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Incoherent coded aperture correlation holographic imaging with fast adaptive and noise-suppressed reconstruction

Yuhong Wan, Chao Liu, Teng Ma, Yi Qin, and Sheng lv

Open Access Open Access

Abstract

Fast and noise-suppressed incoherent coded aperture correlation holographic imaging is proposed, which is utilized by employing an annular sparse coded phase mask together with adaptive phase-filter cross-correlation reconstruction method. Thus the proposed technique here is coined as adaptive interferenceless coded aperture correlation holography (AI-COACH). In AI-COACH, an annular sparse coded phase mask is first designed and generated by the Gerchberg-Saxton algorithm for suppressing background noise during reconstruction. In order to demonstrate the three-dimensional and sectional imaging capabilities of the AI-COACH system, the imaging experiments of 3D objects are designed and implemented by dual-channel optical configuration. One resolution target is placed in the focal plane of the system as input plane and ensured Fourier transform configuration, which is employed as reference imaging plane, and moved the other resolution target to simulate different planes of a three-dimensional object. One point spread hologram (PSH) and multiple object-holograms without phase-shift at different axial positions are captured by single-exposure sequentially with the annular sparse CPMs. A complex-reconstruction method is developed to obtain adaptively high-quality reconstructed images by employing the cross-correlation of PSH and OH with optimized phase filter. The imaging performance of AI-COACH is investigated by imaging various type of objects. The research results show that AI-COACH is adaptive to different experimental conditions in the sense of autonomously finding optimal parameters during reconstruction procedure and possesses the advantages of fast and adaptive imaging with high-quality reconstructions.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

A complete imaging chain is now generally considered to be composed of illumination, object, optical system, detection and reconstruction. Any link of the imaging chain can be designed and optimized in order to improve the imaging performance aiming at certain potential applications. Many technologies have been developed by modulating illumination [1–3], coding wavefront [4,5], coding aperture [6,7] or developing reconstruction algorithms [8,9]. A new type of incoherent digital holography called coded aperture correlation holography (COACH) was proposed in 2016 [10]. The technique of COACH is originated from Fresnel incoherent correlation holography (FINCH) [11,12], but possess the advantages of higher axial-resolution, and spectral-resolution. In COACH system, optical skills are still employed for splitting wave to form a point-object hologram by self-interference of two-waves. With further investigation of COACH, it is founded that the interference of two-wave is not necessary and the 3D position information of the object can be encoded by recording holograms of point-objects at different axial-positions which matched coded phase mask [13]. Interferenceless coded aperture correlation holography (I-COACH) with relatively simple optical set-up [14], even though lensless I-COACH with aberration-free [15], larger field of view has been sequentially proposed [16].

As incoherent digital holographic imaging is concerned, the reconstruction algorithm is a critical link of imaging chain. In order to obtain reconstruction images with good quality, cross-correlation reconstruction algorithm together with phase-shifting technique, image averaging technology [13], and non-linear adaptive reconstruction algorithm have been also developed [17]. Various methods have been proposed to improve time resolution and imaging quality of I-COACH [17,18], but usually with the sacrifice some other properties of imaging field of view (FOV) [19] or consuming a lot of off-line calculation time [13]. The method of improving signal-to-noise ratio (SNR) and time resolution of I-COACH through selecting point spread function sparseness combined bipolar hologram cross-correlation reconstruction or single exposure non-linear reconstruction are some typical techniques [20–24]. However, those still take a lot of off-line calculation time to obtain a reconstructed image with higher SNR, which limits the further application of I-COACH technology.

In any wise, the reconstruction mathematics algorithms, as an inverse problem of imaging forward model is expected to be not ill posed. Therefore, it is necessary and very important to treat recording and reconstruction as interrelated or indivisible when developing new digital or computational imaging methods. Inspiring by annular coded aperture imaging and the effect of PSF sparseness on imaging, we introduced annular sparse distribution as coded phase mask to interferenceless coded aperture correlation holographic imaging in order to improve imaging characteristics by engineering the point spread function of the system. We proposed here a fast and noise-suppressed noninterference coded aperture holographic imaging method by adaptive phase filtering reconstruction algorithm combined with annular sparse of coded phase mask and coined this technique as adaptive interferenceless coded aperture correlation holography (AI-COACH). In our proposed AI-COACH, an annular sparse coded phase mask is first designed and generated by modified GS algorithm to suppress imaging background noise. Only one PSH and multiple holograms without phase-shift of simulated 3D extended object are captured by single-exposure sequentially with the annular sparse CPMs. Non-linear reconstruction algorithm are employed to optimize adaptively the phase filter parameter during reconstruction process. All the measures taken in AI-COACH work together resulting in a fast and noise-suppressed incoherent holographic 3D imaging technique which is adapt adaptively to different experimental conditions.

