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Abstract
Differential phase contrast (DPC) imaging relies on computational analysis to extract quantitative phase information from phase gradient images. However, even modest noise level can introduce errors that propagate through the computational process, degrading the quality of the final phase result and further reducing phase sensitivity. Here, we introduce the noise-corrected DPC (ncDPC) to enhance phase sensitivity. This approach is based on a theoretical DPC model that effectively considers most relevant noise sources in the camera and non-uniform illumination in DPC. In particular, the dominating shot noise and readout noise variance can be jointly estimated using frequency analysis and further corrected by block-matching 3D (BM3D) method. Finally, the denoised images are used for phase retrieval based on the common Tikhonov inversion. Our results, based on both simulated and experimental data, demonstrate that ncDPC outperforms the traditional DPC (tDPC), enabling significant improvements in both phase reconstruction quality and phase sensitivity. Besides, we have demonstrated the broad applicability of ncDPC by showing its performance in various experimental datasets.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantitative phase imaging (QPI) enables non-invasive analysis of 3D structure of living cells [1–3]. Nowadays, many QPI methods, such as holography [4,5], transport-of-intensity equation (TIE) [6,7], Fourier ptychographic microscopy (FPM) [8,9], rely on computational analysis to extract phase information from digital images. In the field of QPI, differential phase contrast (DPC) [10–13] has emerged as a powerful approach for producing high-resolution phase images from biological samples in vitro. It involves an asymmetric illumination and measuring the resulting phase gradient patterns with a camera sensor. Subsequently, the quantitative phase is constructed through one-step deconvolution of several images and the system’s phase transfer function (PTF). Although originally applied in the measurement of relatively thin samples, recent studies have extended the application of DPC to thicker samples [14–16]. The advantages of higher imaging efficiency, a resolution up to the incoherent resolution limit, and the flexibility, make DPC particularly attractive for biomedical study [17,18].
Despite its advantages, the DPC performance fundamentally depends on the quality of the raw image data, in particular the level of associated noise in DPC images. The noise mainly comes from the detection of photon counts on a selected camera sensor, which still subjects to Poissonian noise even under ideal conditions. The shot noise combined with other major sources, such as camera readout noise and fixed pattern noise [19,20], can induce spatial fluctuations of pixel values that are not related to the sample, thus affecting the quantitative analysis of phase results. Meanwhile, the PTF in DPC is band-limited and has incomplete coverage in the frequency domain. Then the noise in images can introduce errors to the recovered phase after the deconvolution-based reconstruction process. The common ringing artifact pattern appears superimposed on the reconstructed image. Also the background noise can lead to the appearance of white-cloud effects in the phase pattern [21].
Furthermore, the presence of camera noise determines the phase sensitivity [22,23] of DPC. The phase sensitivity is defined as the phase value that generates an intensity image whose amplitude equals that of the noise [22]. Ideally, this sensitivity determines the minimum detectable phase in DPC. On the other hand, it only describes the intensity difference due to this phase. It is not clear whether this minimum phase can be accurately recovered, especially under challenging noise condition. Note that, the phase sensitivity can be improved by increasing the power of the illumination source or using a lock-in camera [23]. However, these methods do not effectively remove the noise in many practical cases. Therefore, denoising becomes the next step for further pushing the detection ability of DPC into higher sensitivity.
In this paper, we introduce the noise-corrected DPC (ncDPC) with the high accuracy and stability for phase recovery. ncDPC combines camera physics and traditional DPC (tDPC) to describe the most relevant noise sources in DPC. The fixed-pattern noise of the camera and the non-uniform illumination of the tDPC source are firstly corrected using a map of the offset and relative corrected factor. Then ncDPC performs an automatic estimation of the remaining Poisson-Gaussian noise variance using frequency analysis, which results in a robust and efficient estimation for camera shot noise and readout noise. Furthermore, we use block-matching 3D (BM3D) [24] method to achieve DPC image restoration with the estimated noise variance. Finally, the denoised images are used for phase retrieval based on the common Tikhonov inversion [10]. Using this method, we have demonstrated significant improvements in both phase reconstruction quality and phase sensitivity in a wide range of simulated and experimental data.
