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Abstract
Compressive holography is a relatively time-consuming image estimation in convex optimized problem. We propose an efficient block-wise algorithm to limit the searching space and reduce the calculation time while keeping the reconstruction quality. The effective anti-aliasing boundary of the sub-hologram is located to determine the block size for compressive reconstruction in the total-variation two-step iterative shrinkage/thresholding algorithm. Padded sub-holograms could be reconstructed in parallel by using multi-core processors. Compared with the traditional compressive holography, the block-wise algorithm could take approximately 1/50 of the reconstruction time and achieve an improved reconstruction quality.
© 2017 Optical Society of America
1. Introduction
Coherent lensless imaging [1, 2] is an effective approach to reconstruct a high-resolution image from measured diffraction patterns. Digital holography, as a kind of coherent lensless imaging, has been widely applied in phase imaging [3–5] and topography measurements [6]. Lensless in-line digital holography (LIDH) is a potential solution to maintain high space-bandwidth product (SBP) without optical lenses, which can enlarge the viewing angle compared with off-axis holography [7, 8]. LIDH records an interference pattern between the scattered light from objects and the non-scattered background light. It is a compact and effective imaging set-up and can realize numerical focusing. LIDH can support a sizeable distance between object and sensor for 3-D imaging with a large volume. However, the number of sampling pixels is much smaller than the number of voxels in 3-D object datacube. The 3-D object reconstruction from one-shot 2-D digital hologram is an ill-posed problem [9], which cannot be solved by the conventional methods. It is a complicated and critical issue to suppress high-order noises and the twin imaging. Fortunately, compressive sensing (CS) has been proposed and verified for solving the ill-posed problems effectively. CS has been applied in quantum imaging [10], spectral imaging [11], optical encryption [12] and ultrafast photography [13]. CS has shown its great power to solve the inverse problem of optical propagation in LIDH [14–18].
In compressive holography proposed by Brady et al. [19], compressive measurements were successfully applied to reconstruct a 3-D layered object from a 2-D hologram based on the sparsity constraints. The crosstalk noises among layers were effectively suppressed by using the two-step iterative shrinkage/thresholding (TwIST) algorithm [20]. But the reconstruction cost about 4 hours. Wang et al. combined compressive holography and coded exposure techniques to achieve 4-D reconstruction in spatial and temporal domains from one shot coded captured hologram [21]. High speed video could be acquired by adopting the temporal compressive method of coded exposure with a DMD while it required as long as 7 hours. Wu et al. used the 4f amplified system in the in-line compressive holography to enhance the axial resolution of the reconstruction [22], which required more than one hour to reconstruct a two-layer object. Leportier et al. presented an experimental set-up for compressive optical measurements of phase-shifted holograms based on a single-pixel sensor in more than one hour [23]. In the above works, one to seven hours were required to accomplish the reconstruction. Compressive holography requires large calculation amount to search the global optimal solution, especially for 3-D data cube which usually has millions of variables. For the applications requiring fast reconstruction, the size of the object and the number of the layers have to be greatly reduced, which results to a limited imaging resolution. Endo et al. used the graphics processing unit (GPU) to obtain about 20 times faster calculation speed [24]. The operations of mass 2D-FFT, multiplications of the transfer functions and sum of the depth layers are accelerated based on the advanced hardware. However, there was no consideration of the optical field recording which is multiplex in holography.
Diffraction fringes of the 3-D object only reaches a fraction of the full hologram and detected correctly by the image sensor due to the limited pixel pitch. The reconstruction of the hologram beyond the effective zone means the useless searching space, resulting in the massive waste of reconstruction time. In this work, a block-wise compressive holography (BCH) algorithm is proposed to reduce calculation time by using multi-core processors. The effective anti-aliasing boundary in the block-wise hologram is investigated, which could determine the parameters of zero padding in compressive reconstruction. We implement the BCH algorithm to reconstruct the dandelions seed parachutes’ holograms [19]. The time consumption is compared with that of the traditional compressive holography algorithm. A high SBP LIDH platform is experimentally demonstrated.
