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

CHAPTER THREE DEVELOPMENT OF THEORY 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   m2 n2  Γ(m, n) = C exp  jπλ d  2 + 2   N ∆y    M ∆x  M −1 N −1   2π mk 2π nl   × ∑ ∑ H (k ∆x, l ∆y ) exp  j  +  N  k = l =0   M  (3.1) 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) 56 CHAPTER THREE ϕ (m, n) = arctan DEVELOPMENT OF THEORY Im [ Γ(m, n) ] Re [ Γ(m, n) ] I (m, n) = Γ(m, n) (3.2) (3.3) where Im and Re denote the imaginary and real parts of the reconstructed complex amplitude, respectively In the conventional calculation of a phase difference map (or wrapped phase map), digital phase subtraction method as described in Eq (2.42) is commonly used Since the extracted wrapped phase map is highly contaminated by speckle noise, the subsequent phase unwrapping operation may not succeed One strategy to overcome this problem is that a filter algorithm is employed to process the original wrapped phase map before phase unwrapping However, in many practical applications, the direct filtering of the wrapped phase map will not succeed due to some reasons, such 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 Γ(m, n, 2) at the second (or deformed) state by the conjugate of Γ(m, n,1) at the initial state Without a loss of generality, only a given pixel (m, n) is considered in this case study 57 CHAPTER THREE DEVELOPMENT OF THEORY A(m, n) = Γ(m, n, 2)Γ* (m, n,1) = a(m, n, 2) a(m, n,1) exp { j[ϕ (m, n, 2) − ϕ (m, n,1)]} (3.4) = a '(m, n) exp { j [∆ϕ (m, n)]} where * denotes the complex conjugate, the amplitude a ' (m, n) = a (m, n,2)a (m, n,1), and the phase difference map ∆ϕ (m, n) = ϕ (m, n,2) − ϕ (m, n,1) As the complex phasor method is used, a filtering algorithm can be used to filter the imaginary and real parts of the complex phasor (Ströbel, 1996; Ghiglia and Pritt, 1998) A schematic for an explanation of the complex phasor concept is shown 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 ∆ϕ (m, n) = arctan ∑ Im[Γ(m, n, 2)Γ (m, n,1)] ∑ Re[Γ(m, n, 2)Γ (m, n,1)] ∗ (3.5) * where a 3× window in the average filter is used Imaginary part Noise Complex phasor Real part Figure 3.1 A schematic for an explanation of the complex phasor concept 58 CHAPTER THREE DEVELOPMENT OF THEORY 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 ∆ϕ ( m, n) = arctan ∑ S (m, n) ∑ C (m, n) (3.6) where S (m, n) = sin [ ∆ϕ (m, n) ] , C (m, n) = cos [ ∆ϕ (m, n)] , and ∆ϕ ( m, n) denotes a filtered wrapped phase map In Eq (3.6), a 3× window in the average filter is also employed It is noteworthy that not all values in the wrapped phase map are filtered 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 59 CHAPTER THREE DEVELOPMENT OF THEORY 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 ∆ϕ ( m, n) are respectively described by S (m, n) = sin [ ∆ϕ (m, n)] (3.7) C (m, n) = cos [ ∆ϕ (m, n) ] (3.8) Hence, an exponential signal can be expressed by f ( m, n) = C ( m, n) + jS ( m, n) = exp { j [ ∆ϕ ( m, n) ]} (3.9) Two-dimensional short-time Fourier transform of the exponential signal f ( m, n) is described by Sf (u, v, ξ ,η ) = ∫ +∞ −∞ ∫ +∞ −∞ f (m, n) g (m − u, n − v) exp(− jξ m − jη n) dm dn (3.10) where (u , v ) and (ξ ,η ) denote time and frequency, and g ( m, n) represents a window ( ) ( ) 2 function The function is described by g ( m, n) = exp  − m / 2σ m − n / 2σ n  ,   where σ m and σ n 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 60 CHAPTER THREE DEVELOPMENT OF THEORY 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  Sf (u , v, ξ ,η )  Sf (u , v, ξ ,η ) =  0  if Sf (u , v, ξ ,η ) ≥ thrd if Sf (u , v, ξ ,η ) < thrd (3.11) 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 f (m, n) = +∞ +∞ ηu ξu Sf (u , v, ξ ,η ) 4π −∞ −∞ l l × g (m − u , n − v) exp( jξ m + jη n) d ξ dη dudv ∫ ∫ ∫η ∫ξ (3.12) where ξ u and ξ l denote the upper and lower limits of ξ , and η u and η l denote the upper and lower limits of η , respectively The ranges of ξ and η can be approximately estimated by