2. Methodology

2.1 Synthesis of the coded phase mask

In the AI-COACH system, the phase mask (as shown in Fig. 1, an enlarged view of the center of the mask image) is a combination of CPM and quadratic phase mask (QPM). The CPM is computed using a modified GS algorithm [25], while the QPM with the refractive lens L0 are used to satisfy the Fourier Transform relation between the CPM plane and the sensor plane. The schematic of the CPM generation is shown in Fig. 1. The modified GS algorithm is initialized with a random phase and uniform amplitude on the CPM plane. The complex valued CPM is then Fourier transformed to the Fourier plane (i.e., sensor plane) and the amplitude of the transformed CPM is replaced by the predefined annular sparse distributed dots instead of random sparse distributed dots over a predefined area on the sensor plane but the phase is kept unchanged, and then the inverse Fourier Transform of CPM is performed. Repeat the calculation until the difference between consecutive CPMs is negligible.

 figure: Fig. 1.

Fig. 1. Synthesis of annular sparse CPM schematic by GS algorithm.

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In our GS algorithm, the light scattering degree σ of CPM in the spectral domain is defined as the ratio of the constrained spectral bandwidth a and the maximum spectral bandwidth A as σ = (a/A). Symmetrical annular intensity dots with different widths are set in pixels to realize the sparse operation of system response. Under a certain scattering degree, different annular widths have different sparse dots. Therefore, the CPMs with different sparse dots under different scattering degrees can be obtained by GS algorithm, which in turn will affect the imaging characters of the system and an optimized CPM with expected scattering degree and sparse dots can be selected.

2.2 Recording mechanism in AI-COACH

The optical setup of the AI-COACH system is shown in Fig. 2. The object or pinhole is illuminated critically with incoherent light to guarantee complete incoherence between any two points on the object. The diffracted light of the object is collected and collimated by the lens L1 located at a distance of zs from the object. The light diffracted from the object is polarized by a polarizer P along the active axis of the spatial light modulator (SLM) and totally modulated by the mask loaded onto the SLM which is synthesized by a CPM which generated by the GS algorithm and a QPM with focal length fs. The light modulated by the SLM is recorded by charge coupled device (CCD) located at a distance of zh from the SLM. A pinhole is firstly placed on the front focal plane of lens L1and f1 is expressed as the distance from the point object to the lens. Only one point spread hologram and a series of holograms of different depth sections of the 3D object were recorded by using the same CPM. The reconstructions of different depth sections of the 3D object are obtained by cross-correlation operation between the PSH and OHs. The mathematical model of recording mechanism and reconstruction theoretical analysis are demonstrated here.

 figure: Fig. 2.

Fig. 2. Optical setup of AI-COACH system.

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As shown in Fig. 2, the complex amplitude of light passes through the lens L1 and polarizer P, and then modulated by the mask on the SLM. The light diffracted from the SLM propagates by a distance of zh before it reaches the CCD. The object light is collimated by the lens L1. When a pinhole is placed on the position of $({{x_s},{y_s},{z_s}} )$, the intensity pattern Ipsh on the sensor plane is given as [13],

$$\begin{array}{l} {I_{psh}}({\overline {{r_h}} ;\overline {{r_s}} ,{z_s}} )= {\left|{\sqrt {{A_s}({\overline {{r_s}} ;{z_s}} )} \textrm{C}L\left( {\frac{{\overline {{r_s}} }}{{{z_s}}}} \right)Q\left( {\frac{1}{{{z_s}}}} \right)Q\left( { - \frac{1}{{{f_1}}}} \right){\rm{exp}} [{i\phi ({\overline r } )} ]Q\left( {\frac{1}{{{f_{s\textrm{lm}}}}}} \right) \ast Q\left( {\frac{1}{{{z_h}}}} \right)} \right|^2}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{ = }{\left|{\sqrt {{A_s}({\overline {{r_s}} ;{z_s}} )} CL\left( {\frac{{\overline {{r_s}} }}{{{z_s}}}} \right)Q\left( {\frac{1}{{{z_1}}}} \right){\rm{exp}} [{i\phi ({\overline r } )} ]\ast Q\left( {\frac{1}{{{z_h}}}} \right)} \right|^2}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{ = }{I_{psh}}\left( {\overline {{r_h}} - \frac{{{z_h}}}{{{z_s}}}\overline {{r_s}} ;0,{z_s}} \right) \end{array}$$
where the sign * is a 2D convolution, ${z_1} = {{{z_s}{\kern 1pt} {f_\textrm{1}}{f_{slm}}} / {({{f_\textrm{1}}{f_{slm}} + {z_s}{f_\textrm{1}} - {z_s}{f_{slm}}} )}}$, $\sqrt {{A_s}({\overline {{r_s}} ;{z_s}} )}$ is the amplitude of the object point at $({{x_s},{y_s},{z_s}} )$. C is a complex constant. $\overline {{r_s}} = ({{x_s},{y_s}} )$ is the transverse location vector. $\overline {{r_h}} = ({{x_h},{y_h}} )$ is the transverse location vector on the CCD. ${{{z_h}} / {{z_s}}}$ is the transverse magnification. The function $\phi ({\overline r } )$ represents the phase of the pseudorandom calculated by the GS algorithm. L and Q are a linear and a quadratic phase functions, given by $L({{{{{\overline r }_s}} / z}} )= {\rm{exp}} [{{{i2\pi ({{b_x}x + {b_y}y} )} / {\lambda z}}} ]$ and $Q(b )= {\rm{exp}} [{{{\textrm{ - }i\pi b({{x^2} + {y^2}} )} / \lambda }} ]$, respectively.

