Smart Computational Imaging (SCI) Lab

智能计算成像实验室

Transport of Intensity (TIE) Phase Imaging

We are dedicated to discover the key theories and explore related new methods associated with noninterferometric phase retrieval and quantitative phase contrast microscopy imaging by using the transport of intensity equation (TIE).  With only intensity measurements at several distances along its propagation direction, the phase can be quantitatively retrieved by solving the TIE deterministically, without resorting to interferometry or separated reference wave. Our new technology endows the conventional microscope with the ability of three-dimensional (3D) quantitative phase imaging, enabling us to obtain 3D morphology and optical properties of phase specimens with nanometer-scale and millisecond temporal resolution.

Although phase objects cannot be observed directly, they incessantly, and implicitly manifest their existence: the twinkling of stars in the night, the distorted scene outside the window in the rain, the network of bright lines at the bottom of a swimming pool in the sunshine. These are all manifestations of phase, implying the inextricably ties between phase and intensity of light wave. In 1983, Teague first establish the quantitative relationship between the longitudinal intensity variation and phase of transporting light with use of a second-order elliptic partial differential equation, so called transport of intensity equation (TIE). With only intensity measurements at several distances along its propagation direction, the phase can be quantitatively retrieved by solving the equation deterministically, without resorting to interferometry or separated reference wave.

The TIE outlines the relation between object-plane phase and the first order derivative of intensity with respect to the optical axis in the near Fresnel region. With appropriate boundary conditions, the solution to TIE is known to exist and be unique. That is, the phase can be uniquely determined by solving TIE with determined intensity and longitudinal intensity derivative (by simply capturing a minimum of two intensity images at different z planes and using them to estimate dI/dz).  Though everything looks quite easy and straightforward, there are several important technical issues needs to be solved in order to put this promising technique into practical use:

• Fast, exact numerical solution and the boundary conditions problem
  

• Axial intensity derivative estimation and the noise problem
  

• Extension of transport of intensity equation to partially coherent fields
  

• High speed through-focus stack collection for dynamic imaging

A deep and systematic research has been conducted by our group to discover the key theories and related new methods associated with noninterferometric phase retrieval and quantitative phase contrast microscopy imaging by using the TIE. It is expected to address several key issues in both the theoretic aspects as well of the practical application of the TIE, so that developing the TIE from a theortical equation into a powerful metrology tool. Our new technology endows the conventional microscope with the ability of three-dimensional (3D) quantitative phase imaging, enabling us to obtain 3D morphology and optical properties of phase specimens with nanometer-scale and millisecond temporal resolution. This technology provides the following unique advantages：

• It is non-interferometric, works with partially coherent illumination, like the built-in halogen lamp in a conventional microscope.

• It does not require a complicated optical system, can well be compatible with the conventional bright field microscope.

• It is a single beam method and less stringent to environmental instability.

• It directly recovers the continuous phase without 2π discontinuity, so there is no need to phase unwrapping.

• It can provide high-quality, real-time phase measurement with high resolution in full four-dimension (lateral (x and y) and vertical (z) and temporal (t)).

Fast Discrete Cosine Transform (DCT)-based TIE solver
The TIE is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. We consider the TIE phase retrieval as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green’s function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.

References:

• C. Zuo, Q. Chen, and A. Asundi, "Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform," Optics Express 22, 9220-9244 (2014). [code available]

• C. Zuo, Q. Chen, H. Li, W. Qu, and A. Asundi, "Boundary-artifact-free phase retrieval with the transport of intensity equation II: applications to microlens characterization," Optics Express 22, 18310-18324 (2014).

Optimum derivative estimation using Savitzky-Golay differentiation filter
Several existing strategies for estimating the axial intensity derivative in the transport-of-intensity equation (TIE) from multiple intensity measurements have been unified by the Savitzky-Golay differentiation filter - an equivalent convolution solution for differentiation estimation by least-squares polynomial fitting. The different viewpoint from the digital filter in signal processing not only provides great insight into the behaviors, the shortcomings, and the performance of these existing intensity derivative estimation algorithms, but more important, it also suggests a new way of improving solution strategies by extending the applications of Savitzky-Golay differentiation filter in TIE. Two novel methods for phase retrieval based on TIE are presented - the first by introducing adaptive-degree strategy in spatial domain and the second by selecting optimal spatial frequencies in Fourier domain. Numerical simulations and experiments verify that the second method outperforms the existing methods significantly, showing reliable retrieved phase with both overall contrast and fine phase variations well preserved.

References:

• C. Zuo, Q. Chen, Y. Yu, A. Asundi, "Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter-theory and applications," Optics Express 21, 5346-5362 (2013). [code available]

Generalized TIE in phase-space for partially coherent fields
A generalized transport-of-intensity equation (TIE) is derived in the joint space-spatial frequency domain using Wigner distribution functions. It explicitly covers various optical fields with arbitrary spatial and temporal coherence, and reduces to the conventional TIE derived by Teague in the limiting case of perfect coherence. The new equation clarifies the physical meaning of the phase of partially coherent fields, and provides a powerful tool to analyze the effect of the partially coherent illumination and the imaging system on phase retrieval. The correspondence between the Wigner distribution function and the light field in geometric optics further enables TIE to become a simple yet effective approach to realize high-resolution light field imaging for slowly varying phase specimen, in a purely computational way.

