracted. Figs. 1(c) and 2(c)
show two examples, which can be obtained by either WFF
or WFR with very similar results. The functions of WFF
and WFR are illustrated in Figs. 6 and 7. Suggestion for
the selection of frequency ranges is given in Table 1.
Following are some notations.

(1)
The accuracy of WFF and WFR for fI, fII and fIII was
investigated in Ref. [25]. It is typically below one fiftieth
of a wavelength for both WFF and WFR.
(2)
The accuracy of WFF for fIV was investigated in
Ref. [41]. The relative error is 11% for speckle noise.
(3)
The accuracy of WFR for fIV was investigated in Ref.
[26]. The background of aðx; yÞ should be removed
from fIV before WFR is applied to fIV. The accuracy of
both phase and phase derivatives are high at high-
frequency parts, while they are low at low-frequency
parts. This gives a hint that at least some parts of fIV
can be accurately demodulated. However, whether a
frequency is high or low is rather subjective. A
guideline is that in a local area, bðx; yÞ exp½jjðx; yÞ�
and bðx; yÞ exp½�jjðx; yÞ� do not overlap in the fre-
quency domain. For a Gaussian window function with
sx ¼ sy, empirically a high local frequency should at
least satisfy o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2x þ o2y

q
X1:5=sx.

ARTICLE IN PRESS

Table 1

Selection of the frequency region

Input Algorithms xl xh Zl Zh Examples
a

f I WFF and WFR o0 40 o0 40 Fig. 1, ½�0:5; 0:5� � ½�0:5; 0:5�

f III WFF and WFR oocx 4ocx oocy 4ocy Fig. 2, ½�0:2; 0:2� � ½1:0; 2:1�

f IV WFF o0 40 o0 40 Fig. 3, ½�0:5; 0:5� � ½�0:5; 0:5�

f IV WFR ¼ 0 40 o0 40 Fig. 4, ½0; 0:5� � ½�0:5; 0:5�
o0 40 ¼ 0 40

a(1) Image size of examples: 256� 256; (2) the coordinate is in MATLABs convention and (3) all the regions given are based on WFR. Additional
region expansion is needed for WFF.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317310

(4)

As WFF and WFR are local processors, the accuracy
of the results around image borders is low due to
border effects, though sometimes it is tolerable.
(5)
WFF is usually faster than WFR, as will be mentioned
in Section 4.
(6)

Fig. 8. Removal of phase ambiguity in a close fringe pattern: (a) phase

fromWFR that is reprinted from Fig. 4; (b) phase ambiguity is removed in

the high-frequency part and (c) The low-frequency part is filled using RPT.

Generally speaking, for phase retrieval, WFF is
preferred for fI, fII and fIII while WFR is preferred
for fIV. For local frequency retrieval, WFR is preferred.
However, for each practical problem, both algorithms
can be attempted and compared.

Fig. 9. Demodulation of a single fringe pattern: (a) the original fringe

pattern; (b) phase from WFR; (c) phase ambiguity is removed in the high-

frequency part and (d) The low-frequency part is filled using RPT.

Here demodulation of a single closed fringe pattern will
be discussed a little further. As mentioned previously, when
WFR is applied to fIV, both local frequencies and phase
distribution are ambiguous, which can be observed in
Fig. 4 and is also shown in Fig. 8(a). The phase ambiguity
of the high-frequency parts can be easily removed by
enforcing the continuity of local frequencies. In detail,
consider a pixel ðu; vÞ as initial pixel. For its adjacent pixel,
say, ðuþ 1; vÞ, if

joxðuþ 1; vÞ � oxðu; vÞj þ joyðuþ 1; vÞ � oyðu; vÞj
pjoxðuþ 1; vÞ þ oxðu; vÞj
þ joyðuþ 1; vÞ þ oyðu; vÞj,

take oxðuþ 1; vÞ, oyðuþ 1; vÞ and jðuþ 1; vÞ as the
solution; otherwise change their signs. Other pixels are
processed in the same manner. Fig. 8(b) shows the phase of
Fig. 8(a) with phase ambiguity removed. Finally, the low-
frequency part can be filled using RPT, as shown in
Fig. 8(c). One more example is shown in Fig. 9 which
contains a saddle point. The phases are obtained in the
same way. Further, in recently proposed techniques for
demodulation of a single fringe pattern, WFR can serve as
an initialization for refinement and propagation [45] while
WFF can serve as an effective filtering method in extreme
map based approach [46].

