1. Introduction

Optical systems have been extensively studied for object recognition [1–8]. With rapid advances in CCD array sensors, computers and software, digital holography can be performed efficiently as an optical system for 3-D object recognition. Digital holography has also been a subject of great interest in various fields [9–16]. In digital holography, the off-axis scheme has been widely used since it is simple and requires only a single exposure in separating the desired real image term from the undesired DC and conjugate terms [9]. However, 3-D object recognition by use of the off-axis scheme has inherent limitations in terms of robustness to the variation of the 3-D object position due to the superposition of the real image with undesired terms. The tolerance problem to the variation of the 3-D object positioning can be resolved by employing an on-axis scheme. Recent studies on 3-D object recognition by use of phase shifting digital holography have provided a feasible approach for implementing the on-axis 3-D object recognition system [6–8].

The phase shifting digital holographic method requires multiple hologram recordings [10], however. Therefore, the 3-D object recognition system based on the on-axis phase shifting approach has inherent constraints in the sense that it needs a vibration-free environment that is convenient only in a laboratory. Also, it is sensitive to recording a moving object. It was proposed that such limitations of 3-D object recognition can be overcome by use of single exposure on-axis digital holography [17]. In this paper, we present a distortion-tolerant 3-D object recognition method based on single exposure on-axis digital holography. The main subject in this paper is to show that the single exposure on-axis based 3-D object recognition can provide a distortion-tolerant 3-D object recognition capability compared with that obtainable by use of on-axis phase shifting digital holography, while maintaining practical advantages of the single exposure on-axis scheme, such as robustness to environmental disturbances and tolerance to moving targets.

2. Single exposure on-axis digital holography

The basic experimental setup for the single exposure on-axis digital holographic system is based on a Mach-Zehnder interferometer as shown in Fig. 1. B/S means a beam splitter and M represents a plane mirror. Polarized light from an Ar laser source tuned to 514.5 nm is expanded and divided into object and reference beams. The object beam illuminates the object and the reference beam propagates on-axis with the light diffracted from the object. In order to match the overall intensity level between the reference wave and the reflected wave from the 3-D object, a tunable density filter is used in the reference beam path. The interference between the object beam and the reference beam are recorded by a CCD camera. In contrast to on-axis phase shifting digital holography, no phase shifting components are required. The conventional on-axis phase shifting digital holography can be implemented by inserting a quarter wave plate in the reference beam path. For on-axis phase shifting digital holography, the hologram recorded on the CCD can be represented as follows.

Here, [O(x,y)] and [R(x,y)] represent the Fresnel diffraction of the 3-D object and the reference wave functions, respectively. θ is an induced phase shift. In on-axis phase shifting digital holography based on multiple holograms, the desired object wave function [O(x,y)] can be subtracted by use of two holograms having λ/4 phase difference and two DC terms |O(x,y)|² and |R(x,y)|² as follows:

Here, $u_i^M$ (x,y) represents the synthesized hologram obtained by on-axis phase shifting digital holography which requires multiple recordings.

 

Fig. 1. Experimental setup of the distortion-tolerant 3-D object recognition system based on single exposure on-axis scheme.

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Likewise, the following expression represents the synthesized hologram obtained by use of single exposure on-axis digital holography:

Here, N is the windowed hologram size used for image reconstruction. Δx and Δy represent the directional spatial resolution of the CCD. The reference wave can be assumed to be a known function because it is not changed while the object intensity distribution varies according to the input scene. The DC term |O(x,y)|² generated by the object can be reduced effectively by employing an averaging over the whole hologram as described in the third term of Eq.(3) [14]. Thus, the index $u_i^s$ (x,y) can be regarded as a synthesized result obtained by use of only a single hologram. The single exposure on-axis approach suffers from the superposition of the conjugate image which degrades the reconstructed image quality. However, for 3-D object recognition, the inherent conjugate term in single exposure on-axis digital holography may not substantially degrade the recognition capability since the conjugate image term also contains information about the 3-D object [17].

