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We propose an effective reconstruction method for correcting the joint misplacement of the sub-holograms caused by the displacement error of CCD in spatial synthetic aperture digital Fresnel holography. For every two adjacent sub-holograms along the motion path of CCD, we reconstruct the corresponding holographic images under different joint distances between the sub-holograms and then find out the accurate joint distance by evaluating the quality of the corresponding synthetic reconstructed images. Then the accurate relative position relationships of the sub-holograms can be confirmed according to all of the identified joint distances, with which the accurate synthetic reconstructed image can be obtained by superposing the reconstruction results of the sub-holograms. The numerical reconstruction results are in agreement with the theoretical analysis. Compared with the traditional reconstruction method, this method could be used to not only correct the joint misplacement of the sub-holograms without the limitation of the actually overlapping circumstances of the adjacent sub-holograms, but also make the joint precision of the sub-holograms reach sub-pixel accuracy.
©2009 Optical Society of America
1. Introduction
In digital holography (DH) [1–3], the recording and reconstruction process of the object wave is completely numerical, which is propitious to the quantitative analysis and digital display of the recorded object. However, restricted by the finite size of CCD (or CMOS), the resolution of the reconstructed image in DH is usually lower than that in traditional optical holography and the reconstructed area is also quite limited. An effective way to solve this problem is increasing the numerical aperture (NA) of the optical system used for recording the digital hologram. Therefore, synthetic aperture digital holography (SADH) is proposed. At present there are two typical approaches for SADH. One of the approaches is the angular synthetic aperture digital holography [4–7], in which multiple object or reference beams with different illumination directions are introduced to record the high frequency components of the object wave in a single digital hologram and thus the resolution of the reconstructed image could be effectively improved. However, it often complicates the recording system and reconstruction algorithm. The other approach is the spatial synthetic aperture digital holography [8–13], in which multiple sub-holograms are recorded by moving the CCD transversely and then composed to a larger synthetic aperture digital hologram for reconstruction. It can improve the resolution and increase the area of the reconstructed image effectively, but the quality of the reconstruction result strongly depends on the joint accuracy of the sub-holograms. The joint misplacement of sub-holograms, which is commonly caused by the displacement tolerance of the apparatus used for controlling the movement of CCD, will result in the quality degradation of the synthetic reconstructed image.
For correcting the joint misplacement of the sub-holograms, a feasible method is that, determine the relative position relationships of sub-holograms by looking for a peak in the cross-correlation function of the overlapping portions of the adjacent sub-holograms and then compose all of the sub-holograms to a large digital hologram by image mosaic according to the calculation result [8]. However in some cases, this method couldn’t correct the joint misplacement of the sub-holograms, because it has to satisfy one condition that the adjacent sub-holograms must have the actually overlapping portions, or else the cross-correlation operation of sub-holograms would be meaningless and thus the relative position relationships of the sub-holograms couldn’t be determined according to the calculation result. In this paper, we propose an effective reconstruction method for spatial synthetic aperture digital Fresnel holography, which could be used to not only correct the joint misplacement of the sub-holograms without the limitation of the actually overlapping circumstances of the adjacent sub-holograms, but also make the joint precision of the sub-holograms reach sub-pixel accuracy, so that the quality degradation of the synthetic reconstructed image caused by the joint misplacement of sub-holograms could be avoided.
2. Principles
Figure 1 shows the recording principle of the spatial synthetic aperture digital Fresnel holography, where the object plane (x 0, y 0) and the recording plane (x, y) are parallel and have an interval of d. Every grid in the recording plane corresponds to a frame of the sub-holograms, which are recorded by CCD with different displacement in x or y directions. The synthetic aperture digital Fresnel hologram is composed according to the relative position relationships of the sub-holograms, so that an object field with higher resolution and larger range can be recorded due to the increasing of the spatial bandwidth product of the hologram.
