EVALUATION OF DIGITAL HOLOGRAPHIC RECONSTRUCTION TECHNIQUES FOR USE IN ONE-SHOT MULTI-ANGLE HOLOGRAPHIC TOMOGRAPHY

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1 EVALUATION OF DIGITAL HOLOGRAPHIC RECONSTRUCTION TECHNIQUES FOR USE IN ONE-SHOT MULTI-ANGLE HOLOGRAPHIC TOMOGRAPHY Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Master of Science in Electro-Optics By Haipeng Liu Dayton, Ohio August, 2014

2 EVALUATION OF DIGITAL HOLOGRAPHIC RECONSTRUCTION TECHNIQUES FOR USE IN ONE-SHOT MULTI-ANGLE HOLOGRAPHIC TOMOGRAPHY Name: Liu, Haipeng APPROVED BY: Partha P. Banerjee, Ph.D. Advisor Committee Chairman Professor and Director Electro-Optics Graduate Program Joseph W. Haus, Ph.D. Committee Member Professor Electro-Optics Graduate Program Imad Agha, Ph.D. Committee Member Assistant Professor Physics John G. Weber, Ph.D. Associate Dean School of Engineering Eddy M. Rojas, Ph.D., M.A., P.E. Dean School of Engineering ii

3 Copyright by Haipeng Liu All rights reserved 2014 iii

4 ABSTRACT EVALUATION OF DIGITAL HOLOGRAPHIC RECONSTRUCTION TECHNIQUES FOR USE IN ONE-SHOT MULTI-ANGLE HOLOGRAPHIC TOMOGRAPHY Name: Liu, Haipeng University of Dayton Advisor: Dr. Partha P. Banerjee Tomography is a technique to reconstruct the 3-dimensional (3D) profile of the object. In multi-angle holographic tomography (MAHT), data from different projections with different angles are recorded by an in-line Gabor hologram setup, and the 2-dimensional (2D) shapes of the object for different angles are reconstructed and combined to form a true 3D profile. A one-shot MAHT setup with 3 angles is built and the main issues involved in this technique are discussed. Compressive sensing (CS) is a new alternative to the conventional Fresnel approach for digital holographic reconstruction for sparse objects, and can show a good performance with respect to the depth of focus and working range. The performance of the Fresnel approach, the transfer function approach and the CS approach for holographic reconstruction is compared with respect to the depth of focus, twin-image effect and the computation time, which helps us determine the optimum approach to use in one-shot MAHT. iv

5 ACKNOWLEDGMENTS I would like to express my special gratitude to Dr. Partha P. Banerjee, my advisor, for providing the time and equipment for my thesis, and for his patience and expertise. I would also like to thank Dr. Joseph Haus and Dr. Imad Agha who agreed to serve on my committee. I would like to acknowledge the assistance from Dr. George Nehmetallah and Dr. Logan Williams who have guided me with several of the concepts and programming required for this work. I also want to express my appreciation to Ujitha Abeywickrema, who has helped me set the experimental setup; Zhicheng Xiao and Qian Cao, who have helped me with the programming; and Dr. Russell Hardie and Chen Cui, who have helped me with the image processing. All my friends and professors have offered me great help in class and research. They have encouraged me, discussed with me, and have shared their knowledge and insights with me. I would not have accomplished this work without them and I deeply appreciate the support from them. v

6 TABLE OF CONTENTS ABSTRACT... iv ACKNOWLEDGMENTS... v LIST OF FIGURES... viii LIST OF TABLES... xii LIST OF ABBREVIATIONS... xiii LIST OF SYMBOLS... xv CHAPTER 1 INTRODUCTION... 1 CHAPTER 2 THEORETICAL BACKGROUND Introduction Recording of Digital Holograms Fresnel Approach in DH Non-paraxial and Paraxial Transfer Function Approach in DH CS Approach in DH Multi-angle Holographic Tomography Conclusion vi

7 CHAPTER 3 EXPERIMENTAL EVALUATION OF DIFFERENT DIGITAL HOLOGRAPHIC RECONSTRUCTION APPROACHES Introduction Experimental Setup Experimental Results Conclusion CHAPTER 4 ONE-SHOT MULTI-ANGLE HOLOGRAPHIC TOMOGRAPHY Introduction Experimental Setup Sources of Errors Final Experimental Results Conclusion CHAPTER 5 CONCLUSION BIBLIOGRAPHY vii

8 LIST OF FIGURES Figure 2.1: Diagram of a Gabor setup... 8 Figure 2.2: Diagram of a Mach-Zehnder setup Figure 2.3: Diagram showing the relation between the object plane, the hologram plane and the image plane Figure 2.4: A MAHT setup by rotating the object Figure 2.5: (a) All lofted 3D volumes at different angles; (b) 3D reconstruction result Figure 3.1: Gabor setup for DH. The object is a dandelion seed parachute with few wings Figure 3.2: (a) Recorded hologram of the object; (b) recorded reference without the object; (c) the hologram used for reconstruction after subtraction of (b) from (a) Figure 3.3: (a) Reconstruction using paraxial transfer function approach at 39 mm; (b) reconstruction using non-paraxial transfer function approach at 39 mm Figure 3.4: Reconstruction using the non-paraxial back-propagation approach at (a) 19.3 mm, (b) 100 mm, and (c) mm Figure 3.5: (a) Original hologram when recording distance d is around 6.6 mm; (b) reconstruction using the Fresnel approach; (c) reconstruction using the non-paraxial back-propagation approach; (d) reconstruction using TwIST viii

9 Figure 3.6: (a) Recorded hologram at d=251.2 mm; (b) reconstruction using the Fresnel approach; (c) reconstruction using the non-paraxial back-propagation approach; (d) reconstruction using TwIST Figure 3.7: Diagram showing the concept for the depth of focus. The holograms are reconstructed at different reconstruction distances z. d is the recording distance. z, the DOF for recording distance d, is the variation of z where the interplane intensity ratio (defined below) is between 0.5 and Figure 3.8: (a) Reconstruction at mm using hologram recorded at mm; (b) detailed image around row 400 in the CS reconstruction at mm; (c) detailed image around row 400 in the CS reconstruction at mm Figure 3.9: Plot of the normalized DOF vs the normalized recording distance Figure 3.10: (a) The plot of the non-paraxial back-propagation approach and the CS TwIST approach; (b) the plot of the Fresnel approach and the CS TwIST approach Figure 4.1: Diagram of the one-shot MAHT setup (top-view) Figure 4.2: Equivalent in-line setup (top-view) Figure 4.3: Recorded hologram in one-shot MAHT setup Figure 4.4: Diagram illustrating the tilt factor Figure 4.5: (a) Recorded hologram with two dandelions at 255 mm and 307 mm, respectively; (b) reconstruction at 255 mm; (c) reconstruction at 307 mm Figure 4.6: (a) Recorded hologram of the wire with some overlapping; (b) reconstruction at 120 mm; (c) reconstruction at 395 mm; (d) reconstruction at 565 mm ix