the analysis of Fourier transform of the original phase map Since the spectrum extends infinitely along ξ and η axes, the estimated ranges 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 61 CHAPTER THREE ∆ϕ (m, n) = arctan DEVELOPMENT OF THEORY Im f (m, n) Re f (m, n) (3.13) 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) η' ξ' (a) (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) 62 CHAPTER THREE DEVELOPMENT OF THEORY Ψ ( m, n) = A '( m, n) A* ( m, n) = c ( m, n ) c ( m + δ ξ ' , n ) + d ( m, n ) d ( m + δ ξ ' , n ) (3.14) + j [c ( m, n) d ( m + δ ξ ' , n) − d ( m, n) c ( m + δ ξ ' , n)] where δ ξ ' denotes a shearing in the ξ ' direction, A(m, n) = c(m, n) + j d (m, n), and shifted complex amplitude A '(m, n) = c(m + δ ξ ' , n) + jd (m + δ ξ ' , n) It is noteworthy that there is no necessity to consider the correspondingly shifted part during the calculation process The resultant complex amplitude Ψ (m, n) can also be converted into a complex exponential signal whose real amplitude and phase are described by a ''(m, n) = [c(m, n) c(m + δ ξ ' , n) + d (m, n)d (m + δ ξ ' , n)]2 +[c(m, n) d (m + δ ξ ' , n) − d (m, n) c(m + δ ξ ' , n)]2 ∆∆ϕ (m, n) = arctan c(m, n)d (m + δξ ' , n) − d (m, n) c(m + δ ξ ' , n) c(m, n) c(m + δ ξ ' , n) + d (m, n) d (m + δ ξ ' , n) (3.15) (3.16) An average filter can also be used, and the phase difference map in Eq (3.16) can be rewritten as m +1 n +1 ∑ ∑ ∆∆ϕ (m, n) = arctan i = m −1 j = n −1 m +1 n +1 ∑ ∑ i = m −1 j = n −1 c(i, j )d (i + δ ξ ' , j ) − d (i, j ) c(i + δ ξ ' , j )    (3.17)  c(i, j ) c(i + δ ξ ' , j ) + d (i, j )d (i + δ ξ ' , j )    where an average filter with a 3× window is also used Not all given values are filtered by the 3× window, so Eq (3.17) can also be modified correspondingly In practical cases, zero padding extension technique can be used to ensure that all given 63 CHAPTER THREE DEVELOPMENT OF THEORY 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 ∆∆ϕ u (m, n) can be expressed by ∆∆ϕu (m, n) ≈ 4π ∂w(m, n) δξ ' λ ∂ξ ' (3.18) where w( m, n) denotes the displacement of the test specimen, and ∆∆ϕ u (m, n) denotes an unwrapped phase map An analogue theoretical expression can be developed for the determination of displacement derivative ∂w ∂η ' Different from shearography method (Hung, 1997), the displacement derivative is numerically obtained by using the proposed method in digital holography Since the shearing value is digitally determined, measurement flexibility and accuracy are ensured 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 64 CHAPTER THREE DEVELOPMENT OF THEORY 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 Ψ '(m, n) by the conjugate of the original complex amplitude Ψ (m, n) Ω( m, n) = Ψ '( m, n) Ψ * ( m, n) { } = a ''( m + δ ξ ' , n) a ''( m, n) exp j  ∆∆ϕ ( m + δ ξ ' , n) − ∆∆ϕ ( m, n)    = a '''( m, n) exp ( j {∆[ ∆∆ϕ ( m, n)]}) (3.19) where a '''(m, n) = a ''(m + δ ξ ' , n)a ''(m, n), ∆[∆∆ϕ (m, n)] = ∆∆ϕ (m + δ ξ ' , n) − ∆∆ϕ (m, n) An average filter is also applied, and the phase difference map in Eq (3.19) can be described by ∑ Im[Ψ '(m, n)Ψ (m, n)] ∑ Re[Ψ '(m, n)Ψ (m, n)] * ∆[∆∆ϕ (m, n)] = arctan * (3.20) where a 3× window in the average filter is employed The unwrapped phase map corresponding to the second-order displacement derivatives can be described by  4π  ∂w ∂w 4π ∂ w(m, n) ∆[∆∆ϕ (m, n)] ≈ ( m + δξ ' , n ) − ∂ξ ' (m, n)  δξ ' ≈ λ ∂ξ '2 δξ ' λ  ∂ξ '   (3.21) Similarly to Eq (3.21), analogue theoretical expressions can also be developed for the curvature ∂ w ∂η '2 and the twist ∂ w ∂ξ ' ∂ η ' The shifting direction for the 65 CHAPTER THREE DEVELOPMENT OF THEORY 3.3 Extension of the depth of focus In recent years, digital holography is widely applied to the particle field measurement Coёtmellec et al (2002) analyzed the diffraction patterns of a particle field hologram using a fractional Fourier transform Soulez et al (2007) proposed a micro-particle detection scheme in digital holography with a simplified model of the images