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Abstract
Noise suppression is one of the most important tasks in imaging through inhomogeneous mediums. Here, we proposed a denoising approach based on compressive in-line holography for imaging through an inhomogeneous medium. A reference-beam-free system with a low-cost continuous-wave laser is presented. The suppression against the noise, which is brought by the scattering photons, is presented in simulations using the proposed algorithm. The noise immunity is demonstrated in lensless imaging behind a random phase mask with an optical depth of 1.42 by single exposure, as well as behind a ground glass with an optical depth of 6.38 by multiple exposures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical imaging through an inhomogeneous medium is of great use, but has a challenging problem. It could be applied in target detection through fog [1,2], non-invasive investigation in living tissue [3] or imaging under turbid water [4]. However imaging quality is severely degraded because the propagation direction of the photons changes irregularly. There are various solutions for imaging through inhomogeneous media, such as the ballistic imaging method [5], transmission matrix method [6], speckle correlated method [7], wavefront compensation method [8] or guide-star method [9], with each method having its own pros and cons. For imaging through a static diffusing medium, prior characterization of the medium transmission matrix could be applied [10]. However, for imaging through turbid media, turbidity is dynamic and prevents one from using prior characterization, which makes the problem more challenging [1,11,12]. The ballistic imaging method aims to automatically sort and discard scattered radiation from unperturbed photons which has been widely applied for its high spatial resolution [13]. Different physical characteristics between the image-bearing unscattered photons and scattered photons are studied to screen the unscattered photons out, such as flight time [14], polarization [15], direction of propagation [16], and coherence [17].
Holography is a coherence gating method which can extract the image-bearing unscattered photons [17,18]. Some holographic methods have been studied to enhance the imaging quality through inhomogeneous medium [19], although these methods have limitations when applied. The holographic methods based on the wavefront compensation require calibration, which are not suitable for time-varying inhomogeneous medium [20,21]. The holographic methods based on the auto-correlation are hard to image the object embedded into inhomogeneous medium [4,19]. The holographic methods based on numerical phase conjugation is demonstrated to image through a subsequent misalignment of the diffuser but with one calibration [20]. The ballistic-photons based holographic methods have been developed recently [22]. Holography is a kind of redundancy coding which has potential ability of the noise resistance [23,24]. The large-size objects were recorded through the smoke using far-infrared digital holography [25]. The equipment with the low-cost continuous-wave laser is adopted by fire department and in other industrial fields. When the smoke is thick and mass photons are scattered, the multi-look techniques with averaging a set of holographic reconstructions are applied to enhance the quality of the reconstructions. When the N totally uncorrelated holograms are captured, the noise reduction trend would be bounded by the ideal curve $1/\sqrt N $ [26]. Multi-look techniques are adopted to enhance the reconstruction quality in digital Fresnel holographic interferometry [27]. The dynamic imaging through turbid media based on the off-axis setup is achieved by using the holographic autofocus and the spatial filtering techniques [17]. The Brownian motion of the colloidal particles could be exploited to acquire a set of uncorrelated holograms. The bovine spermatozoa in the turbid microfluidics are recovered and the speckle noise is greatly reduced by multiple holographic acquisitions [28,29]. The different signal-to-noise ratios (SNR) of single-shot digital holograms are analyzed by different denoising algorithms [30]. The holographic imaging behind the scattering bacteria layer is demonstrated to take advantage of turbidity to improve the image quality [31]. The loss of the portions of the hologram is allowed without any detectable deterioration in the reconstruction. There are few models and discussions for the noise suppression by using in-line holography in imaging through inhomogeneous medium in previous works [27–31]. Recently, convolutional neural networks and other deep learning algorithms have been used for reconstructions based on input-output dataset. The training of the network provides a new, simple yet effective solution for ill-conditioned problems [32].
Compressive sensing (CS) [33,34] is a relatively new signal reconstruction framework which works well for ill-conditioned problems [35,36] and noise suppression problems [37,38]. Multi-pixel encoding of holography meets the application conditions of ‘incoherence’ mathematically in CS. Compressive holography (CH) which combines the Gabor holography and CS was proposed to suppress the twin-image noise [39]. It was successfully applied to reconstruct a 3-D layered object from a 2-D hologram based on the sparsity constraints. CS has shown the ability of noise suppression in imaging through inhomogeneous media by using the single-pixel imaging method [40] and the transmission matrix method [7]. To the best of our knowledge, the noise immunity in compressive holographic imaging through inhomogeneous media has not been discussed before.