3.2. Strain estimation in moiré interferometry

Though strains in moiré interferometry and local
frequencies (instantaneous frequencies) in image processing

are different concepts in mechanics and signal commu-
nication, it was recognized that they are the same except for
a constant [18]. Thus strain can be estimated through local
frequency extraction from moiré interferograms using

ARTICLE IN PRESS

Fig. 10. WFR for strain extraction: (a) a moiré fringe pattern; (b) strain

contour in x direction using moiré of moiré technique and (c) strain field

by WFR.

Fig. 11. Unwrap a phase map with ‘‘bad’’ pixels: (a) a wrapped phase; (b)

wrapped phase filtered by WFF and (c) unwrapped phase of (b). Reprint

from Ref. [50] with permission from Elsevier.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317 311

WFF or WFR. It is a trivial extension of the application in
Section 3.1. A real fringe moiré pattern (Fig. 10(a)) is
processed as an example. The strain contour can be
produced using moiré of moiré technique [47]. It is
obtained by shifting the moiré fringe and overlapping it
with the original one, which is shown in Fig. 10(b). The
strain in x direction using WFR is given in Fig. 10(c),
which provides the whole field information with good
quality.

Fig. 12. Unwrap a phase map with ‘‘bad’’ regions: (a) wrapped phase

map; (b) amplitude of filtered exponential field; (c) angle of filtered

exponential field and (d) phase unwrapping only within ‘‘good’’ regions.

Note that (a) is from synthetic aperture radar (SAR), not from an optical

interferometer, but the phase unwrapping problem is the same. Data of (a)

is from [48] with permission from John Wiley and Sons, Inc.

3.3. Phase unwrapping

Phase unwrapping is frequently needed to construct a
continuous phase map jðx; yÞ from a wrapped phase map
jwðx; yÞ [48]. The only difference between jðx; yÞ and
jwðx; yÞ is 2p jumps. Hence, the phase can be unwrapped
by scanning the phase map line by line and compensating
the jumps [49]. However this method is usually unsuccess-
ful when applied to a noisy wrapped phase map
(Fig. 11(a)). Two widely used strategies to overcome the
problem are circumventing the ‘‘bad’’ pixels and approx-
imating the phase in a least-squares sense [50]. A third
strategy is to remove the noise before phase unwrapping,
which can be effectively fulfilled by WFF [50]. Fig. 11(b) is
the wrapped phase filtered by WFF, from which the phase
unwrapping becomes trivial, as shown in Fig. 11(c).

Sometimes not only ‘‘bad’’ pixels but also ‘‘bad’’ regions,
such as shown in Fig. 12(a), need to be handled. This
problem can again be solved by WFF. Recall that wrapped
phase fII is first converted to an exponential field fI, which
has a unit amplitude. Since the spectrum in the ‘‘bad’’
regions is usually broader than in other places, more energy
will be removed in WFF and the amplitude of the filtered
exponential field, jf̄ Ij, becomes smaller. This can be seen
from Fig. 12(b), which shows the amplitude of the filtered
exponential field of Fig. 12(a) and can be used for ‘‘bad’’
region identification. The angle of the filtered exponential
field is shown in Fig. 12(c), which can be used for phase
unwrapping. The phase is unwrapped only within ‘‘good’’
regions using flood algorithm [48]. The unwrapped phase is
shown in Fig. 12(d), where ‘‘bad’’ regions are shown as
black. One more example is shown in Fig. 13, where the
phase map contains both ‘‘bad’’ pixels and regions.

