Theoretical Point Spread Function Research Articles

Adaptive sparse reconstruction for lensless digital holography via PSF estimation and phase retrieval.

In-line lensless digital holography has great potential in multiple applications; however, reconstructing high-quality images from a single recorded hologram is challenging due to the loss of phase information. Typical reconstruction methods are based on solving a regularized inverse problem and work well under suitable image priors, but they are extremely sensitive to mismatches between the forward model and the actual imaging system. This paper aims to improve the robustness of such algorithms by introducing the adaptive sparse reconstruction method, ASR, which learns a properly constrained point spread function (PSF) directly from data, as opposed to solely relying on physics-based approximations of it. ASR jointly performs holographic reconstruction, PSF estimation, and phase retrieval in an unsupervised way by maximizing the sparsity of the reconstructed images. Like traditional methods, ASR uses the image formation model along with a sparsity prior, which, unlike recent deep learning approaches, allows for unsupervised reconstruction with as little as one sample. Experimental results in synthetic and real data show the advantages of ASR over traditional reconstruction methods, especially in cases where the theoretical PSF does not match that of the actual system.

Taking molecular pathology to the next level: Whole slide multicolor confocal imaging with the Pannoramic Confocal digital pathology scanner.

The emergence and fast advance of digital pathology allows the acquisition, digital storage, interactive recall and analysis of morphology at the tissue level. When applying immunohistochemistry, it also affords the correlation of morphology with the expression of one or two specific molecule of interest. The rise of fluorescence pathology scanners expands the number of detected molecules based on multiplex labeling. The Pannoramic Confocal (created by 3DHistech, Hungary) is a first-of-the-kind digital pathology scanner that affords not only multiplexed fluorescent detection on top of conventional transmission imaging, but also confocality. We have benchmarked this scanner in terms of stability, precision, light efficiency, linearity and sensitivity. X-Y stability and relocalisation precision were well below resolution limit (≤50 nm). Light throughput in confocal mode was 4-5 times higher than that of a point scanning confocal microscope, yielding similar calculated confocal intensities but with the potential for improving signal to noise ratio or scan speed. Response was linear with R2 ≥ 0.9996. Calibrated measurements showed that using indirect labeling ≥2000 molecules per cell could be well detected and imaged on the cell surface. Both standard-based and statistical post-acquisition flatfield corrections are implemented. We have also measured the point spread function (PSF) of the instrument. The dimensions of the PSF are somewhat larger and less symmetric than of the theoretical PSF of a conventional CLSM, however, the spatial homogeneity of these parameters allows for obtaining a specific system PSF for each optical path and using it for optional on-the-fly deconvolution. In conclusion, the Pannoramic Confocal provides sensitive, quantitative widefield and confocal detection of multiplexed fluorescence signals, with optical sectioning and 3D reconstruction, in addition to brightfield transmission imaging. High speed scanning of large samples, analysis of tissue heterogeneity, and detection of rare events open up new ways for quantitatively analyzing tissue sections, organoid cultures or large numbers of adherent cells.

Extended focal depth Fourier domain optical coherence microscopy with a Bessel-beam - LP02 mode - from a higher order mode fiber.

We present a robust fiber-based setup for Bessel-like beam extended depth-of-focus Fourier-domain optical coherence microscopy, where the Bessel-like beam is generated in a higher order mode fiber module. In this module a stable guided LP02 core mode is selectively excited by a long period grating written in the higher order mode fiber. Imaging performance of this system in terms of lateral resolution and depth of focus was analyzed using samples of suspended microbeads and compared to the case where illumination is provided by the fundamental LP01 mode of a single mode fiber. Illumination with the LP02 mode allowed for a lateral resolution down to 2.5 µm as compared to 4.5 µm achieved with the LP01 mode of the single mode fiber. A three-fold enhancement of the depth of focus compared to a Gaussian beam with equally tight focus is achieved with the LP02 mode. Analysis of the theoretical lateral point spread functions for the case of LP01 and LP02 illumination agrees well with the experimental data. As the design space of waveguides and long-period gratings allows for further optimization of the beam parameters of the generated Bessel-like beams in an all-fiber module, this approach offers a robust and yet flexible alternative to free-space optics approaches or the use of conical fiber tips.