The system of AI-COACH is a linear space invariant system. Assuming that the three-dimensional sample is composed of K two-dimensional plane, and the intensity pattern obtained in the CCD plane is the convolution of the 2D object in the axial position of zi = zs and the system point spread hologram. Therefore the intensity pattern of the object on the image sensor is given as,

$$\begin{array}{l} {I_{oh}}({\overline {{r_h}} ;{z_s}} )= {O_{{z_i}}} \ast {I_{psh}} = \sum\limits_j^N {{a_j}\delta ({\overline {{r_h}} - {{\overline r }_j}} )} \ast {I_{psh}}\left( {\overline {{r_h}} - \frac{{{z_h}}}{{{z_s}}}{{\overline r }_j};0,{z_s}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_j^N {{a_j}} {I_{psh}}\left( {\overline {{r_h}} - \left( {1 + \frac{{{z_h}}}{{{z_s}}}} \right){{\overline r }_j}; 0,{z_s}} \right){\kern 1pt} = \sum\limits_j^N {{a_j}{I_{psh}}\left( {\overline {{r_h}} - \left( {1 + \frac{{{z_h}}}{{{z_s}}}} \right){{\overline r }_j}; 0,{z_s}} \right)} \end{array}$$
where ${O_{{z_i}}}$ is the input 2D complex amplitude of the object in the axial position of zi = zs.

According to the published work [17–24], the object reconstruction is done by cross-correlating ${I_{oh}}({\overline {{r_h}} } )$ with the phase-only filtered [26,27] version of ${I_{psh}}({\overline {{r_h}} } )$ given by ${I_{psh}}{({\overline {{r_h}} } )^{\prime}} = F.T{.^{ - 1}}\{{{\rm{exp}} [{i \cdot \arg ({F.T.({{I_{psh}}({\overline {{r_h}} } )} )} )} ]} \}$, the reconstructed object can be written as [17],

$$\begin{array}{l} {I_{re}} = {I_{oh}}({\overline {{r_h}} } )\otimes {I_{psh}}{({\overline {{r_h}} } )^{\prime}} = F.T{.^{ - 1}}\{{{{\overline I }_{oh}}({\overline {{r_h}} } )\cdot {{\overline I }_{psh}}{{({\overline {{r_h}} } )}^{\prime}}^ \ast } \}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = F.T{.^{ - 1}}\{{|{{{\overline I }_{oh}}} |{\rm{exp}} ({i \cdot \arg ({{{\overline I }_{oh}}({\overline {{r_h}} } )} )} ){\rm{exp}} ({ - i \cdot \arg ({{{\overline I }_{psh}}({\overline {{r_h}} } )} )} )} \}\end{array}$$
where the sign ${\otimes}$ is 2D cross-correlation. F.T. denotes a 2D Fourier transform and F.T.-1 denotes a 2D inverse Fourier transform. ${\overline I _{oh}}({\overline {{r_h}} } )$ and ${\overline I _{psh}}{({\overline {{r_h}} } )^{\prime}}$ are the 2D Fourier transforms of ${I_{oh}}({\overline {{r_h}} } )$ and ${I_{psh}}{({\overline {{r_h}} } )^{\prime}}$. Based on the convolution theorem [18,24], each object point is reconstructed by autocorrelation of the two complex holograms. Although the original object can be retrieved by the above reconstruction algorithm [shown as Eq. (3)], due to use of a phase-only SLM, the constraint in the CPM spatial spectrum domain just was satisfied in the phase, and the magnitude in the spatial spectrum domain can only be approximated and far from the constraint of being uniform, therefore the sidelobes of point image are higher and the background noise is greater.