3.4. Phase-shifter calibration

In optical interferometry, it is often required to
determine the phase-shift from two consecutive interfero-
metric fringe patterns. For example, a phase-shifter has to
be calibrated to get the relationship between the input of
the phase-shifter and the resulting phase-shift between the
consecutive fringe patterns [1]. Another example is in
‘‘generalized phase-shifting interferometry’’ where the
phase-shifts are arbitrary and need to be determined before
phase extraction can proceed [51,52]. One approach uses
phase-shifting technique [53]. Using phase-shifting techni-
que before phase-shifter calibration is somewhat awkward.
Another approach uses carrier technique with Fourier

ARTICLE IN PRESS

Fig. 13. Unwrap a phase map with ‘‘bad’’ pixels and regions: (a) wrapped

phase map; (b) amplitude of filtered exponential field; (c) angle of filtered

exponential field and (d) phase unwrapping only within ‘‘good’’ regions.

Note that (a) is from magnetic resonance imaging (MRI), not from an

optical interferometer, but the phase unwrapping problem is the same.

Data of (a) is from [48] with permission from John Wiley and Sons, Inc.

Fig. 14. phase-shift determination by WFR: (a) and (b), two phase-shifted

fringe patterns and (c) location of ‘‘good’’ parts shown in gray.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317312

transform [54]. However, it is generally too restrictive to
enforce a carrier in the phase-shifted fringe patterns.

Though, generally, there is no carrier in a fringe pattern,
it is reasonable to expect that it contains ‘‘carrier-like’’
features in some parts of fringe patterns. As indicated in
notation (3) in Section 3.1, the phase and local frequencies
extracted by WFR are accurate where the local frequencies
are high. Hence, the phase-shift can be determined only
from these ‘‘good’’ parts as follows [55],

(1)
Given two phase-shifted fringe patterns, f1 and f2, apply
WFR to obtain their respective phase and local
frequencies as j1ðu; vÞ, ox1ðu; vÞ, oy1ðu; vÞ, j2ðu; vÞ,
ox2ðu; vÞ and oy2ðu; vÞ.
(2)
Find the locations S where o1ðu; vÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2x1ðu; vÞ þ o2y1ðu; vÞ

q
is large. The image boarders are

excluded, as mentioned in notation (4) in Section 3.1.

(3)

Phase-shift can be determined by computing in S the

average value of

Dðu; vÞ ¼ jj2ðu; vÞ � j1ðu; vÞj. (21)

It was mentioned in Section 2.3 that the phase by WFR
should be corrected due to the second-order phase
derivatives. However, j1ðu; vÞ and j2ðu; vÞ share the same
second derivatives, this correction is unnecessary since they
will be eliminated in Eq. (21).

As an example, two closed fringe patterns with a phase-
shift of p/2 are simulated and shown in Figs. 14(a) and (b),
respectively. The location for phase-shift determination can
be found and is shown in Fig. 14(c). The phase-shift error is
less than 0.011 for noiseless fringe patterns and is less than
11 for speckle fringe patterns of Figs. 14(a) and (b).

3.5. Fault detection

Detection of faults from interferometric fringe patterns is
useful for condition monitoring, industrial inspection,
nondestructive testing and evaluation (NDT and NDE)
[56–59]. A fault occurs at location where the phase of the
fringe changes abruptly. Accordingly there is a sudden
change in the fringe density, or local frequencies, at that
location. In the previous work [56–59], faults are detected
from a single fringe pattern, though sequences of dynamic
fringe patterns are often available [56]. In real applications,
such as condition monitoring, a fringe sequence evolving
over time is generally used to monitor the damages or
faults. Moreover, the appearances of faults are very
complicated. For example, different carrier frequencies
give rise to different fault appearance, although the faults
are the same. Hence it would be reasonable to monitor the
temporal changes of the local frequencies. This can be
realized by WFR as follows [60]:

(1)
From the first fringe pattern f1, extract local frequencies
ox1ðu; vÞ and oy1ðu; vÞ using Eq. (15). The ridge value
r1ðu; vÞ can be recorded using Eq. (16) as
r1ðu; vÞ ¼ jSf 1½u; v;ox1ðu; vÞ;oy1ðu; vÞ�j, (22)
where Sf 1ðu; v; x; ZÞ is the WFT spectrum of f 1.
This means that the first fringe pattern in a local area
around pixel ðu; vÞ is most similar to WFT element
gu;v;oxðu;vÞ;oyðu;vÞðx; yÞ, with r1ðu; vÞ as its similarity
measure.
(2)
Compute the similarity between the second fringe
pattern f 2 in the local area around pixel ðu; vÞ and the
WFT element gu;v;oxðu;vÞ;oyðu;vÞðx; yÞ as
r2ðu; vÞ ¼ jSf 2½u; v;ox1ðu; vÞ;oy1ðu; vÞ�j, (23)
where Sf 2ðu; v; x; ZÞ is WFT spectrum of f 2.
(3)
If the value of r2ðu; vÞ=r1ðu; vÞ decreases below a certain
threshold, a fault is likely to occur at ðu; vÞ since the
similarity between f 1 and f 2 around ðu; vÞ is low.

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Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317 313

The effectiveness of this approach was analyzed theore-
tically in Ref. [60]. A simulated example and a real example
are given in Figs. 15 and 16, respectively. The defective
regions are successfully detected and highlighted with
higher gray levels.

3.6. Edge detection and fringe segmentation

Till now it is assumed that the processed fringe has a
continuous phase field, which is not always correct. For
example, discontinuities exist when the tested objects, such
as MEMS devices, have different components. Typical
edge detectors, such as Canny edge detector [61], were
developed for intensity discontinuities and are not suitable
for phase discontinuities. As an example, Canny edge
detector is applied to the fringe pattern in Fig. 17(a). The
detected edges are overlaid on the original fringe pattern
and shown in Fig. 17(b), which is obviously not satisfac-
tory. Hence there is a need to handle the discontinuous
phase field, which has been investigated using Gabor filters
[43], Gabor wavelet [44,62] and Wigner–Ville distribution
[63]. Gabor wavelet treats high and low frequencies
unequally [62] and complex preprocessing is necessary to
assist Wigner–Ville distribution [63]. Similar to Gabor
filters [43], WFR can be used to detect the discontinuities
simply and effectively for fringe patterns.

To detect the fringe edges caused by frequency disconti-
nuities, the local frequencies oxðx; yÞ and oyðx; yÞ is
calculated first using WFR. Then Canny edge detector is

applied to either ox, oy, or o ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o2x þ o2y

q
to locate the

edges. Alternatively, o and y ¼ tan�1ðoy=oxÞ can also be
used. Examples of some basic fringe patterns are shown in
Figs. 18(a), (b), (d) and (e), where discontinuities of ox or
oy are simulated. The detected edges are overlaid on the

Fig. 15. Fault detection in simulated fringe patterns: (a) frame 1; (b)frame

2 and (c) fault alert in frame 2. Reprint from Ref. [60] with permission

from Institute

of Physics (IOP).

Fig. 16. Fault detection in real fringe patterns: (a) frame 1; (b)frame 2 and (c) f

of Physics (IOP).

original fringe patterns. For fringe edges caused by phase
discontinuities, high-frequency components appear due to
these discontinuities and consequently regions around the
edges can be isolated by thresholding the local frequencies.
The edges can be found by thinning the isolated regions.
Figs. 18(c) and (f) are two examples with a simulated
phase-shift of p/2 in the right half of the fringe patterns.
The detected edges are overlaid on the original fringe
patterns. In the same manner, the fringe segmentation can
also be realized. Fig. 19(a) shows a fringe pattern obtained
using a Mach-Zender interferometer for measuring the
refractive index of solution, where a potassium chloride
(KCl) crystal is dissolved [64]. The fringes are segmented
by retaining the regions where local frequencies are high
(Fig. 19(b)).

