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Generation of Binary Off-axis Digital Fresnel Hologram with Enhanced Quality
Generation of Binary Off-axis Digital Fresnel Hologram with Enhanced Quality
ICT Express. 2014. Jan, 1(1): 26-29
Copyright © 2014, The Korea Institute of Communications and Information Sciences
This is an Open Access article under the terms of the Creative Commons Attribution (CC-BY-NC) License, which permits unrestricted use, distribution and reproduction in any medium, provided that the original work is properly cited.
  • Received : August 25, 2014
  • Accepted : September 20, 2014
  • Published : January 30, 2014
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About the Authors
Peter Wai Ming Tsang
Department of Electronic Engineering, City University of Hong Kong

Abstract
The emergence of high resolution printer and digital micromirror device (DMD) has enabled real, off-axis holograms to be printed, or projected onto a screen. As most printers and DMD can only reproduce binary dots, the pixels in a hologram have to be truncated to 2 levels. However, direct binarizing a hologram will lead to severe degradation on its reconstructed image. In this paper, a method for generating binary off-axis digital Fresnel hologram is reported. A hologram generated with the proposed method is referred to as the “Enhanced Sampled Binary Hologram” (ESBH). The reconstructed image of the ESBH is superior in visual quality as compare with the one obtained with existing technique, and also resistant to noise contamination.
Keywords
1. Introduction
A hologram is a complex image which, when illuminated with a coherent beam, will reconstruct an observable three-dimensional object scene with full depth and disparity information. Modern research has also shown that a hologram can be computed numerically and stored in a digital file [1] . It will be desirable if the digital hologram can be reproduced on a physical media through a commodity printer [2] , or a digital micromirror device (DMD) [3] . However, it is well known that commodity printers and DMD can only write real value pixels, and generally as binary (i.e. black and white) dots. An effective solution to address this issue, which is less complicated than using multiplexing multiple diffusers [3] , is to convert the complex hologram into a real image, and quantizing the intensity of the each pixel intensity into a binary value. The above process can be briefl y described as follows. To start with, consider an object scene with the brightness of each object point denoted by the intensity image. The perpendicular distance of an object point to the hologram is represented by the depth map. The complex hologram can be generated from the scene according to the Fresnel diffraction equation as [1]
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where
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is the Euclidean distance between an object point at co-ordinate ( x, y ) in the 3D scene, and a pixel at ( u,v ) on the hologram. λ is the wavelength of the optical beam. The hologram and the object scene are assumed to have the same horizontal and vertical extents, comprising of X columns and Y rows of pixels. The complex hologram derived from Eq. (1) can be converted into a real off-axis digital Fresnel hologram H 1 ( u,v ) by multiplying the hologram with an inclined planar wave R ( v ), and dropping the imaginary part, i.e.,
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Subsequently, the hologram H 1 ( u,v ) can be binarized with sign-thresholding, setting 1 (white) and 0 (black) values to pixels of positive and negative polarity, respectively. However, direct binarizing a hologram with sign-thresholding will lead to severe degradation on shaded (i.e. smooth or homogeneous) region of the reconstructed image. Although a hologram can be binarized with iterative means [4] [5] , the computation could be rather intensive. To overcome this problem, Tsang et al [6] had proposed to down-sampled the intensity image with a lattice of gridcross patterns before generating the hologram. The visual quality of the reconstructed image derived from the downsampled image is favorable, but the shaded region is masked with a coarse granular pattern. In this paper, a method to address the above-mentioned problem is proposed. An off-axis hologram generated with the proposed scheme is referred to as the “Enhanced Sampled Binary Hologram” (ESBH). The visual quality of the reconstructed image of a ESBH is superior to the one generated with the existing methods, and also resistant to noise contamination. In the remaining parts of this paper, detail of the proposed method is presented in Section 2. Experimental evaluation is provided in Section 3, and a conclusion will be given at the end of the paper summarizing the essential findings.