2. Theory and method
DPC employs a linearized forward model to describe the image formation. This linearization is achieved based on a weak phase assumption on the sample’s transmission function $o(r)$, i.e. $o(r)\approx 1-\mu (r)+i\phi (r)$. $\mu$ and $\phi$ are the absorption and the phase of the sample, respectively. $r$ is the transverse spatial coordinate. Under this assumption, the Fourier transform of the intensity at the camera plane under oblique illumination can be expressed as [10]
Therefore, the obtained intensity at the camera plane is the sum of a background term independent of the sample and a phase term:
where $\mathcal {F} ^{-1}$ denotes an inverse Fourier transform, and $r_c$ is the spatial coordinates at the camera plane. Equation (4) is general and can be applied to any source and pupil function. In this paper, we use traditional half-circle illumination and an ideal pupil function of the used objective lens.2.1 Sensitivity in DPC
Ideally, any phase sample can generate a raw pattern of a certain intensity by Eq. (4). Note that the background term does not generate useful information and is subsequently removed in the process of quantitative phase reconstruction. If the sample is too weak, the phase term $\mathcal {F} ^{-1}\{ H_{\mathrm {ph}}(u) \cdot \tilde {\phi }(u)\}$ would be small enough compared with the background term. This makes the presence of noise component of the raw DPC images limit the accuracy of phase reconstruction (see Supplement 1). The sensitivity in DPC describes the minimum phase value that generates an intensity pattern whose contrast-to-noise ratio (CNR) equals one. CNR in DPC is defined as [22]
As shown in Eq. (5), CNR depends on the phase spectrum of the sample and the shape of $H(u)$. For a given sample and $H(u)$, CNR can be improved by increasing the incident power of the illumination source or decreasing the noise. Obviously, it is impossible to keep increasing power because there is a maximum power to make the pixels saturation. Therefore, we need to further minimize the influence of the noise on the phase reconstruction in DPC.
2.2 Camera noise and non-uniform illumination in DPC
Regrading the noise in DPC, the camera noise is the most significant contributor. Currently, complementary metal oxide semiconductors (CMOS) camera has been widely used in DPC [14,25–30] with the advantages of higher frame rates and a wider field-of-view (FOV). Physically, in both CCD and CMOS camera, the light in Eq. (4) hits the camera sensor and is further accumulated at a given exposure time. The light is converted into photoelectrons and then into the voltage. After the analog-to-digital (AD) conversion, the output digital number (DN) is obtained from the voltage produced by each pixel.
The whole process of acquired image for the camera is mainly influenced by three types of noise: fixed pattern noise (FPN), shot noise and readout noise [19]. FPN can be further divided into gain FPN and offset FPN. The gain FPN considers the pixel-dependent photo-response non-uniformity. The offset FPN accounts for the pixel-dependent variations in the absence of light [31]. The shot noise considers the arrival difference of photons at the pixels and is signal-dependent. The readout noise usually represents all types of signal-independent camera noise [32]. Meanwhile, the used illumination source in DPC is approximately spherical, which usually results in the non-uniform of the intensity across FOV. Therefore, the acquired image from the camera under a oblique illumination can be modeled as [20]
When there is no incident light, the output value of the pixel only depends on the readout noise and the offset as shown in Eq. (6). Because the readout noise has a mean value of 0, the pixel-dependent offset $\beta (r_c)$ can be estimated by averaging 500 dark frames. Meanwhile, we get averaged output image $\overline {Z_{\mathrm {DN}}}(r_c)$ from 500 DPC frames without any sample in place. The shot noise and readout noise are removed in $\overline {Z_{\mathrm {DN}}}(r_c)$. Then the relative corrected factor $s\alpha (r_c)$ for each pixel can be calculated by (see Supplement 1)
2.3 Noise estimation and image denoising in DPC
The remaining noise term is composed of two mutually independent parts, a Poisson component $\sigma _p$ and a Gaussian component $\sigma _g$. The overall standard deviation $\sigma _{\mathrm {noise}}$ has the form of $\sigma _{\mathrm {noise}}^2=\sigma _p^2+\sigma _g^2$. We use the method in Ref. [20] to achieve a joint estimate of the noise $\sigma _{\mathrm {noise}}$ by frequency analysis. This method is based on fact that the Poisson component becomes a good approximation of the Gaussian distribution when the photon flux is greater than 3 photons per pixel [32]. Note that, this condition on the photon flux is always satisfied for DPC due to a powerful background term in Eq. (4).