Assuming the accurate joint distance (actual distance between the centers of the two adjacent sub-holograms) and the size of the sub-holograms are D 0 and L, respectively. Figure 2 shows all of the possible cases of the relative position relationships of the two adjacent sub-holograms in spatial synthetic aperture digital holography. Figures 2(a), (b) and (c) demonstrate the cases of D 0 <L, D 0 = L and D 0 >L, respectively. It could be seen that when D 0 is greater than or equal to L, there is no actually overlapping portions between the sub-holograms, which means that the joint misplacement of the sub-holograms couldn’t be determined and then corrected by using the traditional reconstruction method.
In order to solve the problem above, we must look for another way by which we could find out the accurate joint distance between the adjacent sub-holograms without the limitation of the actually overlapping circumstances of the sub-holograms. Based on this, we reconstruct the holographic image from every sub-hologram individually according to its position in the reconstruction coordinates of the synthetic aperture digital hologram, and then acquire the synthetic reconstructed image by superposition of the reconstruction results of the sub-holograms. Figure 3 shows the position relationship between the synthetic aperture digital hologram (I) and the sub-hologram (In). The complex amplitude distribution of the reconstruction image is related to the position of the corresponding sub-hologram in the reconstruction coordinates. Thus, the position difference of the sub-hologram in the two reconstruction coordinates will lead to the reconstructed image u n'(x', y') of the sub-hologram in Fig. 3(b) couldn’t agree with the corresponding component u n(x', y') of the synthetic reconstructed image reconstructed from the same sub-hologram in Fig. 3(a). Assuming the distances between P and Q (centers of the corresponding hologram) in the x and y directions are a and b, respectively, then the complex amplitude distribution of u n(x', y') and u n'(x', y') obtained by Fresnel diffraction integral are as follows
where, λ is the recording wavelength; k is the wave number (k = 2π/λ); F{} represents the Fourier transform operation; C = exp(jkd)exp[jk(x'2 + y'2)/2d]/jλd, is the constant factor irrespective with integral operation; G(fx', fy') = F{g(x, y)}, g(x, y) = I n(x, y)exp[jk(x 2 + y 2)/2d], is the Fourier transform function in the reconstruction process of synthetic aperture digital hologram; fx ' = x'/λd, fy ' = y'/λd are the transverse and longitudinal spatial frequencies in the image plane and fa = a/λd, fb = b/λd are the frequency displacements in the corresponding directions, respectively. By comparing Eq. (1) and (2), it is shown that u n'(x', y') is different from u n(x', y') both in position and phase distribution, which will make the synthetic reconstructed image obtained by the superposition of the reconstructed images u n'(x', y') of the sub-holograms inaccurate.
Fig. 3 Reconstruction coordinates of the hologram. (a) Synthetic aperture digital hologram; (b) Sub-hologram.
To ensure that the reconstructed image of the sub-hologram consists with the corresponding component of the synthetic reconstructed image, according to the phase shift principle of Fourier transform, the reconstructed image of the sub-hologram In should be
The difference between u n*(x', y') and u n(x', y') is only a phase shift factor. That is to say, the corresponding component u n(x', y') of the synthetic reconstructed image could be obtained by multiplying u n*(x', y') with a phase correction factor exp[j2π(fx ' a + fy ' b)]. Therefore, the synthetic reconstructed image could be obtained by adopting the following steps. Firstly, according to the assumed position (a, b) in the hologram plane, all of the sub-holograms are multiplied with corresponding quadratic phase factor exp[jk((x-a)2 + (y-b)2)/2d] (here a and b are variable parameters), and then reconstructed in turn by Fourier transform operation. Secondly, the reconstructed images from the sub-holograms are separately multiplied with the corresponding phase correction factor exp[j2π(fx ' a + fy ' b)]. Finally, the synthetic reconstructed image could be obtained by the superposition of the sub-reconstructed-images after phase correction. By setting the location parameters of the corresponding sub-holograms, we could reconstruct the adjacent sub-holograms with different joint distances and then obtain the corresponding synthetic reconstructed images of them.
