Computational phase imaging for light microscopes

During its 400 years of development, the light microscope has become an essential visualization tool for biological and biomedical applications. Recently, a desire to understand live cell dynamics has rekindled interest in label-free phase contrast microscopy. This technology may enable acquisition of more information than is currently obtained through commonly used fluorescence modalities, such as confocal, multi-photon, and stimulated emission depletion microscopy imaging.

It is possible to precisely quantify the phase of optically transparent biological cells and tissues using digital holographic microscopy.¹ Historically, visualization of phase has been achieved non-quantitatively, with contrast methods such as Zernike phase contrast and differential interference contrast (DIC). Such complete acquisition of scalar field (amplitude and phase) enables lens-free imaging, abberation compensation, and numerically focusing images at different depths.² Nevertheless, in most cases, interferometry and digital holography rely on coherent illumination, and are therefore plagued by the problem of speckle, which prevents the formation of high-quality images. Therefore, a method to obtain quantitative phase in a more noise-insensitive and easy-to-implement manner is highly desirable.

To this end, we have been working on the development of specialized computational methods for the extraction of quantitative information (for phase and beyond) non-interferometrically, by solving the so-called transport-of-intensity equation (TIE). The TIE is a 2D second-order elliptic partial differential equation that must be solved under appropriate boundary conditions. It was originally derived by Teague from the Helmholtz equation under paraxial approximation more than 30 years ago,³ when its main application fields were adaptive optics, electron microscopy, and x-ray imaging.

Recently, technological advancements in optical microscopy and digital signal processing have brought TIE back to the forefront of biomedical microscopic imaging. Specifically, TIE links the phase to variations in intensity induced by wave propagation, which enables retrieval of the phase simply by measuring the intensity along the propagation direction at multiple positions. Using an optimum finite difference scheme to estimate an intensity derivative,⁴ and combining this with the boundary-artifacts-free TIE solver based on a fast discrete cosine transform,^{5, 6} we can directly and accurately reconstruct the continuous phase of the optical field. This is without the need for phase unwrapping (removing discontinuities in the extracted phase), as is common for many interferometric methods. In addition, we can apply TIE to a partially coherent light source, even though it has been initially derived for coherent illumination.^{7, 8} This makes it possible to upgrade a conventional light microscope with an add-on module to a real-time quantitative phase microscope. By introducing an additional image-relay system and a 2D spatial light modulator⁹ or an electrically tunable lens (ETL)¹⁰ located at the objective rear aperture plane (the so-called pupil plane coding, as in Figure 1), we can acquire 2D images of the sample at multiple depths in rapid succession (or even in a single shot) with no moving parts. Such systems enable real-time quantitative phase imaging with nanometer-scale and millisecond temporal resolution.

We have successfully applied these systems for investigations of drug-induced morphology changes and phagocytosis of macrophages,⁹ imaging of cellular dynamics of breast cancer cells,¹⁰ and characterization of micro-optical elements.¹¹ Figure 2 shows the 3D rendering of a phase map of a single macrophage at different stages of phagocytosis. Furthermore, the complete scalar field acquisition provides tremendous flexibility to realize various microscopy modalities computationally, such as dark field, Zernike phase contrast, and DIC, without requiring specialized hardware components. Recently, we reformulated the TIE in the phase space to provide an elegant description of TIE phase-retrieval and computational imaging under partially coherent illumination.⁸ The phase-space formulation of the TIE further enables computational light-field imaging at full sensor resolution, without spatio-angular resolution trade-offs, which are often inevitable in conventional microlens array-based light-field imaging.

Figure 1. Schematic of the electrically tunable lens (ETL)-based transport-of-intensity (TIE) equation quantitative phase microscope (TL-TIE).¹⁰ OL/ETL: ETL combined with an offset lens. L₁, L₂: Fourier lenses. Relay optics (with focal length 4f), with OL/ETL located at the Fourier plane, are attached to a conventional microscope.