10 Figure 4.7: Recorded hologram of a coiled spring with large recording distance Figure 4.8: (a) Reconstruction at 178 mm; (b) reconstruction at 548 mm; (c) reconstruction at 1311 mm Figure 4.9: Cropped regions of interest from (a) Figure 4.8(a); (b) Figure 4.8(b); (c) Figure 4.8(c) Figure 4.10: Result of single thresholding of Figure 4.9(a) Figure 4.11: (a) Result of best single thresholding; (b) the ambiguous parts; (c) disconnected ambiguous parts; (d) final binary image after image processing Figure 4.12: Binary image of the part of interest of the reconstruction at 548 mm Figure 4.13: (a) Recorded hologram of a bigger spring with large recording distance; (b) reconstruction for this bigger spring at 1266 mm Figure 4.14: Diagram of the Mach-Zehnder setup for one-shot MAHT Figure 4.15: (a) Recorded hologram using Mach-Zehnder one-shot MAHT setup; (b) reconstruction at 1135 mm Figure 4.16: Diagram of the cross section difference Figure 4.17: 3D reconstruction of the twisted wire with an inappropriate object orientation Figure 4.18: Recorded hologram in one-shot MAHT of a twisted wire Figure 4.19: (a) Reconstruction at 120 mm; (b) reconstruction at 395 mm; (c) reconstruction at 565 mm Figure 4.20: 2D profile binary images of the regions of interest from (a) Figure 4.19(a); (b) Figure 4.19(b); (c) Figure 4.19(c) x

11 Figure 4.21: The best 3D profile reconstruction of the twisted wire xi

12 LIST OF TABLES Table 3.1: Average computation times of different reconstruction approaches Table 4.1: Data of the angles and recording distances of different projections Table 4.2: Data of the angles and recording distances of different projections xii

13 LIST OF ABBREVIATIONS 2D 3D CCD CGH CS CT DH DHI DHM DHT DOF HI IIR IPIR MAHT ROR two-dimensional three-dimensional charged coupled device computer generated holography compressive sensing computed tomography digital holography digital holographic interferometry digital holographic microscopy digital holographic tomography depth of focus holographic interferometry interplane intensity ratio interplane interference rejection ratio multi-angle holographic tomography reference to object ratio xiii

14 SHOT-MT single-beam holographic tomography based multiplicative reconstruction technique TV TwIST total variation two-step iterative shrinkage/thresholding xiv

15 LIST OF SYMBOLS A d E h area recording distance performance difference of the DOF optical field hologram function h prop spatial impulse response for propagation H prop transfer function for propagation I intensity k 0 wavenumber or propagation constant λ wavelength μ sys system error N x, N y φ Δx, Δy z pixel number along x and y direction, respectively phase pixel size reconstruction distance z R Rayleigh range xv

16 CHAPTER 1 INTRODUCTION Holography was invented by Gabor in 1948 to record and reconstruct the amplitude and the phase of a wave field. He named it holography because the process can record and reconstruct the whole optical field [1]. Although holograms can be recorded by synthesizing a spatially coherent source, it was truly developed after a truly coherent light source, such as a laser, was used to provide high quality interference contrast. Traditionally, holograms have been recorded on photographic plates or films. However, it is also possible to perform real-time holography using photorefractive and other nonlinear optics materials. A hologram records the interference pattern between a scattered wave from an object and a coherent reference wave, and it contains the information of the entire three-dimensional (3D) field of the object, although the hologram is just a two-dimensional (2D) film. When the hologram is illuminated with the same reference wave as that used for recording, the original object wave can be reconstructed. In this way, holography enables the 3D features of an object to be displayed. In Gabor's original setup, the object wave and the reference wave share the same optical path, which is the basis for the so called in-line hologram, and the object is mostly transparent or has 1

17 a dimension which is small compared to the reference beam width. In this case, the reconstruction is expected to have a real image superimposed by a twin-image and the dc term or the zero-order term on the optical axis, and these extraneous components introduce error in the reconstruction. To eliminate this, Leith and Upatnieks created off-axis holography where the reference wave and the object wave nominally travel in different directions when they fall on the recording medium. In this way, during reconstruction, the zero-order and the two twin-images can be separated, and the quality of the reconstruction is highly improved [2]. Instead of optically generating the hologram, which is an interference pattern, holograms can be generated on a computer and loaded onto a spatial light modulator. Therefore, in computer generated holography (CGH), the hologram is generated numerically using a computer, and optically reconstructed. CGH has found various applications, e.g., in the testing of aspheric surfaces, 3D display, etc. [3]. Alternatively, holograms can be optically recorded and numerically reconstructed. Goodman and Lawrence [4] initiated numerical hologram reconstruction. Enlarged parts of in-line and Fourier holograms were recorded on a photographic plate and reconstructed numerically. Later, Onural and Scott improved the reconstruction algorithm and applied it for different measurements [5]. Digital holography (DH) was first introduced by Schnars and Jüptner, who directly recorded Fresnel holograms with charged coupled devices (CCDs) [6]. This was a great development in holography because it enables fully digital recording of the holograms without any photographic recording as an intermediate step. It also makes the holography easier and faster, so that 2

18 near-real-time holographic imaging is possible. DH has been applied in many applications such as digital holographic interferometry (DHI) [7], digital holographic microscopy (DHM) [8], digital holographic tomography (DHT) [9,10], etc. Holographic interferometry (HI) is one of the most important applications of holography. It was developed by Powell et al. [11]. With this technique, it is possible to map the displacements of rough surfaces with an accuracy of a micrometer [12]. The advantages of HI are its high accuracy, and also that it is non-contact and non-destructive. Then, with DH, the displacements, which are the optical path length variations, can be related to the actual surface profile with respect to the reference surface. Phase unwrapping plays an important role in DH and DHI. This is because the calculated phase is always limited between 0 and 2π. The actual phase must be found through phase unwrapping since phase gives the information of the depth of the object. Many phase unwrapping algorithms have been created to obtain better performance and minimize error, such as an algorithm based on network programming [13] and phase unwrapping via graph cuts [14], etc. Various numerical algorithms are used for DH reconstruction. Conventionally, the Fresnel approach [15] or the convolution approach [16] is used to reconstruct the optical field at the object plane. The non-paraxial transfer function approach for back-propagation (which is similar to the angular spectrum approach) can also be used for reconstruction. Recently, a technique based on compressive sensing (CS) has received a lot of interest in DH reconstruction [17,18]. The Fresnel approach is the most commonly used in DH reconstruction since it is simple, 3

19 fast and accurate. The convolution approach can introduce an image magnification in the reconstruction, and the pixel size in the reconstruction is independent on the reconstruction distance [16]. The transfer function for propagation also performs well for DH reconstruction, and is discussed in more details in the thesis. CS has been a rapidly growing signal acquisition technique ever since it was introduced in 2004 [19]. It is a method that can reconstruct the data from a substantially smaller number of measurements than those required by the Shannon-Nyquist sampling theorem. Therefore, it is possible to gather more information with the existing imaging system and create more efficient imaging configurations with CS technique. Brady et al. is the first to introduce CS into holography [20]. Since then, CS has been widely applied in DH, and many applications have been demonstrated in recent years, like compressive Fresnel holography [21], off-axis holography of diffuse objects [18], holographic microscopy in low illumination conditions [22], video rate compressive holographic microscopic tomography [23], etc. While holography is said to record and reconstruct the 3D image of an object, it is, however, restricted to reconstruction of the part of the object which emanates scattered waves when illuminated by a coherent source. If the scattered waves from all around the object are analyzed, one can obtain true 3D reconstruction. Tomography is a technique that can yield true 3D information. It has been widely used in biology, medical and other areas for imaging and testing. Tomography can be implemented using different sources along with different imaging systems, such as holographic tomography [10,24], computed tomography (CT) [25], optical coherence 4