In this work, we propose a denoising approach based on compressive in-line holography for imaging through inhomogeneous media. The ballistic photons are regarded as an image-bearing signal to form the forward propagation model. The scattered photons caused by the non-uniformity of microstructure are regarded as the noise. The numerical focusing based on the angular-spectrum propagation is used to construct the forward propagation model of compressive holography. Imaging behind a random phase mask is demonstrated for a single-shot hologram by using the proposed algorithm. Multi-look techniques are adopted to enhance the reconstruction quality for imaging behind a ground glass. Because CS-based methods reconstruct the object by solving the optimization problem, the noise suppression for imaging behind a ground glass could be achieved by the averaging of a set of captured holograms. A low-cost continuous-wave laser is used without a sophisticated shutter or an expensive light-pulse generator. The reference-beam-free system supports the compact configuration. The capability of the noise immunity for the different holograms with different SNRs is analyzed. The noise, brought by the scattering photons, is suppressed in simulations and in experiments by using the compressive sensing algorithm.
2. Ballistic-photons based on compressive in-line holography
When the collimated light propagates through inhomogeneous media, as shown in Fig. 1, some photons experience random scattering and lose their coherence [17,21,41]. At the same time, some unscattered photons, called ballistic photons, possibly pass through in straight lines without scattering, which keep the coherence. Based on the configuration of the classical Gabor holography, the part of unscattered photons are modulated by the object and become the image-bearing photons. The unscattered photons travel through the inhomogeneous media directly. They interfere with the image-bearing photons and a Gabor hologram is generated in the detector plane.
The elimination of the scattered photons could enhance the SNR of the hologram to obtain a high-quality reconstruction since the ballistic photons carry the true signal. Based on Beer’s law, the exponential relationship between the remaining ballistic photons and the input transmitted photons can be expressed as
where, ${P_i}$ and ${P_\textrm{b}}$ are expressed as the number of photons before and after going through inhomogeneous medium. $\mu $ is the medium extinction factor which is the summation of the scattering coefficient and absorption coefficient. d is the distance at which light travels through the medium. The optical depth ${d_o} = \mu d$ is used to describe the proportion of ballistic photons in the captured hologram [42]. The larger the optical depth is, the less the number of remaining ballistic photons are, and the larger the percentage of noise caused by the scattered photons in the captured hologram is. When the optical depth is large for biological tissue or ground glass, mass random scattered photons would seriously degrade the hologram, which results in the failure of the reconstruction. The time-gating method based on laser pulse generators and ultrafast synchronized shutters has shown excellent performance in capturing ballistic photons and filtering scattered photons. However, the sophisticated equipment is rather expensive, which could be saved in this work. We focus on the improved reconstruction of compressive holography through the weak inhomogeneous medium using the continuous-wave laser.The captured intensity ${I_\textrm{c}}$ through inhomogeneous medium could be regarded to be made up of the interference fringes ${I_h}$ contributed by the ballistic photons, the independent Poisson noise ${n_s}$ brought by the scattered photons and the speckle multiplicative noise ${n_m}$.
The fringes in Gabor holography are formed in the interference between the diffractive field ${U_s}$ contributed by the imaging-bearing photons and the reference field ${U_R}$ contributed by the unscattered reference photons. The interference fringes are expressed asCompressive holography could reconstruct a twin-image-free object from the Gabor hologram by minimizing the target function $\Gamma (s )$.
Two types of inhomogeneous medium are considered. One is the random phase mask, which would only slightly perturb the holographic encoding. A CS-based algorithm could suppress the noise with a single-shot captured hologram. The other is ground glass, which would severely perturb the holograms, where the averaging of multiple holograms is adopted to enhance the SNR of measurements. It is quite different from majority methods which average multiple holographic reconstructions [27–28]. The proposed algorithm base on CS requires the averaging of multiple holograms so that the interference fringes could be enhanced. As shown in Fig. 2, the single-shot hologram through a random phase mask or the averaged hologram through ground glass is preprocessed. The hologram is rebuilt by using the back propagation method [47] in order to obtain an initial estimate. In the iteration loop, the estimate object ${s_\textrm{i}}$ is optimized by the denoising operation to obtain the intermediate smooth estimate $s$. The two-step iteration is used to obtain a new estimate ${s_{\textrm{i + 1}}}$ based on the last two estimates and intermediate smooth estimate. Then, it is put in the forward model to obtain the estimate hologram. The residual of the target function is calculated. The monotonically decreasing is judged to guarantee that the target function is convergent. The step of gradient descent and the number of iterations would be updated to locate the minimum of the target function. The loop would not stop until it gets to the defined maximal iteration number or an acceptable residual value. The final estimate is obtained and evaluated when the loop stops. The flow chart of the algorithm is presented below.