The joint misplacement of the sub-holograms will make the wavefronts of the reconstructed image fields misplaced with each other, which would result in the illegibility of the synthetic reconstructed image and thus degrade the image quality. Therefore, for every two adjacent sub-holograms along the motion path of CCD, we reconstruct them with different joint distances and then find out the accurate joint distance between them by evaluating the quality of the corresponding synthetic reconstructed images. In order to assure the accuracy of the evaluation result, we adopt two different but commonly used evaluation approaches, which assess the quality of the reconstructed image from the perspective of the spatial and the frequency domain, respectively. In the spatial domain, the quality of the reconstructed image is assessed according to the image variance [14], which should reach the maximum when the reconstructed image is clearest. In the spatial frequency domain, the quality of the reconstructed image is assessed according to the energy distribution of its spectra [15], and the energy ratio between the high and the low spectral components should also reach the maximum when the reconstructed image is clearest. It is considered that the accurate joint distance between each two adjacent sub-holograms, which is corresponding with the clearest synthetic reconstructed image, could be found out if the assessment results of the two approaches are consistent with each other. Then the accurate relative position relationships of the sub-holograms could be confirmed according to all of the identified joint distances, with which the accurate synthetic reconstructed image could be obtained by superposing the reconstruction results of the sub-holograms.
Compared with the traditional reconstruction method proposed in Ref [8], this method can reconstruct the sub-holograms with arbitrary joint distances and thus can be implemented even when the adjacent sub-holograms have no actually overlapping portions. Moreover, the joint distance between two adjacent sub-holograms could be adjusted with an arbitrary precision by setting the location parameters of the sub-holograms, which could make the joint precision of the sub-holograms reach sub-pixel accuracy.
3. Experiment results
Figure 4 shows the experimental setup used for recording the synthetic aperture digital Fresnel hologram. The object is a 5# test target with an area of 2.45cm × 2.45cm. The thin beam from a laser (λ = 632.8nm) is divided into two beams by a beam splitter. Then the two beams are separately spatially filtered and collimated to generate the reference and object wave, respectively. The CCD is a black-white type with 1392H × 1040V pixels and 4.65μm × 4.65μm pixel size. A two-dimensional precision translation stage with a resolution of 1.25μm is used for controlling the motion of CCD in the hologram recording plane. The recording distance d is 30.6cm.
Fig. 4 Experimental setup for recording synthetic aperture digital Fresnel hologram. BS: beam splitters; M: mirrors; MO: microscope objective; PH: pinholes; L: lens; CCD: charge coupled device.
The motion path of CCD is shown in Fig. 5(a) , the default horizontal and vertical translation intervals are chosen as 6472.5μm and 4835μm, respectively, which are nearly equal to the horizontal and vertical size of the sub-hologram recorded by CCD (L x = 1392 × 4.65μm = 6472.8μm, L y = 1040 × 4.65μm = 4836μm). Therefore, the displacement tolerance of CCD will result in that the accurate joint distance D 0 between every two adjacent sub-holograms might be less or greater than the size L of the sub-hologram. So the accurate synthetic reconstructed image couldn’t be obtained by using the traditional reconstruction method for it is only suitable for the case of that D 0 <L. 4H × 5V sub-holograms are recorded with the motion of CCD and then patched up to a synthetic aperture digital hologram with 5568H × 5020V pixels by image mosaic according to the default relative position relationships in the hologram recording plane (Fig. 5(b)).
Fig. 5 Recording process of the synthetic aperture digital hologram. (a) Motion path and translation interval of CCD; (b) Synthetic aperture digital hologram obtained by image mosaic.