20 tomography [26], optical diffusion tomography [27] and optical projection tomography [28], etc. X-ray CT, ultrasound imaging, and some of the optical tomography methods use the set of angular and distance data for varying source positions to calculate the 3D coordinates of the object. The signal is detected one pixel at a time and the 3D image is reconstructed by scanning the three dimensions pixel by pixel. In our multi-angle holographic tomography (MAHT), 3D imaging can be performed using a fixed detector through rotation of either the object or the illumination beam, and the 3D image can be obtained by superimposing each 2D image with different angles at the object position. In this thesis, the theory of different DH reconstruction approaches and the multiplicative tomography technique are introduced in Chapter 2. In Chapter 3, the performance of the Fresnel approach, the non-paraxial and paraxial transfer function approach, and the CS approach used in DH reconstruction is experimentally evaluated, for the first time to the best of our knowledge. Most of this work has been published as an SPIE paper and has been presented in SPIE Photonics West 2014 [29]. Then, based on the evaluation of different DH reconstruction approaches, the one-shot MAHT setup is built, and the 3D profile of a twisted wire is successfully reconstructed. This is presented in Chapter 4. Chapter 5 concludes the thesis, along with suggestions for future work. 5

21 CHAPTER 2 THEORETICAL BACKGROUND 2.1 Introduction In this Chapter, the theory of the digital hologram recording, different holographic reconstruction approaches and MAHT are introduced. Specifically, in Section 2.2, the basic mathematics behind hologram construction is presented along with two of the commonly used recording schemes. Sections 2.3 and 2.4 pertain to commonly used digital holographic reconstruction techniques, based on the Fresnel approach and the transfer function approach, respectively. In Section 2.5, the basics of CS and its application to holographic reconstruction is summarized. Finally, In Section 2.6, the concept of MAHT and its application to true 3D holographic reconstruction is discussed. 2.2 Recording of Digital Holograms The hologram generation is an interference process, and holograms are usually recorded with a laser light source. The wave containing the information of the object is called the object wave, and the other 6

22 interfering wave is the reference wave. For simplicity, the reference wave is sometimes taken to be a uniform plane wave. In general, the object wave at the recording plane can be described as E O (x, y) = a O (x, y)exp (iφ O (x, y)), (2.1) where a O is the amplitude and φ O is the phase of the object wave. The reference wave at the recording plane can similarly be described as E R (x, y) = a R (x, y)exp (iφ R (x, y)), (2.2) where a R is the amplitude and φ R is the phase of the reference wave. Then, the recording intensity I(x, y), which is proportional to the hologram function h(x, y) in DH, can be expressed as h(x, y) I(x, y) = E o (x, y) + E R (x, y) 2 = (E O (x, y) + E R (x, y))(e O (x, y) + E R (x, y)) = E O (x, y) 2 + E R (x, y) 2 + E O (x, y)e R (x, y) + E R (x, y)e O (x, y), (2.3) which contains the dc term (the first two terms), as well as information of the virtual image (the third term) and the real image (the fourth term), respectively. Usually, there are various experimental arrangements to generate a digital hologram. Two of these will be discussed here, viz., the Gabor setup and the Mach-Zehnder setup. The Gabor setup is an in-line setup, and is shown in Figure 2.1. When a collimated beam is incident on the object, the optical field behind the object depends on the nature of the object. The interacting fields forming a hologram on the recording plane comprise the diffracted field from the object and the part of the reference bypassing the object. For opaque objects, the Gabor setup registers the hologram of the edge of the object. In some cases, if the object is transmissive and also a weak phase object of the form e iφ, φ 1, the optical field immediately 7

23 behind the object can be approximately expressed as 1 + iφ, where the first term can be considered the reference and the second the object. Figure 2.1: Diagram of a Gabor setup. The Mach-Zehnder setup can be either in-line or off-axis, and is shown in Figure 2.2. The original collimated beam passes through a beam-splitter, so that the object beam and the reference beam can be separated. The object is usually a reflective object, and the object beam contains the information of the surface of the object. The depth or height information of that surface, which modulates the phase of the reflected light, is encoded in the interference pattern or hologram. Reconstruction of the hologram, along with proper phase unwrapping, can yield the exact surface profile of the object. Therefore, the Mach-Zehnder setup is a typical setup for DHI. In Figure 2.2, the setup is in-line if the object beam and the reference beam travel nominally in the same direction to the recording or CCD plane. If the beam splitter before the CCD is rotated by an angle θ, the reference beam will be reflected to another direction, and an angle 2θ will be applied between the reference beam and the object beam, which is an off-axis setup. It is worthwhile to mention that in some cases, the Mach-Zehnder setup suffers from the 8

24 disadvantage that the object is titled with respect to the incident light, and therefore, the obliquity factor has to be taken into account after reconstruction. Furthermore, shadow effects sometimes show up due to the fact that the light is not normal to the surface. To alleviate these effects, a Michelson setup is sometimes used; however, it has the disadvantage of having extra reflections for the light before interference on the CCD plane. Figure 2.2: Diagram of a Mach-Zehnder setup. 2.3 Fresnel Approach in DH In this Section, the most commonly used reconstruction approach for DH, viz., the Fresnel reconstruction approach, is summarized. Figure 2.3 shows the commonly used coordinate system for the object plane, the hologram plane and the image plane during reconstruction. 9

25 y' y y'' x' x x'' Object plane (virtual image) d Hologram plane d Image plane (real image) Figure 2.3: Diagram showing the relation between the object plane, the hologram plane and the image plane. Generally, the reconstruction of a hologram can be achieved by the Fresnel-Kirchhoff integral [30]: where + + E (x, y ) = i λ h(x, y)e R (x, y) e i 2π λ ρ ρ dxdy, (2.4) ρ = (x x ) 2 + (y y ) 2 + d 2 (2.5) represents the distance between a point in the hologram plane and a point in the reconstruction plane (here, for instance, the object or virtual image plane), λ is the wavelength, and d represents the recording distance, which equals the reconstruction distance. E R (x, y) denotes the reconstruction wave, which, for a plane wave, can be modeled simply as a real constant. Equation (2.5) can be expanded to a Taylor series: ρ = d + (x x ) 2 2d + (y y ) 2 2d 1 8 [(x x ) 2 +(y y ) 2 ] 2 d 3 +, (2.6) where the fourth term can be neglected if it is small compared to the wavelength. Therefore, if the recording distance d can satisfy the following condition: or 1 [(x x ) 2 +(y y ) 2 ] 2 8 d 3 λ, (2.7) 10