2.3 Adaptive reconstruction method

In order to suppress the background noise and obtain high-quality reconstructed images adaptively through the single-exposure PSH and the OH in the AI-COACH system, we developed the phase filter cross-correlation reconstruction method as shown in Eq. (4),

$$\begin{array}{l} {I^{\prime}}_{re} = {I_{oh}}({\overline {{r_h}} } )\otimes {I_{psh}}{({\overline {{r_h}} } )^{\prime}}\textrm{ = }\left\{ {\left. {\sum\limits_j^N {{a_j}} {I_{ps{h^{\prime}}}}\left( {\overline {{r_h}} - \left( {1 + \frac{{{z_h}}}{{{z_s}}}} \right){{\overline r }_j}; 0,{z_s}} \right)} \right\}} \right. \otimes {I_{psh}}{({\overline {{r_h}} } )^{\prime}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = F.T{.^{ - 1}}\left\{ {\left. {F.T.\left[ {\sum\limits_j^N {{a_j}\delta ({\overline r - {{\overline r }_j}} )} } \right] \cdot |{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |{\rm{exp}} ({i \cdot \arg ({{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} )} )\cdot \frac{{{\rm{exp}} ({ - i \cdot \arg ({{{\overline I }_{psh}}({\overline {{r_h}} } )} )} )}}{{|{{{\overline I }_{psh}}({\overline {{r_h}} } )} |+ \xi }}} \right\}} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\limits_j^N {{a_j}\delta ({\overline r - {{\overline r }_j}} )} \ast F.T{.^{ - 1}}\left( {\frac{{|{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |}}{{|{{{\overline I }_{psh}}({\overline {{r_h}} } )} |+ \xi }}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {O_{{z_i}}}({\overline r } )\ast F.T{.^{ - 1}}(\kappa )\cong {O_{{z_i}}}({\overline r } )\end{array}$$
where $\kappa \textrm{ = }\frac{{|{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |}}{{|{{{\overline I }_{psh}}({\overline {{r_h}} } )} |+ \xi }}$, $\xi = {\raise0.7ex\hbox{${{\xi _o}}$} \!\mathord{\left/ {\vphantom {{{\xi_o}} {{\xi_n}}}} \right.}\!\lower0.7ex\hbox{${{\xi _n}}$}}$ is the ratio of signal power to noise power of reconstruction image. In the cross-correlation calculation of Eq. (4), we substituted $\kappa \textrm{ = }\frac{{|{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |}}{{|{{{\overline I }_{psh}}({\overline {{r_h}} } )} |+ \xi }}$ for $|{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |$ to avoid the background noise generated by the distribution of the real function of the inverse Fourier transform of $|{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |$ in the spectral domain. The denominator $|{{{\overline I }_{psh}}({\overline {{r_h}} } )} |+ \xi$ in $\kappa$ is introduced to weakening the background noise generated by the inverse Fourier transform of $|{{{\overline I }_{ps{h^{\prime}}}}({\overline {{r_h}} } )} |$, and the parameter ξ can also prevent the value of the denominator in $\kappa$ from being small and causing the abnormal value of $\kappa$.

Generally, in Eq. (4), the signal power ξo and noise power ξn in parameter ξ are unknown. In order to determine the value of parameter ξ quickly and adaptively under different experimental conditions in AI-COACH, a referenced reconstruction image is required. Non-linear reconstruction (NLR) is used to obtain the referenced reconstruction image, thus to calculate the value of parameter ξ. The parameter ξ can be rewrite as Eq. (5) and NLR reconstruction equation is described as Eq. (6),

$$\xi = \frac{{{\xi _o}}}{{{\xi _n}}}\textrm{ = }\frac{{{{\left[ {\sum\limits_M {\sum\limits_N {{\raise0.7ex\hbox{${{f_{re}}}$} \!\mathord{\left/ {\vphantom {{{f_{re}}} {\max ({{f_{re}}} )}}} \right.}\!\lower0.7ex\hbox{${\max ({{f_{re}}} )}$}}} } } \right]}^2}_{o = 1;p = 1}}}{{{{\left[ {\sum\limits_M {\sum\limits_N {{\raise0.7ex\hbox{${{f_{re}}}$} \!\mathord{\left/ {\vphantom {{{f_{re}}} {\max ({{f_{re}}} )}}} \right.}\!\lower0.7ex\hbox{${\max ({{f_{re}}} )}$}}} } } \right]}^2}_{o = 0;p = 0}}}$$
$${f_{re}}^{\prime} = F.T{.^{ - 1}}\{{{{|{{{\overline I }_{oh}}({\overline {{r_h}} } )} |}^o}{\rm{exp}} [{i \cdot \arg ({{{\overline I }_{oh}}({\overline {{r_h}} } )} )} ]{{|{{{\overline I }_{psh}}({\overline {{r_h}} } )} |}^p}{\rm{exp}} [{ - i \cdot \arg ({{{\overline I }_{psh}}({\overline {{r_h}} } )} )} ]} \}$$
where o and p in Eq. (6) are the adjustment parameters of the frequency domain amplitude of OH and PSH respectively, −1≤o ≤1 and −1≤p≤1.