2. Proposed method for generation of the Enhanced Sampled Hologram
For the sake of clarity of explanation, the concept for generating binary off-axis Fresnel hologram through downsampling is shown in Fig. 1 , and outlined as follows. To begin with, an object scene is modeled by an intensity image I ( x, y ) and a depth map D ( x, y ) . The intensity image is first down-sampled by a lattice S(x, y) as given by
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Method for generating binary off-axis Fresnel hologram
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The lattice S ( x, y ) is a two-dimensional image with the sample points set to unity, and the non-sample points set to zero. Next, Eq. (1) is applied to convert the down-sampled image Id ( x, y ) and the depth map D ( x, y ) into a complex hologram H ( x, y ) . The latter is multiplied with a reference plane wave R ( v ) , and the real part of the product is quantized to a binary off-axis hologram Hb ( x, y ) through sign-thresholding. In the method reported in [6] , a grid-cross down-sampling lattice with the sampling points positioned along regularly spaced horizontal, vertical, and diagonal lines is employed. The reconstructed image of the grid-cross sampled hologram has favorable visual quality, and capable of preserving the intensity distribution at the shaded areas. However, the intensity at the crossing points of the sampled lines is found to be relatively stronger than the rest of the areas, resulting to a granular and unnatural appearance. To overcome the above mentioned problem, a new downsampling scheme is proposed in this paper. In essence, a uniform grid down-sampling lattice with the sample points removed near the crossing points of the line segments is adopted. Mathematically, the new down-sampling lattice can be expressed as
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In Eq. (4), Sc ( x, y ) is a grid down-sampling lattice comprising of regular spaced horizontal and vertical lines as given by
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where M is the down-sampling factor. The other term N ( x, y ) is a two-dimensional mask where all the entries within a τ ×τ square region centered at each crossing point of the grid down-sampling lattice is set to zero, and the remaining elements are all set to unity. A small section of the grid sampling lattice Sc ( x, y ) is shown in Fig. 2(a) to illustrate the formation of the new down-sampling lattice S ( x, y ). The mask N ( x, y ) is shown in Fig. 2(b) , and the new down-sampling lattice that is derived from Eq. (4) is shown in Fig. 2(c) . It can be seen that S ( x, y ) is comprising of regular spaced horizontal and vertical lines without the crossing points. As will be shown in the later part of this paper, the elimination of the sampling points around the crossing points will reduce the granular effect, resulting in a more natural visual quality on the reconstructed image of the binary hologram.
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The grid sampling lattice Sc (x, y)
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The mask N(x, y)
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The down-sampling lattice S(x, y) formed by the product of Sc(x, y) and N(x, y)
3. Experimental evaluation
A 512×512 source image “Lena” ( Fig. 3 ) is employed to demonstrate the performance of the proposed method, and its comparison with the existing method in [6] . The source image is parallel to the hologram, and separated by an axial distance of 0.3m. To evaluate the method in [6] , a grid-cross lattice having a down-sampling factor of M = 9 is applied to down-sampled the source image. An off-axis binary hologram of size 1024×1024 , and with a square pixel size of 8 μm , is then generated from the down-sampled image. The wavelength of the optical beam, and the angle of incidence of the reference wave, are 633 nm and 1.2˚, respectively. The reconstructed image at the focused plane is shown in Fig. 4 . It can be seen that although the quality of the image is favorable, it is overlaid with a coarse granular texture. Next, the proposed method is evaluated. Eqs. (4) and (5) are applied to generate a down-sampling lattice S ( x, y ) with M = 9 and τ = 1. As in the previous test, the source image is down-sampled with S ( x, y ), and an ESBH is generated. The reconstructed image at the focused plane is shown in Fig. 5(a) . We observed that the texture pattern is finer, and the visual quality is enhanced as compare with the result in Fig. 4 . To test the noise resistant capability of the ESBH, 25% of the pixels are selected randomly, and the value of each selected pixel is complemented. The reconstructed image is shown in Fig. 5(b) , showing that even at such a high noise level, the quality of the image is only mildly affected.
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The source image "Lena"
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Reconstructed image at the focused plane from the binary hologram generated with the method in [6].
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The reconstructed image at the focused plane from the ESBH.
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The reconstructed image at the focused plane from the ESBH that has been added with 25% noise
4. Conclusion
Past research has proven that a binary off-axis Fresnel hologram can be obtained by down-sampling the source image with a grid-cross lattice prior to the generation and binarization of the hologram. However, the reconstructed image is masked with a coarse granular pattern. This paper reports a method for generating the Enhanced Sampled Binary Hologram (ESBH) by adopting a modified grid down-sampling lattice. Experimental evaluation reveals that the granular pattern in the reconstructed image of the ESBH is less prominent, and also highly resistant to noise contamination of the hologram. This feature is particular attractive in a holographic projection or display system, as it implies that large amount of dead pixels could be tolerated in the display device. In addition, part of the ESBH can be utilized to embed external data or pictorial information, a function that can be readily applied in encryption and watermarking.
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