26 3 d 1 8 [(x x ) 2 +(y y ) 2 ] 2 λ, (2.8) the distance ρ can be expressed as ρ = d + (x x ) 2 2d + (y y ) 2. (2.9) 2d The condition expressed in Eq. (2.7) or Eq. (2.8) is called the Fresnel approximation. By putting Eq. (2.9) into Eq. (2.4), + E (x, y ) = i λd e i2π λ d e i π λd (x 2 +y 2 ) [h(x, y)er (x, y)e i π λd (x2 +y 2) ] e i2π + λd (xx +yy ) dxdy, (2.10) which is mathematically similar to the Fourier transform of the integrand in Eq. (2.10) within square brackets. In this way, the optical field at the object plane can be reconstructed. The intensity of the reconstructed object field is I(x, y ) = E (x, y ) 2, (2.11) and the phase is φ(x, y ) = arctan Im[E (x,y )] Re[E (x,y )]. (2.12) In DH, the recording is done on a CCD camera which has a certain pixel size. In digital reconstruction, if Δx and Δy represent the pixel size in the hologram plane, the pixel size in the reconstruction plane Δx and Δy are calculated as Δx = λd and N x Δx Δy = λd, (2.13) N y Δy where N x is the number of the pixels in the x-direction and N y is the number of the pixels in the y-direction. This introduces a pixel magnification factor in the Fresnel reconstruction. If only the intensity map is desired, this effect can be eliminated by resizing the result. However, if the phase is also required, this effect has to be carefully taken into account. 11

27 The discretized version of Eq. (2.10) is E (m, n) = i 2π λd e i N y 1 l=0 λ d e iπλd( m2 N 2 x Δx 2+ n2 N 2 y Δy 2) N [h(k, l)e R (k, l)e i π λd ( x2 k 2 + y 2 l 2) ]e i2π(km x 1 k=0, (2.14) Nx + ln Ny ) where m and n are the coordinates of the pixels in the reconstruction plane, and k and l are the coordinates of the pixels in the hologram plane. The sum part of this expression is the discrete Fourier transform of the components within square brackets, which can be easily operated using computer, and the factor in front of the sum only affects the overall phase and can be neglected if only amplitude retrieval is desired. 2.4 Non-paraxial and Paraxial Transfer Function Approach in DH An alternative to DH reconstruction using the Fresnel approach is the transfer function approach, which can be used to back-propagate the hologram to reconstruct the object field [31]. With a uniform plane wave as a reference, the non-paraxial transfer function approach (also called the angular spectrum approach) can be used to reconstruct the object through the relation E (x, y ) = F 1 {H(k x, k y )e i k 0 2 k x 2 k y 2 d }, (2.15) where H(k x, k y ) is the Fourier transform of the hologram function h(x, y), k 0 = 2π λ, and k x and k y denote spatial frequencies in x and y directions, respectively. Also in Eq. (2.15), d is the back-propagation (or reconstruction) distance, and F 1 denotes the inverse Fourier transform operator. The exponential in Eq. (2.15) denotes the non-paraxial transfer function for (back)-propagation, and Eq. (2.15) is also called the non-paraxial back-propagation approach. 12

28 If the recorded hologram does not contain spatial frequencies large enough such that the square root in the exponential can be approximated using the binomial expansion, the paraxial transfer function approach can also be used, and Eq. (2.15) simplifies to E (x, y ) = F 1 {H(k x, k y )e i(k x 2 +k y 2 )d/2k 0 }. (2.16) It is to be pointed out that in Fresnel reconstruction, a Fourier transform is taken only as a means to numerically perform the integration in Eq. (2.10), which has been expressed in discretized form in Eq. (2.14). In the transfer function approach, because a forward and inverse Fourier transform operation is applied to the hologram, there is not any effect of the pixel magnification factor as mentioned above. Also, since there is no approximation involved in the non-paraxial transfer function approach, it should provide the most accurate reconstruction. 2.5 CS Approach in DH A different approach for DH reconstruction is by using the concept of CS. Traditionally, CS was developed to ensure an accurate reconstruction for multiplex encoders under a sufficient condition called the restricted isometry property (RIP) [18]. The sparsity and the l 1 norm, which is defined as c 1 = i c i, are important for CS. An S-sparse signal can be defined as a signal with only S non-zero components. A matrix P R M N is considered as satisfying S-RIP with constant δ s (0,1) for any S-sparse f if (1 δ s ) f 2 2 P f 2 2 (1 + δ s ) f 2 2, (2.17) where T represents the set of indices where the S-sparse signal is supported, and 2 represents the Euclidean norm, which is defined as c 2 = ( c i 2 i ) 1/2. This condition 13

29 implies that the corresponding sub-matrix P of P composed of S columns of P has to form a nearly isometry transformation for a good reconstruction. If o represents an object with an N-dimensional real valued vector being imaged or transformed to h, which is an M-dimensional vector, the sensing or transformation process can be written as h = Φo, (2.18) where h is the measurement, and Φ is an M N matrix, known as the measurement matrix. Note that M has to be smaller than N for h to be an under-sampled version of o. With a basis Ψ, where the signal can be sparsely represented, b is a S-sparse representation of o and related to h as h = Φo = ΦΨb, with o = Ψb. (2.19) CS requires incoherence between the sensing and sparsifying operators, where the mutual coherence μ is bounded by 1 μ N [32]. The mutual coherence is defined as μ = Nmax φ i, ψ j, (2.20) ij where N is the length of the column vector and φ i, ψ j represent the column vectors of Φ and Ψ respectively. Then, we can accomplish the reconstruction by solving the optimization problem: b = argmin b 1 b such that h = Φo = ΦΨb. (2.21) The hologram, which is an interference pattern of the scattered optical field of the 3D object and the reference wave, can be expressed as Eq. (2.3). Under uniform plane wave illumination, E R (x, y) can be simply replaced by unity, so that the actual recorded hologram for a Gabor setup 14

30 is h(x, y) I(x, y) = E O (x, y) + E O (x, y) + = 2Re{E O (x, y)} + e(x, y), (2.22) where Re{ } denotes the real term operator, E O (x, y) is the scattered field of the object at the CCD plane and e(x, y) represents the error, which includes the remaining dc terms in Eq. (2.3). In optics, the scattered field E O (x, y) can be expressed in terms of the convolution integral: E O (x, y, z) = o(x, y, z )h prop (x x, y y, z z )dx dy dz, (2.23) where o(x, y, z ) represents the object field and h prop (x, y, z ) is the spatial impulse response which can be expressed as h prop (x, y, z ) ik 0 e ik 0 x 2 +y 2 +z 2. (2.24) 2π x 2 +y 2 +z 2 Equation (2.22)-(2.24) represent the relation between the 3D object scattering potential and the measured 2D data, and a linear transformation of the holographic measurement can be expressed as h = 2Re {F 1 {H prop F{o}}} + e = 2Re{Φo} + e, (2.25) where h is the 2D measurement at the CCD plane, F 1 denotes the inverse discrete 2D Fourier transform operator, F denotes the forward discrete 2D Fourier transform operator, H prop is the Fourier transform of the spatial impulse response h prop, o is the sampled signal, and Φ is the measurement matrix [9]. The linear transformation model, Eq. (2.25), can be inverted by decompressive inference either by enforcing a sparsity constraint on the total variation (TV), or by selecting a basis, typically a particular wavelet basis where o may be assumed to be sparse, and they can be achieved by minimizing an objective function [33]. 15