Enhancing Direct Exoplanet Spectroscopy with Apodizing and Beam Shaping Optics

Direct exoplanet spectroscopy aims to measure the spectrum of an exoplanet while simultaneously minimizing the light collected from its host star. Isolating the planet light from the starlight improves the signal-to-noise ratio (S/N) per spectral channel when noise due to the star dominates, which may enable new studies of the exoplanet atmosphere with unprecedented detail at high spectral resolution (>30,000). However, the optimal instrument design depends on the flux level from the planet and star compared to the noise due to other sources, such as detector noise and thermal background. Here we present the design, fabrication, and laboratory demonstration of specially-designed optics to improve the S/N in two potential regimes in direct exoplanet spectroscopy with adaptive optics instruments. The first is a pair of beam-shaping lenses that increase the planet signal by improving the coupling efficiency into a single-mode fiber at the known position of the planet. The second is a grayscale apodizer that reduces the diffracted starlight for planets at small angular separations from their host star. The former especially increases S/N when dominated by detector noise or thermal background, while the latter helps reduce stellar noise. We show good agreement between the theoretical and experimental point spread functions in each case and predict the exposure time reduction (∼33%) that each set of optics provides in simulated observations of 51 Eridani b using the Keck Planet Imager and Characterizer instrument at W. M. Keck Observatory.

Two simple criteria to estimate an objective’s performance when imaging in non design tissue clearing solutions

Tissue clearing techniques are undergoing a renaissance motivated by the need to image fluorescent neurons, and other cells, deep in the sample without physical sectioning. Optical transparency is achieved by equilibrating tissues with high refractive index (RI) solutions. When the microscope objective is not perfectly matched to the RI of the cleared sample, aberrations are introduced. We present two simple-to-calculate numerical criteria predicting: (i) the degradation in image quality (brightness and resolution) from optimal conditions of any clearing solution/objective combination; (ii) which objective, among several available, achieves the highest resolution in a given medium. We derived closed form approximations for image quality degradation versus RI mismatch and other parameters available to the microscopist, validated them with computed and measured aberrated point spread functions and by imaging fluorescent neurons in high RI solution. These approximations apply to the widefield fluorescent microscope but are also relevant to more advanced microscopes. Currently, to accurately predict the impact of RI mismatch-induced aberrations on imaging, the life scientist must examine theoretical or experimental point spread functions (PSFs) obtained under the optical configuration of interest. These criteria can be used to select a suitable objective for the chosen clearing method (particularly when subject to budget constraints) or to tweak a clearing solution RI to the available objectives. Even with a nominally optimal objective, one may wish to assess the impact of any small unavoidable mismatches.

Deconvolution of light sheet microscopy recordings

We developed a deconvolution software for light sheet microscopy that uses a theoretical point spread function, which we derived from a model of image formation in a light sheet microscope. We show that this approach provides excellent blur reduction and enhancement of fine image details for image stacks recorded with low magnification objectives of relatively high NA and high field numbers as e.g. 2x NA 0.14 FN 22, or 4x NA 0.28 FN 22. For these objectives, which are widely used in light sheet microscopy, sufficiently resolved point spread functions that are suitable for deconvolution are difficult to measure and the results obtained by common deconvolution software developed for confocal microscopy are usually poor. We demonstrate that the deconvolutions computed using our point spread function model are equivalent to those obtained using a measured point spread function for a 10x objective with NA 0.3 and for a 20x objective with NA 0.45.

The deconvolved conventional beamforming for conformal array

As the conformal array has the advantages over the large array aperture, superior aeroacoustic dynamics, and wide angle cover range, it is widely used in radar and sonar system. The deconvolved conventional beamforming (dCv) can yield not only the improved direction of arrival (DOA) estimation accuracy in terms of a narrower beamwidth and suppressed side lobes but also the higher directivity index than the CBF. This paper uses an extended R-L deconvolution algorithm and applies the dCv in conformal array. Simulation and experiment results show that the extended R-L can be applied to an arbitrary conformal array with a shift-variant point spread function (PSF). Beyond the high-resolution and high array gain performance advantages, the extended R-L algorithm is shown to be suitable for coherent signals. The results also show that it is better to use the actual PSF than the theoretical PSF when there exit position errors.

