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Abstract
An off-axis digital holographic reconstruction method with fractional Fourier transform domain filtering is proposed. The theoretical expression and analysis of the characteristics of fractional-transform-domain filtering are given. It is proven that the filtering in a lower fractional-order transform domain can utilize more high-frequency components than that in a conventional Fourier transform domain under the same size of filtering regions. In simulation and experiment, the results demonstrate that the reconstruction imaging resolution can be improved by filtering in the fractional Fourier transform domain. The presented fractional Fourier transform filtering reconstruction provides a novel (to our knowlede) optional way for off-axis holographic imaging.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The Fourier spectrum domain filtering is typically used to reconstruct the sample’s amplitude and phase distributions in off-axis digital holography [1,2]. Due to the existence of an off-axis interference angle, the object information is loaded on the high frequency carrier to be recorded in the hologram. So, the signal term and the zero-order term are effectively separated from each other in the Fourier transform domain, because their carrier frequencies are different.
The Fourier spectrum domain filtering has been widely applied in off-axis digital holographic reconstruction, while it still has some insufficiency [3–5]. The noise term and the signal term both occupy a certain frequency band in the Fourier transform domain. For the signal term filtering, if the object frequency band is not covered completely by a filtering window, the resolution of reconstruction imaging will inevitably be reduced. On the other hand, if the filtering window is taken too large, which extends into the band region of the zero-order spectrum, the strip noise will appear in the reconstructed images. In addition, for different objects, the band occupied by each term in the hologram is unknown in advance. Therefore, as compared to the in-line digital holographic reconstruction, the filtering in off-axis digital holographic reconstruction may bring about more limitations on resolution.
The various methods have been proposed successively to deal with these problems. In the spectrum filtering schemes, for the improvement of reconstruction resolution, the filtering range of an object spectrum can be extended by removal or suppression of the zero-order term of the hologram. The phase-shifting method can effectively remove the zero-order and conjugate terms from the hologram [6,7], but it also make some advantages in off-axis digital holography lost. With multiplexing specific parameters in a recording system or by nonlinear filtering, the zero-order term can be obviously suppressed [8,9]. The other methods to eliminate or to suppress the zero-order terms also include iterative operations [10,11] and image processing techniques [12,13]. In another way, the holographic reconstruction can be realized directly by establishing a functional relationship between the hologram and the reconstructed complex amplitude distribution [14]. The maximum utilization of the hologram frequency band can also be achieved by angular complexation [15], and the adaptive-optimizing filtering range by using convolutional neural networks is an effective way as well [16].
A Fractional Fourier transform is a generalized form of a Fourier transform. For an image in space domain or a signal in time domain, it still keeps the original pure space or time distribution via the zero-order Fourier transform, and it is converted as the pure spatial-frequency or time-frequency distribution in its Fourier spectrum domain via the 1$^{\rm st}$-order Fourier transform. But, via a fractional Fourier transform with the non-integer order between 0-1, it will become a mixed signal containing both space/time domain information and frequency domain information. Although this mathematical transform is mainly applied in the communication field [17], it is concerned by researchers in the field of 2D image processing in recent decades. The fractional Fourier transform is also used in digital image encryption, watermarking, and other information security applications [18–21]. In the field of diffractive optics, the fractional Fourier transform is mostly used to simulate the propagation of the optical field [22], since using fractional Fourier transform to simulate diffraction propagation in the near field does not cause under-sampling as using the Fresnel diffraction formulation. Based on this feature, Talbot effect can be described under the theoretical system of the fractional Fourier transform [23], and the parameters of Newtonian rings can be estimated via fractional Fourier transform [24]. Moreover, the fractional Fourier transform can be implemented experimentally in a specific optical system [25]. A joint fractional Fourier transform correlator based on a fractional transform system has been reported [26,27]. In the imaging field, the fractional Fourier transform has been used in an in-line digital holography [28]. With a fractional Fourier transform and iterative phase recovery algorithms, the high-quality quantitative phase imaging can be achieved in a coherent diffraction imaging system [29]. In addition, the sampling theorem in the fractional Fourier transform domain has been discussed in recent studies [30].