31 By selecting a basis Ψ where o may be assumed to be sparse. o can be estimated as o = argmin [1 2 h 2Re{Φo} l ωγ(o)] o = argmin [1 h 2Re{Φo} 2 l ω Ψo l1 ], (2.26) o where Γ(o) is a regularizer function and ω [0, + ) is the regularization parameter. Additional information is introduced to solve an ill-posed problem to prevent overfitting. Minimizing Eq. (2.26) is a compromise between the lake of fitness of a candidate estimate o to the observed data h, which is measured by h Φo 2 and the degree of undesirability Ψo l1. The parameter ω controls the relative weight of these two terms. By enforcing a sparsity constraint on the TV domain as defined by Rudin et al. [34], we need to find an o which minimizes the TV, and o can be estimated as where o V is defined as o = argmin [1 2 h 2Re{Φo} l ωγ(o)] o = argmin [1 h 2Re{Φo} 2 l ω o V], (2.27) o o V = k i j (o k,i+1,j o k,i,j ) 2 + (o k,i,j+1 o k,i,j ) 2, (2.28) where o k represents a 2D plane of the 3D object datacube, and i, j are the coordinates representing the sampling points. The two-step iterative shrinkage/thresholding (TwIST) algorithm, firstly introduced by Bioucas-Dias and Figueiredo [35], is often applied to solve the above optimization problem, and MATLAB code is available from the site The number of iterations is important for the quality of the reconstruction, and it is set as 10 through our 16

32 reconstruction simulations. 2.6 Multi-angle Holographic Tomography The reconstruction techniques described above yield the information about the object, but only the part that has been illuminated with laser light. Often, when the object is opaque, the reconstructed image is only pseudo-3d, since it only yields information about the edges. Tomography helps generate a true 3D image by illuminating the object from different angles. In the single-beam holographic tomography based multiplicative reconstruction technique (SHOT-MT), the 3D profile of the object can be found by combining the 2D reconstructions at different angles [10]. The angles can be chosen as θ k = 0, 30, 45, 60, 90, 135, where k = 1,2,3,4,5,6 representing the order. The 3D reconstruction can be more accurate with more angles, but usually 5 angles are enough to obtain a fairly accurate result. The data of different angles can be achieved by rotating either the object or the reference beam. Each 2D reconstruction process of specific angle is an in-line holographic reconstruction. The reconstruction distance z for each angle is obtained by reconstructing around the ideal object plane and finding the one with the clearest edge. Therefore, the Gabor setup can be applied in MAHT, and a typical setup which records the data of different angles by rotating the object is shown in Figure 2.4. However, rotating the object will introduce a time delay between different projections, which can introduce error in the 3D reconstruction. Therefore, we use the one-shot technique to capture all the information of the object at one time by rotating the reference beam. Note that the rotational axis of the object has to be numerically moved to the center in each 2D 17

33 reconstruction before 3D multiplication reconstruction. Figure 2.4: A MAHT setup by rotating the object. Some of the technical details for SHOT-MT are outlined below. First, the 2D reconstruction needs to be converted into a binary image by setting a threshold to only represent the 2D profile at different angles. The threshold is usually half of the maximum value in the reconstruction. For each 2D binary reconstruction, it is lofted into a 3D volume and rotated by the corresponding angle, and the 3D volume of the object can be reconstructed by multiplying the multiple lofted 2D reconstructions with different angles: Q V 3D = k=1 V k, (2.29) where Q represents the number of all recorded angles, k is the order of projections at different angles, and V k is the related lofted 3D volumes. Figure 2.5 shows the 3D reconstruction process of a ball with 4 angles, 0º, 45º, 90º, and 135º, respectively. Figure 2.5(a) shows all lofted 3D volumes (which are, really, cylinders) at different angles, and Figure 2.5(b) is the product of all lofted 3D volumes, which is the overlap of all projections in Figure 2.5(a), and it offers us a 3D reconstruction of the ball. Note that since such a process requires a lot of memory in the computer, the 2D reconstruction may be sub-sampled (in our case, a factor of

34 has been used) before lofting to save memory and computation time. (a) Figure 2.5: (a) All lofted 3D volumes at different angles; (b) 3D reconstruction result. (b) It is clear that multi-angle tomography can reveal additional axial details without ambiguity, so as to form a more accurate 3D shape of the object. However, because this reconstruction depends on the 2D projection profile, the inside structure which is blocked in any projection applied in the measurement cannot be detected or reconstructed. 2.7 Conclusion In this Chapter, holographic construction and reconstruction approaches have been discussed. The commonly used reconstruction approaches are the Fresnel approach, the transfer function approach, and lastly, the CS approach. Fundamentals of tomography using multiplicative technique have been discussed, and it is shown that MAHT can yield true 3D reconstruction of the object. In Chapter 3, different DH reconstruction techniques are applied, evaluated, and compared experimentally. 19

35 CHAPTER 3 EXPERIMENTAL EVALUATION OF DIFFERENT DIGITAL HOLOGRAPHIC RECONSTRUCTION APPROACHES 3.1 Introduction In the last Chapter, different holographic construction and reconstruction methods have been discussed. Holographic construction methods include Gabor type setup and Mach-Zehnder configuration, etc. Reconstruction approaches include the Fresnel approach, paraxial and non-paraxial transfer function approach, and CS approach. In this Chapter, the performance of the CS approach using TwIST algorithm is compared with the Fresnel approach and the non-paraxial transfer function approach, based on experimental results of hologram reconstruction. The holograms are recorded in an in-line Gabor configuration and the object is a dandelion seed parachute with few wings. In the process, it is also shown that the non-paraxial and the paraxial transfer function approaches yield identical results over the range of recording distances used in our experiments. The main results of this work have been published as an SPIE paper [29]. 20

36 3.2 Experimental Setup In our experiment, a simple Gabor setup is used, as shown in Figure 3.1. Light from a He-Ne laser at nm passes through a microscope objective, pinhole, and collimating lens to generate a collimated uniform plane wave, and then illuminate the object, which is placed between the CCD camera and the lens. The distance between the object and the CCD plane is the recording distance d, and the reconstruction distance should be the same as the recording distance to achieve best reconstruction, which will be discussed in the following section. The object, as mentioned above, is a dandelion seed parachute with few wings. This object is very sparse, so that the CS approach can be applied. By changing d, different holograms with different recording distances are recorded, and the data is downloaded to a computer to be processed. Figure 3.1: Gabor setup for DH. The object is a dandelion seed parachute with few wings. 21

37 3.3 Experimental Results A typical recorded hologram is shown in Figure 3.2(a). It is clear that there are unwanted interference rings in the recorded hologram, possibly originating from multiple reflections within the glass cover of the CCD, and there is no way to get rid of these interference rings by modifying the setup. In order to eliminate the extraneous effect of these rings, a reference without the object is recorded, as shown in Figure 3.2(b), which also contains the noise information from the CCD itself. By subtracting these two images, a clean hologram is obtained, that only contains the information of the object, as shown in Figure 3.2(c). The dc term in the hologram is also eliminated in this way, which is important for any DH reconstruction. Because the laser is not perfectly stable, which involves speckle noise, thermal noise, vibration and other noises, these unwanted interference rings keep changing slowly. Therefore, to perfectly eliminate these rings, the reference has to be recorded immediately after the hologram recording. (a) (b) (c) Figure 3.2: (a) Recorded hologram of the object; (b) recorded reference without the object; (c) the hologram used for reconstruction after subtraction of (b) from (a). The dandelion we use as the object has wing size w of approximately 30 µm (corresponding to a Rayleigh range of approximately z R = πw2 ~5.2 mm). Our minimum recording distance is λ 7 mm. Figure 3.3(a) is the reconstruction using the paraxial transfer function approach with 22