which incorporate the size and location of a diffracting particle Yang and Kang (2008) proposed a correlation coefficient method for the determination of particle positions in digital holography Wu et al (2009) applied digital holographic technique for the measurement of small particles A disadvantage of these particle field measurements is the small depth of focus, and micro-objects existing in the different recording distances can not be in focus simultaneously With an increase of lateral resolution using an objective lens in digital holographic microscopy, the field of view decreases nonlinearly and the size of the resolvable volume is greatly reduced (Sheng et al., 2006; Kim and Lee, 2007; Antkowiak et al., 2008) In this thesis, a new method using entropy concept is proposed to overcome the above problem through extending the depth of focus for a particle field using a single-shot digital hologram 3.3.1 Entropy method It is demonstrated in Section 2.4.2 that in digital holography, both intensity and phase distributions can be determined from the reconstructed complex amplitude Γ(ξ ',η ') The reconstructed intensity I (ξ ',η ') in the image plane is described by I (ξ ',η ') = {Im[Γ(ξ ',η ')]}2 + {Re[Γ(ξ ',η ')]}2 (3.50) where Im and Re denote the imaginary and real parts, respectively 82 CHAPTER THREE DEVELOPMENT OF THEORY The most obvious advantage of digital holographic technique is numerically focusing with a flexible reconstruction distance z, so a series of reconstruction intensity maps can be obtained In this thesis, a method based on an entropy concept (Gillespie and King, 1989) is applied to extend the depth of focus for a particle field measurement in digital holography In the reconstructed intensity map I z (ξ ',η ') at a distance z, the entropy for a point (k, l) is calculated by Τ z (k , l ) = − k +( n −1)    l + ( n −1)     I z (ξ ',η ')  I (ξ ',η ')     log  z   Ωz ξ '= k −( n −1)  η '=l −( n −1)    Ωz       where Ω z = ∑ k + ( n −1)    ∑ ∑ l +( n −1)    ∑ ξ '= k −( n −1)  η '= l −( n −1)      (3.51) I z (ξ ',η ') and n × n denotes a defined block size for the point In the entropy method, the intensity distribution I z (ξ ',η ') should be first normalized Therefore, the minimum entropy value for each point along the axis of reconstruction distances can be obtained, and a focal distance for the point is determined Finally, a depth map Dm(k , l ) is extracted, and an extended focused image (EFI) is obtained by EFI (k , l ) = I Dm ( k ,l ) (k , l ) (3.52) where in Eq (3.52), I Dm ( k ,l ) (k , l ) is not normalized and is calculated by Eq (3.50) 3.3.2 Wavelet modulus maxima algorithm After the EFI is obtained, a wavelet modulus maxima algorithm (Mallat and Zhong, 1992) is proposed to detect the edges Using a combination of EFI and edge maps, the 83 CHAPTER THREE DEVELOPMENT OF THEORY dimensions of particles can be determined more effectively In the wavelet modulus maxima algorithm, two wavelets are obtained by 1 ξ ' η' ψ (ξ ',η ') =  i  ψ  i , i  2  2  (3.53) 1 ξ ' η' ψ (ξ ',η ') =  i  ψ  i , i  2  2  (3.54) 2i 2 2i where 2i denotes a scale, i denotes an integer, and ψ (ξ ',η ') and ψ (ξ ',η ') are wavelets respectively determined by the partial derivatives of a 2D Gaussian function along the ξ ' and η ' axes Hence, wavelet transform of EFI (ξ ',η ') with a scale of 2i contains two components defined by W1 EFI (2i , ξ ',η ') = EFI (ξ ',η ') ⊗ψ 2i (ξ ',η ') (3.55) W2 EFI (2i , ξ ',η ') = EFI (ξ ',η ') ⊗ψ 2i (ξ ',η ') (3.56) where ⊗ denotes a 2D convolution operation The modulus and angle maps are respectively given by MEFI (2i , ξ ',η ') = W1 EFI (2i , ξ ',η ') + W2 EFI (2i , ξ ',η ')  W EFI (2i , ξ ',η ')  AEFI (2i , ξ ',η ') = arctan   i  W1 EFI (2 , ξ ',η ')  (3.57) (3.58) Conversions, such as subtraction or addition of 2π or π , can be used in the calculation of the angle map Wavelet modulus MEFI (2i , ξ ',η ') is compared, and 84 CHAPTER THREE DEVELOPMENT OF THEORY local maximum points are determined in the direction of