For non-linear reconstruction, it is necessary to determine an optimized combination of o and p by large number of reconstructions and the evaluations, thus taking a lot of off-line calculation time. According to the published work [24], analysis of NLR shows that when the modulation parameters o and p=0, the amplitudes of Ioh and Ipsh in the frequency domain have a uniform constraint value, and the reconstructed image satisfies the uniform constraint of the phase distribution on SLM. However, since the parameters o and p=0 will amplify the background noise of the reconstructed image, the reconstructed image when the parameters o and p=0 in the non-linear reconstruction are approximately regarded as the noise signal. When the modulation parameter o = p=1, the reconstructed image does not contain any constraints. Therefore, the reconstructed image with parameters o and p=1 in non-linear reconstruction is approximated as object information. In short, fre′ with the parameters of o=1, p=1 is regarded as the signal and the fre′ with the parameters of o=0, p=0 in our proposed method here, which avoid the optimization process of o and p. Once ξ is determined, the reconstructed images with high-quality can be retrieved quickly by Eq. (4).

3. Experiments and results

In order to demonstrate the imaging performance of AI-COACH system and verify the advantages of our proposed technique, three groups of different experiments were designed and carried out. In the first experiment, a transmitted 2D object was imaged, and the imaging quality of AI-COACH was compared with that of lens direct imaging, I-COACH with non-linear reconstruction and with phase filtering cross-correlation reconstruction. The experiment 2 and experiment 3 verified the adaptive imaging capabilities of AI-COACH for three-dimensional transmitted objects and the reflective objects, respectively.

3.1 Imaging of 2D transmitted object by AI-COACH

The experimental setup of AI-COACH system is shown in Fig. 3. In experiment 1, only channel 1 worked. The object was illuminated by the incoherent light emitting diode (LED) (Thorlabs LED 625L4, 700 mW, center wavelength of λ=625 nm, Δλ=17 nm) placed at the front focal plane of the lens L0 with a diameter of 45 mm and focal length f0=150 mm. The light from beam splitter BS1 was collimated by the lens L0 and passed through a polarizer P. The polarizer P polarizes the light along the orientation of the active axis of the spatial light modulator (SLM, Holoeye PLUTO, 1080×1920pixels, 8µm pixel pitch, phase-only modulation) located at a distance of 55 mm from the L0. On the SLM, a phase mask was displayed whose phase is the combine of the CPM which generated by GS algorithm and the quadratic phase mask with a focal length fs=150 mm. The light modulated by the SLM was collected by a charge coupled device (CCD, Thorlabs CS235MU,1200×1920pixels, 5.86µm pixel pitch, and monochrome) located at a distance of zh=171 mm from the SLM.

 figure: Fig. 3.

Fig. 3. Experimental setup of the AI-COACH.LED1, LED2and LED3-Identical light emitting diodes; L0, L1, L2 and L3-Refractive lenses; BS1 and BS2-Beam splitters; P-Polarizer; SLM-Spatial light modulator; CCD-Charge Coupled Device.

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In the experiment, we optimize the selection for different σ and annular width $\varpi$ to obtain the CPM with the best imaging performance, firstly. The CPM corresponding to different σ and annular width $\varpi$ in combination with QPM of focal length of fs=150mm was displayed on the SLM. In channel 1, a pinhole with a diameter of 20µm was used as point-object to record the PSH corresponding to different numbers of sparse dots under different σ on the camera plane. Then the NBS transmission target (NBS, 1963A 1X R2L2S1N) was used as the object and illuminated the elements 32lp/mm and 36lp/mm to recording the hologram by using the same CPM. The reconstructed images were obtained by Eq. (4). The reconstructed images and image evaluation results are shown in Fig. 4, and the blue numbers in the figure indicate the sparse dots. It can be seen from Fig. 4(a) that, under the different σ, the imaging quality decreases with the increase of the number of sparse dots, and the imaging quality increases with the increase of σ. Meanwhile, the Fig. 4(b) shows that the AI-COACH with annular sparse distribution can obtain the best imaging when the annular width is $\varpi \textrm{ = }1\textrm{pix}$ under the scattering degree σ = 0.167.

 figure: Fig. 4.

Fig. 4. Comparison of reconstruction image quality with different modulated CPM. (a) the reconstruction of AI-COACH with annular sparse CPM; (b) SSIM of (a) reconstructions [28].

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Under the same scattering degree σ = 0.167, the PSH and OH recorded by CPM with annular with $\varpi \textrm{ = }1\textrm{pix}$ and the number 610 of random sparse dots are taken as examples. The PSH, OH and reconstruction images which obtained by used the proposed method are shown in Fig. 5. Comparing Figs. 5(a), 5(b), and 5(d) 5(e), we can see that the system response of AI-COACH and the object hologram no longer have the form of random distribution, but the form of annular distribution. As shown in Figs. 5(c) and 5(f), both annular sparse distribution and random sparse distribution can obtain high quality reconstructed images by the proposed method. Meanwhile, as shown in Fig. 5(c), the AI-COACH with annular sparse CPM has lower background noise than random sparse CPM. Therefore, we choose scattering degree σ = 0.167 and annular width $\varpi \textrm{ = }1\textrm{pix}$ as the optimal parameters of CPM in GS algorithm.

 figure: Fig. 5.