2. Multi-layer compressive holography model
Compressive holography reconstructs a 3-D layered object from a snapshot 2-D hologram. The multi-layered object is illuminated by a plane wave. The hologram recorded on the image sensor is considered as the interference between the scattered field U of the 3-D object and the non-scattered light A which is the reference wave. The captured intensity I(x,y) by the sensor is
where A is defined as 1. * donates complex conjugate. Re expresses the operation of acquiring the real component. n is the model error which includes high-order noises. And the scattered field U in the hologram plane can be expressed under the first Born approximation aswhere donates the convolution. PSF(x,y;zi) is the point spread function in free space. PSF(x,y;zi) is determined by the propagation distance zi. The scattered field U in the imaging plane could be described with the angular spectrum diffraction, when [18]where is the number of pixel along transverse direction, is the size of pixel pitch and is the wavelength of the illumination. The discrete formula is given bywhere is the angular spectrum representation of , and is inverse 2D-FFT operation. is the discrete formula of angular-spectrum transfer function. Through the matrix processing, Eqs. (2) and (4) can be rewritten aswhere S is the vectorization of the 3-D layered object s with the Nz layers along the z direction, the number of pixels Nx along the x direction and the number of pixels Ny along the y direction. The size of S is Nx × Ny × Nz. The measured intensity g with the size of Nx × Ny is the vectorization of the captured intensity I. F and F−1 with the size of (Nx × Ny × Nz) × (Nx × Ny × Nz) are the block diagonal matrix which represents the operations of the 2D-FFT and the inverse 2D-FFT. P with the size of (Nx × Ny × Nz) × (Nx × Ny × Nz) is the matrix of the angular-spectrum transfer function. T with the size of (Nx × Ny) × (Nx × Ny × Nz) represents the summation of the complex field at different layers. With neglecting high-order noises, a relation between the measured intensity g and the 3-D layered object S can be expressed as the forward propagation model g = 2Re(HS) + n, where the measurement matrix H has the size of (Nx × Ny) × (Nx × Ny × Nz). The 3-D layered object S with a large data volume can be down-sampled into a 2-D hologram g.The large data elements are reconstructed from a few data elements in the forward propagation model, the reconstruction of S is an ill-posed inverse problem. The TwIST algorithm could be applied to estimate the 3-D object by solving
where is the l2 norm. is the regularization parameter. is the total variation (TV) function as the sparsity basis because the object is expected with sharp edges. In this work, the nonisotropic discrete TV regularizer is used, which is given bywhere and denote horizontal and vertical first-order local difference operators on the 2-D lattice without boundary corrections.A simulation is designed to show the power of compressive holography. The objects with specific patterns at different layers are shown in Fig. 1(a). The object is assumed as a 64 × 64 × 5 pixel datacube from Layer 0 to Layer 4. The size of measurement matrix is (64 × 64) × (64 × 64 × 5). The lateral pixel pitch is 10 μm. The axial coordinate goes from Layer 0 (z = 1 mm) to Layer 4 (z = 10.2 mm) and the interlayer spacing is 2.3 mm. The in-line holographic measurements are designed with the angular spectrum method at the wavelength of 350 nm. The phases of the propagation kernels at different layers are shown in Fig. 1(b). Optical wave propagation in free space at different distances has different propagation kernels. The distributions of scattered object waves at different layers are projected onto the layer of sensor under different propagation kernels. The objects at different layers are reconstructed by using traditional back-propagation (TBP) algorithm, as is shown in Fig. 1(c). Due to the essential defect of 2D-2D solver with the neglect of the defocus effect, the reconstructed layers are blurred with out-of-focus noise. The estimations are reconstructed by using the TCH algorithm as shown in Fig. 1(d). All the layers are reconstructed accurately at the correct depth with high signal-to-noise ratio (SNR). The out-of-focus noises are almost filtered. The TCH algorithm is proven to be effective for reconstructing layered 3-D objects.
Fig. 1 (a) 3-D distribution of the layered object. (b) Phase of propagation kernel at different layers. (c) Reconstructed layers with traditional back-propagation (TBP) algorithm. (d) Reconstructed layers with traditional compressive holography (TCH) algorithm.