Point spread function for diffuser cameras based on wave propagation and projection model.

Compared to traditional imaging system, diffuser camera is an easy-built imaging system to capture the light field with small form factor. Its imaging target can be reconstructed by deconvolving the sensor images with the point-spread-function (PSF) at the corresponding depth. However, the existing method to obtain the PSFs is generally relied on measuring point-source-responses at several depths which presents high complexity and low reliability. In this paper, we propose a theoretical PSF model for the diffuser camera by estimating the diffuser's phase based on projection model and deriving the image response at any depth by forward Fourier optics to enable the reconstruction of the object at any depth. The experimental results demonstrate the effectiveness of the proposed model in terms of the correlation between the captured PSFs and our model derived PSFs, the correlation between the ground truth image and reconstructed images under different depths, and the correlation between the images reconstructed by real captured PSFs and the images reconstructed by our model derived PSFs. The objects at different depths can be correctly reconstructed by our theoretically derived PSFs, which benefits the application of diffuser camera for much lower complexity.

Modeling Point Spread Function in Fluorescence Microscopy With a Sparse Gaussian Mixture: Tradeoff Between Accuracy and Efficiency.

Deblurring is a fundamental inverse problem in bioimaging. It requires modeling the point spread function (PSF), which captures the optical distortions entailed by the image formation process. The PSF limits the spatial resolution attainable for a given microscope. However, recent applications require a higher resolution and have prompted the development of super-resolution techniques to achieve sub-pixel accuracy. This requirement restricts the class of suitable PSF models to analog ones. In addition, deblurring is computationally intensive, hence further requiring computationally efficient models. A custom candidate fitting both the requirements is the Gaussian model. However, this model cannot capture the rich tail structures found in both the theoretical and empirical PSFs. In this paper, we aim at improving the reconstruction accuracy beyond the Gaussian model, while preserving its computational efficiency. We introduce a new class of analog PSF models based on the Gaussian mixtures. The number of Gaussian kernels controls both the modeling accuracy and the computational efficiency of the model: the lower the number of kernels, the lower the accuracy and the higher the efficiency. To explore the accuracy-efficiency tradeoff, we propose a variational formulation of the PSF calibration problem, where a convex sparsity-inducing penalty on the number of Gaussian kernels allows trading accuracy for efficiency. We derive an efficient algorithm based on a fully split formulation of alternating split Bregman. We assess our framework on synthetic and real data, and demonstrate a better reconstruction accuracy in both geometry and photometry in point source localization-a fundamental inverse problem in fluorescence microscopy.

Deconvolution using CLEAN-SC for acoustic source identification with spherical microphone arrays

The point spread function (PSF) based deconvolution algorithms may fall short due to the inconformity between the actual beam pattern of acoustic sources and the theoretical PSF. CLEAN-SC, originally developed for delay and sum beamforming with planar microphone arrays, can circumvent the inconformity. This paper devotes to adapting CLEAN-SC to spherical harmonics beamforming (SHB). The core of adapting CLEAN-SC to SHB is to express the output of SHB in the spatial domain as a specific form of matrix operation completely. We solve the issue and establish CLEAN-SC with spherical microphone arrays. Its performance is analyzed and compared with PSF based CLEAN. CLEAN-SC with spherical microphone arrays can improve spatial resolution and suppress sidelobes more effectively, as well as quantify the sound pressure contribution accurately. Moreover, it has better convergence, higher computational efficiency, and stronger robustness to interference such as background noise and frequency response mismatch of microphone and measurement channel. This study provides an alternative approach for acoustic source identification in three-dimensional space.

Experimental validation of a customized phase mask designed to enable efficient computational optical sectioning microscopy through wavefront encoding.