the gradient given by AEFI (2i , ξ ',η ') As the maximum modulus are obtained, normalization and threshold operations are carried out In addition, a similar approach using Canny algorithm (Canny, 1986) is also used to detect the particle edges, and the result using Canny algorithm is demonstrated in Section 5.3 3.4 Complex-valued object reconstruction With the rapid progress of material and biological sciences, phase retrieval from intensity distributions, especially x-ray diffraction intensity, is becoming more and more important in the structural determination of micro or nano-objects (Miao et al., 2003a; Zou et al., 2003; Pfeifer et al., 2006; Caro et al., 2007) However, in practice, only diffraction intensity distribution can be recorded by the CCD camera, and information about phase distribution is lost Note that this diffraction recording method might also be considered as in-line digital holography Recently, some phase retrieval algorithms, such as Gerchberg-Saxton algorithm (Gerchberg and Saxton, 1972), error reduction approach (Fienup, 1982), hybrid input-output method (Fienup, 1982), oversampling phasing method (Zou et al., 2003; Miao et al., 1998, 1999, 2003b), moving aperture technique (Faulkner and Rodenburg, 2004), aperture-plane modulation (Zhang et al., 2007) and shifting illumination (Rodenburg and Faulkner, 2004), have been proposed to recover the test objects However, in many cases, it seems that a stronger support constraint is usually required for a complex-valued object than that for a real or non-negative object (Liu et al., 2008; Johnson et al., 2008) In this thesis, a simple but effective iterative method with position shift of a phase mask is developed for the reconstruction of a complex-valued object 85 CHAPTER THREE DEVELOPMENT OF THEORY In the proposed method, there are three planes, i.e., object plane, phase mask plane and CCD plane A phase mask is located between object plane and CCD plane A collimated plane wave is used to illuminate the object, and a diffraction intensity distribution is recorded by a CCD camera When the phase mask is respectively located at positions 1, and (described in Section 4.4), three diffraction intensity distributions are recorded The detailed experimental setup for the proposed method is also shown in Section 4.4 In this study, the coordinates for mask plane and CCD plane are (ξ ,η ) and ( x, y ), respectively In addition, the coordinates for the phase mask in positions 1, and are denoted as (ξ1 ,η1 ), (ξ ,η2 ) and (ξ3 ,η3 ), respectively Angular spectrum algorithm (Goodman, 1996) is used to describe wave propagation, and complex amplitude in the CCD plane is given by F ( x, y ) = FFT −1 O( fξ , fη ) T ( fξ , fη )  = ASAz [ O(ξ ,η ) M (ξ ,η )]   (3.59) where f ξ and fη are spatial frequencies, O(ξ ,η ) is a complex-valued object wave just before the phase mask, M (ξ ,η ) denotes a known random-phase mask, FFT −1 denotes a 2D inverse fast Fourier transform, O( f ξ , f η ) denotes a 2D Fourier transform of O(ξ , η ) M (ξ , η ), ASAz is angular spectrum algorithm with a distance z, and T ( f ξ , fη ) is a transfer function described in Section 2.4.2.4 Similarly, wave propagation between object plane and phase mask can also be calculated by Eq (3.59) Advantages of angular spectrum algorithm are that unlike Fresnel approximation algorithm (Goodman, 1996), no minimum distance is required and the pixel size can maintain constant during wave propagation In this thesis, a new iterative method is proposed using only three recorded diffraction intensity maps 86 CHAPTER THREE DEVELOPMENT OF THEORY  I ( x, y ), I ( x, y ) and I ( x, y )  to recover the original complex-valued object The   proposed phase retrieval algorithm proceeds as follows: (1) Assume an initial random or constant distribution On (ξ ,η ) (iteration number n=1, 2, 3, ……) just before position of the phase mask plane; (2) multiply a phase mask M (ξ ,η ) : ' On (ξ ,η ) = On (ξ ,η ) M (ξ ,η ); (3) propagate to CCD plane with