Accurate Quantitative Phase Imaging: Iterative Deconvolution Differential Phase Contrast Breaks Through Weak Object Approximation

In optical imaging, "phase" is one of the most important components of the light field (strictly speaking, monochromatic coherent light field). Especially in the field of optical microscopic imaging, most objects are phase objects with weak absorption, where the amplitude of light passing through the object (e.g., a cell) is almost constant, while the phase of transmitted light contains crucial information about the sample, such as 3D morphology and refractive index distribution. The acquisition of phase information is particularly significant, has driven the emergence of "phase measurement" technology. At present, "phase measurement" as a major research direction in the field of optical microscopic imaging, and has been widely applied in the fields of pathology, biological cytology and drug development.

Because of the invention of lasers, computers and charge-coupled devices (CCDs), classical phase measurement techniques are proposed based on the principle of interferometric imaging. It encodes invisible phase information into the interference fringe by superimposing additional coherent reference beams and the original object beam, and demodulates quantitative phase information of the sample by combining algorithms such as fringe analysis, numerical propagation and phase unwrapping. This technique provides reliable optical metrology for accurate measurement of samples' phase and enables precise quantitative observation of transparent samples. However, interferometric phase measurement has not been widely used in microscopic imaging because it is highly susceptible to scattering noise and it requires a highly coherent illumination source and a precisely calibrated interferometric optical path.

Quantitative phase imaging (QPI) under partially coherent illumination (transport of intensity equation, differential phase contrast) opens up new possibilities for higher quality "non-interferometric" phase measurements. It has the advantages of robustness, simple and easy-to-implement imaging configuration, imaging algorithm without phase unwrapping and higher lateral/axial resolution of imaging results. However, unlike the linear relationship between object distribution and complex amplitude in interferometric imaging, the intensity and object transmission under partially coherent imaging is "bilinear", which makes it impossible to establish the intensity-phase solution mechanism directly.

Typically, QPI techniques under partially coherent imaging require the introduction of a weak object approximation model to linearize the intensity distribution and then establish an explicit expression of the acquired intensity and object phase so as to derive a phase inversion algorithm. However, the introduction of weak-object approximation models brings convenience while also leading to the inherent drawbacks of such techniques: 1) weak-object approximation models are described as qualitative, vague "objects with small phases" that lack a clear physical or mathematical definition; 2) the imaging performance of the QPI algorithms is limited by how well the real sample matches the approximation model. These two limitations result in QPI losing its proudest "quantitative" properties and no longer providing credible evidence for subsequent analyses such as cell morphology and cell mass measurement.

To solve the above problems, the group of Prof. Qian Chen and Chao Zuo from Nanjing University of Science and Technology has carried out exploratory work on the "quantitative nature" of partially coherent QPI, giving the first strict numerical definition of weak object approximation and proposing an iterative deconvolution algorithm based on the pseudo-weak object approximation model. The algorithm achieves accurate QPI without any additional data, extending the applicability of DPC QPI from weak-phase objects to large-phase objects. The relevant research results were published in Photonics Research, Volume 11, No. 3, 2023 (Yao Fan, Jiasong Sun, Yefeng Shu, Zeyu Zhang, Qian Chen, Chao Zuo. Accurate quantitative phase imaging by differential phase contrast with partially coherent illumination: beyond weak object approximation[J]. Photonics Research, 2023, 11(3): 442).

To address the problem that the weak object approximation lacks a clear physical or mathematical definition, the researchers analyzed the intensity generation model under partially coherent imaging and found that the weak object approximation is an approximate model for the observed sample, but the nonlinear intensity error under this approximation is affected by the object phase distribution and the illumination aperture. This implies that both factors need to be considered to determine a strict definition of the weak object approximation. To obtain a rigorous definition of the weak object approximation, the researchers imposed the abrupt object as well as the annular illumination aperture constraint to produce the maximum nonlinear intensity error and determined the first strict mathematical definition of the weak object approximation through numerical simulations. This conclusion offers a theoretical explanation of the conditions that need to be met for the "quantitative" nature of quantitative phase imaging and a theoretical basis for measuring the accuracy of phase reconstruction.