Fig. 5. Typical experimental results of PSH and OH with different modulated CPM. (a) and (b) the intensity of point spread hologram and object hologram with annular sparse CPM; (c) reconstructed results of AI-COACH for annular sparse CPM; (d) and (e)the intensity of point spread hologram and object hologram for random sparse CPM; (f) reconstruction results of AI-COACH for random sparse CPM.

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In order to reflect the improvement in imaging quality of the AI-COACH adaptive fast imaging than other method, the cross-correlation reconstructed images of PSH and OH which recorded by annular sparse CPM and random sparse CPM with the same scattering degree σ = 0.167 and annular width $\varpi \textrm{ = }1\textrm{pix}$(dots=610) were obtained by the AI-COACH, non-linear reconstruction (NLR) and phase filtering (PF), respectively. The reconstruction results are shown in Fig. 6.and Fig. 7, respectively. Compared with Figs. 6(d), 6(e), 6(f) and Figs. 7(d), 7(e), 7(f), the reconstructed images with annular sparse CPM have lower background noise than those with random sparse CPM under the same reconstruction algorithm. Meanwhile, it can be seen that the AI-COACH reconstruction method can obtain the reconstructed image with lower background noise than NLR and PF methods in the both case of annular sparse CPM and random sparse CPM. Simultaneously, as shown in Fig. 6(d), AI-COACH annular sparse CPM imaging has lower background noise than other imaging methods. In terms of imaging time resolution, the AI-COACH requires only 4.272s to obtain the optimal reconstructed image. Compared with the 356.298s required by the NLR method (the change step of modulation parameters o and p is set to 0.1) in Ref. [13], the imaging time resolution is more than 80 times faster [29,30].

 figure: Fig. 6.

Fig. 6. Comparisons of COACH Imaging with annular sparse CPM but different reconstruction algorithms. (a) and (b) the holograms of the NBS target and the 20µm pinhole, respectively; (c)regular lens imaging; (d) reconstruction results of AI-COACH with the adaptive parameter ξ=63.29; (e) non-linear reconstruction with the optimal parameters o=-0.3 and p=0.7; (f) phase filtering cross-correlation reconstruction.

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 figure: Fig. 7.

Fig. 7. Comparisons of I-COACH Imaging with random sparse CPM but different reconstruction algorithms. (a) and (b) the holograms of the NBS target and the 20µm pinhole, respectively; (c)regular lens imaging; (d) reconstruction results of AI-COACH with the adaptive parameter ξ=63.29; (e) non-linear reconstruction with the optimal parameters o=-0.3 and p=0.7; (f) phase filtering cross-correlation reconstruction.

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3.2 Imaging of 3D transmitted object by AI-COACH

In order to maintain the best imaging ability of AI-COACH system and suppress the imaging background noise, the object or pinhole in the experiment was located in the front focal plane of input lens L0 to satisfy the Fourier relationship between object plane and CCD plane. In the experiment, CPM corresponding to σ = 0.167 and $\varpi \textrm{ = }1\textrm{pix}$ in combination with QPM of focal length fs=150 mm was displayed on the SLM.

At the beginning, only PSH of 20µm pinhole in channel 1 was recorded on the front focal plane of input lens L0, and then the two United States Air Force (USAF (GO Edmund optics, USAF 1951 1X and Newport, USAF 1951 RES-1)) target were placed on the front focal plane of input lens L0 of channel 1 and channel 2 respectively, and the USAF target group of 3 was illuminated. As shown in Fig. 2, the object of channel 1 was placed on the front focal plane of the input lens L0. At the same time, the object of channel 2 was moved to different axial positions in steps of Δz=2mm to simulate different axial depths of 3D objects and OHs of different depths of objects were recorded respectively. Similarly, OHs of different depths of channel 2 objects were recorded in the same way. In the OHs recording process, different layers of the 3D object were moved in front of the focal plane of the input lens L0 to ensure the Fourier relationship between the object plane and the CCD plane to reduce background noise. Therefore, only one PSH needs to be recorded in AI-COACH, and the OHs of different axial positions of the 3D object were recorded by axial scanning and the OH library was obtained. The reconstructed image of 3D object with different axial positions was obtained by AI-COACH cross-correlation calculation from PSH and OHs corresponding to the axial position. The experimental results are shown in Fig. 8, it can be seen from the results that AI-COACH can adaptively obtain the parameter ξ value at each axial position, thereby obtaining high-quality reconstructed images. Therefore, the AI-COACH method proposed in this paper can adaptively and quickly obtain high-quality reconstruction image of 3D object at different layers through the PSH and OH of the single exposure.

 figure: Fig. 8.

Fig. 8. Comparison of experimental results of 3D objects direct lens imaging and AI-COACH imaging.