In order to apply the distortion-tolerant 3-D object recognition, we first need to numerically reconstruct 3-D object images. 3-D multi perspective section images on any parallel plane can be reconstructed by using the following inverse Fresnel transformation:

Here, F denotes the 2-D Fourier transformation. u_i (x,y) represents the synthesized hologram which can be either $u_i^M$ (x,y) in Eq. (2) or $u_i^S$ (x,y) in Eq. (3). u_o (x’,y’) is the reconstructed complex optical field at a parallel plane which is at a distance d from the CCD. By changing d, we can reconstruct multiple section images from the single hologram without using any optical focusing. Different regions of the digital hologram can be used for reconstructing different perspectives of the 3-D object with the angle of view (α,β) as depicted in Fig. 2. u_i (x,y; a_x,a_y ) denotes the amplitude distribution in the window of the hologram used for the 3-D object reconstruction. Here, a_x and a_y are central pixel coordinates of the hologram window.

 

Fig. 2. Three-dimensional multiple image sectioning and multi perspectives by digital holography.

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3. Composite filter design

The reconstructed complex wave u_o (x’,y’) in Eq. (4) can be directly used to construct a variety of correlation filters [1–4] for 3-D object recognition [17]. However, direct use of the reconstructed image for constructing a matched filter may prevent us from recognizing correctly distorted true class targets [8]. In order to improve the distortion-tolerant recognition capability, removing phase terms in the reconstructed complex field decreases high sensitivity. Also, we can employ some pre-processing such as averaging and median filtering to remove speckle. The main goal of the proposed distortion-tolerant 3-D object recognition by use of single exposure on-axis digital holography is to recognize a distorted reference 3-D object as a true input while correctly rejecting false class objects. For this purpose, a filter must be designed to provide high performance distortion-tolerant 3-D object recognition capability.

We now describe the detailed process of creating the composite filter by use of single exposure on-axis digital holography. We measure M different holograms as we change the perspectives on a 3-D reference object. After capturing each single hologram, we can make a synthesized hologram $u_i^S$ (x,y) which can be used for reconstructing the complex wave function u_o (x’,y’). Then, in order to decrease high sensitivity of the reconstructed wave function, we remove phase terms in the reconstructed complex field and apply some pre-processing techniques of averaging and median filtering. By use of such pre-processed M training images represented by s₁ (x,y), s₂ (x,y), …, s_M (x,y), we can generate p-dimensional column vectors s₁ , s₂ , …, s_M by taking the rows of the m×n matrix and linking their transpose to create a p=m×n vector. We compute the Fourier transformation of each training image and obtain a new set of column vectors S₁, S₂, …, S_M. Then, we nonlinearly modify the amplitude of each column vector by use of the power law nonlinearity by which those column vectors become ${{\mathbf{S}}_{\mathit{1}}}^{\mathit{k}}$${{\mathbf{S}}_{\mathit{2}}}^{\mathit{k}}$,…, ${{\mathbf{S}}_{\mathit{M}}}^{\mathit{k}}$ [18]. This power-law operation is defined for any complex vector v as follows:

Now, we define a new matrix S^k=[${{\mathbf{S}}_{\mathit{1}}}^{\mathit{k}}$, ${{\mathbf{S}}_{\mathit{2}}}^{\mathit{k}}$ , …, ${{\mathbf{S}}_{\mathit{M}}}^{\mathit{k}}$ ] with vector S_i as its ith column. Using such notation, the k-law based non-linear composite filter h ^k can be defined in the spatial domain as follows:

Here (S^k )⁺ is the complex conjugate transpose of S^k . The vector c contains the desired cross-correlation output origin values for each Fourier transformed training data S_i and the notation c* denotes the complex conjugate of c [18].