In this paper, wavefront-encoded (WFE) computational optical sectioning microscopy (COSM) using a fabricated square cubic (SQUBIC) phase mask, designed to render the system less sensitive to depth-induced aberration, is investigated. The WFE-COSM system is characterized by a point spread function (PSF) that does not vary as rapidly with imaging depth compared to the conventional system. Thus, in WFE-COSM, image restoration from large volumes can be achieved using computationally efficient space-invariant (SI) algorithms, thereby avoiding the use of depth-variant algorithms. The fabricated SQUBIC phase mask was first evaluated and found to have a 75% fidelity compared to the theoretical design; it was then integrated in a commercial wide-field microscope to implement a WFE-COSM system. Evaluation of the WFE-COSM system is demonstrated with comparative studies of theoretical and experimental PSFs and simulated and measured images of spherical shells located at different depths in a test sample. These comparisons show that PSF and imaging models capture major trends in experimental data with a 99% correlation between forward image intensity distribution in experimental and simulated images of spherical shells. Our experimental SI restoration results demonstrate that the WFE-COSM system achieves more than a twofold performance improvement over the conventional system of up to a 65 μm depth below the coverslip investigated in this study.

CALIBRATION OF THE NuSTAR HIGH-ENERGY FOCUSING X-RAY TELESCOPE

We present the calibration of the \textit{Nuclear Spectroscopic Telescope Array} (\nustar) X-ray satellite. We used the Crab as the primary effective area calibrator and constructed a piece-wise linear spline function to modify the vignetting response. The achieved residuals for all off-axis angles and energies, compared to the assumed spectrum, are typically better than $\pm 2$\% up to 40\,keV and 5--10\,\% above due to limited counting statistics. An empirical adjustment to the theoretical 2D point spread function (PSF) was found using several strong point sources, and no increase of the PSF half power diameter (HPD) has been observed since the beginning of the mission. We report on the detector gain calibration, good to 60\,eV for all grades, and discuss the timing capabilities of the observatory, which has an absolute timing of $\pm$ 3\,ms. Finally we present cross-calibration results from two campaigns between all the major concurrent X-ray observatories (\textit{Chandra}, \textit{Swift}, \textit{Suzaku} and \textit{XMM-Newton}), conducted in 2012 and 2013 on the sources 3C\,273 and PKS\,2155-304, and show that the differences in measured flux is within $\sim$10\% for all instruments with respect to \nustar.

Hyperpolarized (129) Xe imaging of the rat lung using spiral IDEAL.

To implement and optimize a single-shot spiral encoding strategy for rapid 2D IDEAL projection imaging of hyperpolarized (Hp) (129) Xe in the gas phase, and in the pulmonary tissue (PT) and red blood cells (RBCs) compartments of the rat lung, respectively. A theoretical and experimental point spread function analysis was used to optimize the spiral k-space read-out time in a phantom. Hp (129) Xe IDEAL images from five healthy rats were used to: (i) optimize flip angles by a Bloch equation analysis using measured kinetics of gas exchange and (ii) investigate the feasibility of the approach to characterize the exchange of Hp (129) Xe. A read-out time equal to approximately 1.8 × T2* was found to provide the best trade-off between spatial resolution and signal-to-noise ratio (SNR). Spiral IDEAL approaches that use the entire dissolved phase magnetization should give an SNR improvement of a factor of approximately three compared with Cartesian approaches with similar spatial resolution. The IDEAL strategy allowed imaging of gas, PT, and RBC compartments with sufficient SNR and temporal resolution to permit regional gas exchange measurements in healthy rats. Single-shot spiral IDEAL imaging of gas, PT and RBC compartments and gas exchange is feasible in rat lung using Hp (129) Xe. Magn Reson Med 76:566-576, 2016. © 2015 Wiley Periodicals, Inc.

Back-propagation beamformer design for motion estimation in echocardiography.