4. Implementations of WFF and WFR

Implementation issues are addressed in this section so
that the results in this paper are reproducible and the
algorithms can be readily tested by the readers [65]. Define

hx;Zðx; yÞ ¼ g0;0;x;Zðx; yÞ ¼ gðx; yÞ exp jxxþ jZyð Þ, (24)

it is easy to rewrite the important equations in Section 2
equivalently as follows:

Sf ðu; v; x; ZÞ ¼ ½f ðu; vÞ � hx;Zðu; vÞ� expð�jxu� jZvÞ, (90)

f ðx; yÞ ¼ 1
4p2
Z 1
�1

Z 1
�1

½f ðx; yÞ � hx;Zðx; yÞ�

� hx;Zðx; yÞdxdZ, ð100Þ

ault alert in frame 2. Reprint from Ref. [60] with permission from Institute

Fig. 17. A fringe pattern (a) and its intensity edges (b).

ARTICLE IN PRESS

Fig. 18. Discontinuity detection in some basic fringe patterns.

Fig. 19. A real fringe pattern with a crystal (a) and the segmented fringes (b).

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317314

f̄ ðx; yÞ ¼ 1
4p2
Z Zh
Zl
Z xh
xl
½f ðx; yÞ � hx;Zðx; yÞ�

� hx;Zðx; yÞdxdZ, ð130Þ

½oxðu; vÞ;oyðu; vÞ� ¼ arg max
x;Z
jf ðu; vÞ � hx;Zðu; vÞj, (150)

rðu; vÞ ¼ jf ðu; vÞ � hoxðu;vÞ;oyðu;vÞðu; vÞj, (160)

jðu; vÞ ¼ angle½f ðu; vÞ � hoxðu;vÞ;oyðu;vÞðu; vÞ�, (170)

where the symbol � denotes a 2-D continuous convolution.
All the above equations are continuous and should be

discretized for implementation with a computer. It is
assumed that the input fringe pattern f ðx; yÞ has already
been discretized into pixels when it was captured by a CCD
camera and consequently x and y here are integers.
Accordingly x and y in hx;Zðx; yÞ are also assumed to be
integers and the unit of x and y is pixel. Now the symbol �
should be understood as a 2-D discrete convolution.
Finally, x and Z should also be discretized, which can be
simply done as follows:

x ¼ px0; p ¼ xl=x0; . . . ;�1; 0; 1; . . . ; xh=x0 , (25)

Z ¼ qZ0; q ¼ Zl=Z0; . . . ;�1; 0; 1; . . . ; Zh=Z0 , (26)

where x0 and Z0 are sampling intervals; p and q are integers.
For WFF, it is shown by frame theory that x0 ¼ 1=sx and
Z0 ¼ 1=sy are recommended [41], while for WFR, x0 and Z0
should be selected according to the desired frequency
resolution and usually high resolution is preferred. Thus
WFF is usually faster than WFR. Note that after
discretization, the frequency range of ½x; Z� must belong
to ½�p;p� � ½�p;p�, instead of ½�1;1� � ½�1;1� [30,31].
Two algorithms of WFF and WFR are integrated into a

MATLABs function ‘wft2’, which is provided in Fig. 20
and is used to produce all the results in this paper. The
codes are also available on request. The input and output
arguments are as follows,

�
type means the algorithm to be used. Choose ‘wff’
for WFF and ‘wfr’ for WFR, respectively;

�

f is an input fringe pattern. It can be fI, fIII and fIV. fII is

converted to fI as input;

�

sigma corresponds to sx and sy for Gaussian window

function and sx ¼ sy ¼ 10 is recommended [41] ;

�

wxl, wxi and wxh (resp., wyl , wyi and wyh)

correspond to xl, x0 and xh (resp., Zl, Z0 and Zh). The
choice of xl and xh (resp., Zl and Zh) for WFF and WFR
are explained in Sections 2.2 and 2.3, respectively.
wxi ¼ wyi ¼ 1/sigma is recommended for WFF [41];

�

thr is the threshold for WFF in Eq. (14), which is not

needed for WFR;

�

g is the output of the function. It is a 2-D matrix for

WFF, while it is a structure for WFR with the local
frequencies, phase and ridges stored in g.wx, g.wy,
g.phase and g.r, respectively. Note that g.phase is
the phase from ridges and has not been corrected.