In this paper, an off-axis digital holographic reconstruction method based on fractional Fourier transform domain filtering is proposed. The filtering characteristics in fractional Fourier transform domains are analyzed and described in expression, which shows the capability for resolution improvement.
2. Methods
2.1 Fractional Fourier-transform reconstruction of off-axis hologram
The intensity distribution of an off-axis digital hologram can be expressed as:
The fractional Fourier-transform of a hologram can be expressed as:
Further, the fractional Fourier transform of the signal term can be written as:
It should be noted that the center point of the distribution $T(u,v)$ is just at the center of the $p$-order transform domain. Thus, $T(u+f_x{\rm sin}\alpha,v+f_y{\rm sin}\alpha )$ in Eq. (3) means that its distribution center is changed to the position $(-f_x{\rm sin}\alpha, -f_y{\rm sin}\alpha )$, due to a spatial-frequency shifting caused by an off-axis angle. In other words, as the transform order $p$ varies from 0 to 1, the distance between the distribution center of the function $T$ and the center of the transform domain increases from 0 to $(f_x^2+f_y^2)^{1/2}$. So, the center of the first-order term in typical Fourier-transform domain ($p$=1) is farthest apart from the center of the zero-order term.
The intensity distributions in fractional Fourier transform domains with different $p$-orders are shown in Fig. 1. In simulation, the computer-generated off-axis digital hologram (CGH) uses a Pepper pattern as an amplitude distribution and a pattern of USAF-1951 resolution-test-target as a phase distribution. According to Eq. (2), the fractional Fourier transform of the CGH can be performed, and its corresponding distributions in the $p$-order Fourier transform domain are obtained, when $p\in [0,1]$. Figure 1 shows the distributions in the $p$-order Fourier-transform domain when $p$=0, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.
Fig. 1. Distributions in the fractional Fourier transform domains of the CGH when $p$=0, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1, respectively.
As shown in Fig. 1(a), the zero-order Fourier transform of the hologram is still itself. The distribution obtained by performing the 1$^{\rm st}$-order Fourier transform is just the spatial-frequency spectrum of the hologram, i.e. commonly termed as Fourier spectrum. Further, when the fractional order increases from 0 to 1, the centers of the signal term and the conjugate term are gradually separated from the center of the zero-order term, as seen in Figs. 1(a)–1(h). The separated spacing become farer with the greater $p$, which can be explained according to $T(u+f_x{\rm sin}\alpha, v+f_y{\rm sin}\alpha )$ in Eq. (3). The spacing reaches the maximum when $p$=1, which is just the case of conventional Fourier spectrum of a hologram.
In the fractional order method, a filtering interception is firstly performed in the $p$-order Fourier transform domain of the hologram. Next, the $(1-p)$-order Fourier transform on the filtering-intercepted distribution is conducted, which transforms the $p$-order domain into the frequency domain, i.e. the 1$^{\rm st}$-order Fourier transform domain. Then, the signal term loaded on the high carrier frequency can be shifted in the center of the frequency domain to eliminate the carrier. Thus, the subsequent reconstruction procedure can be performed as similar as that in conventional holographic reconstruction.
2.2 Suppression of zero-order noise and compensation for edge loss
Figure 2 shows the fractional Fourier-transform domain filtering when $p$=0.75 and the reconstructed images of the CGH. The intensity distribution in the 0.75-order Fourier transform domain of the CGH is shown in Fig. 2(b), in which the red frame indicates the filtering region used for imaging reconstruction. The amplitude and phase images reconstructed with the fractional order filtering are shown in Figs. 2(c) and 2(d), respectively.
Fig. 2. (a) Computer-generated off-axis digital hologram; (b) its intensity distribution in the 0.75-order Fourier transform domain, in which a red frame indicates the filtering region used for imaging reconstruction; (c) amplitude distribution and (d) phase distribution, reconstructed after filtering in the 0.75-order Fourier transform domain.
As seen in the dash–framed areas of Figs. 2(c)–2(d), there exist some strip noises caused by the zero-order term in the reconstructed amplitude and phase images. According to Eq. (3), if the filtering is performed in a lower order fractional transform domain, the zero-order noise in the reconstructed images will be more serious. Such zero-order noise can be removed by using our presented zero-order elimination method in Ref. [31].