4. Experimental results

We compare the distortion tolerant 3-D object recognition capability of the single exposure scheme with that of the multiple exposure on-axis phase shifting holographic approach. Experiments have been conducted with a 3-D object as the reference object which is 25×25×35 mm and several false 3-D objects with similar size. The reference car is located at a distance d=880 mm from the CCD camera. We use an argon laser of 514.5 nm. The CCD has a pixel size of 12×12 µm and 2048×2048 pixels. However, throughout the experiment, we use only 1024×1024 pixels for each reconstruction process since multi perspectives are required from a single hologram in making a composite filter.

A. Rotation tolerance

First, we acquire nine synthesized holograms of the reference true class object labeled from 1 to 9 (synthesized holograms: #1~#9) for on-axis phase shifting approach as well as single exposure on-axis scheme. For every synthesized hologram capturing step, the reference object is rotated by 0.2° around the axis orthogonal to Fig. 1. Figure 3(a) and (b) show the reconstructed image of the reference 3-D object from one of the synthesized holograms by use of single exposure on-axis digital holography and on-axis phase shifting digital holography, respectively. They are reconstructed by using the central window area of 1024×1024 pixels from the total synthesized hologram of 2048×2048 pixels. As can be seen, even the reconstructed image with only amplitude data contains high frequency speckle patterns. In order to smooth the high frequency amplitude distribution, we need to employ pre-processing techniques. Specifically, we use the window of 6×6 pixels for averaging which is followed by a median filtering with the window size of 5×5 pixels. After the averaging process, 1024×1024 pixels is reduced to 171×171 pixels since we reconstruct a new image with averaged values calculated for each averaging window. After the median filtering process, the image size of 171×171 pixels remains unchanged.

Among the nine synthesized holograms of the true class reference object, we use three (synthesized holograms: #3, #6 and #9) to construct a non-linear composite filter. For each of those three synthesized holograms, we select one centered window of 1024×1024 pixels and two different windows of the same size to reconstruct three training images with different perspectives from each synthesized hologram. Figure 4(a)–(c) depict three pre-processed training images reconstructed from the synthesized hologram #6. Figure 4(a) and (b) have a viewing angle difference of 0.4°. The viewing angle difference between Fig. 4(a) and (c) is 0.8°. Figure 4(d)–(f) represent the three views obtained by use of phase-shifting on-axis digital holography. In the same way, we can obtain nine training images from the three synthesized holograms (synthesized holograms: #3, #6 and #9). Then, we construct a non-linear composite filter by following the procedures described in Section 3. We use a non-linear factor k=0.1 in Eq. (6) to improve the discrimination capability of the nonlinear filter. Figure 5(a) and (b) depict the final non-linear composite filters that can be used for the single exposure approach and the multiple exposure approach, respectively.

 

Fig. 3. Reconstructed images of the reference true target prior to pre-processing by use of (a) single exposure on-axis digital holography and (b) multiple exposures on-axis phase shifting digital holography.

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Fig. 4. (a)–(f): Three reconstructed training images from the synthesized hologram #6 by use of (a)–(c) single exposure on-axis digital holography and (d)–(f) multiple exposures on-axis phase shift digital holography.

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Once we construct the composite filter, we perform the distortion-tolerant 3-D object recognition process for various input objects. We acquire more synthesized holograms for several false class objects labeled from 10 to 15 (synthesized hologram: #10~#15). Now, we have fifteen synthesized holograms consisting of true and false class objects. We can reconstruct fifteen images by using the central window area of 1024×1024 pixels from the total synthesized hologram of 2048×2048 pixels and perform the pre-processing.

 

Fig. 5. Constructed non-linear composite filters for distortion-tolerant 3-D object recognition by use of (a) single exposure on-axis digital holography and (b) on-axis phase shift digital holography.

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Finally, those fifteen pre-processed images (input images: #1~#15) are put in the input scene for distortion-tolerant 3-D object recognition. Figure 6 shows some of the false class objects (images: #10, #13 and #15). Six input images (images: #1, #2, #4, #5, #7 and #8) among the nine true class inputs (images: #1~#9) are used as our non-training distorted true class objects.