Furthermore, to break the limitation that the weak object approximation is difficult to be applied to large phase samples, the group introduced the pseudo-weak object approximation model into optical microscopy. The pseudo-weak object approximation was first proposed by Fanghua Li in 1982, which firstly elucidated the relationship between image intensity and crystal thickness, revealed the variation law of image intensity for atoms with different weights, and extended the interpretation of the contrast degree of high-resolution electron microscope images to a much more tolerable thickness than the weak-phase object approximation. Different from the single-layer phase distribution under traditional weak object approximation, pseudo-weak object approximation describes the large phase object as a superposition of a multilayer weak phase structure, which could be seen as a more accurate model.

Based on this model, the phase reconstruction is no longer achieved through a direct deconvolution algorithm, but by solving an optimization problem that needs iterative reconstruction. The basic idea of the algorithm can be described as: the nonlinear components in the intensity are continuously eliminated by the intensity difference between Abbe's model and the transfer function model. The ideal linear distribution is approached by the intensity used in deconvolution reconstruction to realize the update and refinement of the quantitative phase. This algorithm can achieve accurate quantitative phase reconstruction of large phase objects without additional data acquisition, breaking through the limitation of weak object approximation and extending the partial coherent quantitative phase imaging algorithm to large phase objects.

The accurate QPI capability of the iterative deconvolution algorithm was validated in the observation experiment of MCF-7 human breast cancer cells, and the results are shown in Figure 4. Compared to the traditional one-step deconvolution algorithm, one-step deconvolution (blue curve) provides too small cell morphological features, and iterative deconvolution (red curve) accurately recovers quantitative features of the sample, demonstrating its accurate 3D morphological data. The results show that the technique provides more accurate 3D morphological data of cells, which is expected to provide an effective technical solution for early cancer diagnosis and treatment.

The quantitative definition of the weak object approximation provides a basis for measuring the reconstructed phase accuracy of the DPC technique, the researchers of the group said. The proposed iterative deconvolution extends the range of applicable objects for DPC and obtains more accurate 3D morphological data of cells, which is expected to be an effective technical solution for early cancer diagnosis and treatment. In the future, the research group will conduct research on the biological application of this technology to realize the transition from technological innovation to practical application.

精确定量相位成像

在光学成像中，“相位”是光场（严格来说是单色相干光场）中最重要的组成部分之一。尤其是光学显微成像领域，大部分被测物体属于弱吸收的相位物体，光通过物体（如细胞）时，其振幅几乎不变，而透射光的相位则包含了关于样品的重要信息，如三维形貌与折射率分布。因此，相位信息的获取就显得尤为重要，推动了“相位测量”技术的出现。目前，“相位测量”已成为光学显微成像领域一大主要的研究方向，并在生物细胞学、病理学以及药物研发等领域进行了广泛的应用。

得益于激光、电荷耦合器件（CCD）和计算机的发明，经典的相位测量技术基于干涉成像的原理被提出。其通过叠加附加的相干参考光束和原始物体光束，将不可见的相位信息编码到条纹图案中，结合条纹分析、数值传播和相位解包裹等算法解调样品的定量相位信息。这一技术为精确测量待测样品的相位提供了一套可靠的光学计量方法，可实现对透明样品的精确定量观测。然而，由于其所需相干性极高的照明光源和精密校准的干涉测量光路，且极易受到散斑噪声的影响，干涉相位测量并未得到广泛的应用。

部分相干照明下的定量相位成像（光强传输方程、差分相衬）为实现更高质量的“非干涉”相位测量开辟了新的可能。其具有成像系统配置简单、更易实现，成像算法无需相位解包裹，成像结果具有更高的横向/轴向分辨率以及鲁棒性等优势。然而，不同于干涉成像物体分布与复振幅的线性关系，部分相干强度与物体传输呈“双线性”特征，这一复杂的关系使得无法直接建立强度-相位的求解机制。