Therefore, the noise is the sum of two independent Gaussian components and is still a Gaussian-distribution. Then, the noise has a constant power spectral density (PSD) and is present at every frequency. Meanwhile, the sample signals are bounded within the PTF $H_{\mathrm {ph}}(u)$ as shown in Eq. (4). The corresponding cutoff frequency $f_\mathrm {DPC}$ depends on objective’s numerical aperture and the illumination wavelength. Furthermore, the used camera in DPC acts as a low pass filter and also has a cutoff frequency $f_\mathrm {camera}$. Usually, $f_\mathrm {camera}>f_\mathrm {DPC}$, which satisfies the Nyquist-Shannon sampling theorem. Therefore, the pixel fluctuation $\sigma _c$ between $f_\mathrm {DPC}$ and $f_\mathrm {camera}$ can be estimated and is only due to the noise. Finally, $\sigma _{\mathrm {noise}}$ can be obtained by (see Supplement 1) [20]
Once $\sigma _{\mathrm {noise}}$ is known, we can use the methods of image denoising to filter it out. Image denoising has been a widely researched problem for decades and has made continued progress in recent years [34,35]. The representative block-matching 3D (BM3D) [24] method effectively combines the non-local filtering and transform domain filtering, which shows outstanding denoising performance. In this paper, we use BM3D method to achieve quantitative image restoration with the estimated $\sigma _{\mathrm {noise}}$.
2.4 Phase reconstruction in DPC
Four measurements in the orthogonal direction are used for phase retrieval in this paper. These measurements are conventional DPC images with half-circle illumination [10]. Then the pixel-wise difference of left-right images and top-bottom images is calculated. It is noted that the DPC sensitivity in Section 2.1 is defined for the single DPC measurement. This subtraction process can improve the CNR by $\sqrt {2}$ times [22]. Meanwhile, the PTF is anti-symmetric and is zeros along the axis of asymmetry. Therefore, the reconstructed phase along the illumination direction from four measurements will have a improved CNR which is $\sqrt {2}$ times that of the single measurement.
The whole process of quantitative phase reconstruction is shown in Fig. 1(a). The denoised DPC images are obtained after fixed pattern noise and non-uniform illumination removal, an estimation of Gaussian-distribution noise and BM3D denoising. Then the quantitative phase can be recovered by deconvolving the denoised DPC images with the common Tikhonov regularization [10]. Note that traditional DPC uses fixed pattern noise and non-uniform illumination removed images to reconstruct quantitative phase, which will be compared with our proposed method.
Fig. 1. Principle of our proposed ncDPC. (a) Flowchart for ncDPC to improve phase sensitivity. (b) Schematic of tDPC imaging system.
3. DPC setup
The schematic diagram of tDPC is shown in Fig. 1(b). It consists of a 4-f imaging system in which a programmable TFT panel is placed on the front plane of the condenser to generate half-circle illumination pattern. The illumination LED source has a central wavelength of 530 nm. The light exiting the sample is collected by an Objective lens (Plan N 10$\times$, 0.25NA, Olympus) and directed to a CMOS cameara (pixel size of 2.6 $\mu$m, 12bit, USC180, Oplenic) via a tube lens (SWTLU-C, Olympus). 4 tDPC images from traditional half-circle illumination are captured.
The sample of a USAF phase resolution target (Benchmark Technologies) which contains periodic pattern of continuous spatial frequencies is considered here. This target has different heights and the height of 50 nm is imaged in this paper. The refractive index of this target at the wavelength of our LED source is 1.52. For further reducing the phase difference, this target is immersed in the refractive index liquids (Series AA or A, Cargille Laboratories) whose refractive index at our wavelength is 1.4529 (Series AA) or 1.5043 (Series A). Then the height of 50 nm results in 39.8 or 9.3 mrad peak-to-valley. Meanwhile, the liquid is sensitive to temperature changes, and the experiment is performed in a temperature controlled ($25^{\circ }\mathrm {C}$) room.
4. Results
4.1 Simulation results
To demonstrate the performance of our proposed method, we calculate simulated DPC images. The corresponding parameters are chosen to match our experimental setup. We use an ideal pupil function of the used objective lens to obtain $H_{\mathrm {ph}}$ by Eq. (3). For a given phase sample, we calculate the intensity image without the noise at the camera plane by Eq. (4). Note that we assign the background intensity $B$ with the value of 0.75 (the intensity of all images in this paper is normalized between 0 and 1) to improve CNR as shown in Eq. (5). For all images, we add synthesized noise using a joint Poissonian-Gaussian noise as shown in Eq. (6). The fixed pattern noise of the camera and non-uniform illumination are not considered in the simulation. We use the method of Ref. [36] to obtain an automatic estimate of the parameters of the used camera that define the Poissonian noise $a=4.74\times 10^{-5}$ and Gaussian noise $b=1.16\times 10^{-5}$. Then we add the noise to the image $I(r_c)$ with the overall standard deviation of $\sigma _{\mathrm {noise}}(r_c)=\sqrt {aI(r_c)+b}$ as shown in Figs. 2(a) and 3(a). Meanwhile, we do not saturate the maximum image intensity.