In addition, as seen in Fig. 2, the obvious edge loss surrounds the sides of the reconstructed amplitude and phase images, and the range of FOV loss at the edge is uncertain. The loss of the field of view can be explained by the displacement theorem of fractional order Fourier transform. If the gray value at the center pixel of the hologram is denoted as $I(0,0)$, there has a fractional Fourier transform relationship as:
The gray value of the farthest pixel from the center in the field of view is denoted as $I(M\Delta x, N\Delta y)$. According to the displacement theorem of fractional Fourier transform, we can deduce the expression as:
When the fractional order is taken as $p$=1, the last factor in Eq. (7) changes as $\delta _{1+p}(u, v)$, in which both $M\Delta x{\rm cos}\ \alpha$ and $N\Delta y{\rm cos}\ \alpha$ become zero since $\alpha$=$\mathrm{\pi} /2$ when $p$=1. This means that the spatial displacement between the center pixel point and the farthest pixel point completely convert into a phase shift between them in Fourier transform domain. Moreover, two spatially separated pixels of the hologram are at the same pixel point in their Fourier spectrum domain. Accordingly, the conventional filtering in Fourier spectrum domain is always able to reconstruct a complete FOV, no matter how small the filtering range is. When the fractional order $p$ is less than 1, the effect of the spatial displacement relationship is not completely gone, so the limitation of a filtering range will bring about the loss of FOV.
We propose to perform spatial-domain zero-padding on the hologram for compensating the edge loss that is caused by a fractional Fourier transform domain filtering. After the FOV of the hologram is artificially expanded by zero-padding in the spatial domain, the edge loss will act on the zero-padding boundary region where no object information is contained. Figure 3 shows the reconstruction results in the 0.75-order Fourier transform domain filtering by performing the zero-order term elimination only, the spatial domain zero-padding only, and both of the zero-order elimination and the spatial-domain zero-padding meanwhile.
Fig. 3. (a) Resultant CGH after performing the zero-order elimination only; (b) CGH after performing the spatial-domain zero-padding only; (c) Resultant CGH after performing both the zero-order elimination and the spatial-domain zero-padding; (d)-(f) intensity distributions in the 0.75-order Fourier transform domain respective to (a)-(c), in which the red frames mark out the filtering regions for reconstruction imaging, respectively; (g)-(i) reconstructed amplitude maps; (j)-(l) reconstructed phase maps.
In the case of only performing zero-order suppression before filtering, the resultant hologram with the zero-order suppression and its intensity distribution with the 0.75-order Fourier transform are shown in Figs. 3(a) and 3(d). The reconstructed amplitude and phase maps are shown in Figs. 3(g) and 3(j), in which the strip noises are suppressed completely, but the edge loss around the border is obviously visible. For solving the edge loss of FOV, the spatial domain zero-padding is applied on the original hologram. After only zero-padding processing by extending the periphery of the original hologram outward and setting the intensity value in the extended area as zero, the zero-padding hologram in Fig. 3(b) is obtained. Figure 3(e) is the intensity distribution in its 0.75-order Fourier transform domain, in which the zero-order portion also exists. As compared with the maps in Figs. 2(c) and 2(d), the lost information at the edges of the field of view in the reconstructed amplitude and phase maps have been recovered by this spatial domain zero-padding, as shown in Figs. 3(h) and 3(k), but the strip noise still stays thereon. Further, both of the image resolution and the integrity of FOV can be improved by performing the zero-order suppression and the zero-padding processing at the same time. Figure 3(c) is the resultant CGH with the zero-order suppression and the spatial-domain zero-padding. Its reconstructed amplitude and phase maps are shown in Figs. 3(i) and 3(l). It can be seen that the integrity of the FOVs in the reconstructed amplitude and phase maps is achieved effectively, as well as the merit of high resolution with the presented fractional Fourier transform filtering method is kept.