We take the sub-holograms A and B as examples to illustrate the reconstruction process of the synthetic aperture digital hologram. Firstly, to avoid the vision aberration and assure the resolution of the reconstructed image, each original sub-hologram (1392H × 1040V) is transformed to a square array (4096H × 4096V) by padding zeros [16], as shown in Figs. 6(a) and 6(b), respectively. Then, the two zero-padding sub-holograms are separately reconstructed with different joint distances D by employing the proposed method and the corresponding gradual change process of the reconstructed image with the variety of D is shown in Fig. 6(c) (Media 1). The variety range and stepping precision of D are chosen as 6472.5 ± 10μm and 1μm, respectively. Figure 6(d) (Media 2) shows the gradual change process of the spectra of the reconstructed image vs. D, where the zero-order of the spectra is chosen as the low spectral component. The relationship of the joint distance D with the variance γ of the reconstructed image and the energy ratio τ between the high and low spectral components are shown in Figs. 6(e) and 6(f), respectively. It is found that both of γ and τ achieve the maximum when D = 6467.5μm and the corresponding reconstructed image is shown in Fig. 6(g). Figure 6(h) shows the reconstructed image obtained under the default joint distance D = 6472.5μm. Figures 6(i) and 6(j) are the magnification of the marked area in Figs. 6(g) and 6(h), respectively. It is shown that the accurate reconstruction image could be obtained when the two adjacent sub-holograms are reconstructed with the identified joint distance by employing the proposed method.
Fig. 6 Reconstruction process of the sub-holograms A and B. (a) and (b) Zeros-padding holograms of A and B; (c) and (d) Gradual change process of the reconstructed image (Media 1) and the spectra (Media 2) vs. the variety of the joint distance D; (e) and (f) Relationship of D with variance γ of the reconstructed image and energy ratio τ between the high and the low spectral component; (g) and (h) Reconstructed image obtained under D = 6467.5μm and 6472.5μm; (i) and (j) Magnification of the marked area in (g) and (h)
Similarly to the reconstruction process of the sub-holograms A and B, along the motion path of CCD, we reconstruct every two adjacent sub-holograms with different joint distances and then find out the accurate joint distance between them by evaluating the quality of the corresponding synthetic reconstructed images. Then the accurate relative position relationships of the sub-holograms could be confirmed according to all of the identified joint distances, with which the accurate synthetic reconstructed image could be obtained by superposing the reconstruction results of the sub-holograms. Figure 7 shows the reconstruction results of the synthetic aperture digital Fresnel hologram. The reconstruction process along the motion path of CCD by using the proposed method is shown in Fig. 7(a) (Media 3). The corresponding synthetic reconstructed image (4096H × 4096V) and the reconstructed image of Fig. 5(b) (5568H × 5200V) are shown in Figs. 7(b) and 7(c), respectively. Figures 7(d), (e), (f) and (g) are the magnifications of the marked area in Figs. 7(b), (c), (d) and (e), respectively. It could be seen that the influence of the joint misplacement of sub-holograms on the quality of the synthetic reconstructed image has been well corrected by using the proposed method.
Fig. 7 Reconstruction result of the synthetic aperture digital Fresnel holographic image. (a) Reconstruction process along the motion path of CCD (Media 3); (b) Reconstruction result obtained by the proposed method; (c) Reconstructed image of Fig. 5(b); (d), (e) and (f), (g) Magnification of the marked area in (b), (c) and (d), (e), respectively.
4. Conclusions
The quality of the reconstructed image in synthetic aperture digital holography strongly depends on the joint accuracy of the sub-holograms. The joint misplacement of the sub-holograms, which is commonly caused by the displacement tolerance of the apparatus used for controlling the movement of CCD, will result in the illegibility of the synthetic reconstructed image and thus degrade the image quality. The traditional reconstruction method, which is based on the cross-correlation analysis for the sub-holograms and the image mosaic technique, couldn’t determine and then correct the joint misplacement of the sub-holograms when the adjacent sub-holograms have no actually overlapping portions. Therefore, we proposed a method which could be used to not only correct the joint misplacement of the sub-holograms without the limitation of the actually overlapping circumstances of the adjacent sub-holograms but also make the joint precision of the sub-holograms reach sub-pixel accuracy at the same time. The numerical reconstruction results are in agreement with the theoretical analysis. It should be noticed that, although here we demonstrate this method only with a transparent recorded object experimentally, it would be ensured in principle that a diffusely reflecting object could also be recorded by spatial synthetic aperture digital Fresnel holography and then effectively reconstructed by using this method.
Acknowledgements
This work is supported by the Science Foundation of Aeronautics of China under Grants No 2006ZD53042.
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