Development of spatial and temporal phase evaluation techniques in digital holography 3

After a wrapped phase map is determined by the complex phasor method, another algorithm, i.e., two-dimensional short-time Fourier transform, is also proposed to overcome the above proble

CHAPTER THREE DEVELOPMENT OF THEORY

3.1 Spatial phase evaluation

3.1.1 Wrapped phase extraction

It is described in Section 2.4.2 that complex amplitude for the test specimen can be numerically reconstructed in digital holography With the reconstructed complex amplitude, both intensity and phase distributions of the test specimen can be directly determined If pure Fourier transform reconstruction method is applied, a discrete representative of Eq (2.33) can be expressed as

where ( , )Γ m n is a matrix of M×N points, C denotes a complex constant, d denotes

the reconstruction distance, λ is laser wavelength, j= −1, and ∆x and y∆ denote pixel sizes at the hologram (or CCD) plane For simplicity, the factors ∆ξ' and ∆η'

of pixel sizes in the reconstruction (or image) plane are omitted, and the pixel in the

reconstruction plane is denoted as (m, n)

Hence, phase map ϕ(m,n) and intensity distribution I(m,n) can be directly extracted from the reconstructed complex amplitude ( , )Γ m n

as dense fringes and high-level noise It is well known that phase itself is not a signal but rather a property of the signal (Ghiglia and Pritt, 1998) Hence, it is necessary to develop a new method which can effectively avoid the direct filtering of the wrapped phase map

In this research work, a new method based on the concept of complex phasor is proposed Since the result by using a reconstruction algorithm can be considered as a complex value, the reconstructed result calculated by Eq (3.1) can be described as a complex exponential signal A complex phasor ( , )A m n is calculated by multiplying

( , , 2)m n

Γ at the second (or deformed) state by the conjugate of ( , ,1)Γ m n at the initial

state Without a loss of generality, only a given pixel (m, n) is considered in this case

study

in Fig 3.1 In the presence of noise, the filtered result relies on the point whose amplitude is higher It means that the phase value is more reliable if the amplitude is higher Another distinctive advantage of the proposed method is that phase manipulation is effectively avoided, so better results can be expected If an average filter is employed, the wrapped phase map in Eq (3.4) can be expressed by

Since the complex phasor method is proposed, direct filtering of the wrapped phase map is avoided If the wrapped phase map is contaminated by speckle noise, conventional sine/cosine transformation with an average filter may be further applied which can be described by

( , ) 9( , ) arctan

by their neighboring nine points In the sine/cosine transformation method, it usually requires several iterative cycles to produce satisfactory results In the first iterative cycle, the phase values are converted into sine and cosine formats, and then an average filter is applied In the second iterative cycle, an average filter is again applied to the filtered phase values obtained in the first cycle The procedure is required until all the preset iterative cycles are completed The final filtered wrapped phase map is obtained by using the arc-tangent operation

However, as the iterative cycle in sine/cosine transformation method with an average filter is large, the dense fringes may be smeared out Hence, in many practical cases, the iterative cycle is preset as a small value After a wrapped phase map is determined by the complex phasor method, another algorithm, i.e., two-dimensional short-time Fourier transform, is also proposed to overcome the above problem The short-time Fourier transform is also known as windowed Fourier transform (Mallat, 1999) Compared with the conventional global Fourier transform, short-time Fourier

transform has a localization characteristic, so it is able to prevent the propagation of errors more efficiently

Since exponential signals can be considered as the source signals of the practical measurement, the extracted wrapped phase map is converted into an exponential signal before implementing short-time Fourier transform Similarly to sine/cosine transformation method, sine and cosine formats of the wrapped phase ( , )m n

where ( , )u v and ( , )ξ η denote time and frequency, and ( , )g m n represents a window

where σm and σn control the extension of the Gaussian window which can provide the smallest Heisenberg box (Mallat, 1999) In this study, (m,n) and ( ', ')ξ η are

interchangeably used as no confusion is raised, and the meaning of ( , )ξ η is different from those in Fig 2.12 (Chapter 2)

As the signal is converted into the spectral domain, noise usually has small coefficients Coefficients smaller than a preset threshold are eliminated, so the noise is efficiently reduced In addition, short-time Fourier transform is performed over a local area, so the transform of a signal does not influence pixels at other positions The filtered spectrum Sf(u,v,ξ,η) is described by