3. Filtering characteristics in fractional Fourier transform domains
All of the spatial frequency components contained in the digital hologram can be discretized into $2M\times 2N$ discrete frequencies with an equal interval, i.e. the adjacent frequencies have the differences as $\Delta f_x$ at the $x$ direction and $\Delta f_y$ at the $y$ direction, respectively. After being decomposed with Fourier analysis, the signal term $OR{\rm exp}[-{\rm j}2\mathrm{\pi} (f_xx+f_yy)]$ of the hologram can be written as:
The fractional Fourier transform of the component with the serial number $(m,n)$ is expressed as:
According to Eq. (10), the spacing between the highest frequency component and the zero frequency component of the signal term in a fractional transform domain when $p<1$ is smaller than that in the Fourier spectrum domain. Thus, more high-frequency components can be collected by a fractional Fourier transform filtering.
Figure 4 shows the intensity distributions of spatial-frequency spectrum of the CGH after preforming both the $p$-order and the $(1-p)$-order Fourier transforms in order. The regions for filtering interception used in the 0.7-order transform, 0.8-order transform, 0.9-order transform, and 1$^{\rm st}$-order Fourier transform are the same. According to the above-mentioned way of performing the $p$-order firstly and then the $(1-p)$-order, the 0.3-order, 0.2-order and 0.1-order transforms are correspondingly conducted, respectively. Thus, the spatial-frequency distributions with 0.7-order then 0.3-order transforms, 0.8-order then 0.2-order transforms, 0.9-order then 0.1-order transforms, and 1$^{\rm st}$-order Fourier transform are obtained, as shown in Figs. 4(a)–4(d), where their filtering interception sizes are the same. The solid frames in the figure indicate their filtering regions. As the frequency-spectrum intensities along the blue dashed lines in Figs. 4(a)–4(d) are taken logarithmically, the curves of spatial-frequency distribution along the horizontal direction in their individual frequency domains when $p$=0.7, 0.8, 0.9 and 1 are shown in Fig. 4(e). The two vertical dashed lines in Fig. 4(e) mark out the same filtering interception range in the horizontal direction under the different orders. If the filtering interception is performed when $p<1$, as seen the corresponding curves in Fig. 4(e), the higher spatial-frequency components than the cut-off frequency of 1${\rm st}$-order Fourier transform are acquired. It means that more high-frequency components, which are greater than the cut-off frequency intercepted by filtering in the conventional Fourier transform domain, can be intercepted by filtering in a lower order transform domain. This also implies that the off-axis holographic reconstruction by using fractional transform domain filtering is capable of improving its reconstruction resolution. On the other hand, when the order of fractional Fourier transform is taken too small, for instance when $p<0.7$ in our case, the signal term is unable to be effectively separated from the conjugate term, as seen in Figs. 1(b)–1(d).
Fig. 4. (a)-(d) Spatial-frequency spectrums of the hologram after filtering in the fractional Fourier transform domains when $p$=0.7, 0.8, 0.9 and 1, respectively; (e) Logarithmic curves of spectrum intensity versus a horizontal coordinate along the dashed lines in (a)-(d).
4. Experiment and results
The off-axis digital holographic recording system is equipped as shown in Fig. 5. A linear-polarized laser with the wavelength 633 nm is used as light source. The pixel size of the CCD is 6.45$\times$6.45 µm$^2$. As seen in the optical setup, the object beam reflected via the PBS is $s$-polarized, and the transmitted reference beam is almost $p$-polarized. A half-wave plate HWP$_2$ is used to adjust the reference wave into arbitrary linear polarization orientation.
Fig. 5. Off-axis digital holographic recording system: BEC, beam expander and collimation; HWP, half wave plate; PBS, polarized beam splitting prism; M, mirror; L, lens.
In order to remove the zero-order noise in reconstruction images, two holograms are each recorded by interference of two different linear-polarizations of the reference wave with the $s$-polarized object wave. By performing the subtraction of the two holograms, we can obtain one resultant hologram, of which the zero-order term is effectively eliminated [31]. In the holographic reconstruction experiments with the fractional order filtering, the zero-order strip noise in the reconstructed amplitude and phase images are removed.
In the experiment of amplitude reconstruction, using a USAF-1951 resolution test target as an object, two holograms are recorded by setting the HWP$_2$ at two arbitrary positions, and then are subtracted. Thus, the zero-order-eliminated hologram is obtained. Then, the holographic reconstruction in a fractional Fourier transform domain is performed from this resultant hologram. The resultant hologram and the filtering processing in fractional Fourier transform domains are shown in Fig. 6.