To assess the characteristics of calculation time in the TCH varying with the number of reconstructed voxels of the 3-D object datacube, the curve is given as shown in Fig. 2(a). The curve is obtained through the least square fitting method, which is approximately linear with the square of correlation coefficient of 0.9996. The stopping criterion of the loops is set to a fixed number of iteration, which is defined as 500 in this work. It means that the calculation time with the same iteration number varies linearly with the size of the number of reconstructed voxels. In the TwIST algorithm with TV function, there are four core operations, which are gradient calculation, denoising operation, iterative estimation and residual calculation. Each operation has different consuming time. The stacked chart of the calculation time is shown in Fig. 2(b). The denoising operation consumes a large proportion of total time which is equal to 71.8%. The gradient calculation, iterative estimation, residual calculation consume 6.3%, 13.6% and 1.4% of the total time, respectively. The rest operations consume 6.9% of the total time.
Fig. 2 (a) Time of reconstruction versus the number of reconstructed voxels of 3-D object datacube by using the TwIST algorithm. (b) Stacked chart of consuming time for 500 iterations.
The TCH algorithm as a convex optimized problem is limited by the tremendous amount of the reconstruction time. The amount of computations is reflected by the size of the measurement matrix. When the number of reconstructed voxels increasing, the size of the measurement matrix and the searching space would be enlarged accordingly, which leads to the increase of the calculation time.
3. Hologram segmentation based on sampling theorem
For holographic imaging, the diffraction wavefront of an object is transformed into a hologram by the image sensor. Because the pixel pitch of the sensor is limited, usually a few microns, only part of the wavefront can be detected correctly according to the sampling theorem. The wavefront consists of plane waves with different angles , as is shown in Fig. 3(a), for different spatial frequencies
When recording, only the frequency componentcan be detected correctly, where is the highest frequency that could be recorded by the image sensor. is the pixel pitch. Only the wavefront whose propagation angel forcan be detected correctly. So the maximum effective diffraction angle isOnly the field located inside the angle can be detected correctly.Fig. 3 (a) Maximum recording angle of image sensor determined by the pixel pitch. (b) Maximum diffraction angle of the hologram when reconstruction. (c) The two-points object with the size of 1 μm. (d) The hologram sampled by an image sensor with the pixel size of 1 μm. (e) The hologram sampled by an image sensor with the pixel size of 8 μm. (f) The reconstructed results from the hologram with the pixel size of 1 μm without occluding the rectangular region. (f) The reconstructed results from the hologram with the pixel size of 1 μm with occluding the rectangular region. (h) The reconstructed results from the hologram with the pixel size of 8 μm with occluding the rectangular region.
When reconstruction, due to the limited pixel pitch of the image sensor, the recorded digital hologram can only reconstruct the information from a certain viewing frustum, as is shown in Fig. 3(b). Aliasing would appear when the recorded information beyond the viewing frustum is used to reconstruct the object [25]. According to the grating equation, the maximum diffraction angle of the image sensor is given by:
where is the half-angle of the viewing frustum. For a certain sub-hologram, the corresponding reconstructed object must be restricted into a fixed area in order to prevent aliasing. The effective anti-aliasing boundary of one pixel can be calculated as:where z is the distance between the object plane and the hologram plane. By comparing the procedure of recording and reconstruction, it is evident that using sub-holograms to reconstruct corresponding objects is self-consistent.To verify the validity of the theoretical analysis, we use a part of the hologram to reconstruct the corresponding part of the object. As shown in Fig. 3(c), a two-point image with the size of 500 × 500 is used as the object. The size of two rectangular points is 1 μm. The distance from the object plane to the hologram plane is 1 mm and the propagation is calculated by angular spectrum method. The illumination wavelength is 532 nm. The hologram with the pixel size of 1 μm is sampled as shown in Fig. 3(d). The corresponding one-dimensional intensity distribution shows the dramatic variation, which illustrates that high-frequency parts are remained. And the hologram with the pixel size of 8 μm is sampled as shown in Fig. 3(e). The variation of intensity is smooth and the fringe patterns only in a small region are retained. The hologram in Fig. 3(d) could be correctly reconstructed, as shown in Fig. 3(f). When the hologram in Fig. 3(d) is occluded by the rectangular region, the reconstruction is shown in Fig. 3(g). The reconstructions keep the high fidelity and the full width at half maximum (FWHM) of the reconstructions is 1 μm for both points. However, when the same region is occluded in the hologram in Fig. 3(e), the reconstruction is shown in Fig. 3(h). Only the point on the left side is reconstructed with the FWHM of 12 μm. The gray-scale depth of the right point is missed entirely. Thus, for the object with a specific size, the contribution area is only located at the limited region of the hologram. The hologram with the effective anti-aliasing boundary is enough for reconstructing the corresponding zone in the object plane. The hologram could be divided into multiple sub-holograms for parallel reconstruction processing.