( , , , ) if ( , , , )( , , , )

where thrd denotes a preset threshold

After the filtering process, a filtered signal ( , )f m n is obtained by an inverse

short-time Fourier transform

of ξ and η should be adjusted to focus on the most of energy using the selected window size (Qian, 2007) After the implementation of short-time Fourier transform,

a filtered wrapped phase map is obtained by

Im ( , )( , ) arctan

where ∆ϕ( , )m n represents the filtered wrapped phase map

3.1.2 Determination of displacement derivative

3.1.2.1 First-order displacement derivative

In practice, since the displacement derivative is of more interest, digital shearography technique has received wide applications (Hung, 1997) In this research work, much effort has also been made to directly determine displacement derivative but by using digital holographic technique As a complex phasor is determined by Eq (3.4), the complex amplitude can be numerically shifted in the 'ξ or 'η direction (described in

Fig 2.12) The shifting direction is also demonstrated in Figs 3.2(a) and 3.2(b)

Figure 3.2 (a) A shift direction for numerical calculation of displacement derivative∂ ∂w ξ'; (b) a shift direction for numerical calculation

of displacement derivative ∂ ∂w η'

Hence, a new complex amplitude Ψ( , )m n is calculated by multiplying the shifted complex amplitude A'(m,n) by the conjugate of the original complex phasor ( , ).A m n

values are filtered by nine points, and different filtering window sizes can also be applied In addition, the iterative cycles are considered in the complex phasor method

with the filtering algorithm, and the meaning of j in Eq (3.17) is different from that

in Eq (2.5) (in Chapter 2)

Similarly, the shift in the other directions [such as 'η direction indicated in Fig 2.12 and Fig 3.2(b)] can also be easily realized The continuous phase distribution

3.1.2.2 Second-order displacement derivatives

Since flexural and torsional moments are related to second-order derivatives of the displacement, the measurement of curvature and twist is also an important aspect in the study of out-of-plane displacement of an object (Rastogi, 1996; Chau and Zhou, 2003; Liu, 2003) In this thesis, the determination of curvature and twist is also studied, but digital holographic technique is used to determine these parameters After the determination of first-order displacement derivative in Section 3.1.2.1, the

complex amplitude corresponding to first-order displacement derivative is further shifted in order to extract the information about the second-order displacement derivatives A new complex amplitude Ω(m,n) that corresponds to the second-order displacement derivatives is determined by multiplying a shifted complex amplitude '( , )m n

Ψ by the conjugate of the original complex amplitude Ψ( , ).m n

where a 3×3 window in the average filter is employed

The unwrapped phase map corresponding to the second-order displacement derivatives can be described by

determination of the curvature follows the same definition in Fig 3.2 The shifting direction for the determination of the twist 2

∂ ∂ ∂ is shown in Fig 3.3 As shown in Fig 3.3, the shifting along the 'ξ direction is first carried out, and then the shifting along the 'η direction is employed so as to determine the twist ∂2w ∂ ∂ξ η' '

As described in Section 3.1.1, the proposed post-processing methods, such as sine/cosine transformation method and short-time Fourier transform, can be further employed if the extracted phase map is still contaminated by speckle noise After the extraction of a filtered wrapped phase map, the extracted wrapped phase map should also be processed using a phase unwrapping algorithm, such as a branch cut method (Ghiglia and Pritt, 1998)

Similar to the determination of first-order displacement derivative, since the shifting value is digitally selected, high measurement flexibility and accuracy can also

be achieved during the determination of curvature and twist Figure 3.4 shows a flow chart for the determination of curvature and twist of a deformed object using the proposed method in digital holography Through the calculation of the second-order displacement derivatives, the first-order displacement derivative can also be obtained Hence, this flow chart is also useful for the illustration of the calculation procedure of the first-order displacement derivative in digital holography