References:

• C. Zuo, Q. Chen, and A. Asundi, "Light field moment imaging: comment," Optics Letters 39, 654-654 (2014). (Selected for the Virtual Journal for Biomedical Optics);

• C. Zuo, Q. Chen, W. Laura, L. Tian, A. Asundi, "Transport of intensity phase retrieval andcomputational imaging for partially coherent fields: the phase space perspective" Opt Laser Eng 67, 176-181 (2015).

  
  

Electrically tunable lens based TIE (TL-TIE)
We present a high-speed transport-of-intensity equation (TIE) quantitative phase microscopy technique, named TL-TIE, by combining an electrically tunable lens with a conventional transmission microscope. This permits the specimen at different focus position to be imaged in rapid succession, with constant magnification and no physically moving parts. The simplified image stack collection significantly reduces the acquisition time, allows for the diffraction-limited through-focus intensity stack collection at 15 frames per second, making dynamic TIE phase imaging possible. The technique is demonstrated by profiling of microlens array using optimal frequency selection scheme, and time-lapse imaging of live breast cancer cells by inversion the defocused phase optical transfer function to correct the phase blurring in traditional TIE. Experimental results illustrate its outstanding capability of the technique for quantitative phase imaging, through a simple, non-interferometric, high-speed, high-resolution, and unwrapping-free approach with prosperous applications in micro-optics, life sciences and bio-photonics.

Noninterferometric single-shot quantitative phase microscopy (SQPM)
We present a noninterferometric single-shot quantitative phase microscopy technique with the use of the transport of intensity equation (TIE). The optical configuration is based on a Michelson-like architecture attached to a nonmodified inverted transmission bright field microscope. Two laterally separated images from different focal planes can be obtained simultaneously by a single camera exposure, enabling the TIE phase recovery to be performed at frame rates that are only camera limited. Precise measurement of a microlens array validates the principle and demonstrates the accuracy of the method. Investigations of chemical-induced apoptosis and the phagocytosis process of macrophages are then presented, suggesting that the method developed can provide promising applications in the dynamic study of cellular processes.

TIE phase retrieval using spatially modulated illumination
We propose a method for retrieving the phase of a wavefront from the diffraction patterns recorded when the object is sequentially illuminated by spatially modulated light. For wavefronts having a smooth phase, the retrieval is achieved by using the deterministic method TIE. Unlike the traditional TIE, which records the object wave intensities in two or more axially spaced planes, this method records the intensity patterns in a single plane. In the traditional TIE, the phase is retrieved from the second-order phase derivatives, and knowledge of the boundaries is needed. Instead, the proposed method reconstructs the phase from the first-order derivatives and avoids the difficulty of determining the boundary conditions. The traditional TIE produces artifacts when the object wavefront has low-frequency modulation, while the modulated illumination used here produces a high frequency and thus improves the SNR of the phase imaging. When the phase has discontinuities, an iterative process is used for the retrieval and enhancement of the spatial resolution. Both the deterministic and iterative phase reconstructions are demonstrated by experiments.
References:

• P. Gao, G. Pedrini, C. Zuo, and W. Osten, "Phase retrieval using spatially modulated illumination," Optics Letters 39, 3615-3618 (2014).

Micro-optics characterization using the TIE
We demonstrated that the TIE-based phase retrieval approach allows for optical inspection of micro-optical components. It is implemented based on a conventional bright-field microscope and use the partially coherent illumination. Thus it possesses better spatial resolution and less suffers from the speckle noise than interferometry. Furthermore, with regard to the state of the art in phase-retrieval by means of the TIE, the technique has another two major advantages: First, it completely erases the boundary artifacts by simply introducing an aperture in the intermediate image plane, without requiring any assumptions or a prior knowledge about the test object and the setup. Second, in contrast to existing approaches, with use of an ETL, the measurement time is considerably reduced and no further mechanical adjustment is needed during the measurement. The validity and practicality of the proposed method have been demonstrated with a wide variety of microoptical components, suggesting that it is a simple, fast, and accurate measurement technique for the micro-optics inspection.

References:

• C. Zuo, Q. Chen, H. Li, W. Qu, and A. Asundi, "Boundary-artifact-free phase retrieval with the transport of intensity equation II: applications to microlens characterization," Optics Express 22, 18310-18324 (2014).

Comparsion and combination between DH and TIE
In the past couple of decades, digital holography (DH) has emerged as a front-runner for phase imaging by providing quantitative phase measurements of the wave field with high accuracy and in near real-time. Recently, however, direct phase retrieval from intensity measurements using the Transport-of-Intensity Equation (TIE) has gained increasing attention. We conduct a comparative study between these two approaches and  propose a new way to demodulate continuous phase of digital holography directly by solving the transport of intensity equation, achieving complementary advantages for both approaches.

References:

• C. Zuo, Q. Chen, W. Qu, A. Asundi, "Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation", Opt Commun 309, 221-226 (2013).
  

• C. Zuo, Q. Chen, and A. Asundi, "Comparison of Digital Holography and Transport of Intensity for Quantitative Phase Contrast Imaging," in Fringe 2013 (Springer Berlin Heidelberg, 2014), pp. 137-142.