通常情况下，部分相干成像下的定量相位成像技术需要引入弱物体近似模型来线性化其强度分布，建立采集强度和物体相位的显式表达，从而推导相位反演算法。然而，弱物体近似模型的引入在带来便利的同时也导致这类技术固有的缺陷：1）弱物体近似模型被描述为定性的、含糊的“相位较小的物体”，缺乏物理或数学上的明确定义；2）相位重构的精确度受限于真实物体与近似模型的匹配程度。这两个局限性将使定量相位成像失去其最引以为傲的“定量”特性，无法再为后续分析（如细胞形态学和细胞干质量测量）提供精确的、可靠的数据。

为解决上述问题，南京理工大学陈钱教授和左超教授课题组开展部分相干定量相位成像探索性工作，首次给出了弱物体近似的严格数值定义，并引入赝弱物体近似模型，提出迭代反卷积算法，无需任何额外数据实现了精确的定量相位成像，使差分相衬成像算法的适用范围从弱相位物体扩展到大相位物体。相关研究成果发表于Photonics Research 2023年第3期（Yao Fan, Jiasong Sun, Yefeng Shu, Zeyu Zhang, Qian Chen, Chao Zuo. Accurate quantitative phase imaging by differential phase contrast with partially coherent illumination: beyond weak object approximation[J]. Photonics Research, 2023, 11(3): 442）。

针对弱物体近似缺乏物理或数学上的明确定义的问题，研究人员分析了部分相干成像下的强度生成模型，发现尽管弱物体近似是对观测样品的近似模型，但是该近似下的非线性强度误差受到物体相位分布和照明孔径的影响，这就意味着探究弱物体近似的定义需要同时考虑这两个因素。为得到弱物体近似的严格定义，研究人员施加突变物体以及环形照明孔径约束以产生最大的非线性强度误差，通过仿真不同相位数值下的定量相位重构结果并通过均方根误差数值作为相位“定量化”的评判标准，首次给出了弱物体近似的严格数学定义。

弱相位近似模型要求物体相位不大于0.5 rad，此时可对任意物体分布和照明孔径都可以实现精确的定量相位重构；而当物体相位大于0.5 rad时，弱相位近似不被满足，重构结果与物体真值不再一致，其成像结果不再可靠。这一结论从来理论上说明了定量相位成像的“定量化”特性所需要满足的条件，为衡量相位重构的准确性提供了理论基础。

为了应对弱物体近似下定量相位成像技术难以实现对大相位物体准确重构的问题，研究团队将电子显微镜中的赝弱物体近似模型引入光学显微成像领域。赝弱物体近似最初于1982年由李方华院士提出，其首次阐明了像强度与晶体厚度之间的关系，揭示了轻重不同原子像强度的变化规律，将高分辨电子显微像的衬度解释拓展到比弱相位物体近似宽容得多的厚度。

算法的基本思想可以描述为：通过阿贝模型和传递函数模型下的强度差分来不断消除强度中的非线性成分，使用于反卷积重构的强度逼近理想线性分布，实现物体定量相位的更新和细化。该算法无需额外的采集数据即可实现对大相位物体的精确定量相位重构，突破了弱物体近似的限制，将部分相干的定量相位成像算法拓展到了大相位物体。

上述迭代反卷积算法的精确定量相位成像能力在MCF-7人乳腺癌细胞的观察实验中得到了验证，结果如图3所示。与传统一步反卷积算法相比，对于具有大相位值的有丝分裂细胞，一步反卷积（蓝色曲线）提供了过小的细胞形态特征，迭代反卷积（红色曲线）准确地恢复了样品的定量特征，展示了其准确的三维形态数据。结果表明，该技术提供了更加准确的细胞三维形态学数据，有望为早期癌症确诊治疗提供有效的技术方案。

研究团队表示：“弱物体近似的定量化定义为衡量差分相衬技术的重构相位准确性提供依据。所提出的迭代反卷积拓展了差分相衬的适用物体范围，提供了更加准确的细胞三维形态学数据，有望为早期癌症确诊治疗提供有效的技术方案。未来，研究团队将针对这一技术的生物学应用展开研究，以实现技术创新到实践应用的转变。”