Figure 3.3 A schematic for an illustration of the shift direction

in the calculation of the twist 2

Figure 3.4 Flow chart for the measurement of curvature or twist

using the proposed method in digital holography

Phase map corresponding to order displacement derivative

Sine/cosine transformation method or short-time Fourier transform method

Continuous phase map

Branch cut method

Curvature or twist

3.1.3 Fringe density estimation

After the phase map is determined at each state, a phase difference map (or called wrapped phase map) can be directly determined by a digital phase subtraction method

as illustrated in Eq (2.42) Since the phase difference map obtained is noisy, a filtering algorithm is usually applied to reduce the noise before phase unwrapping operation Subsequently, a phase unwrapping algorithm is required to correct the 2 π

jumps of the filtered wrapped phase map In many cases, a priori knowledge about

the fringe density distribution is useful for the selection or development of the filtering algorithm and phase unwrapping algorithm (Quan et al., 2005)

Marroquin et al (1998) presented a technique for recovering the phase from a single fringe-pattern image The proposed method is based on the estimation of the local frequency of the pattern by successive decoupled estimation of the local orientation, direction, and magnitude of the frequency field The phase is recovered from the complex output of an adaptive quadrature filter Marklund (2001) proposed a method to estimate local fringe density and direction in the wrapped phase map The fringe density distribution obtained would facilitate the phase unwrapping operation (Stephenson et al., 1994) It is also demonstrated that the information about fringe density distribution can also be used to construct phase-jump-preserving filtering strategies and to perform robust segmentation of phase data Recently, Quan et al (2005) proposed a fringe-density estimation method by using continuous wavelet transform algorithm However, the results are highly influenced by speckle noise In this thesis, a novel method is proposed for the fringe density estimation of a wrapped phase map in digital holographic interferometry using short-time Fourier transform The improved short-time Fourier transform mentioned in Section 3.2.1 can also be applied

In digital holographic interferometry, the wrapped phase map is determined by digital phase subtraction method Subsequently, short-time Fourier transform method

is employed to process the wrapped phase map Because of its localization characteristic, short-time Fourier transform method has high noise-immunity, so it has drawn much attention in the optical image processing field (Qian, 2007)

One-dimensional short-time Fourier transform of a signal f (x) can be expressed as

* ,

As short-time Fourier transform is employed to process the signal, there usually exists an element that gives the highest similarity, which is called a ridge The value

of ξ that maximizes the similarity is taken as the local frequency, which can be described by

'( )u arg max Sf u( , )

ξ

∆ = (3.24)

where ∆ϕ' u( ) is defined as instantaneous frequency of the signal Because the local

fringe density can be described by instantaneous frequency, the fringe-density

estimation for the wrapped phase map can be considered equivalently to the

measurement of instantaneous frequency on the ridge

In the short-time Fourier transform, the ridge can be determined by

r u = Sf u ∆ϕ u (3.25)

For a noisy wrapped phase map, the extracted fringe-density distribution might

not be smooth, and the conventional mean filter can be applied

( 1) 2 ( 1) 2

where ∆ϕ' u( ) denotes the filtered instantaneous frequency of the signal, and K

denotes the number of points which are involved in the filtering for a given value In

the mean filter not all given values can be filtered by the K points, and K is an odd

value

3.2 Temporal phase evaluation

3.2.1 Improved short-time Fourier transform

Fourier transform method discussed in Section 2.6.1 can be considered as one of the

most popular methods for the extraction of a phase map In Fourier transform method,

the first harmonic of the Fourier spectrum should be accurately extracted, but it is usually contaminated by other higher harmonics Hence, the extracted phase map may not be accurate In addition, Fourier transform method performs globally, which is suitable for the analysis of stationary signals but not suitable for the analysis of non-stationary signals A non-stationary signal is defined as follows: its instantaneous frequency varies with space, such as a fringe pattern with singular points, but varies slowly in other places (Zhong and Zeng, 2007) Conventional advanced analysis methods, such as continuous wavelet transform and short-time Fourier transform, can partially overcome the above problem However, in wavelet transform, the capability

of suppressing noise and errors is not good (Qian, 2007), while in the conventional short-time Fourier transform, the width of the window should be manually chosen (Mallat, 1999; Qian, 2007) Hence, an optimal window for the signal is not easy to be determined in the conventional short-time Fourier transform Moreover, a preset window size is usually used for all the sequences in the analysis of dynamic situations

In this thesis, an improved short-time Fourier transform is proposed in order to effectively overcome the above problems