38 recording distance being 39 mm and Figure 3.3(b) is the reconstruction using the non-paraxial transfer function approach for the same hologram. The paraxial transfer function approach and the non-paraxial transfer function approach have nearly identical performance over the range of recording distances involved. The non-paraxial transfer function approach has been used (unless stated otherwise) for back-propagation reconstruction, since it is mathematically more accurate and its computation time is nearly identical to the paraxial transfer function approach. And it is also called the non-paraxial back-propagation approach in the thesis. (a) (b) Figure 3.3: (a) Reconstruction using paraxial transfer function approach at 39 mm; (b) reconstruction using non-paraxial transfer function approach at 39 mm Reconstruction results A well-focused intensity reconstruction shows a clear object with sharp edges and a clean background. The reconstruction is supposed to be the best if the reconstruction distance matches the recording distance perfectly, and this is called the in-focus situation. However, the measured recording distance is generally not accurate enough to use as the in-focus reconstruction distance, since the optical path cannot generally be measured so precisely. 23

39 Therefore, reconstruction has to be performed for several values around the measured recording distance to find the best one, which is the in-focus reconstruction. Figures 3.4(a-c) show the reconstruction using the non-paraxial back-propagation approach at 19.3 mm, 100 mm, and mm, respectively. The object (the wing of the dandelion) is quite clear in all reconstructions. However, there are many fringes remaining beside the object, which are due to the out-of-focus twin-image resulting from the in-line recording setup. A careful analysis of Figures 3.4(a-c) shows that the fringes have increasingly larger periods and decrease in intensity with increasing recording distance. Hence, it is reasonable to conclude that non-paraxial back-propagation offers us a good reconstruction with a sharp object image, although the fringes resulting from the out-of-focus twin-image are present. It is later shown that this twin-image effect is also present in every reconstruction approaches used, although it is the most prominent when using the non-paraxial back-propagation approach. (a) (b) (c) Figure 3.4: Reconstruction using the non-paraxial back-propagation approach at (a) 19.3 mm, (b) 100 mm, and (c) mm. Figure 3.5(a) shows a hologram at a recording distance around d = 6.6 mm, which is approximately 1.27z R. Figure 3.5(b) is the reconstruction using the Fresnel approach. For the Fresnel approximation, given by Eq. (2.8), to hold d > 100 mm, the Fresnel approach does not 24

40 yield accurate results, as expected. Figure 3.5(c) is the reconstruction using the non-paraxial back-propagation approach, and Figure 3.5(d) is the reconstruction using the CS approach with TwIST algorithm. Both these two approaches offer faithful reconstructions. However, twin-image effect is observed in both cases. (a) (b) (c) (d) Figure 3.5: (a) Original hologram when recording distance d is around 6.6 mm; (b) reconstruction using the Fresnel approach; (c) reconstruction using the non-paraxial back-propagation approach; (d) reconstruction using TwIST. 25

41 (a) (b) (c) (d) Figure 3.6: (a) Recorded hologram at d=251.2 mm; (b) reconstruction using the Fresnel approach; (c) reconstruction using the non-paraxial back-propagation approach; (d) reconstruction using TwIST. Figure 3.6(a) shows a hologram recorded at around 250 mm. Figure 3.6(b) is the reconstruction using the Fresnel approach. Now the recording (and reconstruction) distance is d~48z R, where the Fresnel approximation is valid (see the range of d above), and thus the Fresnel approach yields an accurate reconstruction. Figure 3.6(c) is the reconstruction using the non-paraxial back-propagation approach, and Figure 3.6(d) is the reconstruction using the CS approach. The twin-image effect exists in all approaches, but is the strongest in the non-paraxial back-propagation approach. Note that the mounting bracket holding the dandelion is not in focus at this distance for any reconstruction approach, since it is an extended object spanning a 26

42 relatively long z dimension compared to the dandelion. Also, the dandelion wings are generally not perpendicular to the base, so that all parts of the wing are not in-focus simultaneously; this is why the reconstructed dandelion wings are not of uniform intensity. A note on computation times: Table 3.1 shows the average times required to compute a reconstruction using the Fresnel approach, the non-paraxial back-propagation approach, and the CS approach using TwIST. It is clear that the CS TwIST approach requires about 30 times the computation time than that required for the Fresnel approach and the non-paraxial back-propagation approach. Table 3.1: Average computation times of different reconstruction approaches. Approach Fresnel approach Non-paraxial CS TwIST approach back-propagation Average time taken for each reconstruction (s) Reference to object ratio The reference to object ratio (ROR) is defined as ROR = R, (3.1) O where R has the connotation of the mean intensity of an area in the reference and O represents the mean intensity of the same region in the recorded hologram. Rivenson et al. have shown that a higher ROR yields better reconstruction quality for CS [36]. However, in the Gabor setup, the ROR is fixed for a given object since the object wave and the reference wave share the same path, and dense and opaque objects are expected to give a considerably lower ROR. In our experiments, a higher ROR is not especially ensured to get a better performance for the CS 27

43 approach Depth of focus Except for the exact in-focus reconstruction, if the reconstruction distance differs from the ideal recording distance by some value, one may still obtain a good reconstruction. We define the depth of focus (DOF) as the variation of the reconstruction distance that still offers a relatively good reconstruction (as shown in Figure 3.7). z reconstruction planes z CCD d Figure 3.7: Diagram showing the concept for the depth of focus. The holograms are reconstructed at different reconstruction distances z. d is the recording distance. z, the DOF for recording distance d, is the variation of z where the interplane intensity ratio (defined below) is between 0.5 and 1. Each hologram is now reconstructed at different reconstruction distances z, which is centered at the recording distance d as shown in Figure 3.7. In order to quantitatively assess the DOF, Rivenson et al. have instead used the interplane interference rejection ratio (IPIR) to evaluate the performance at different reconstruction distances, which is judged by the fundamental frequency amplitude of the grating object used [36]. Our object is a dandelion with few wings, where the wings are the parts of interest, and it does not have a periodic structure, which means that it is hard to identify the object in the frequency domain. Therefore, a different figure of merit has been employed, based on the intensity of the reconstructed image. In the reconstructed image 28

44 intensity matrix, let r(m,n1,n2,z) represent the vector representing a horizontal cross section of the reconstruction in row m, column n1 to n2 at a reconstruction distance z; and let n(m,n1,n2,z) represent the vector containing all the components of r(m,n1,n2,z) except for the components around the maximum value of r(m,n1,n2,z) which represent the object, so that the mean value of n(m,n1,n2,z) can represent the background of r(m,n1,n2,z). I(z) are the maximum values of r(m,n1,n2,z) at different reconstruction distances z, and they are adjusted to the same mean background for all the values of z that are used. I max is the maximum value of I(z) for a fixed recording distance d. The interplane intensity ratio (IIR) is defined as IIR(z) = I(z) I max. (3.2) The DOF z is the variation of reconstruction distance z where the IIR is between 0.5 and 1. In our experiment, since the dandelion is inclined, one row (row 400 in the reconstructed intensity matrix) is evaluated for each reconstruction. Figure 3.8(a) shows a slightly out-of-focus reconstruction where IIR is around 0.5 using the hologram from Figure 3.6(a). Figure 3.8(b) is the cropped area of Figure 3.8(a), detailing the out-of-focus reconstruction. Figure 3.8(c) is the identical cropped area of Figure 3.6(d), which details the in-focus situation where IIR equals 1. In both Figures 3.8(b) and Figure 3.8(c), row 400 is indicated. The difference is clear, and it is reasonable to propose that an IIR of 0.5 can give an adequate estimate of the DOF. 29