 

Figure 2. Dynamic TIE phase imaging of the macrophage phagocytosis.⁹(a) Color-coded phase profiles at different stages of phagocytosis. (b) Phase maps of the nuclear region of the macrophage—the black square in (a)—during the internalization stage of phagocytosis. (c) Phase/thickness variation with time of three points, indicated by the dots in (b): (A) red, (B) green, and (C) blue, and the average of the whole square region (bottom, black curve). Scale bar: 10μm.

Computational phase imaging also opens up the possibility to transform the microscope into a miniaturized device by discarding its lenses and other bulky optical components. By exploiting wavelength-dependent diffraction in the Fresnel domain, we have developed a new lensless microscope that is capable of producing high-resolution quantitative phase images as well as 3D tomographic images of microscopic objects.¹² The lensless microscope is composed of just a programmable color LED matrix and a detector array, so that it is small enough to fit in the palm of the hand (see Figure 3). Based on multi-wavelength TIE phase retrieval and multi-angle illumination diffraction tomography, this microscope offers high-quality, depth-resolved images with a lateral resolution of 3.72μm and an axial resolution of 5μm, across a wide field of view of 24mm². Furthermore, we can digitally adjust the focus of the image (the ‘ z’ parameter) during the reconstruction process, a feature that extends the device's depth-of-field compared with conventional optical microscopes (which use objectives). We used the lensless microscope to image a slice of the uterus of Parascaris equorum (an equine roundworm), resulting in high-resolution 3D tomographic images of fertilized egg cells (see Figure 4). Such a powerful and miniaturized imaging device may offer a cost-effective tool for telemedicine applications, or point-of-care diagnostics in locations where advanced laboratory facilities are unavailable.

Figure 3. Lensless microscope with an LED matrix.¹² (a) Schematics explaining the principle of lensless imaging. Typical values: L=32.5mm, z=300μm∼1.6mm. (b) Photograph of the microscope.

Figure 4. 3D tomographic imaging of a slice of the uterus of Parascaris equorum. (a) Refractive index. (b) Absorption depth sections at z=–10.6, 0, and 26.4μm. The second row shows the corresponding x–z views of 3D stacks (scale ratio of z to x axis is 1:2). The arrows point out the dust particle located at a higher layer (at z=168.4μm). (c) 3D rendering of the refractive index. (d) 3D rendering of the absorption distribution. Scale bar: 400μm.

Furthermore, conventional lens-based light microscopes can also benefit from the LED matrix illumination approach by obtaining greater control over light for the design. Computational illumination enables flexible image coding in ways that are not possible by only modifying the imaging optics. Recently, we built a multi-modal computational microscope, the Smart Computational microscope (SCscope), by combining illumination coding (programmable LED illumination) with pupil plane coding (an electrically tunable lens). The SCscope configuration enabled a wide variety of imaging modalities, including bright-field, dark-field, Zernike phase-contrast, differential phase-contrast,¹³ Rheinberg illumination, stereo, full resolution light-field, quantitative TIE phase,¹⁰ super-resolution Fourier ptychographic,¹⁴ and real-time extended depth-of-field microscopy.¹⁵ All these approaches can be dynamically controlled by computer software with no moving parts.

Figure 5. SCscope (Smart Computational microscope) enables a wide variety of imaging modalities realized by combining illumination coding (programmable LED illumination) with pupil plane coding (an electrically tunable lens).

In summary, computational phase imaging helps to realize smart microscopes using a combination of optics coding and digital processing to acquire a vast amount of information that cannot be captured with traditional microscopes. The technology enhances the imaging capability and flexibility of light microscopes, while also reducing their size and cost.

In future work, we will exploit the power of the SCscope in a more thorough way, by implementing more imaging methods on the same platform, including real-time 3D convolution microscopy, high-dynamic-range imaging, and 3D diffraction tomographic microscopy.

The author acknowledges financial support from the National Natural Science Fund of China (grants 11574152, 61505081), Fundamental Research Funds for the Central Universities (30915011318), Open Research Fund of Jiangsu Key Laboratory of Spectral Imaging and Intelligent Sense (3092014012200417), and the ‘Zijin Star’ program of Nanjing University of Science and Technology.