4. Block-wise compressive holography
The hologram used in this section is obtained from the Duke Imaging and Spectroscopy Program [19]. The wavelength of the illumination is 632 nm. Two dandelions seed parachutes were placed at the distance of 15 mm and 56 mm away from the sensor. A 1024 × 1024 measurement array with 10 bit accuracy was captured using a Lumenera LU100 camera, which was then down-sampled to a 512 × 512 array with equivalent pixel pitch of 10.4 μm. The measurement array was zero-padded with 100 pixels in both sides to remove the effect of circular convolution caused by using FFT. The ten-layer 3-D estimations of 712 × 712 × 10 are reconstructed successfully with the TwIST algorithm. The size of the measurement matrix is (712 × 712) × (712 × 712 × 10). The reconstruction with the TCH algorithm needs 5196 s. The reconstruction was performed on a personal computer with Intel Core i7-6700K at 4 GHz and 32 GB of RAM. Matlab R2016a with 64-bit application was used for algorithm implementation. All computations were implemented under the same computer hardware and software.
The BCH algorithm is proposed to reduce the reconstruction time of compressive holography, as is shown in Fig. 4. The hologram of 512 × 512 pixels is divided into four sub-holograms of 256 × 256 pixels. Each sub-hologram is expanded with 50 pixels on its four sides to cover the anti-aliasing boundary. The expanded zones are padded with surrounding hologram or zeros for those beyond the hologram. The sub-holograms are used in compressive holographic algorithm to reconstruct corresponding ten-layer object. Since a smaller blocked hologram of 356 × 356 pixels is applied, the size of the measurement matrix is reduced to (356 × 356) × (356 × 356 × 10).
Single-Program-Multiple-Data (SPMD) is a technique employed to achieve parallelism. Tasks are decomposed and processed simultaneously on multiple work-pool (local physical core) with different inputs. In the TCH, the whole hologram is used in each iteration. However, only the information within the viewing frustum can be effectively recorded, as the pixel pitch of the digital sensor is limited. For the sake of time reduction, the hologram is divided into small sub-holograms. Each sub-hologram is used to reconstruct the multi-layer object in the corresponding area independently. Then, the reconstructed results of small sub-hologram can be spliced into the whole multi-layer 3-D scenes. In BCH algorithm, four sub-holograms are reconstructed respectively in different work-pool as shown in Fig. 5. The reconstructions with different sub-holograms in different work-pool run in parallel. The reconstructed time of four sub-holograms is 1287s, 1279s, 1289s and 1292s, respectively. The total reconstructed time of the BCH algorithm is 1377s, including the time of memory allocation and data communication. Four reconstructed ten-layer object are spliced into a full ten-layer 3-D scene. The dandelions seed parachutes are successfully reconstructed in the correct distances of 15 mm at Layer 3 and 56 mm at Layer 8, respectively. The reconstructed images in Figs. 6(c) and 6(d) have the same fidelity level with the reconstructed images in Figs. 6(a) and 6(b) using the TCH algorithm. The total calculation time using the BCH algorithm is only one fourth of the reconstruction time using the TCH algorithm.
Fig. 5 The parallel frame of Single-Program-Multiple-Data based on block-wise compressive holography (BCH).
Fig. 6 The compressive holographic reconstructions. (a) z = 16 mm, with TCH algorithm, (b) z = 56 mm, with TCH algorithm, (c) z = 16 mm, with proposed BCH algorithm, (d) z = 56 mm, with proposed BCH algorithm.