45 (a) (b) (c) Figure 3.8: (a) Reconstruction at mm using hologram recorded at mm; (b) detailed image around row 400 in the CS reconstruction at mm; (c) detailed image around row 400 in the CS reconstruction at mm. Note that when using the Fresnel approach, there is a pixel magnification factor that must be applied after reconstruction. To ensure that the reconstruction using each approach is in the same image size, the Fresnel reconstruction is resized to equal the pixel size of the other reconstruction approaches. However, this resizing process is not perfect and the pixels in one row may jump to the next row over some reconstruction distances (i.e. sampling error). As a result, the I(z) plot of one particular row may be saw-shape instead of a continuous line. This problem can be solved by examining the I(z) plots of some adjacent rows centered at the desired row. When the pixel row of interest jumps to an adjacent row, it can be captured in the other I(z) plot of the adjacent row and attached to the saw-shape position. By tracing the row of interest, a continuous I(z) plot can be rebuilt and used to measure the DOF at that particular recording distance for that reconstruction approach. In the CS TwIST approach and the non-paraxial back-propagation approach, the DOF of adjacent rows at the same recording distance may not be the same. Therefore, the DOF for 5 adjacent rows centered at the desired one is measured and the average is taken to judge the mean DOF at that particular recording distance. 30

46 In order to express a more general result, the recording distance in our experiment is converted and expressed in terms of the object's Rayleigh range. The DOF results for several recording distances using different reconstruction approaches are shown in Figure TwIST Fresnel Non-paraxial back-propagation DOF (DOF/z R ) Recording distance (d/z R ) Figure 3.9: Plot of the normalized DOF vs the normalized recording distance. The performance difference of the DOF is defined as = DOF x DOF CS DOF CS, (3.3) where DOF CS represents the DOF of the CS TwIST approach at some recording distance, and DOF x represents the DOF of the Fresnel approach or the non-paraxial back-propagation approach at the same recording distance. A negative (positive) value for indicates a superior (inferior) performance as compared to CS TwIST approach, since DOF x < DOF CS, indicating better rejection of out-of-focus components during reconstruction. The performance difference of the DOF between the non-paraxial back-propagation and the CS TwIST approach is shown in Figure 3.10(a), and the performance difference of the DOF between the Fresnel approach and the 31

47 CS TwIST approach is shown in Figure 3.10(b). Note that the DOF of all approaches increases with recording distance increasing. For the CS TwIST approach and the Fresnel approach, when the recording distance starts to satisfy the Fresnel approximation, the CS TwIST approach has a much better DOF performance (i.e. narrower DOF) than the Fresnel approach, where is initially around 0.4. However, between the CS TwIST approach and the Fresnel approach decreases when the recording distance gets larger, implying that they have quite similar DOF performance in the far-field. For the CS TwIST approach and the non-paraxial back-propagation approach, they have similar DOF performance in the near-field, and the CS TwIST approach is slightly better. However, the DOF performance of the non-paraxial back-propagation is consistently better than both the CS approach and the Fresnel approach as recording distance increasing, which agrees with our expectation in that there is no approximation involved in the non-paraxial back-propagation approach DOF difference DOF difference Recording distance (d/z R ) Recording distance (d/z R ) (a) (b) Figure 3.10: (a) The plot of the non-paraxial back-propagation approach and the CS TwIST approach; (b) the plot of the Fresnel approach and the CS TwIST approach. 32

48 3.4 Conclusion The CS approach (TwIST algorithm) works well in digital holography but it is quite time consuming. Unlike the Fresnel approach, the CS TwIST approach and the non-paraxial back-propagation approach both work well in the near-field region where the Fresnel approximation is not valid. The CS approach has better DOF performance than the Fresnel approach for small recording distances, although they have similar performance in the far-field. While the DOF performance of the CS approach is similar to that of the non-paraxial back-propagation approach in the near-field, the non-paraxial back-propagation approach has better (i.e. narrower) DOF performance than both of the other two approaches in the far-field. Since an in-line setup has been used, the twin-image effect is dominant, producing unwanted side fringes in the reconstruction for all approaches. The non-paraxial back-propagation approach has the strongest twin-image effect, while the CS TwIST approach can reduce this effect to a certain extent. In the next Chapter, one-shot MAHT is used to determine the true 3D profile of an object, using Fresnel reconstruction. Fresnel reconstruction is used since it has less effect of twin-images as compared to the transfer function approach, and it is much faster than CS approach with similar performance in the far-field. 33

49 CHAPTER 4 ONE-SHOT MULTI-ANGLE HOLOGRAPHIC TOMOGRAPHY 4.1 Introduction In the last Chapter, different digital reconstruction approaches for holograms have been compared. It has been found that while the non-paraxial transfer function approach is the most accurate, it also has the most contamination from twin-images; also, CS approach is computationally time-intensive. In this Chapter, a one-shot MAHT setup is developed, and the 3D profile of a twisted wire is reconstructed. Experimental results are presented and the conditions that may cause error are discussed. The objective is to determine the best possible recording conditions for one-shot MAHT. From Chapter 3, it is clear that the Fresnel approach is simple, fast and accurate when Fresnel approximation is valid. Since our object is not in the near-filed, the Fresnel approach can always be applied in our case. 4.2 Experimental Setup A conventional MAHT setup can be the same as a Gabor setup, as shown in Figure 2.4, where the 2D profile of different angles can be obtained by rotating the object and capturing the 34

50 holograms separately. In our one-shot MAHT setup, the data of 2D profiles of 3 different angles are obtained by reflecting the reference beam, and all information of the object can be captured at the same time, as shown in Figure 4.1. Mirrors 1 and 2 are used for alignment purposes only. The microscope objective, pinhole and lens are used to generate collimated beam. Mirrors 3-6 are used to reflect the reference beam to obtain 3 projections with 3 angles, which are 0º, 90º, and 225º, respectively. This setup can be theoretically converted into an equivalent unfolded typical in-line setup, as shown in Figure 4.2. Also the objects 1-3 are the same object with different orientations. Figure 4.1: Diagram of the one-shot MAHT setup (top-view). Figure 4.2: Equivalent in-line setup (top-view). Figure 4.3 is a typical hologram recorded at the CCD plane. There are three individual 35

51 holograms located at different parts of the entire CCD plane. By reconstructing the three different holograms, the 2D profiles of the object at different angles of illumination can be found, and a 3D profile can be generated using these 2D data via Eq. (2.29). Figure 4.3: Recorded hologram in one-shot MAHT setup. 4.3 Sources of Errors It is to be pointed out that there are several potential sources of errors in the experiment, and they are discussed below Tilt factor of the reference beam If the reference beam is not perpendicular to the object, and the CCD plane is adjusted to be perpendicular to the reference beam, a tilt factor will be introduced. As shown in Figure 4.4, there is an angle θ between the optical axis and the normal of the object plane. The ideal length to be recorded is AB, while the actual recorded length is AC, and they have the following relation: AC = AB cosθ. (4.1) 36