a distance of ' z : Fn3 ( x, y ) = ASAz2 On (ξ3 ,η3 )  ; (4) apply a support constraint in CCD   12 plane: Fn3 ( x, y ) =  I ( x, y )    and  Fn3 ( x, y ) Fn3 ( x, y )  ; (5) propagate back to position   { } multiply M * (ξ ,η ) : On (ξ3 ,η3 ) = ASAz−21  Fn3 ( x, y )  M *(ξ3 ,η3 ),   and the asterisk denotes complex conjugate; (6) propagate to position using On (ξ ,η ) , repeat steps 2-5 with corresponding parameter modification, and obtain On (ξ ,η ); (7) propagate to position using On (ξ ,η ) , repeat steps 2-5 with corresponding parameter modifications, and obtain On (ξ1 ,η1 ) ; (8) propagate to position again with a distance of ( z + z ), and calculate measurement error If measurement error is not larger than a preset threshold ( σ ), iteration operation stops Otherwise, On (ξ ,η ) in step is replaced by complex amplitude obtained in the step for next iteration (n=n+1) Note that the phase mask in three positions is the same, and ASA −1 denotes angular spectrum algorithm using an inverse axial distance To evaluate the accuracy of the proposed method, a sum-squared error (SSE) is calculated, which can be described by 2  SSE = ∑  X (m, n) − X (m, n)      m,n   2 ∑  X (m, n)      m,n  (3.60) 87 CHAPTER THREE DEVELOPMENT OF THEORY where X (m, n) and X (m, n) are original and recovered complex-valued objects in the object plane Once the iterative step reaches position of the phase mask, wave propagation to object plane is carried out 3.5 Optical image encryption With the rapid development of modern communication techniques, the unauthorized distributions of important information become a serious problem, and optical encryption is an important topic for information security or data transmission (Refregier and Javidi, 1995; Javidi, 1997) Some methods, such as double randomphase encoding (Refregier and Javidi, 1995), digital holography (Chen and Zhao, 2006) and virtual optics (Peng et al., 2002), have been proposed In addition, a phase retrieval method based on an iterative algorithm (Situ and Zhang, 2004) was proposed, and to avoid the iteration operation a simpler algorithm using a wave-superposition approach (interference principle) without any iteration (Zhang and Wang, 2008) has also been proposed As a monochromatic light is used to illuminate a color image, the real color information of a decrypted image is lost Since the color information of an image is useful in many practical applications (Zhang and Karim, 1999; Chen and Zhao, 2006; Joshi et al., 2008), optical color image encryption should be studied In this thesis, two novel strategies based on interference principle are proposed to investigate optical color-image encryption, and one more method using a bit-plane separation is proposed in order to enhance the security level in an optical image encryption system using phase-shifting digital holographic technique The basic principles of the proposed methods are described in the following sections 88 CHAPTER THREE DEVELOPMENT OF THEORY 3.5.1 Arnold transform and interference A schematic numerical experimental arrangement is shown in Fig 3.7 Collimated plane light sources respectively illuminate two retrieved phase-only masks at each channel, and the two beams are combined in the image plane To obtain the original color image at the image plane, a digital approach should be first used to embed each channel of an original input image into two phase-only masks In addition, a different distance is preset for each channel to enhance the security level, but within a channel the two phase-only masks are placed at the same distance from the image plane z1 z2 z3 Beam splitter cube λ1 λ2 λ3 Image plane Green Blue channel Red channel channel mask mask M5 mask M1 M3 Red Green Blue channel mask M6 Blue Green channel mask M4 A color input image Red channel mask M2 λ1 λ2 λ3 Figure 3.7 A schematic numerical experimental arrangement Before an original color image is hidden, Arnold transform (ART) is employed to process the color image For simplicity, only