Compensation of aberration and speckle noise in quantitative phase imaging using lateral shifting and spiral phase integration

We present a simple and effective method to eliminate system aberrations and speckle noise in quantitative phase imaging. Using spiral integration, complete information about system aberration is calculated from three laterally shifted phase images. The present method is especially useful when measuring confluent samples in which acquisition of background area is challenging. To demonstrate validity and applicability, we present measurements of various types of samples including microspheres, HeLa cells, and mouse brain tissue. Working conditions and limitations are systematically analyzed and discussed.

depicts the working principles of the present method. The quantitative phase image (x, y) retrieved from a hologram can be decomposed into an ideal aberration-free phase image 0(x, y) and an aberration term (x, y) as (x, y) = 0(x, y) + (x, y), as shown in Figs. 1(a)(b). In the background subtraction method [26], aberration term (x, y) is obtained by measuring the background hologram of an area without a sample [ Fig. 1(c)]. Then, (x, y) is subtracted from the measured (x, y) in order to retrieve 0(x, y), [ Fig. 1(d)]. However, the background subtraction method has difficulty in finding clean background regions, especially for confluent cells or tissue slices.
The present method does not require finding a background region to measure (x, y). Instead, the method precisely retrieves (x, y) from three holograms with lateral shifts even in the presence of confluent samples. One hologram of a sample is recorded, and two other holograms are recorded after shifting the sample in orthogonal directions. From three measured holograms, original and shifted phase images are obtained. Subtracting the original phase image from shifted phase images, two differential phase images x(x, y) and y(x, y) are calculated as x(x, y) = 0(x + x, y) -0(x, y) and y(x, y) = 0(x, y + y) -0(x, y), In order to retrieve the ideal phase image from two differential phase images, we utilized spiral phase integration [31]. According to the Fourier shift theorem, the Fourier transforms of x(x, y) and y(x, y) are (e 2ixu  1)0(u,v) and (e 2iyv  1)0(u,v), respectively, where 0(u,v) = FT[0(x, y)]. Next, we define G(u,v) and H(u,v) as follows, Dividing G(u,v) by H(u,v), we obtain 0(u,v) and subsequently the ideal phase image 0(x, y). See Appendix A for detailed derivations.

Optical setup
The experimental setup is presented in Fig. 2. A coherent plane-wave beam from an He-Ne laser (λ = 633 nm, HNL050R, Thorlabs Inc.) is split by a beam splitter into a reference and a sample beam. Samples were mounted on a motorized scanning stage (MLS203-1, Thorlabs Inc.) for the automatic shift. The light diffracted from the samples was collected by a high numerical aperture (NA) objective lens (NA=1.2, water immersion, UPLSAPO 60XW, 60×, Olympus, Inc., Japan). The sample beam was further magnified by a factor of two and interfered with a reference beam at the image plane. Interference pattern was recorded using a complementary metal-oxide semiconductor camera (DCC3240M, Thorlabs Inc.).

Fig. 2 Optical Setup.
Sample is mounted on a motorized stage. Sample and reference beam interfere and generate a hologram, which is recorded by the camera.

Sample preparation
Silica (SiO₂) beads (n = 1.4570 at λ = 633 nm) with diameter of 3 μm were immersed in water and sandwiched by two cover slips. Polystyrene beads (n = 1.5875 at λ = 633 nm) with a diameter of 10 μm were immersed in index matching oil (n = 1.5279 at λ = 633 nm) and sandwiched by two cover slips. HeLa cells (human cervical cancer cell line) were cultured in DMEM (Dulbecco's modified Eagle's media) with 10% of FBS (fetal bovine serum) and 1% of penicillin-streptomycin, and fixed with He-Ne Laser Motorized stage 4% formaldehyde. Fixed cells were prepared in a Petri dish and immersed in PBS (Phosphate-buffered Saline). Mouse brain tissue was obtained from a 22-year-old male mouse. After fixation and dehydration, brain tissue was sliced and sandwiched between two cover slips with a mounting medium (n = 1.355).