In compressive holography for imaging through inhomogeneous media, scattered photons lose the correct propagation directions and generate a random intensity field. The random intensity field no longer matches the forward model of Gabor holography. The estimated signal contributed by the random intensity field also does not satisfy the sparsity regularization function. In order to filter out the noise components contributed by the randomly scattered photons and the squared term, the filter layer is set in the compressive holographic model, as shown in Fig. 3(a). The filter layer provides a pool to deposit noise that have little correlation with the forward model. Sparse regularization acts as a filter that evaluates the sparsity of the estimates. When the estimated object s is put in the forward model, the algorithm would determine how much percentage of the estimates should be placed in which object layer or filter layer based on the amount of correlation between the measurements and the estimated field.
Fig. 3. (a) Regularization mechanism for noise suppression; (b) the reconstruction by compressive holography without the filter layer; (c) the reconstruction by compressive holography with the filter layer; (d) PSNR for the location of the filter layer.
Here, a degraded Gabor hologram is considered. The collimating beam at a wavelength of 633 nm illuminates the target object with the letters of ‘THU’. The imaging-bearing diffractive field interferes with the unscattered reference beam and the Gabor hologram is recorded by the sensor array. The distance between the target object and the sensor array is set as 4 mm. The pixel pitch of the Gabor hologram is 10 µm. The pixel number of the Gabor hologram is set as 64×64. The random Poisson noise has been added into the captured hologram. The index of the signal-to-noise ratio (R1) of Gabor hologram is defined as
The location of the filter layer is also analyzed in Fig. 3(d). The filter layer is set at different distances away from the sensor array. The reconstructions are demonstrated with the same regularization parameter and the same number of iterations. The results are evaluated by using the peak signal-to-noise ratio (PSNR).
3. Analysis of noise immunity
The inhomogeneous medium located in the propagation path could cause random scattered photons and the captured Gabor hologram would degrade. The compressive holography with a filter layer at the sensor plane could effectively suppress the noise which does not match the forward model and sparse constraint.
The Gabor holograms with different SNRs are reconstructed and the results are analyzed. A simple object of letter ‘T’ with a size of 25 µm×25 µm is illuminated. The Gabor holograms are captured by the sensor array with pixel pitch of 5 µm. The distance between the object and the sensor array is 2 mm. The diffuser as inhomogeneous medium is placed in the propagation path. The captured holograms added with the random Poisson noise are generated to simulate that the random scattered photons degrade the Gabor hologram. As shown in Fig. 4(a), the diffraction in the Gabor holography is a multi-pixel encoding where the object patterns are encoded into the interference fringes. The redundancy of the Gabor holography provides the robustness for the reconstruction of the degraded hologram. Different captured holograms with different R1 are shown in Fig. 4(a). With the increase of the random Poisson noise, the details of the interference fringes are decaying with the noise. When R1 is equal to -15 dB, the interference fringes are nearly indistinguishable. The corresponding reconstructions using the classical back propagation method (BPM) and the compressive holography method (CHM) are shown in Figs. 4(b) and 4(c), respectively. Both methods show the robustness against the added noise. The CHM reveals better performance in noise suppression due to the sparse constraint and the added filter layer. The target object could still be reconstructed for the degraded Gabor hologram when R1 is equal to -15 dB. PSNR and SSIM are used to evaluate the reconstruction recovered by different R1 of the Gabor hologram as shown in Fig. 5. The constants in the SSIM index formula are set as 0.01 and 0.03. The local Gaussian window with the size of 11×11 and the standard deviation of 1.5. The PSNRs are improved by nearly 10 dB by using CHM instead of by using BPM in our simulation. The index of SSIMs by using the CHM are also better than the results by using the BPM. The reconstruction quality could have great improvements when the Gabor hologram degrades weakly.
Fig. 4. (a) The captured Gabor hologram added by Poisson noises; (b) the reconstructions using the back-propagation method; (c) the reconstructions using the compressive holography method.
4. Experiments
To demonstrate our model against the noise by compressive holography, two optical imaging experiments are presented here. The imaging system in Figs. 6(a) and 6(b) contains a He-Ne laser source, a beam expander, an object of lithographic mask with Greek letter of π, a diffuser, and a CMOS (xiQ, MQ013MG-E2) with the pixel pitch of 5.3 µm. The distance between the diffuser and the object is 20 mm. The distance between the diffuser and the sensor array is 10 mm. Two types of diffusers are considered in this work. One diffuser in Fig. 6(a) is a random phase mask with an optical depth of 1.42, where the correspond proportion of the remaining ballistic photons ${\textrm{N}_\textrm{b}}$ to the total input photons ${\textrm{N}_\textrm{T}}$ is 24.17%. The other diffuser in Fig. 6(b) is a ground glass (Thorlabs, GD10-220) with an optical depth of 6.38, where the corresponding proportion of the remaining ballistic photons ${\textrm{N}_\textrm{b}}$ to the total input photons ${\textrm{N}_\textrm{T}}$ is 0.17%. The measurement method of the optical depth is presented in the Appendix.