 

Fig. 6. Reconstucted images of some false class objects (images: #10, #13 and #15) by use of (a)–(c) single exposure on-axis digital holography and (d)–(f) on-axis phase shift digital holography.

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Figure 7(a)–(b) and (c)–(d) represent the normalized correlation obtained by use of single exposure on-axis digital holography and phase shifting on-axis digital holography, respectively. All plots are normalized by the correlation peak value obtained when the training image #3 is used as input scene. As can be seen in Fig. 4(a)–(c) and Fig. 6(a)–(c), for the single exposure on-axis digital holography, the reconstructed image quality degrades due to the presence of its conjugate image. The 3-D object recognition by use of single exposure on-axis digital holography, which employs both the phase and amplitude information, is capable of 3-D object recognition. However, the distortion-tolerant capability of the single exposure on-axis scheme somewhat degrades after the averaging and median filtering since the 3-D object information contained in the conjugate image term is disappeared due to the pre-processing. The degradation of the recognition capability could be from the existence of the conjugate term in the single exposure approach.

 

Fig. 7. (a)–(d): Normalized correlation when input image #8 of the non-training true targets and input image #13 among the false class objects are used as input scenes for (a)–(b) single exposure on-axis digital holography and (c)–(d) on-axis phase shift digital holography.

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Fig. 8. Normalized correlation distribution for various input images of true targets (#1~#9) and false class objects (#10~#15) for single exposure on-axis digital holography (□) and on-axis phase shifting digital holography (*).

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Figure 8 illustrates the distortion tolerant 3-D recognition capability of the single exposure on-axis digital holographic approach. The degradation in recognition capability is not substantially less that that of on-axis phase shifting digital holography.

B. Longitudinal shift tolerance

We have examined the longitudinal shift tolerance of the single exposure scheme in order to compare with that of phase shifting digital holography. For the practical use of digital holography in distortion-tolerant recognition, it must also be less sensitive to longitudinal shifts along the z axis for recognizing a true class input target. It provides benefits in terms of the reduced number of reconstructions that must be performed for recognizing the input 3-D object. For this longitudinal shift tolerance, we construct another non-linear composite filter which includes a number of out-of focus reference reconstruction images. For constructing this filter, we use only one hologram (synthesized hologram: #3) among the three used for constructing the composite filter in rotational distortion tolerance. We use the same three reconstruction windows for generating training images. However, for each window, we also use three reconstructed images with a defocus of -20 and 20 mm as well as the reconstructed image of the reference object in the focus plane. Likewise, nine views are used to construct the new non-linear composite filter for longitudinal shift tolerance.

 

Fig. 9. Normalized correlation peak values versus longitudinal shift along the z axis for single exposure on-axis digital holography (□) and on-axis phase shifting digital holography (*).

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Figure 9 represent the normalized correlation peak values versus longitudinal shift for both the single exposure on-axis scheme and the phase shifting digital holographic approach when a distorted reference target is in the input scene. It shows that the single exposure on-axis scheme is less sensitive to longitudinal shift than the on-axis phase shifting approach. The reason that the sensitivity is less for the single exposure scheme compared with that of the phase-shifting approach may be due to the inherent image degradation of the single exposure scheme as stated earlier. The single exposure on-axis approach seems to be somewhat more robust to the variation of longitudinal shift.

5. Conclusions

We have presented a distortion-tolerant 3-D object recognition system using single exposure on-axis digital holography. The proposed method requires only a single hologram recoding for the distortion-tolerant 3-D object recognition while the on-axis phase shifting based approach requires multiple recordings. The main benefit of the proposed single exposure method is that it can provide more robustness to environmental noise factors such as vibration since it uses only a single hologram, which can be measured and analyzed in real time. Experimental results show that the distortion-tolerant recognition capability of the single exposure approach is somewhat worse than that of the phase-shifting digital holographic method. Nevertheless, the proposed single exposure method makes the recording system simpler and more tolerant to object parameters such as moving targets while maintaining a distortion-tolerant 3-D object recognition capability.