Fig. 2. Performance of tDPC and ncDPC on a simulated USAF target. (a) Simulated tDPC measurements and corresponding denoised images in ncDPC for a given phase ground truth (GT) pattern. (b) Calculated CNR for different phases between 1 mrad and 100 mrad. (c) Comparison of reconstructed phase accuracy between tDPC and ncDPC. The reconstructed phase difference of the region marked with orange solid-box is calculated. The reconstructed phase profiles along the solid lines in (c) are compared with GT.
In particular, two different samples are considered in the simulation. One sample is the phase pattern of a local binary resolution target as shown in Fig. 2(a). This simulated target has the same spatial frequencies with the Elements 4-6 of Group 6 (G6E4-E6) in real USAF resolution target. The phase can be set to the amplitude scaled between 1 mrad and 100 mrad. For a given phase, 4 raw images in tDPC and corresponding denoised images are shown in Fig. 2(a). The raw image has a size of $429\times 429$ pixels. The CNR of the varying phase is calculated by Eq. (5). Note that the target is placed along the direction of illumination and the edges are parallel to it. Then the obtained CNR in Fig. 2(b) is $\sqrt {2}$ times that of Eq. (5) [22]. To assess the performance of our proposed method, we quantitatively compare the reconstructed phase to those obtained in tDPC as shown in Fig. 2(c). Obviously, ncDPC has a better reconstructed quality compared to tDPC. Meanwhile, the local difference of reconstructed phase between ncDPC and tDPC is calculated, which shows obvious ringing artifact pattern due to the noise. The mean of this phase difference is close to zero. It is difficult for tDPC to recognize the shape of the target from the quantitative phase profiles in Fig. 2(c) when the phase is set to 9.3 mrad and the corresponding CNR is 1.25. By contrast, ncDPC makes the correct quantitative phase reconstruction even the phase is 7.6 mrad.
Similarly, the simulated results for a smoother circular object are shown in Fig. 3. The phase varies with the distance to the center of the circle as shown in Fig. 3(a). The CNR of the phase between 10 mrad and 400 mrad is calculated in the same manner. From a quantitative point of view, Fig. 3(c) shows ncDPC is able to reconstruct correct phase even when the CNR is 0.61. Meanwhile, the noise in tDPC has been also transferred to the reconstructed phase, which greatly reduces phase fidelity.
Fig. 3. Performance of tDPC and ncDPC on a simulated smoother circular object. (a) Simulated tDPC measurements and corresponding denoised images in ncDPC for a given phase ground truth (GT) pattern. (b) Calculated CNR for different phases between 10 mrad and 400 mrad. (c) Comparison of reconstructed phase accuracy between tDPC and ncDPC. The reconstructed phase difference of the region marked with orange solid-box is calculated. The reconstructed phase profiles along the solid lines in (c) are compared with GT.
4.2 Experimental results
The reconstructed phase of a 39.8 mrad USAF target in both tDPC and ncDPC are shown in Fig. 4. The whole region of G8 and G9, as well as G9E3-E6, are enlarged in Figs. 4(a) and 4(b) for better visualization. Obviously, tDPC suffers from resolution degradation due to the noise. By contrast, ncDPC provides better resolution and all elements in G9 are resolved (like as the simulated results, see Supplement 1). Meanwhile, a region of $10.66\times 10.66 \mu m^2$ ($41\times 41$ pixels) where no features are present is selected and shown in Figs. 4(a) and 4(b). The standard deviation of the reconstructed phase over this region is calculated. tDPC has a larger phase fluctuation due to the noise that has been transferred to the phase. Therefore, the reconstructed phase along the lines of G6E2-E4 has stable phase contrast in ncDPC, and is consistent with the GT phase as shown in Figs. 4(c) and 4(d).