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3.3 Imaging of reflective objects by AI-COACH

In this section, two experiments were carried out for imaging reflective objects. The first experiment proved the imaging ability of binary reflection objects the experimental setup is shown in Fig. 3, instead of illuminating the targets in a transmission mode, they were illuminated by the reflective light. The object and pinhole (20µm) were positioned at the front focal plane of the input lens L1, respectively. The channel 1 illumination pinhole was used to record the point spread hologram, and the channel 3 illumination USAF reflective target group of 3 (Edmund Industrial Optics, USAF 1951 1X) was used to record the object hologram. The rest of the experimental parameters were kept the same as described in the previous sections. CPM corresponding to σ = 0.167 and $\varpi \textrm{ = }1\textrm{pix}$ in combination with QPM of focal length of fs=150mm was displayed on the SLM. The hologram of the USAF target and the point spread hologram of pinhole are shown in Fig. 9(a). The results of direct imaging and AI-COACH reconstruction with the adaptive parameter ξ=41.07 are shown in Fig. 9(b) and 9(c), respectively. It can be seen that AI-COACH can adaptively obtain high-quality reconstructed images of binary objects.

 figure: Fig. 9.

Fig. 9. Imaging results of various objects all illuminated in reflective mode. (a) the hologram of the USAF target (group 3) and the point spread hologram of 20µm pinhole; (b)direct imaging; (c) reconstruction results of AI-COACH with the adaptive parameter ξ=41.07;(d) Reflective object; (e) The hologram of the object and the point spread hologram of 20µm pinhole; (f) direct imaging of the object; (g) reconstruction results of AI-COACH with the adaptive parameter ξ=217.65.

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In the other experiments, the ability of AI-COACH to image diffusely reflective object was demonstrated. Experiential parameters, pinhole, CPM and QPM of these experiments were kept the same as that of the first experiment. The diffuse reflecting object is shown in Fig. 9(d). The object was positioned at the front focal plane of lens L1 in channel 1, and the incoherent light source LED3 (Thorlabs LED 627L3, 270mW, center wavelength of λ=632nm, Δλ=15nm) of channel 3 illuminates the area shown by the red spot in the Fig. 9(d). The hologram of the reflective objects and the point spread hologram of pinhole are shown in Fig. 9(e). Reconstruction result for direct imaging and AI-COACH result with the adaptive parameter ξ=217.65 are shown in Figs. 9(b) and 9(c), respectively. It can be seen that AI-COACH still has good adaptive imaging ability for diffuse reflection objects

4. Summary and conclusions

In this paper, we proposed an adaptive imaging method with annular sparse point spread function in the interferenceless coded aperture correlation holography and named it AI-COACH. In AI-COACH, the GS algorithm is modified to generate the coded phase mask with annular sparse distribution. Meanwhile, the sparse dots of the system are controlled by the annular width to reduce the background noise in reconstructions, thus improve the imaging signal-to-noise ratio of reconstructed images. By introducing an adaptive parameters ξ to phase filter cross-correlation reconstruction method, the AI-COACH technology can obtain high-quality reconstructed images through only one single exposure of PSH and OH. The reconstruction process can be done in a few seconds to obtain the optimal reconstructed image, which is faster dozens of times than that of reconstruction by the mentioned NLR. The imaging results of transmission and reflection objects demonstrate adequately that AI-COACH possess the capability of fast, adaptive and high-quality imaging. Therefore, the AI-COACH technology improves the imaging performance of the I-COACH technology effectively.

Funding

Natural Science Foundation of Beijing Municipality (4182016); National Natural Science Foundation of China (61575009).

Disclosures

The authors declare no conflicts of interest.

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Figures (9)
Fig. 1.
Fig. 1. Synthesis of annular sparse CPM schematic by GS algorithm.

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Fig. 2.
Fig. 2. Optical setup of AI-COACH system.

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Fig. 3.
Fig. 3. Experimental setup of the AI-COACH.LED1, LED2and LED3-Identical light emitting diodes; L0, L1, L2 and L3-Refractive lenses; BS1 and BS2-Beam splitters; P-Polarizer; SLM-Spatial light modulator; CCD-Charge Coupled Device.

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Fig. 4.
Fig. 4. Comparison of reconstruction image quality with different modulated CPM. (a) the reconstruction of AI-COACH with annular sparse CPM; (b) SSIM of (a) reconstructions [28].

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Fig. 5.
Fig. 5. Typical experimental results of PSH and OH with different modulated CPM. (a) and (b) the intensity of point spread hologram and object hologram with annular sparse CPM; (c) reconstructed results of AI-COACH for annular sparse CPM; (d) and (e)the intensity of point spread hologram and object hologram for random sparse CPM; (f) reconstruction results of AI-COACH for random sparse CPM.