52 Figure 4.4: Diagram illustrating the tilt factor. In reality, this tilt angle θ is hard to measure. On the other hand, the tilt angle for different parts of the system can be different, which makes the measurements and calculations too complicated. An easy way to prevent this situation is to ensure that the optical axis of the unfolded system is perpendicular to the object plane and the height of the optical axis is fixed, so that the tilt factor will not be involved Hologram overlapping Ideally, for best reconstruction, different holograms should not overlap. However, it is seen that if the objects are sparse, recovery of the objects from overlapping holograms is possible. As an example, holograms of dandelion wings placed at different distances, viz., 255 mm and 307 mm from the recording plane, are shown in Figure 4.5(a). Using reconstruction distances of 255 mm and 307 mm, respectively, it is shown that they can be correctly reconstructed uniquely, as shown in Figures 4.5(b), (c), respectively. 37

53 (a) (b) (c) Figure 4.5: (a) Recorded hologram with two dandelions at 255 mm and 307 mm, respectively; (b) reconstruction at 255 mm; (c) reconstruction at 307 mm. For objects that are not that sparse, like the wire we use in Section 4.4, another composite hologram with some overlapping is shown in Figure 4.6(a). Figure 4.6(b-d) are the reconstructions at different object planes. The reconstructed object in Figure 4.6(d) is not even continuous, which means that this reconstruction is not ideal to use in tomography with multiplicative technique. 38

54 (a) (b) (c) (d) Figure 4.6: (a) Recorded hologram of the wire with some overlapping; (b) reconstruction at 120 mm; (c) reconstruction at 395 mm; (d) reconstruction at 565 mm. The reason for the suboptimal reconstruction in the above case is that some parts of the object wave of the illumination with the largest recording distance are blocked by the object at the other two illumination angles. Since such blocking effect is not negligible, the object wave from this illumination is compromised, and therefore cannot be correctly reconstructed with a plane wave as the reading beam Recording distance If the recording distance is too large, considerable error will occur and this is the most dominant error source. Figure 4.7 is a recorded hologram of a coiled spring with a large 39

55 recording distance for the 0 degree angle. The recording distances and related angles are shown in Table 4.1. Figure 4.7: Recorded hologram of a coiled spring with large recording distance. Table 4.1: Data of the angles and recording distances of different projections. Order Angle (degree) Recording distance(mm) The reconstructions of different angles with different reconstruction distances are shown in Figure 4.8. Since the heading part of the spring is interested, only this part is cropped for further processing. The cropped data is shown in Figure 4.9. Note that the Fresnel reconstruction is inverted with respect to the original hologram; this is merely an artifact of applying the Fourier transform as defined in Eq. (2.14) to implement the Fresnel reconstruction integral. This effect is corrected in our reconstruction by rotating the image by 180 degrees. 40

56 (a) (b) (c) Figure 4.8: (a) Reconstruction at 178 mm; (b) reconstruction at 548 mm; (c) reconstruction at 1311 mm. (a) (b) (c) Figure 4.9: Cropped regions of interest from (a) Figure 4.8(a); (b) Figure 4.8(b); (c) Figure 4.8(c). For the reconstruction of the first angle, where the recording distance is 178 mm, which is around 1 7 of the largest recording distance, the edge of the spring is very clear, and it is quite easy to discern the 2D profile of the spring with our naked eye. However, numerically, we have to use a threshold to convert the 2D reconstruction into a binary image. Although the edge is 41

57 apparently very clear visually, the intensity of some parts of the spring is around the same value of the background, in which case it is impossible to extract the spring profile using a single threshold. In fact, the result of single thresholding is shown in Figure 4.10, which is not suitable for use in tomography reconstruction. Therefore, some image processing is necessary to obtain a correct binary image of the 2D profile of the spring in this case. Figure 4.10: Result of single thresholding of Figure 4.9(a). First, based on the intensity map, a threshold value is chosen to erase the background as much as possible, as shown in Figure 4.11(a). Secondly, another threshold is chosen to eliminate most of the object, leaving the ambiguous parts, as shown in Figure 4.11(b). The morphological opening algorithm (MATLAB code: imopen ) is used to disconnect different parts, as shown in Figure 4.11(c). Thirdly, by comparing Figure 4.11(c) and Figure 4.9(a), the parts representing the background can be distinguished in Figure 4.11(c), and these parts are extracted to make a mask to erase the remaining background in Figure 4.11(a). The holes in the object profile can be filled and a fairly good binary image of the 2D profile of the spring can be obtained, as shown in Figure 4.11(d). Although the image processing cannot perfectly extract the ideal object profile, the error is within several pixels, which is still tolerable. 42

58 (a) (b) (c) (d) Figure 4.11: (a) Result of best single thresholding; (b) the ambiguous parts; (c) disconnected ambiguous parts; (d) final binary image after image processing. For the reconstruction of the second angle, where the recording distance is 548 mm, a single threshold works well to extract the 2D shape of the spring since the contrast of the object is very large, and the result after single thresholding is shown in Figure This is the best situation for our 2D reconstruction to obtain the binary images of the 2D shapes of the object at different angles. 43

59 Figure 4.12: Binary image of the part of interest of the reconstruction at 548 mm. Reconstruction of the third angle, where the recording distance is 1311 mm, is now described. Compared with Figure 4.12, which shows the ideal 2D profile of the spring, it is fair to say that this reconstruction is not applicable since it does not give us the correct 2D profile of the spring. The reason is that the coils of the spring are too close to each other, so that the diffraction of each coil is affected by the diffraction of other coils at such a large recording distance. In addition, this spring is very small, and the spring diffracts considerably with such a large recording distance, where the diffraction pattern is very wide with low intensity. The diffraction limitation is related with the object size, so that the large or small recording distance is highly dependent on the object size. The smaller the object size is, the larger the diffraction effect is, so that the recording distance should be smaller to guarantee the quality of the reconstruction. Figure 4.13(a) is a hologram of a bigger spring recorded at a similar recording distance, which is 1266 mm, as in Figure 4.8(c), and the reconstruction is shown in Figure 4.13(b). It is seen that for the parts where the coils are close, the reconstruction is better than the previous case, but it is still not very good, even though it can be discerned via image processing. For the parts where the coils are not that close, the 2D profile of that parts is 44

60 correctly reconstructed with high contrast, and it can be easily extracted with a single threshold. (a) (b) Figure 4.13: (a) Recorded hologram of a bigger spring with large recording distance; (b) reconstruction for this bigger spring at 1266 mm. A Mach-Zehnder arrangement is set up to enhance the fringe visibility in the hologram, which may help to improve the reconstruction quality for large recording distance, as shown in Figure The extra reference beam has to go through a longer path to compensate the optical path difference with the object beam. The optical path and the angle of the reference beam are carefully adjusted to obtain just one ring at the CCD plane, in the absence of the object, which is the best interference condition that can be achieved. Figure 4.15(a) is the recorded hologram, and Figure 4.15(b) is the reconstruction for the largest recording distance. It is clear that a Mach-Zehnder setup does not help improve the quality of the reconstruction. 45

61 Figure 4.14: Diagram of the Mach-Zehnder setup for one-shot MAHT. (a) (b) Figure 4.15: (a) Recorded hologram using Mach-Zehnder one-shot MAHT setup; (b) reconstruction at 1135 mm. Mathematically, a Mach-Zehnder setup amounts to adding another reference beam to the in-line setup when the object is not a reflective object. Therefore, the intensity map recorded at the CCD plane can be expressed as I(x, y) = E o (x, y) + E R1 (x, y) + E R2 (x, y) 2, (4.2) where E R1 (x, y) and E R2 (x, y) are two reference beams. For plane wave illumination, 46