The LFOCDHM is a cost-effective and compact system without any lenses or other optical elements between the object and the sensor plane, and it places the sample as close as possible to the imaging sensor to obtain a wide-field hologram. Multi-wavelength LFOCDHM uses different wavelengths of light sources to irradiate the sample to record different holograms shown in Figure 1.

The hologram recorded by LFOCDHM under the assumption of a thin sample is given by I n = | H n ( u n ) | 2 + ε n ,   n = 1,2 , … , N , (1) where N denotes the number of measurements, I n denotes the intensity image recorded by the sensor, H n denotes the transmission mode of diffraction, ε n denotes the noise in the imaging, and u n denotes the complex amplitude distribution function of the sample under λ n illumination, which is given by u n = A n e i φ n (2) where A n denotes the amplitude and φ n represents the phase. The absorption of the sample is assumed to be independent of the illumination wavelength, which is a reasonable assumption for most weakly scattering samples. Under different wavelength illumination, the amplitude is the same and the phase varies proportionally, which can be expressed as A 1 = A 2 = ⋯ = A N (3) φ n + 1 = λ n λ n + 1 φ n (4)

This assumption implies that   u 1 , u 2 , ⋯ , u N can be uniformly represented by u . We consider the u 1 as u , then u n can be expressed as: u 1 = u = A e i φ (5) u n = A e i λ 1 λ n φ (6)

Due to the inherent ill-posedness of the phase retrieval problem, finding the desired u is the efficient way to eliminate the twin image. To avoid tedious parameter calibration, we choose the alternating projection method as the iterative model for MWPR, which finds desired u by iterating back and forth between the object and image planes and adding intensity constraints on the image plane. Compared to the reconstruction from a single-intensity image ( N = 1 ) that finds desired u in a single known constraint set C 1 shown in Figure 2A, the reconstruction from some different intensity images ( N > 1 ) is easier to find desired u in the intersection of some known constraint sets   C 1 , C 2 , ⋯ , C N shown in Figure 2B, which is formulated as a feasibility problem: F i n d   u ∈   C 1 ∩   C 2 ∩ ⋅ ⋅ ⋅ ∩   C N (7)

The size of the intersection set is related to the diffraction diversity determined by the difference and number of holograms. When there is sufficient diffraction diversity, the size of the intersection will be small and the MWPR will find the desired u quickly and accurately. Small source-to-sample distances and a small number of measurements make it difficult to produce a sufficient number of holograms with significant differences, resulting in poor elimination effect on LFOCDHM. Therefore, additional constraints are required to be imposed on the object plane to realize clear imaging for LFOCDHM.

The intensity of the hologram is mainly influenced by the intensity of the reference light. Dividing the measured hologram I m e a s u r e by the background image I b a c k g r o u n d measured after removing the sample results in the normalized hologram I n o r m , which will avoid non-uniform illumination of the light source: I n o r m = I m e a s u r e / I b a c k g r o u n d (8)

The basic physical notion of energy conservation requires that absorption will not lead to an increased amplitude following a scattering process. Since the normalized hologram eliminates the effect of the reference light intensity which is equivalent to irradiating the sample with a unit light wave, the amplitude A on the object plane should not exceed 1. Therefore, during the iterative process, the amplitude part of regions with amplitude exceeding 1 are considered as interference from the twin image and should be replaced by 1 while the phase part remains, and the other regions are retained. This process can be expressed as: u ′ ( x , y ) = { u ( x , y ) ,     A ( x , y ) < 1 e i φ ( x , y ) ,     A ( x , y ) ≥ 1 (9) where u ′ denotes the updated complex amplitude distribution function and ( x , y ) is the Cartesian coordinate system on the object plane.

Figure 3 shows the computational flowchart of two common update strategies for MWPR in the k th iteration, where u ′ k , u 1 ′ k ,   u 2 ′ k , ⋯ u N ′ k denotes the updated complex amplitude distribution, A ′ k , A 1 ′ k ,   A 2 ′ k , ⋯ A N ′ k denotes the updated amplitude distribution, and φ ′ k , φ 1 ′ k ,   φ 2 ′ k , ⋯ φ N ′ k denotes the updated phase distribution. The incremental update strategy shown in Figure 3A updates the u n k at the λ n wavelength based on the u n − 1 ′ k that has updated at the λ n − 1 wavelength. The global update strategy shown in Figure 3B updates the u n k at the λ n wavelength based on the u k and takes the average of the updated u n ′ k as a result of the k th iteration.

The incremental update strategy converges quickly. However, it is an error accumulation process, resulting in iterative computation being sensitive to noise in the case of a small number of measurements. As a result, the incremental update strategy leads to unstable convergence and imaging artifacts under noise interference. Inexpensive imaging sensors are prone to introduce a series of disturbances such as scattering noise, dark noise, and readout noise during image acquisition, which is especially challenging for low-cost LFOCDHM. Compared with the incremental update strategy, the averaging operation in the global update strategy can effectively suppress the effect of noise and improves the convergence stability. Therefore, the global update strategy is more suitable for LFOCDHM. However, it converges slowly.