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Fig. 6.
Fig. 6. Comparisons of COACH Imaging with annular sparse CPM but different reconstruction algorithms. (a) and (b) the holograms of the NBS target and the 20µm pinhole, respectively; (c)regular lens imaging; (d) reconstruction results of AI-COACH with the adaptive parameter ξ=63.29; (e) non-linear reconstruction with the optimal parameters o=-0.3 and p=0.7; (f) phase filtering cross-correlation reconstruction.

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Fig. 7.
Fig. 7. Comparisons of I-COACH Imaging with random sparse CPM but different reconstruction algorithms. (a) and (b) the holograms of the NBS target and the 20µm pinhole, respectively; (c)regular lens imaging; (d) reconstruction results of AI-COACH with the adaptive parameter ξ=63.29; (e) non-linear reconstruction with the optimal parameters o=-0.3 and p=0.7; (f) phase filtering cross-correlation reconstruction.

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Fig. 8.
Fig. 8. Comparison of experimental results of 3D objects direct lens imaging and AI-COACH imaging.

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Fig. 9.
Fig. 9. Imaging results of various objects all illuminated in reflective mode. (a) the hologram of the USAF target (group 3) and the point spread hologram of 20µm pinhole; (b)direct imaging; (c) reconstruction results of AI-COACH with the adaptive parameter ξ=41.07;(d) Reflective object; (e) The hologram of the object and the point spread hologram of 20µm pinhole; (f) direct imaging of the object; (g) reconstruction results of AI-COACH with the adaptive parameter ξ=217.65.

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Equations (6)

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I p s h ( r h ¯ ; r s ¯ , z s ) = | A s ( r s ¯ ; z s ) C L ( r s ¯ z s ) Q ( 1 z s ) Q ( − 1 f 1 ) e x p [ i ϕ ( r ¯ ) ] Q ( 1 f s lm ) ∗ Q ( 1 z h ) | 2  =  | A s ( r s ¯ ; z s ) C L ( r s ¯ z s ) Q ( 1 z 1 ) e x p [ i ϕ ( r ¯ ) ] ∗ Q ( 1 z h ) | 2  =  I p s h ( r h ¯ − z h z s r s ¯ ; 0 , z s )
I o h ( r h ¯ ; z s ) = O z i ∗ I p s h = ∑ j N a j δ ( r h ¯ − r ¯ j ) ∗ I p s h ( r h ¯ − z h z s r ¯ j ; 0 , z s ) = ∑ j N a j I p s h ( r h ¯ − ( 1 + z h z s ) r ¯ j ; 0 , z s ) = ∑ j N a j I p s h ( r h ¯ − ( 1 + z h z s ) r ¯ j ; 0 , z s )
I r e = I o h ( r h ¯ ) ⊗ I p s h ( r h ¯ ) ′ = F . T . − 1 { I ¯ o h ( r h ¯ ) ⋅ I ¯ p s h ( r h ¯ ) ′ ∗ } = F . T . − 1 { | I ¯ o h | e x p ( i ⋅ arg ⁡ ( I ¯ o h ( r h ¯ ) ) ) e x p ( − i ⋅ arg ⁡ ( I ¯ p s h ( r h ¯ ) ) ) }
I ′ r e = I o h ( r h ¯ ) ⊗ I p s h ( r h ¯ ) ′  =  { ∑ j N a j I p s h ′ ( r h ¯ − ( 1 + z h z s ) r ¯ j ; 0 , z s ) } ⊗ I p s h ( r h ¯ ) ′ = F . T . − 1 { F . T . [ ∑ j N a j δ ( r ¯ − r ¯ j ) ] ⋅ | I ¯ p s h ′ ( r h ¯ ) | e x p ( i ⋅ arg ⁡ ( I ¯ p s h ′ ( r h ¯ ) ) ) ⋅ e x p ( − i ⋅ arg ⁡ ( I ¯ p s h ( r h ¯ ) ) ) | I ¯ p s h ( r h ¯ ) | + ξ } = ∑ j N a j δ ( r ¯ − r ¯ j ) ∗ F . T . − 1 ( | I ¯ p s h ′ ( r h ¯ ) | | I ¯ p s h ( r h ¯ ) | + ξ ) = O z i ( r ¯ ) ∗ F . T . − 1 ( κ ) ≅ O z i ( r ¯ )
ξ = ξ o ξ n  =  [ ∑ M ∑ N f r e / f r e max ( f r e ) max ( f r e ) ] 2 o = 1 ; p = 1 [ ∑ M ∑ N f r e / f r e max ( f r e ) max ( f r e ) ] 2 o = 0 ; p = 0
f r e ′ = F . T . − 1 { | I ¯ o h ( r h ¯ ) | o e x p [ i ⋅ arg ⁡ ( I ¯ o h ( r h ¯ ) ) ] | I ¯ p s h ( r h ¯ ) | p e x p [ − i ⋅ arg ⁡ ( I ¯ p s h ( r h ¯ ) ) ] }
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