Results
To demonstrate the validity of the present method, we captured phase images of the 3-μm-diameter silica beads. Phase images obtained by background subtraction method and by the present method are presented in Fig. 3. Figure 3(a) displays a raw phase image (x, y) retrieved from a single hologram. In the background subtraction method, an aberration phase image (x, y) is additionally measured. Then, (x, y) was subtracted from (x, y) to obtain an improved phase image 0(x, y), as presented in Fig. 3(c). In contrast, the present method utilizes two shift differential phase images x(x, y) and y(x, y) [ Fig. 3(b)]. Spiral phase integration of the differential phase images yields the reconstructed phase image 0(x, y), in which the aberration phase term (x, y) is automatically canceled out. Based on the reconstructed phase image, we can also retrieve (x, y) by subtracting the reconstructed phase image from the raw phase image [ Fig. 3(c)]. The phase image improved via the conventional method involves high-frequency aberration that originates from a slight variation of the angle between the sample and the reference beam, as shown in Fig. 3(d). On the other hand, the phase image improved by the present method showed relatively cleaner phase images. The standard deviation of the phase values in the flat no-sample area are 0.163 and 0.086 rad for the conventional method and the present method, respectively.
To demonstrate the applicability of the present method to biological samples, phase images of HeLa cells are measured [ Fig. 4]. In the raw phase images, due to the system aberration, the quality of phase images is severely deteriorated [ Fig. 4(a)]. In contrast, when the present method is applied, the background phase is removed and, then, the details of the sample can be clearly seen in the phase images [ Fig. 4(b)]. The overall quality of the aberration-corrected images when using the present method is comparable to that of images measured with white-light QPI techniques [32][33][34]. To further demonstrate the potential of the present method, a phase image of mouse brain tissue slice is presented [Fig. 5]. 25 phase images were measured and aberration was corrected with the present method. Then, the 25 phase images were manually stitched together to construct a large field-of-view phase image [ Fig. 5(a)]. Representative raw and improved phase images are presented in Figs. 5(b)(c). Raw phase images contain significant aberration artifacts, which can clearly be removed using the present method.

Discussion and summary
In this Letter, exploiting the spiral phase integration of differential phase images, we propose and experimentally demonstrate a method of compensation for aberration and speckle noise in QPI. This method does not require the measurement of any sample region, which limits the applicability of QPI for confluent samples such as high-density cell cultures and tissue slices. Instead, the present method measures three laterally translated phase images in the presence of samples, from which only set of sample phase information can be reconstructed; other static noises such as system aberration and speckle patterns are systematically canceled out.
We demonstrate that the present method effectively removes aberrations from various types of samples including silica beads, eukaryotic cells, and tissue slices. We also compare the performance of the present method to that of the conventional background subtraction method. Although both methods successfully remove aberration, remaining error terms differ. The background subtraction method is susceptible to physical vibration or the alteration of the path geometry during the measurement of the no-sample region, leaving aberrations of high spatial frequency from multiple reflections. However, the present method removes most of the aberration, including multiple light scattering, providing two-fold enhancement in phase sensitivity. Nonetheless, the present method produces artifacts of low spatial frequency at the point of abrupt phase change. However, this issue can be alleviated by using a high NA objective lens or by reducing abrupt phase changes by matching phase contrast between a sample and a medium. As shown in Figs. 4 and 5, the present method is also applicable to samples with complex contour, including cells and biological tissues. Moreover, phase images corrected with the present method can be successfully stitched together to reconstruct large field-of-view phase images. We also verify that our method is applicable to phase images that occupy the boundary of the field-of-view, including tissue samples.
One of the limitations of the present method is that it requires the measurement of two additional phase images with lateral shifts in orthogonal directions. However, using a motorized stage, these additional measurements can be performed in a short period of time. Because the present method is not limited by the type of instrumentation but is generally applicable to optical field measurement techniques, it can also be readily applied to various types of setup, ranging from digital holographic microscopy [35,36] to optical diffraction tomography [37,38].
In summary, we present a simple but powerful method of compensation for aberration and speckle noise in QPI. The present method will be applied to various fields, in particular the study of as confluent samples such as adherent biological cells and tissue slices.