5. Experiments
According to the viewing frustum of limited pixel pitch in Eqs. (12) and (13), short propagation distance and short illumination wavelength could reduce the diameter of the effective anti-aliasing boundary. The hologram could be divided into more sub-holograms for parallel reconstruction. The optical set-up is shown in Fig. 7 to verify the proposed algorithm. Three pieces of glass with the thickness of 2.3 mm are stacked above the image sensor array. The optical refractive index of soda-lime glass is 1.52. Designed patterns are etched on the glass with the lithographic techniques. A collimated illumination with the green laser at the wavelength of 532 nm is applied. The equivalent wavelength in the soda-lime glass is 350 nm. An in-line hologram for multi-layer objects was captured by using a CMOS (QHY 163M) with the pixel size of 3.8 μm. The hologram with 1024 × 1024 pixels is shown in Fig. 8(a). The top view and the side view of the experimental set-up are shown in Figs. 8(b) and 8(c), respectively. Three patterns are shown, such as a line with the resolution of 10 μm at Layer 1, a pattern of chessboard with the resolution of 50 μm at Layer 2 and a pattern of π with the resolution of 10 μm at Layer 3. Some dusts with the size of a few microns are located at Layer 4. The reconstructed layered images are shown in Figs. 9(a)-9(c) for TBP, TCH and BCH algorithms, respectively. Both TCH and BCH algorithms have great power of removing out-of-focus noise and keeping high fidelity. The BCH algorithm consumes 191s with 64 block reconstructions in parallel, compared with the time consumption of 10296 s by using the TCH algorithm.
Fig. 8 (a) The captured in-line hologram with multi-layer objects (b) The top view of experimental set-up (c) The side view of experimental set-up.
The reconstruction estimations in Layer 3 are presented in Fig. 10. The reconstructed image and the cross section with the TBP, TCH and BCH algorithms are shown in Figs. 10(a)-10(c), respectively. Three-dimensional profiles of the patterns are also shown in Figs. 10(d)-10(f), respectively. The reconstruction with TBP algorithm has severe crosstalk noise from other layers, while the reconstruction with both TCH and BCH algorithms has very high signal-to-noise ratio with little noise. To quantitatively evaluate the reconstructed layers, image contrast K is used to describe the image quality [26], which is given by
where Imax is the maximum value of the reconstructed image intensity, Imin is the minimum value of the reconstructed image intensity. Table 1 presents the image contrast of different patterns with the three methods. For the line with 10 μm at Layer 1, the side-lobe noise is obvious by using TBP algorithm. The image contrasts of the reconstructions are 0.3197, 0.1110, 0.1281 and 0.1473 for Layer 1, Layer 2, Layer 3 and Layer 4, respectively. The image contrasts of the reconstructions by using TCH are 0.5420, 0.0555, 0.2489 and 0.8141, respectively. By using BCH algorithm, the image contrasts of the reconstructions are 0.9075, 0.6820, 0.5013, and 0.9362, respectively. It means that the BCH algorithm is more robust compared with the TCH algorithm. The BCH algorithm can locate an accurate and small searching space, which results in improved reconstruction quality and high convergence speed.Fig. 10 The cross section of different reconstructions by (a) TBP algorithm, (b) TCH algorithm and (a) BCH algorithm. Profiles of the reconstructed pattern by (d) TBP algorithm, (e) TCH algorithm and (f) BCH algorithm.
Table 1. Image contrasts of reconstruction at different layers with TBP, TCH, and BCH algorithms.
6. Conclusions and outlook
The proposed BCH algorithm can reduce the calculation time greatly compared with the TCH algorithm. According to the viewing frustum of a limited pixel pitch in the image sensor, short propagation distance and short illumination wavelength mean a small diameter of the effective anti-aliasing boundary. Our experiments could achieve 50 × speedup for 64-channel parallelization. The accurate location of the searching space results in an improved reconstruction quality.
Noted that the diameter of the anti-aliasing boundary is dependent on the propagation distance, which causes that the BCH algorithm is on longer feasible to split the hologram into blocks for the object at a long distance. When the object is beyond a certain distance, the quality of the reconstruction in the boundary will reduce as the propagation distance increases. The choice of the size of block and the number of the padding would also affect the reconstruction quality.
The BCH is a new framework based on the key property that any point source in the object plane only influences a limited region on the sensor resulting from the limited pixel pitch of the digital hologram. The advantages of the proposed block-wise approach are not dependent on the TwIST algorithm. It could also be applied in other CS algorithms to save the calculation time. Meanwhile, the GPU could help accelerate the BCH algorithm for sub-second processing, which will be considered in the future work.
Funding
Innovation Method Fund of China (2015IM020500); National Natural Science Foundation of China (NSFC) (61775117, 61327902, 61505095); National Instrument Program (2012YQ 140005); National Basic Research Program of China (2013CB328803).
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