This invention belongs to optical microscopic imaging and measurement technology, in particular, a differential phase contrast quantitative phase imaging method based on the optimal lighting pattern design.
Due to the weak absorption of biological cells, label-free imaging has been a hot topic in microscopic imaging methods. Quantitative phase imaging (QPI) techniques have gradually become the main tool for cell research. In recent years, a variety of quantitative phase imaging techniques have been proposed, and they tend to have different imaging performances in terms of imaging resolution, imaging throughput, and imaging speed. Among these imaging methods, differential phase contrast (DPC) quantitative phase imaging (QPI) only requires a few original images to achieve the quantitative phase recovery, not only achieves twice the lateral resolution of the diffraction limit of the objective lens, but also makes the imaging results have high correctness and stability. These advantages make DPC QPI extremely promising for cellular imaging applications.
DPC converts the invisible specimen phase into measurable intensity information by asymmetrical lighting. In recent years, new light sources, such as programmable LED arrays and LCD displays, have been introduced into microscopic imaging systems, making the lighting method flexible and adjustable. This greatly promotes the development of DPC, and some new DPC imaging mechanisms have been proposed. For example, Tian et al. proposed DPC QPI and successfully used DPC to observe the quantitative phase distribution of unlabeled cells (Tian, L., & Waller, L. (2015). Quantitative differential phase contrast imaging in an LED array microscope. Optics express, 23(9), 11394-11403). In LED-based DPC methods, four images are usually acquired using asymmetric semi-circular lighting patterns in two axes. These images are used to calculate phase gradient images in the two directions, and then obtain the quantitative phase of the specimen by deconvolution. Since only a few acquisition images are required, DPC offers great advantages for live cell imaging applications. However, the phase transfer function (PTF) obtained from the semi-circular lighting exhibits poor transfer response, which makes it difficult to achieve the desired maximum resolution and stable phase contrast. Moreover, the semi-circular lighting corresponds to non-uniform PTF distribution, which results in the phase results with artifacts and errors due to the absence of some frequencies in the case of two-axis lighting.
In order to improve the PTF of DPC imaging, new lighting methods have been investigated to enhance the transfer response of the PTF of DPC QPI. In 2015, a DPC method based on multi-axis lighting for ideal symmetry of the PTF (Tian, L., & Waller, L. (2015). Quantitative differential phase contrast imaging in an LED array microscope. Optics express, 23(9), 11394-11403) obtained isotropic PTF by illuminating the specimen with 12 different axes. Although this multi-axis approach can obtain isotropic PTF, the transfer response of the low-frequency component is still too poor to obtain a high phase contrast. Moreover, 12-axis lighting greatly increases the number of acquired images, which will obstacle the imaging speed of DPC. In order to obtain isotropic PTF with fewer lighting axes, a new lighting of DPC with radial lighting (Lin, Y. Z., Huang, K. Y, & Luo, Y. (2018). Quantitative differential phase contrast imaging at high resolution with radially asymmetric illumination. Optics letters, 43(12), 2973-2976) is proposed to obtain an optimized PTF. However, the PTF obtained in this method is not strictly isotropic, and the low-frequency component is still very poor. Subsequently, a new embedded amplitude gradient mask has been proposed to encode the lighting (Chen, H. H., Lin, Y. Z., & Luo, Y. (2018). Isotropic differential phase contrast microscopy for quantitative phase bio-imaging. Journal of biophotonics, 11(8), e201700364). However, the problem of missing phase low-frequency information remains unresolved. Overall, the optimal lighting pattern corresponding to the strictly isotropic PTF is still not available in the existing methods.
The purpose of this invention is to provide a real-time dynamic, high-correctness, high-resolution DPC QPI microscopy under optimal lighting pattern to solve the problems of serious missing of partial frequency information, slow imaging speed, and noise sensitivity of the system in DPC QPI.
The technical solution of this invention is a method of DPC QPI based on the optimal lighting pattern design with the following process:
Step 1, design optimal lighting pattern: firstly, the lighting pattern corresponding to the isotropic phase transfer function (PTF) in DPC QPI is derived, which is considered to be the optimal lighting pattern. Specifically, the optimal lighting pattern is semi-annular lighting with the lighting numerical aperture (NA) Nill equal to the objective NA NAobj. Consider the NA of the objective is expressed as NAobj, the distance between the single LED on lighting annulus and the light axis is d, and the angle between the light emitted by a single LED and the optical axis is αi, then the lighting NAill obtained by the LED on the annular pattern satisfies the following equation:
The lighting intensity distribution of the optimal lighting pattern varies with the angle of lighting according to the cosine, and the expression in polar coordinates can be expressed:
S(θ)=cos(θ)
where the angle θ increases in a clockwise direction.