Fig. 6. Imaging system through (a) a phase mask and (b) a ground glass; (c) the degraded Gabor hologram captured in (a), the inset gives the size of object; The reconstructions of (c) are obtained using (d) BPM and (e) CHM, the insets show the intensity varying in the red line; The data processing of (b) is shown in (f); Scale bar in (c-e) is 500 µm; Scale bar in (f) is 250 µm.
The degraded Gabor hologram for Fig. 6(a) is shown in Fig. 6(c). The interference fringes are difficult to distinguish. However, the underlying intensity variation encodes the information of the object. Pixel binning and the elimination of background light are applied to enhance the SNR of the captured hologram. The reconstructions using BPM and CHM are shown in Figs. 6(d) and 6(e), respectively. Two methods show the robustness against the noise caused by the scattered photons. CHM performs better in noise suppression than BPM. The cross sections along the red lines are presented. To quantitatively evaluate the reconstructions, image contrast K is used to describe the image quality [46], which is given by
where ${I_{m\textrm{ax}}}$ is the maximum intensity value of the cross section and ${I_{\min }}$ is the minimum intensity value of the cross section. The image contrast for the cross section in Figs. 6(d) and 6(e) is 0.21 and 0.48, respectively.For the ground glass, the scattered photons are so abundant that the Gabor hologram degrades entirely. The captured hologram could not be rebuilt directly. When the ground glass is shifted, the signal contributed by the ballistic photons is a frozen-in distribution and the noise brought by the scattered photons is an independent and random distribution. The signal increases by $\textrm{N}$ times with the exposure number $\textrm{N}$, while the noise increases by $\sqrt {\textrm N} $ times with the exposure number $\textrm{N}$. So the SNR of the captured Gabor hologram could be improved with the shifting of diffuser and the average of multi-exposure acquisition. As shown in Fig. 6(f), the interference fringes in the hologram emerge with increase of average captured holograms. The reconstruction by using CHM is shown in Fig. 6(f). The multiple acquisition of holograms requires that the target keeps static during the shifting of the ground glass.
To quantitatively evaluate the reconstructions by BPM and CHM with the varying of the average captured holograms, the reconstructions are shown in Figs. 7(a) and 7(b) when the number of captured holograms is 130 and 500, respectively. CHM has better performance in noise suppression compared with BPM. The reconstructions under 500 exposures are significantly better than those under 130 exposures. Some inherent patterns along the horizontal direction appear in both reconstructions because the shifting direction of the diffuser is horizontal in the acquisition of the multiple holograms. The cross sections along the black arrows are presented. The intensity in the reconstructions by using BPM is uneven obviously where the intensity in the right side of the curve is higher than that in the left side of the curve. However, the reconstructions using CHM are uniform because of the distortion correction through solving the optimization problem. The image contrast for the cross section is 0.05 for BPM in Fig. 7(a) and is 0.10 for CHM in Fig. 7(b). The image contrast for the cross section is 0.38 for BPM in Fig. 7(c) and 0.44 for CHM in Fig. 7(d). The curves of image contrast of the reconstructions are shown in Fig. 8. When one exposure hologram through the ground glass is adopted to reconstruct the object, both methods cannot recover the target object. With the increase of the number of captured holograms, the image contrast becomes higher. When the number of captured holograms is larger than 250, the image contrast using CHM is superior to that of using BPM.
Fig. 7. Reconstructions using the BPM and the CHM when the number of captured hologram is equal to (a-b) 130 and (c-d) 500.
5. Conclusion
A CS-based method of noise suppression is presented for ballistic-photons based in-line holographic imaging behind an inhomogeneous medium. This method is based on a denoising process that is nested inside an iterative phase-retrieval algorithm. Superior performance is achieved in lensless imaging behind a random phase mask with single exposure and lensless imaging behind a ground glass with multiple exposures. The results of simulations and experiments show the benefits of using CS to attenuate the noise. Multi-look techniques could greatly improve the quality of reconstruction and it require that the target keep static approximately during the multiple acquisition. This work will be beneficial to many of the recent advanced studies in ballistic-photons based imaging through inhomogeneous medium. There would be potential applications in digital holography through turbid microfluidic channels or biological tissue.
Appendix: measurement of the optical depth
Based on Eq. (1), the optical depth ${d_o}$ of the diffuser is defined as
Funding
National Natural Science Foundation of China (61327902, 61827825); China Scholarship Council (201706210290).
Disclosures
The authors declare no conflicts of interest.
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