We further measure the phase target with a 9.3 mrad, and the corresponding results are shown in Figs. 5(a) and 5(b). When the sample’s phase is decreased, the sample’s details is comparable with the noise fluctuations and the reconstructed phase has a loss of image resolution, especially in tDPC. Figures 5(c) and 5(d) respectively show the comparison of the reconstructed phase along the lines in G6E5-E6 and G8E2-E3, highlighting the correct quantitative reconstructed result from ncDPC.
Fig. 4. Experimental results of a 39.8 mrad phase resolution target. Quantitative reconstructed phase using (a) tDPC and (b) ncDPC. All elements in G8 and G9, as well as G9E3-E6 are enlarged for better visualization. For the obtained phase, the standard deviation of a featureless area in (a) and (b) is calculated. Phase values of (c) tDPC and (d) ncDPC along the lines of G6E2-E4.
Fig. 5. Experimental results of a 9.3 mrad phase resolution target. Quantitative reconstructed phase using (a) tDPC and (b) ncDPC. All elements in G8 and G9, G6E4-E6 as well as G8E2-E4 are enlarged for better visualization. The standard deviation of a featureless area in (a) and (b) is calculated. Phase values along the lines of (c) G6E5-E6 and (d) G8E2-E3 for tDPC and ncDPC.
4.3 Results in open datasets
We have demonstrated the ability of ncDPC in our simulated and experimental results. ncDPC relies on computational analysis to correct the noise in DPC raw images, which further increases the signal-to-noise ratio (SNR). Meanwhile, Bonati [23] use a high-end lock-in camera to increase SNR, which further improves the DPC sensitivity. In this section, we demonstrate it is possible to use ncDPC to obtain a similar reconstruction quality as in lock-in DPC.
Figure 6 shows the reconstructed phase of a USAF target from the open-source data in Ref. [23,37]. This data for Fig. 5 in Ref. [23] consists of 20 pairs of raw DPC images with left and right half-circle illumination. This target has a phase difference of 50.6 mrad and is placed with an angle of $25^{\circ }$ with respect to the direction of illumination [23]. The corresponding background images are also open in this data. Note that the raw image has a size of $300\times 300$ pixels and is under-sampling. The standard deviation of the shot noise and readout noise is estimated from a featureless area of raw images. The reconstructed phase in tDPC and ncDPC is based on a pair of images, as shown in Fig. 6(a) and 6(c). In Fig. 6(b), 20 pairs of tDPC frames are averaged to obtain reconstructed phase. The comparison of the cross sections from reconstructed phase in Fig. 6(a)-(c) is shown in Fig. 6(d). Similarly, it is difficult for tDPC to recognize the shape of the target due to the noise, even by averaging 20 frames. The authors in Ref. [23] show that 58 averaged frames in tDPC can achieve a similar reconstruction quality as in lock-in DPC. Obviously, our proposed method can obtain the correct quantitative result of reconstruction only using a pair of images. Note that raw DPC images without any sample in place are not obtained and the results in Fig. 6 are limited by the non-uniform illumination. Therefore, the fixed slash-like artifacts appear in Fig. 6, which can be corrected by the obtained relative corrected factor or better illumination. Besides, the noise model in lock-in camera may be different from CMOS camera. However, our method is still suitable for the DPC measurements from lock-in camera as shown in Fig. 6.
Fig. 6. Reconstructed phase of a 50.6 mrad USAF resolution target from the open-source data in Ref. [23]. The data consists of background images and raw DPC images with left and right half-circle illumination. The target is placed with an angle of $25^{\circ }$ with respect to the direction of illumination. Reconstructed phase in (a) tDPC and (c) ncDPC using a pair of images. Reconstructed phase in (b) tDPC using 28 averaged frames. (d) Phase values along the lines in (a-c).
Figure 7(a) shows the reconstructed phase of human breast basal epithelial cells from the open-source data in Ref. [10,38]. Raw images are captured with LED array illumination using four halves of the bright-field circle on each image. The central wavelength of LED array is 513 nm. Example images are all taken with a 20$\times$ 0.4NA objective. All used parameters are identical to those described in Ref. [10]. Note that this data does not have background images by taking DPC measurements without any sample in place as described in Section 2.2. Or the open DPC images have been divided by the corresponding background image. Two selected zoom-ins of the phase images are shown in Fig. 7(c) and 7(d) to better observe the subcellular structure. Clearly, after ncDPC processing, we observe that noise-correction makes the sample’s details stand out better. Not that a deficiency of many existing denoising methods is the loss of fine detail in the images. The detail features of the cellular structures are not lost in ncDPC (see Supplement 1). Meanwhile, the results in Fig. 4 confirmed that ncDPC does not induce the loss of the resolution. ncDPC provides the trade-off between phase reconstruction accuracy and the resolution. For more biomedical applications, the performance of ncDPC need to be demonstrated in future. Besides, a region of $8.29\times 8.29 \mu m^2$ ($51\times 51$ pixels) where no features are present is selected and shown in Fig. 7(b). ncDPC can correct the ringing artifact pattern, which further reduces the reconstructed phase fluctuation from 25.0 mrad to 8.4 mrad.