Step 2, capture original images under optimal lighting: computer-controlled LCD or high-density programmable LED arrays are used to display four optimal lighting patterns under two-axis lighting, and generate synchronous trigger signals to the camera to acquire an image under each pattern, thus a total of four original images are recored, i.e., Il, Ir, Iu, and Id. Here, Il and Ir represent two images in the left and right axis directions, and Iu and Id represent two images in the up and down axis directions.
Step 3, calculate phase gradient image: according to the formula for DPC imaging, the phase gradient image of the specimen in the left-right direction Ilr=(Il−Ir)/(Il+Ir) and the up-down direction Iud=(Iu−Id)/(Iu+Id) calculated under optimal lighting.
Step 4, solve the phase transfer function (PTF) corresponding to the optimal lighting pattern: derive the PTFs PTFlr(u) and PTFud(u) in the two axis directions in the optimal lighting pattern DPC system based on the weak phase approximation condition.
Step 5, solve the quantitative phase of the sample: perform deconvolution (Tikhonov criterion) using the two-axis PTF and the corresponding phase gradient images to obtain the final reconstructed quantitative phase results.
Compared with the existing methods, the method proposed in this invention has the following advantages: (1) The PTF corresponding to the optimal lighting pattern is isotropic. In the polar coordinate system, let ra be the distance from any point to the center zero frequency in the PTF distribution, and rNA be the distance from the cut-off frequency to the center zero frequency in the frequency domain under the partially coherent imaging, then the PTF in the left-right direction is obtained as follows
The PTF in the up-down direction is:
Their intensity distribution is expressed as:
It can be seen that the transfer response of each frequency component of the PTF is independent of the angle and only related to the frequency, which indicates that the phase transfer characteristics of DPC QPI is isotropic. Compared with the conventional semi-circular lighting, not only the transfer performance of the high frequency phase is enhanced, but also the low-frequency phase can be effectively recovered, ensuring the correctness, information integrity and high resolution of the phase recovery results. (2) From the perspective of imaging speed, the design of the optimal lighting pattern reduces the number of lighting axes of isotropic PTF to a minimum of two-axis lighting, which greatly reduces the number of acquired images required for DPC QPI. This allows DPC QPI to achieve real-time dynamic high-resolution QPI. (3) From the perspective of sensitivity to noise, the optimal lighting pattern greatly enhanced the low and high frequency responses of the PTF, which relaxes the sensitivity of the imaging results to noise, allowing the correct quantitative phase to be obtained with essentially no consideration of regularization. (4) From the perspective the implementation process, the optimal lighting pattern not only ensures the information integrity of the DPC QPI, but also enhances the phase gradient in all directions.
SPECIFIC IMPLEMENTATION
The actual hardware platform of this invention is a microscopic imaging system based on a high-density programmable LED array or LCD display. The entire system includes an industrial camera for image acquisition, a microscope objective, a sample, a carrier table, and a programmable LED array or LCD display as the microscope lighting source. The LED array or LCD display is positioned under the carrier table at a spacing H typically between 20-100 mm and centered on the optical axis of the microscope system. The LED array or LCD display includes a number of point light sources, which are regularly aligned to form a two-dimensional matrix. Each point source can be illuminated in red, green, and blue, with typical wavelengths of 635 nm for red, 525 nm for green, and 475 nm for blue. Each point source has a typical center spacing d of 1-10 mm, and the LED array or LCD display is fixed in position by a fixed substrate.
If LED arrays are used for lighting, the implementation circuit to drive the LED arrays to light each of the point sources can be used existing technologies such as microcontrollers, ARM, or programmable logic devices. The specific implementation method can be found in related paper (Guo, Bao-Zeng, and Chun-Miao Deng. “Design of LED Display Control System Based on FPGA.” Chinese Journal of Liquid Crystals and Displays 25.3 (2010): 424-428).
If the LCD display is used for lighting, it requires the microscope's own halogen lamp and other light sources at the bottom of the LCD display to provide raw lighting, and the LCD is introduced to replace the aperture diaphragm of condenser lens. By displaying different patterns as the spatial light filter, the lighting is modulated to different shapes and color distribution. The technology used in the driving circuit of LCD is basically the same as the LED array, and the specific implementation method can be referred to the related literature 201510631692.4.