red channel of an input image is analyzed For a given image O( x, y ) with N × N pixel number at red channel (nonnegative values), ART of a pixel ( x, y ) in the input image O( x, y ) is expressed as 89 CHAPTER THREE  x'  =  y ' DEVELOPMENT OF THEORY x 1 A   ( mod N ) =   y 1 1 1 1 , 2 where A =  1x    ( mod N ) 2  y  (3.61) mod denotes the modulus after division, and ( x' , y ' ) denotes a new pixel position after the ART operation Hence, a discrete ART of the image can be described by { ART [O( x, y ), N ] = v, ( x ', y ' )    ( x ', y ') T = A ( x, y ) T ( mod N ) , } v, ( x, y )  ∈O( x, y )   (3.62) where T denotes a transpose operation, v denotes a value at the pixel ( x, y ) in the original image, and the terms on the right-hand side of “|” denote the algorithm conditions or operation procedures In the ART method, an Arnold scrambling period is determined by { Period = n {ART [O( x, y), N ] } n = O ( x, y ) } (3.63) where denotes a minimum value, and n is a positive integer which denotes the number of iterations In this study, N is larger than and the period usually satisfies the requirement of Period ≤ N 2 (Dyson and Falk, 1992) After the procedure of pixel scrambling, an interference principle (Zhang and Wang, 2008) is proposed to hide the scrambled color image During the hiding procedure, a digital or numerical method should be employed The complex amplitude for phase-only masks M1 and M2 at the red channel which interfere in the image plane can be expressed as 90 CHAPTER THREE DEVELOPMENT OF THEORY O( x ', y ') exp [ j P( x ', y ') ] = FFT −1 [ H ( fξ , fη ; z1 ) M ( fξ , fη )] (3.64) + FFT −1 [ H ( fξ , fη ; z1 ) M ( fξ , fη )] where O( x' , y ' ) = {ART [O( x, y ), N ] } , P( x' , y ' ) is a map randomly distributed in n [0, 2π ] , j = − 1, f ξ and fη are spatial frequencies, FFT −1 denotes a 2D inverse fast Fourier transform, M ( f ξ , fη ) and M ( f ξ , fη ) denote 2D Fourier transforms of phase masks M (ξ ,η ) and M (ξ ,η ), z1 denotes the distance between the phase mask plane (M1 or M2) and the image plane, and H ( fξ , fη ; z1 ) is a transfer function which is described in Section 2.4.2.4 Note that a different random map P( x' , y ' ) is used for each channel, and the implementation of ART with the iteration number (Period - n) is called an inverse ART After a proper derivation of Eq (3.64), the red channel of the input image is encrypted into two phase-only masks M1 and M2 as ( M (ξ ,η ) = angle FFT −1 { FFT [Ow ( x ', y ')] H ( fξ , fη ; z1 )} { ( ) ) − arccos abs FFT −1{ FFT [Ow ( x ', y ')] H ( fξ , fη ; z1 )}    ( } M (ξ ,η ) = angle FFT −1 { FFT [Ow ( x ', y ') ] H ( fξ , fη ; z1 )} − exp [ jM (ξ ,η ) ] (3.65) ) (3.66) where Ow ( x' , y ' ) = O( x' , y ' ) exp[ j P( x' , y ' )] , FFT denotes a 2D fast Fourier transform, and angle denotes an arc-tangent operation In this case, the values of ( ) abs FFT −1 { FFT [Ow ( x ', y ') ] H ( fξ , fη ; z1 )}  are compared with a threshold   value of 1, and values larger than the threshold value are set as It is illustrated in 91 CHAPTER THREE DEVELOPMENT OF THEORY Eqs (3.65) and (3.66) that the red channel of a color image is encrypted into two phase-only masks Similarly, phase-only masks M3 and M4 for the green channel and M5 and M6 for the blue channel can also be obtained The most obvious advantage of the interference method is that compared with other phase retrieval methods (Situ and Zhang, 2004), it does not require any iterative operations In the proposed method, although the security level is high only with the extracted random phase masks (M1M6), the security level can be further enhanced using the proposed pre-processing ART method For image decryption, a light source with a corresponding wavelength is used to illuminate the two retrieved phase-only masks at each channel As the incident wavelength of each channel is close to the wavelength of the basic color, the original color information can be obtained at the image plane A decrypted real color image can be