Fig. 7. Reconstructed phase of human breast basal epithelial cells from the open-source data in Ref. [10]. (a) Full field of the reconstructed phase by ncDPC. (b) A $51\times 51$ featureless region is selected in the reconstructed phase. Phase results of two selected zoom-ins are shown in (c) and (d).
5. Discussion
In fact, DPC relies on computational analysis to extract phase information from phase gradient images. Usaully, image deconvolution is widely used in DPC. Note that band-limited phase transfer functions (PTF) in DPC have incomplete coverage in the frequency domain, which makes the deconvolution ill-conditioned. Meanwhile, the noise can easily degrade the performance of tDPC by producing deconvolution artifacts. Especially, when the sample phase is smaller, the contrast in the captured phase gradient images will be very low. If the noise level is comparable with this gradient, it is difficult to accurately recover the phase. Here, we propose a denoising method designed for DPC. This is based on a theoretical DPC model which effectively considers multiple camera noise sources and non-uniform illumination. The dominating shot noise and readout noise can be jointly estimated using frequency analysis, and further corrected by BM3D method. Because the deconvolution-based reconstruction process is highly sensitive to the SNR. For this reason, we observe a remarkable reduction of ringing artifacts in deconvolved phase images by employing ncDPC.
Our propose method improves the reconstruction quality and is suitable for all DPC experiments, which does not require any modifications to existing DPC systems. We have demonstrated the broad applicability of ncDPC by showing its performance in different experimental conditions as shown in Fig. 4-7. It involves different samples and camera sensors. Our method is not designed for specific cases and we recommend to apply our method to raw DPC data. Meanwhile, to further demonstrate the generalizability of ncDPC, this method will be applied in a broader range of samples and imaging scenarios in future, such as weak phase details in biological samples where interferometric techniques still dominate.
Besides, our method allows for an acceptable accuracy even at low-light intensity as shown in Fig. 8. Figure 8(a) shows one raw measurement in tDPC and corresponding denoised image in ncDPC. The maximum value of raw measurement is 0.034. The target immersed in the distilled water (its refractive index at our wavelength is 1.3338) is imaged, which introduces a phase difference of 110.4 mrad. We observe that our method can attenuate the detrimental effect of the noise, avoiding loss of image resolution, even at low-light intensity. Meanwhile, the fluctuation of the reconstructed phase due to the noise, which is not related to the sample, is increased at low-light intensity as shown in Fig. 8(b) and 8(c). The mean squared error (MSE) of the reconstructed phase is obtained by averaging along the lines of G8E2-G8E4. The MSE of tDPC and ncDPC is $2.74\times 10^3$ and $1.12\times 10^3 \mathrm {mrad}^2$, respectively. When the sample phase is further decreasing and comparable with this fluctuation, it becomes harder to retrieve. On the other hand, ncDPC can maintain the same quality of the reconstructed phase with a lower intensity (or a shorter exposure time) and a higher sample’s phase. Combined with color-multiplexed DPC (cDPC) [11,28,29], our method can achieve a faster imaging speed without compromising the underlying signal, which will be further studied in future.
Fig. 8. Experimental result of the phase resolution target at the low-light intensity. The target is immersed in the distilled water and has a phase difference of 110.4 mrad.(a) One raw measurement in tDPC and corresponding denoised image in ncDPC. Reconstructed phase in (b) tDPC and (c) ncDPC. The standard deviation of a featureless region in (b) and (c) is calculated. Phase values of (d) tDPC and (e) ncDPC along the lines of G8E2-G8E4.