Transverse oscillation (TO) techniques have shown their potential for improving the accuracy of local motion estimation in the transverse direction (i.e., the direction perpendicular to the beam axis). The conventional design of TOs in linear geometry, which is based on the Fraunhofer approximation, relates point spread function (PSF) and apodization function through a Fourier transform. Motivated by the adaptation of TOs in echocardiography, we propose a specific beamforming approach based on back-propagation (BP) to build TOs in sector-shaped geometry. Numerical simulations and experimental data give a comparison between proposed and conventional beamforming for TOs. The accuracy is first quantified by comparing the generated and theoretical PSF using the root mean square error (RMSE) and shows that BP-based beamforming approximates the desired TOs more closely than the conventional approach. Motion estimation is then evaluated. The axial and lateral displacements are within the range [0-0.6] mm and [0°-6.4°], respectively, which correspond to 0.8 times the axial (0.73 mm) and lateral (8°) wavelengths. The result shows that the proposed method yields a clear improvement for lateral displacements, by reducing the error by 28.6% compared with Fourier transform-based beamforming, while maintaining the same error for axial motion estimation. Experimental measurements are discussed to complete this study and confirm that BP-based beamforming leads to better controlled TO images than conventional Fourier-based beamforming.

Sampling strategies for subsampled segmented EPI PRF thermometry in MR guided high intensity focused ultrasound.

To investigate k-space subsampling strategies to achieve fast, large field-of-view (FOV) temperature monitoring using segmented echo planar imaging (EPI) proton resonance frequency shift thermometry for MR guided high intensity focused ultrasound (MRgHIFU) applications. Five different k-space sampling approaches were investigated, varying sample spacing (equally vs nonequally spaced within the echo train), sampling density (variable sampling density in zero, one, and two dimensions), and utilizing sequential or centric sampling. Three of the schemes utilized sequential sampling with the sampling density varied in zero, one, and two dimensions, to investigate sampling the k-space center more frequently. Two of the schemes utilized centric sampling to acquire the k-space center with a longer echo time for improved phase measurements, and vary the sampling density in zero and two dimensions, respectively. Phantom experiments and a theoretical point spread function analysis were performed to investigate their performance. Variable density sampling in zero and two dimensions was also implemented in a non-EPI GRE pulse sequence for comparison. All subsampled data were reconstructed with a previously described temporally constrained reconstruction (TCR) algorithm. The accuracy of each sampling strategy in measuring the temperature rise in the HIFU focal spot was measured in terms of the root-mean-square-error (RMSE) compared to fully sampled "truth." For the schemes utilizing sequential sampling, the accuracy was found to improve with the dimensionality of the variable density sampling, giving values of 0.65 °C, 0.49 °C, and 0.35 °C for density variation in zero, one, and two dimensions, respectively. The schemes utilizing centric sampling were found to underestimate the temperature rise, with RMSE values of 1.05 °C and 1.31 °C, for variable density sampling in zero and two dimensions, respectively. Similar subsampling schemes with variable density sampling implemented in zero and two dimensions in a non-EPI GRE pulse sequence both resulted in accurate temperature measurements (RMSE of 0.70 °C and 0.63 °C, respectively). With sequential sampling in the described EPI implementation, temperature monitoring over a 192×144×135 mm3 FOV with a temporal resolution of 3.6 s was achieved, while keeping the RMSE compared to fully sampled "truth" below 0.35 °C. When segmented EPI readouts are used in conjunction with k-space subsampling for MR thermometry applications, sampling schemes with sequential sampling, with or without variable density sampling, obtain accurate phase and temperature measurements when using a TCR reconstruction algorithm. Improved temperature measurement accuracy can be achieved with variable density sampling. Centric sampling leads to phase bias, resulting in temperature underestimations.

Ultrafast, accurate, and robust localization of anisotropic dipoles

The resolution of single molecule localization imaging techniques largely depends on the precision of localization algorithms. However, the commonly used Gaussian function is not appropriate for anisotropic dipoles because it is not the true point spread function. We derived the theoretical point spread function of tilted dipoles with restricted mobility and developed an algorithm based on an artificial neural network for estimating the localization, orientation and mobility of individual dipoles. Compared with fitting-based methods, our algorithm demonstrated ultrafast speed and higher accuracy, reduced sensitivity to defocusing, strong robustness and adaptability, making it an optimal choice for both two-dimensional and three-dimensional super-resolution imaging analysis.