obtained by an inverse ART with a correct number of iterations for each channel and the incorporation of three decrypted channels With a proper modification of the experimental arrangement as shown in Fig 3.7, a practical real-time optical implementation of image decryption is feasible In addition, a constant value, such as 2π , can be added to all the retrieved phase masks (M1-M6) to avoid the negative values when a spatial light modulator is used This operation will not affect the finally decrypted image To evaluate the similarity between the original color image O( x, y ) and a decrypted image Or ( x, y ) , correlation coefficient (CC) is calculated by ( CC = cov ( O, Or )  σ O ⋅ σ Or   ) (3.67) where cov(O, Or ) denotes the cross-covariance, and σ denotes standard deviation 92 CHAPTER THREE DEVELOPMENT OF THEORY 3.5.2 Fractional Fourier transform and interference In Section 3.5.1, a pre-processing method, i.e., Arnold transform, is proposed to scramble the original image before the hiding operation with the interference method Another strategy is to apply a post-processing method after the operation of interference method In this proposed post-processing method, the numerical experimental setup is the same as that shown in Fig 3.7 The random phase masks M1-M6 are also extracted by using Eqs (3.65) and (3.66) To enhance the security level, phase-only masks M1-M6 are further processed in fractional Fourier domain based on a concept of virtual optics For simplicity, an one-dimensional Fractional Fourier transform (FRFT) (Ozaktas et al., 2001) is analyzed, and FRFT with an order a is described by FRFTa [ M (ξ ) ] = ∫ +∞ −∞ M (ξ ) exp [ j 2π Q1 (ξ ) ] Ta ( xa , ξ ) d ξ (3.68) where { [ ]}  R exp jπ x a cot (aπ ) + ξ cot (aπ ) − x aξ csc(aπ )  Ta ( x a , ξ ) = δ ( x a − ξ ) δ ( x + ξ )  a if a ≠ 2m if a = 4m , if a = 4m ± m denotes an integer, Q1 (ξ ) denotes the values randomly distributed in [0, 1], and R = − j cot (aπ 2) In this investigation, different random maps of P1 ( x, y ) (inside the interference method) and Q1 (ξ ,η ) are used in different channels For image decryption, FRFT with an order of (–a) is first carried out, and a complex conjugate of exp[ j 2π Q1 (ξ ,η )] is then multiplied A light source with a correct wavelength is used to illuminate the two retrieved phase-only masks at each 93 CHAPTER THREE DEVELOPMENT OF THEORY channel Finally, a decrypted color image is obtained by incorporating the three decrypted channels The methods described above (also including the proposed method in Section 3.5.1) are based on a combination of multiple wavelengths and interference method To avoid the multiple-step operation, another concept using an indexed image (Zhang and Karim, 1999) is applied An original color image is first converted into an image matrix and a color map The image matrix is divided by a constant value before image encryption, and is then encrypted based on the interference method and virtual optics Therefore, only two random-phase masks (in this case, called M7 and M8) before FRFT and a color map are required to hide the original color image For image decryption, FRFT with an order of (–a) is also carried out, and a complex conjugate of exp[ j 2π Q1 (ξ ,η )] is then multiplied A light source is used to illuminate the two retrieved phase masks, and a decrypted color image is obtained by a combination of the decrypted image matrix (multiplied by the constant value) and the color map Correlation coefficient calculated by Eq (3.67) can also be used to evaluate the similarity between the original color image and a decrypted image Since complex amplitudes are obtained by using the FRFT method, the proposed techniques mentioned above are more suitable for a digital usage 3.5.3 Phase-shifting technique and a bit-plane separation As mentioned in Chapter 2, phase-shifting digital holography is one of the most powerful techniques for numerical reconstruction and the phase evaluation In this thesis, by a combination of phase-shifting