The use of acceleration methods can compensate for the disadvantage of slow convergence of global update strategy. Compared with other acceleration methods, the vector extrapolation acceleration method requires only a small amount of information about the iterative algorithm to improve the convergence speed and convergence accuracy, which is nonparametric. This acceleration method is a form of vector extrapolation, which gains a correction step and information for adjusting the step length based on previous iteration. Since IPR is to recover the missing phase, we chose the vector extrapolation acceleration method to update the phase. This process in the k th iteration can be expressed as: φ k + 1 = φ k + α k h k (10) where h k = φ k − φ k − 1 (11) α k = ∑ g k g k − 1 ∑ g k − 1 g k − 1 (12) g k = φ ′ k − φ k (13)

Algorithmic details about the MWPREGV are shown in Figure 4 as follows: (1) initializing the first estimation of u 0 with the average of all holograms back-propagated to the object plane with the angular spectrum method; (2) forward-propagating k th estimation of u n k to recording plane with the angular spectrum method and replacing the computed amplitude distribution of recording plane with the modulus of recorded holograms yet retaining the computed phase; (3) back-propagating this synthesized complex amplitude distribution to the object plane with the angular spectrum method and updating the computed amplitude distribution of object plane based on energy constraint to obtain the updated complex amplitude distribution u n ′ k ; (4) averaging the u n ′ k to obtain a globally updated complex amplitude distribution u ′ k ; (5) updating the phase distribution φ k based on the globally updated phase distribution φ ′ k by the vector extrapolation acceleration method yet retaining the globally updated amplitude distribution A ′ k and thus the result u k + 1 of the k th iteration is obtained; and (6) iteratively running steps (2) to (5) so that the reconstructed accuracy L o s s meets the given requirement ε or the number of iterations k reaches the maximum number k m a x .

To produce sufficient diffraction variations, a virtual object is illuminated by λ 1,2,3 = 623   nm,   523   nm,   460   nm , and its amplitude and phase under λ 1 illumination are shown in Figure 5. The other parameters are listed as follows: (1) the imaging size is 1.1 × 1.1 mm² (512 × 512 pixels); (2) the distance from light source to sample is 100 mm, and the distance from sample to sensor is 1 mm. To quantify the reconstruction quality, we use the relative error (RE) as reconstructed accuracy L o s s , which is defined as R E = ‖ u e s t − u t r u e ‖ 2 ‖ u t r u e ‖ 2 (14) where ‖ · ‖ 2 denotes the Euclidean norm, u e s t denotes the estimated value of u , and u t r u e denotes the ground truth value of u .

We first compared the performance of MWPR and MWPR based on energy constraint (MWPRE). When the distance from the virtual object to sensor is 30 mm, MWPR eliminates most of the twin image after 50 iterations, as shown in Figure 6A. When the distance from the virtual object to sensor is 1 mm, MWPR only partially eliminates the twin image after 50 iterations due to insufficient diffraction variation, resulting in a large deviation between u e s t and u t r u e , as shown in Figure 6B. In contrast, MWPRE after the same number of iterations almost completely eliminates the twin image for the same diffraction variation shown in Figure 6C. And it achieves better convergence accuracy and faster convergence speed, as shown in Figure 6D.

Next, we compared the convergence behavior of the incremental update strategy and global update strategy on the MWPRE (MWPREI and MWPREG). The MWPREI converges much faster than the global update strategy and has a better convergence accuracy when the ideal holograms and background images without noise are used, as shown in Figure 7A. Nevertheless, when adding white Gaussian noise to obtain holograms and background images with a signal-to-noise ratio of 30 dB, we find that the MWPREG achieves better convergence accuracy as expected, as shown in Figure 7B. Therefore, the global update strategy is more suitable for LFOCDHM with a lot of noisy interferences.

Finally, we tested the impact of the acceleration algorithm on the MWPREG, as shown in Figure 8. Compared with the MWPREG, the MWPREGV achieves more efficient elimination and faster convergence, which compensates for the disadvantage of slow convergence due to the global update strategy without tedious parameter calibration.

We tested the proposed method on ant mounts, housefly leg mounts, and bee wing mounts. The experimental setup is shown in Figure 1. An RGB LED (LZ4-04DCA, λ 1,2,3 = 623   nm ,   523   nm,   460   nm ) is utilized as the illumination source to avoid scattering noise, and a CMOS sensor with a pixel size of 2.2 μm (2592 × 1944, MT9P031) is utilized to record the hologram. The distance from the light source to the sample is 100 mm, and the distance from the sample to the sensor is 1.2 mm. Since the ground truth value of u is unknown in the experiment, the L o s s is used to evaluate the speed of convergence rather than the quality of the reconstruction, which is given by: L o s s = ‖ | H 1 ( u ) | − I 1 ‖ 2 (15)

Figure 9B shows full FOV(∼27.78 mm²) holograms of the LFOCDHM captured at the 523 nm wavelength, which has a larger field of view than microscopic images taken with a 4 × times objective lens shown in the Figure 9A and Figure 9C show the reconstructed amplitude distribution by using MWPR after 15 iterations, indicating that only partial elimination of the twin image can be achieved. In contrast, MWPRE eliminates most of the twin image after 15 iterations, as shown in Figure 9D. However, the reconstruction by MWPRE is not satisfactory due to the interference of noise, which is far from the microscopic images taken with a ×4 objective lens. Under the same conditions, MWPREGV realizes more efficient elimination effect after the same number of iterations, making the reconstruction similar to the microscopic images taken with a ×4 objective lens, as shown in Figure 9E. And as shown in Figure 9F, MWPREGV converges to a stable value with a faster rate compared to MWPREG.