From
Step 1, design optimal lighting pattern
The design of the optimal lighting pattern for DPC QPI is to derive the lighting pattern of the isotropic PTF. It can be divided into the following two major steps, that is, for the determination of the optimal lighting shape and the lighting intensity distribution.
(1) The shape of the optimal lighting pattern is determined by the following derivation process:
First, the outer diameter of the optimal lighting pattern is determined, i.e., the maximum lighting angle. In order to research the relationship between the lighting shape and the PTF, the PTF under different aperture lightings are simulated.
Next, the inner diameter of the optimal lighting pattern is further determined by analyzing the PTF. In
Based on the above analysis, it can be determined that the shape of the optimal lighting pattern in DPC should be semi-annular lighting, and its NA of the lighting NAill should be equal to the NA of the objective lens NAobj. The transfer response of any lighting intensity distribution will be enhanced when such a semi-annular lighting is adopted. Thus, it can be determined that if the NA of the objective used for DPC QPI is NAobj, the radius of the optimal lighting annulus is d, the distance between the LEDs and the specimen is h, and the angle between the light emitted by a single LED and the optical axis is αi. The optimal lighting pattern should be satisfy:
(2) The process of determining the intensity distribution of the optimal lighting pattern is as follows:
In polar coordinates, the PTF is defined as PTF (r,θ), where r denotes the distance from a point to the center point of the PTF distribution, θ denotes the angle between the point and the axis direction. Achieving an isotropic PTF requires a minimum of two axes of lighting, and they are required to satisfy:
|PTFlr(r,θ)|2+|PTFud(r,θ)|2Cr
where PTFlr(r,θ) and PTFud(r,θ) denote the PTFs in the left-right and up-down axis directions, respectively. Cr is a constant only related to r.
In polar coordinates, the PTF is a periodic function, so it can perform the Fourier series expansion to yield the following expression:
where an and bn in the coefficient terms independent of r and θ, and n is an arbitrary positive integer. To satisfy |PTFlr(r,θ)|2+|PTFud(r,θ)|2Cr, |PTFlr(r,θ)|2 and |PTFud(r,θ)|2 cannot produce any cross term, then the expression of PTFlr(r,θ) can be obtained as follows:
PTFlr(r,θ)=an cos(nθ) or PTFlr(r,θ)=bn sin(nθ)
The two lighting axes of DPC are perpendicular to each other, so their PTFs differ by π/2 rad. Therefore, when PTFlr(r,θ)=an cos(nθ), we will get PTFud(r,θ)=bn sin(nθ), and when PTFlr(r,θ)=bn sin(nθ), we will get PTFud(r,θ)=an cos(nθ).
When using a light source with a lighting NA equal to the objective NAobj, the intensity distribution of the light source is expressed as S(θ). In the polar coordinate system, S(θ) is required to satisfy three constraints: (a) S(θ) is an even function about the direction axis; (b) S(θ) is a periodic function about the angle θ; (c) S(θ) is symmetric about the point (π/2,θ).
The PTF response of any point can be obtained by integrating the lighting intensity corresponding to the red arc in
PTFlr(r,θ)=∫θ−αθ+αS(θ)dθ
α denotes half of the circular angle corresponding to the red arc. It can be obtained that:
S(θ)=kn cos(nθ)
where kn is an arbitrary constant. From this equation, the optimal lighting pattern is an annular lighting whose lighting intensity varies with the cosine of the lighting angle.
In the process of phase recovery, considering the minimum number of over-zero points, the intensity distribution of the optimal lighting pattern should be determined as:
S(θ)=cos(θ)
here the angle β increases in a clockwise direction.
Step 2, capture original images under optimal lighting:
computer-controlled LCD or high-density programmable LED arrays are used to display four optimal lighting patterns under two-axis lighting, and generate synchronous trigger signals to the camera to acquire an image under each pattern, thus a total of four original images are recored, i.e., Il, Ir, Iu, and Id. Here, Il and Ir represent two images in the left and right axis directions, and Iu and Id represent two images in the up and down axis directions.
LCD or high-density LED array can be used to generate the optimal lighting pattern. In the LCD-based system, the lighting system consists of the light source and LCD. The LCD is used to adjust the lighting pattern. In the LED-based system, the LED array is considered as the lighting source to directly display optimal lighting pattern by computer control.
Step 3: calculate phase gradient image:
According to the formula for DPC imaging, the phase gradient image of the specimen in the left-right direction Ilr=(Il−Ur)/(Il+Ir) and the phase gradient image of the specimen in the up-down direction Iud=(Iu−Id/(Iu+Id) are calculated under optimal lighting.