digital holographic technique and optical encryption technique, a new method is proposed for optical image encryption based on conventional double random-phase masks and FRFT In addition, a bit-plane 94 CHAPTER THREE DEVELOPMENT OF THEORY separation method is proposed to enhance the security level of the designed optical cryptographic system The detailed experimental setup for the proposed method is shown in Section 4.5.2 (Chapter 4) An input image is first divided into eight bit planes including 0th to 7th bit planes, and each bit plane is encrypted independently For 0th bit plane, two phase masks M1 and M2 with complex transmittances exp [ j P ( x1 , y1 )] and exp [ j P2 ( x2 , y2 ) ] are used as two independent functions randomly distributed in [ 0, 2π ] For the sake of brevity, the same phase masks M1 and M2 are also used for other bit planes Phase mask M1 is inserted in the plane just behind the input image, and phase mask M2 is inserted in the fractional Fourier plane Hence, object wave in the hologram (or CCD) plane is described by O( x, y ) = FRFTb1, b { ( FRFT a1, a {O ( x1 , y1 ) exp[ j P1 ( x1 , y1 )]}) exp[ j P2 ( x2 , y2 )]} (3.69) where O( x1 , y1 ) denotes a bit plane of the input image, and FRFTa1, a and FRFTb1, b denote 2D FRFT (described in Section 3.5.2) with orders of (a1, a2) and (b1, b2) The in-line hologram intensities, denoted by I i ( x, y ), are obtained by the interference between an object wave O( x, y ) and a plane reference wave Ri ( x, y ) in the hologram plane I i ( x, y ) = O( x, y ) O* ( x, y ) + Ri ( x, y ) Ri* ( x, y ) + Ri ( x, y ) O* ( x, y ) + Ri* ( x, y )O( x, y ) (3.70) where integer i = 1, 2,3, and * denotes complex conjugate As three intensity maps are recorded by a CCD camera, object wave O ( x, y ) in the hologram plane can be retrieved by 95 CHAPTER THREE O ( x, y ) = DEVELOPMENT OF THEORY 1− j { I1 ( x, y) − I ( x, y) + j [ I ( x, y) − I3 ( x, y)] } 4R = A( x, y ) exp [ j ϕ ( x, y ) ] (3.71) where R denotes real amplitude of the reference wave, j = −1, A( x, y ) denotes real amplitude of object wave, and ϕ ( x, y ) denotes phase distribution of object wave The intensity maps I1 ( x, y ), I ( x, y ) and I ( x, y ) are recorded when the phase shift of the reference wave path is 0, π and π , respectively After the complex amplitude in the hologram plane is retrieved for each bit plane, the ART algorithm (described in Section 3.5.1) is further proposed to process the complex amplitudes Different number of iterations in ART algorithm is set for real and imaginary parts of the complex amplitudes extracted in the hologram plane To decrypt the image, an inverse ART is first carried out on the encrypted complex amplitude for each bit plane Subsequently, each complex amplitude at the fractional Fourier plane is obtained by a FRFT with function orders of (-b1,-b2) and the multiplication of a conjugate of exp[ j P2 ( x2 , y2 )] Then the original image at each bit plane is decrypted using FRFT with function orders of (-a1, -a2) and the multiplication of a conjugate of exp[ j P ( x1 , y1 )] Since only the intensity distribution is of interest, the phase mask M1 can be ignored during image decryption Finally, a decrypted image is obtained by the incorporation of the eight decrypted bit planes Correlation coefficient described in Eq (3.67) is also used to evaluate the quality of a decrypted image 96 ... advantage in using the integration method is that after the phase of each pixel at each instant is determined by Eq (3. 38), phase unwrapping is not required in both temporal and spatial domains This... phase- shifting error using the proposed algorithms in phase- shifting digital holography 81 CHAPTER THREE DEVELOPMENT OF THEORY 3. 3 Extension of the depth of focus In recent years, digital holography. .. procedure to determine phase- shifting error is shown in Fig 3. 6 80 CHAPTER THREE DEVELOPMENT OF THEORY Recording of intensities I1 , I , I O , and I R Definitions of phase- shifting error detection