3.2.1.1 Selection of window size

As mentioned in Section 2.6.4, in the case of short-time Fourier transform, frequency resolution will not vary according to the frequency of interest However, the time-frequency uncertainty principle (Mallat, 1999) can affect the resolution, which leads to a trade-off between time and frequency localization With narrower time window, the temporal resolution will be better, but the resolution in frequency will be poorer This trade-off is the consequence of the uncertainty principle which

time-states that the product of the temporal duration t∆ and frequency bandwidth ∆ ω

should be larger than a constant factor of ∆ ∆ ≥t ω 1 2 The equality in this constant factor can hold only if the window function is Gaussian function Hence, no function can be better localized in both temporal and spectral domains than a Gaussian window (Mallat, 1999)

As short-time Fourier transform is used, the factor of ∆t∆ ω will be always constant in the whole time-frequency plane after one window function is selected This localization is uniform in the entire plane as illustrated in Fig 3.5

Figure 3.5 Time-frequency analysis in the case of the conventional

short-time Fourier transform

If a signal has a transient component with a duration smaller than t∆ , it is

difficult to locate the signal with an accuracy better than t∆ Similarly, phenomenon

can also be observed in the frequency domain If a smaller value of t∆ is chosen, the time resolution is higher, but the frequency resolution ∆ ω will be poorer Hence, conventional short-time Fourier transform is only suitable for analyzing signals that have signal components with similar ranges as the temporal and frequency supports

In this thesis, time bandwidth product (Durak and Arikan, 2003) is considered as the measurement for the time-frequency domain support of the signal, and the corresponding optimal Gaussian window is employed As the optimal Gaussian

Time Frequency

window is used, each pixel along the time axis (or each sequence) can possess a window value of σ The time-frequency domain support of a signal f x( ) is commonly measured by its time-width T x and its frequency-domain bandwidth B x

(Durak and Arikan, 2003) The optimal Gaussian window can be described by

3.2.1.2 Processing of wrapped phase map

In the dynamic situation, the reconstructed complex amplitude for each instant can be obtained using the reconstruction algorithm mentioned in Section 2.4.2 After a series

of reconstructed complex amplitudes is obtained, complex phasor concept (mentioned

in Section 3.1.1) is also applied A complex phasor is calculated by multiplying ( , , )m n t

Γ (t≠0) by the conjugate of ( , , 0)Γ m n at the initial instant

schematic for the processing of complex phasor along the time axis has been shown in Fig 2.22 In digital holography, since the phase map can be extracted directly from a digital hologram, no temporal carrier is required and sign ambiguity is automatically overcome In addition, by using complex phasor method, direct phase manipulation is avoided, so better results can also be obtained

Improved short-time Fourier transform is proposed to process the signals, and short-time Fourier transform of a signal A (t) is also defined as

* ,

−∞

where the kernel g u,ξ( )t =g t( −u) exp(j tξ ), window function g(t)=exp(−t2 2σ2),

σ is used to control the extension of g (t), and uandξ represent the time and

frequency For brevity, the coordinate (m,n) is omitted In this study, time-frequency

domain support of the signal A (t) is measured by its time-width T t and its frequency

domain bandwidth B t Since time bandwidth product is chosen as the measurement

for the time-frequency domain support of the signal, the corresponding optimal Gaussian window (described in Section 3.2.1.1)can be applied

By substituting Eq (3.28) into Eq (3.29), we can obtain

−∞

The real amplitude a(t+u) and the phase difference map ∆ϕ(t+u) both can

be expanded using Taylor series in the local region around u

2

(t u) ( )u t '( )u t ''( ) 2u

If ∆ϕ' t( ) is sufficiently slow varying, we can set ∆ϕ ''(t)≈∆ϕ' ''(t)≈⋅ ⋅⋅=0

In addition, if the real amplitudes of complex phasor have a small relative variation over the support of the window ,g the amplitudes of complex phasor can be assumed constant a (u) In the practical situations, a normalization operation can be carried out onto the complex phasor to keep the real amplitudes a small relative variation over the support of the window g

Based on the above assumptions, shot-time Fourier transform of the signal )

and Re and Im denote the real and imaginary parts

Since 2πB T t/ t is greater than zero, sign( 2πB T t t) is equal to 1