Fig. 6. (a) Resultant hologram of a USAF-1951 target; (b)-(f) its intensity distributions after the fractional Fourier transforms of the orders 0.6, 0.7, 0.8, 0.9 and 1, where the solid-line frames with the same size mark the filtering regions for imaging reconstruction.
The intensity distributions by performing the fractional Fourier transforms on the resultant hologram when $p$=0.6, 0.7, 0.8, 0.9 and 1 are shown in Figs. 6(b) to 6(f), respectively. Further, the reconstructed amplitude images by filtering in the above fractional order transform domains are shown in Fig. 7. As seen in the zoomed images in Figs. 7(a) to 7(e), the line-pair element 5-6 can be clearly distinguished when $p$=0.6 and 0.7, but when $p$=0.8, 0.9 and 1, the element 5-6 is indistinct. According to the nominal resolution of each line-pair element of USAF-1951 target, the resolutions of the amplitude images reconstructed in 0.6-order and 0.7-order transform domains reach up to 8.77 µm, which is higher than the resolutions of the images reconstructed in the orders 0.8, 0.9 and 1. On the other hand, as seen the upper and right edges in Fig. 7(a), the edge loss of the FOV in the amplitude image reconstructed in 0.6-order transform domain is visible. The reconstructed amplitude images in Fig. 7(b) to 7(e) exhibit the complete field of view, while the resolutions when $p$=0.8 and 0.9 are 9.84 µm, and the resolution when $p$=1 is 11 µm.
Fig. 7. (a)-(e) Amplitude maps reconstructed by filtering in the 0.6-order, 0.7-order, 0.8-order, 0.9-order and 1$^{\rm st}$-order Fourier transform domains, respectively; (f) Amplitude curves at the red lines in (a)-(e).
Although the best reconstruction with a fractional Fourier transform in principle is corresponding to the case when the positive and negative first-order terms just separate, it is difficult in practice to exactly determine when the terms are just separated, because there is usually no clear boundary between the terms in the transform domain. For better choice of the fractional order number, we need to have a trade-off between two factors of the best local resolution and the complete field of view. In other words, the optimal fractional order will be determined by considering which of high local resolution and complete field of view in imaging application is more importance. As seen in Fig. 7(f), the fractional order $p$=0.6 can be viewed as an optimal value if the reconstruction image is evaluated only according to its local resolution. But, the loss of edge distribution of the reconstructed amplitude in this case is obvious, as seen the upper and right edges in Fig. 7(a). It is clear that the FOV is not complete. In contrast, the reconstructed amplitude image when $p$=0.7 shows the complete FOV, although the amplitude contrast at the line pair element 5-6 is slightly less than that when $p$=0.6.
5. Conclusion
A filtering reconstruction method in a fractional Fourier transform domain for off-axis digital holograms is proposed, based on the principle of fractional-order Fourier transform. By theoretical derivation and analysis, we prove that the filtering reconstruction in a fractional-order transform domain with the order less than 1 can contain more high-frequency components under the same size of filtering region than that in conventional Fourier transform domain, so that the reconstruction imaging resolution can be improved. For solving the issue of a zero-order noise accompanied with an edge loss in a lower fractional order filtering, the combination of a spatial domain zero-padding with the zero-order term elimination is used in fractional transform filtering reconstruction. The reconstructed images in simulation verify its improvement on imaging quality in effect. Experimentally, the reconstructed results demonstrate that the imaging resolution is obviously improved by using fractional Fourier transform filtering. Since the derived expressions disclose that the filtering in a lower order Fourier transform domain can acquire more high-frequency information, the presented fractional Fourier transform filtering reconstruction provides a novel optional way for off-axis holographic imaging.
Further, it should be noted that in the fractional-order Fourier transform domain filtering reconstruction, the choice of the optimal fractional order should have a trade-off between two factors of the higher local resolution and the complete field of view. When the interested object information is distributed in the center region of the field of view and the higher resolution is required, the lower fractional transform order as possible is a good choice. If the field of view of the reconstructed images is required to be complete, the best choice is to take the minimum of the orders that can achieve the complete FOV.
Funding
National Natural Science Foundation of China (61575009).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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