Step 4, solve the phase transfer function (PTF) of the optimal lighting pattern:
Based on the weak phase approximation, the PTFs in the two axis directions in the optimal lighting pattern DPC PTFlr(u) and PTFud(u) are derived. Consider imaging a thin pure phase specimen with a complex transmittance function of t(r)=e−a(r)+iϕ(r), it is illuminated by a single angle tilted lighting of S(uj), where r=(x, y) indicates the coordinates of the specimen plane, ϕ(r) represents the phase of the specimen. uj represents the phase shift corresponding to a single angular lighting in the frequency domain. In order to analyze the specimen phase and intensity independently, the weak phase approximation is introduced to simplify the complex amplitude distribution of the specimen as t(r)=1+a(r)+iΦ(r). Then, the Fourier transform of the complex amplitude distribution in the camera plane can be expressed as:
Wj(u)=√{square root over (S(uj))}[δ(u−uj)−A(u−uj)+iΦ(u−uj)]P(u)
u denotes the frequency component in the frequency domain, A(u) denotes the amplitude distribution of the specimen in the frequency domain, Φ(u) denotes the phase distribution of the specimen in the frequency domain. P(u) is the pupil function of the objective. The spectral distribution of the intensity image acquired by the camera under a single angular tilt lighting is obtained by convolving it with its conjugate term:
Ij(u)=Wj(u)⊗Wj*(u)=S(uj)δ(u)|P(u)|2−S(uj)A(u)[P*(uj)P(u+uj)+P(uj)P*(u+uj)]+iS(uj)Φ(u)[P*(uj)P(u+uj)−P(uj)P*(u+uj)]
Calculating the intensity distribution in the frequency domain under complex lighting, the intensity spectrum distribution can be divided into three terms (ignoring the higher-order convolution terms of the computational process):
I(u)=Bδ(u)+A(u)ATF(u)+iΦ(u)PTF(u)
B represents the background term B=∫∫S(uj)|P(uj)|2d2uj; ATF(u) denotes the amplitude transfer function, and PTF(u) denotes the PTF. Where
PTF(u)=∫∫S(uj)[P*(uj)P(u+uj)−P(uj)P*(u−uj)]d2uj
This is a general expression for the PTF under complex lighting, applicable to all lighting patterns light pupil functions.
DPC uses asymmetric lighting patterns in the up-down and left-right axis directions to illuminate the specimen, such that the tilted lighting introduces a phase factor that converts the invisible specimen phase into a measurable intensity. A simple differential calculation procedure is used to highlight the phase contrast of the specimen, and the corresponding PTF is expressed as:
Where PTFlr(u) denotes the PTF in the left-right axis directions, Slr(u,) denotes the lighting in the left-right direction. The PTF in the up-down direction PTFud(u) is derived as above process.
In the optimal lighting pattern, the PTF of DPC imaging is required be isotropic. In the polar coordinate system, let ra be the distance from any point to the center zero frequency in the PTF distribution, and rNA be the distance from the cut-off frequency to the center point in the frequency domain under the partially coherent imaging, then the PTF in the left-right direction is obtained as follows:
The PTF in the up-down direction is
Their intensity distribution is obtained as
5. Step 5, solve the quantitative phase of the specimen:
Perform deconvolution (Tikhonov criterion) using the obtained two-axis PTF PTFlr(u) and PTFud(u), and the corresponding phase gradient mage to obtain the final reconstructed quantitative phase results:
ϕ(r) is the reconstructed quantitative phase of the sample. The regularization parameter β in this equation is used to suppress the error caused by excessive amplification by noise. In the optimal lighting pattern, since the PTF is greatly enhanced in entire partially coherent imaging range (from 0 to 2 NAobj), the Tikhonov criterion can be calculated without considering the regularization parameter to obtain the anti-noise reconstructed quantitative phase. Thus, the regularization parameter β is usually set as a very small or zero value.
In order to verify the imaging performance of DPC QPI under the optimal lighting pattern, we simulated the PTFs under four different lighting patterns. As shown in
To further simulate the imaging performance of the optimal lighting patterns in a real system, we added the same level of Gaussian noise to the raw images to simulate the imaging in a real situation. The regularization parameter β is used to suppress the system noise during the deconvolution with the Tikhonov criterion.
Finally, we verified the imaging performance of DPC QPI in optimal lighting pattern by an experiment on unstained Hela cells cultured in vitro. Raw experimental data were collected using an objective with 10×, 0.4 NA, and the recovered quantitative phase is shown in
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