Gaussian function is chosen as the window function.
The window size of ð6sx þ 1Þ � ð6sy þ 1Þ should be

ARTICLE IN PRESS
x
y

(0,0) x

y(1,1)

(a) (b)

Fig. 21. Coordinate systems for theoretical analysis (a) and for

MATLABs (b).

Fig. 20. MATLABs codes for 2-D WFT.

Q. Kemao / Optics and Lasers in Engineering 45 (2007) 304–317 315

selected. To save computation time, it is assigned to be
ð4sx þ 1Þ � ð4sy þ 1Þ in the codes since it gains 91.1%
energy of the Gaussian function and less than 10% of
energy is truncated. Users can adjust it by modifying the
first line of codes to ‘‘s ¼ round(3*sigma);’’ to expand
the window size if computation time is not concerned.

As examples, fringes in Figs. 3 and 4 are obtained by
executing the following commands in MATLABs envir-
onment, respectively,

g ¼ wft2(‘wff’,f,10,-0.7,0.1,0.7,-0.7,0.1,0.7,6);
g ¼ wft2(‘wfr’,f,10,0,0.025,0.5,-0.5,0.025,0.5).

About 2 and 9min are taken to execute the above
commands for a fringe pattern of 256� 256 by a Pentium
IV 3.2GHz desktop.

It can be noted that the 2-DWFT kernel is separable and
can be written as the production of two 1-D kernels:

gu;v;x;Zðx; yÞ ¼ gu;xðxÞgv;ZðyÞ (27)

with

gu;xðxÞ ¼ expð�x2=2s2x þ jxxÞ, (28)

gv;ZðyÞ ¼ expð�x2=2s2y þ jZyÞ. (29)

Thus the 2-D convolution can be realized by a 1-D
convolution along each row, followed by a 1-D convolu-
tion along each column. This is similar to the realization of

2-D FFT by two 1-D FFTs [31]. By this modification, the
execution time of the previous examples reduces to 45 s and
3min, respectively.
Finally it should be noted that, in Sections 1 and 2, it is

comfortable to assume the coordinate system as in
Fig. 21(a), while in the MATLABs codes, it is assumed
that the coordinate system is as in Fig. 21(b), which follows
the convention of MATLABs.

5. Conclusions

Demodulation of fringe patterns is necessary for many
optical metrological systems. Systematic solutions using
time–frequency analysis are provided. Two-dimensional
windowed Fourier transform is chosen for the determina-
tion of phase and phase derivatives and it is compared with
other time–frequency techniques. Two approaches are
developed, one is based on the concept of filtering the
fringe pattern and the other is based on the similarity
measure between the fringe pattern and windowed Fourier
elements. Their applications in determination of phase and
phase derivatives, strain extraction in moiré interferometry,
phase unwrapping, phase-shifter calibration, fault detec-
tion and edge detection and fringe segmentation are given,
showing that windowed Fourier transform is a very useful
technique for various fringe pattern analysis tasks. Finally,
how to implement the algorithms is discussed so that
readers can easily reproduce the results and use these
techniques to solve their own problems.

Acknowledgments

I would like to express my sincere gratitude to Prof. Seah
Hock Soon and Prof. Anand Asundi of the Nanyang
Technological University and Prof. Wu Xiaoping of the
University of Science and Technology of China for their
encouragement and to the reviewers for their helpful
comments.

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Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations

Introduction
Principles of windowed Fourier transform
Windowed Fourier transform
Windowed Fourier filtering
Windowed Fourier ridges
Comparison with other techniques
WFF vs. Fourier and orthogonal wavelet filtering
WFR vs. RPT
WFR vs. wavelet ridges
WFR vs. Gabor filters and wavelets for texture analysis

Applications of WFT
Phase and frequency retrieval by WFF and WFR
Strain estimation in moiré interferometry
Phase unwrapping
Phase-shifter calibration
Fault detection
Edge detection and fringe segmentation
Implementations of WFF and WFR
Conclusions
Acknowledgments
References

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