The performance of ncDPC is intrinsically related to the CNR of the raw images. Meanwhile, the CNR depends on the background intensity of raw DPC images, the sample’s phase, the illumination source, the system’s pupil function and the noise as shown in Eq. (5). For a given source, pupil function and background intensity, the simulation results show that the phase shape of circular object can be accurately recovered by our method even when the CNR is 0.61 as shown in Fig. 3. For the USAF target, ncDPC can obtain an acceptable accuracy of phase reconstruction when the CNR is 1.00, but phase fluctuation due to the noise damages the shape of the target as shown in Fig. 2. By contrast, the noise can easily degrade the performance of tDPC by producing more obvious artifact in reconstructed phase. This artifact affects the accuracy of the reconstructed phase and the shape of the sample. The DPC sensitivity in this paper is defined as the minimum detectable phase that makes the CNR equal to 1. However, the simulation results have demonstrated the minimum detectable phase determined by this sensitivity can not be accurately recovered in tDPC.
The common half-circle illumination is used in this paper. The corresponding PTF suffers a weak response at the high frequency, which results in poor phase contrast and a lower CNR for phase reconstruction at the high frequency. Experimental results show that the noise introduces additional background that covers the target phase, resulting in reducing lateral resolution for tDPC as shown in Fig. 4. Using ncDPC, these deficiencies can be alleviated. When the phase is further decreased to 9.3 mrad, the reconstructed phase in ncDPC also has a loss of lateral resolution as shown in Fig. 5. Meanwhile, more artifacts appear in the reconstructed phase image. In fact, the noise cannot be completely removed in raw images. The performance of ncDPC is reduced due to the very low CNR. Compared with tDPC, ncDPC still has substantially improved quality and accuracy of the reconstructed phase. Using a optimized illumination [39,40] will allow our method to reach better performance.
The DPC sensitivity is defined as the same manner with Ref. [22] in this paper. It describes the relation between the standard deviation of the noise and the difference in intensity of maximum and minimum values introduced from sample’s phase. In fact, this sensitivity is unable to be estimated on normal experimental conditions, due to the absence of prior sample’s phase. Meanwhile, the sensitivity is mainly used to determine the minimum detectable phase for DPC. Our results have demonstrated that ncDPC can improve the minimum detectable phase. Besides, the sensitivity in DPC is also defined as the noise-equivalent phase which is determined by measuring the the standard deviation of the reconstructed phase over a featureless region [29,41]. The experimental results show that ncDPC can also improve this phase sensitivity.
During the last years, numerous algorithms have been extensively applied in the processing of DPC images. Most algorithms are created for Gaussian noise dominated images. Our results show that the Poisson noise is dominated in our raw images (see Supplement 1). ncDPC targets specifically Poisson-Gaussian noise in used camera, and thus performs such estimates by the knowledge of system parameters and one-time correction of the non-uniform illumination. This results in a more robust and efficient performance for the phase reconstruction. The minimum detectable phase can be decreased to 9.3 mrad. Meanwhile, BM3D based image denoising still demonstrates very competitive performance for the denoising tasks in terms of both effectiveness and efficiency in this paper. Recently, learning-based image denoising outperforms traditional method with excellent results [34,35]. However, its application depends on adequate training data, which is challenging for recovering phase information beyond the training. In contrast, ncDPC is effective for any image without prior information.
Furthermore, it should be noted that ncDPC assumes the mixed Poisson-Gaussian noise to be spatially-invariant. The whole image has one estimate of the noise variation. In fact, this can be further mitigated by the use of patch-based processing. ncDPC can provide a different local estimate of the noise variation from different subsets of the raw DPC images. Finally, we should be aware that the noise may not be completely removed in ncDPC and other advanced denoised method can be introduced to DPC in future.
6. Conclusion
In conclusion, we have demonstrated the broad applicability of ncDPC by showing its superior performance in various experimental conditions. It involves different sensors, including a CMOS camera and a lock-in camera, and a diverse range of samples, such as USAF targets with different phases and human breast basal epithelial cells. These results show that the noise correction in raw DPC images can result in an improvement of phase reconstruction quality and accuracy. For the USAF target, ncDPC can obtain an acceptable accuracy of phase reconstruction when the phase is decreased to 9.3 mrad. Besides, our method can be applied for all DPC measurements without any modification to the imaging system, which can be useful for quantitative phase imaging.
Funding
National Key Research and Development Program of China (2021YFC2400100, 2022YFC2404400, zchj2022006L); National Natural Science Foundation of China (62375280); Natural Science Foundation of Shandong Province (ZR2023MF087); Basic Research Program of Suzhou (SJC2022009, SSD2023010, szs2020308); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2023340).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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