3-D airborne ultrasound synthetic aperture imaging based on capacitive micromachined ultrasonic transducers

In this paper, we present an airborne 3-D volumetric imaging system based on capacitive micromachined ultrasonic transducers (CMUTs). For this purpose we fabricated 89-kHz CMUTs where each CMUT is made of a circular single-crystal silicon plate with a radius of 1mm and a thickness of 20μm, which is actuated by electrostatic force through a 20-μm vacuum gap. The measured transmit sensitivity at 300-V DC bias is 14.6Pa/V and 24.2Pa/V, when excited by a 30-cycle burst and a continuous wave, respectively. The measured receive sensitivity at 300-V DC bias is 16.6mV/Pa (−35.6dB re 1V/Pa) for a 30-cycle burst. A 26×26 2-D array was implemented by mechanical scanning a co-located transmitter and receiver using the classic synthetic aperture (CSA) method. The measurement of a 1.6λ-size target at a distance of 500mm presented a lateral resolution of 3.17° and also showed good agreement with the theoretical point spread function. The 3-D imaging of two plates at a distance of 350mm and 400mm was constructed to exhibit the capability of the imaging system. This study experimentally demonstrates that a 2-D CMUT array can be used for practical 3-D imaging applications in air, such as a human–machine interface.

Experimental study of z resolution in acousto-optical coherence tomography using random phase jumps on ultrasound and light

Acousto-optical coherence tomography (AOCT) is a variant of acousto-optic imaging (also called ultrasound modulated optical tomography) that makes possible to get resolution along the ultrasound propagation axis z. We present here AOCT experimental results, and we study how the z resolution depends on time step between phase jumps T(φ), or on the correlation length Δz. By working at low resolution, we perform a quantitative comparison of the z measurements with the theoretical point spread function. We also present images recorded with different z resolution, and we qualitatively show how the image quality varies with T(φ), or Δz.

Localization Precision for Asymmetric Single-Molecule Images in Superresolution Localization Microscopy

We present theoretical localization precision formulae for asymmetric single-molecule images in superresolution localization microscopy. Superresolution localization microscopy, such as photoactivated localization microscopy (PALM) and stochastic optical reconstruction microscopy (STORM), have demonstrated superior performances in cell imaging and enable the investigation of cellular processes at close to the molecular scale. All these techniques rely on the precise localization measurements of single-molecules at the nanoscale by using statistical estimators to fit diffraction-limited single-molecule images with the theoretical point spread function (PSF) of the imaging system, which is commonly approximated as a two-dimensional Gaussian. However, to our best knowledge, all previous theories [e.g., R. E. Thompson et al, Biophys. J. 82, 2775 (2002) and R. J. Ober et al, Biophys. J. 86, 1185 (2004)] on theoretical localization precision are developed for circularly symmetric single-molecule images. In contrast, many of the recent advances in the developments of localization microscopy have demonstrated that astigmatism can occur and result in asymmetric PSFs as a result of optical aberrations in the imaging system [S. Quirin et al. Proc. Natl. Acad. Sci. USA 109, 675 (2012)] and asymmetric molecular emission [K. I. Mortensen et al., Nat Meth 7, 377 (2010)]. Asymmetric PSF has also been implemented in localization microscopy to achieve astigmatic imaging for three-dimensional single-molecule localization techniques [B. Huang et al., Science 319, 810 (2008)]. Therefore, our new theory for asymmetric single-molecule images can be particularly useful where asymmetric PSFs have been used or observed in localization microscopy.

Improved first‐pass spiral myocardial perfusion imaging with variable density trajectories

To develop and evaluate variable-density spiral first-pass perfusion pulse sequences for improved efficiency and off-resonance performance and to demonstrate the utility of an apodizing density compensation function (DCF) to improve signal-to-noise ratio (SNR) and reduce dark-rim artifact caused by cardiac motion and Gibbs Ringing. Three variable density spiral trajectories were designed, simulated, and evaluated in 18 normal subjects, and in eight patients with cardiac pathology on a 1.5T scanner. By using a DCF, which intentionally apodizes the k-space data, the sidelobe amplitude of the theoretical point spread function (PSF) is reduced by 68%, with only a 13% increase in the full-width at half-maximum of the main-lobe when compared with the same data corrected with a conventional variable-density DCF, and has an 8% higher resolution than a uniform density spiral with the same number of interleaves and readout duration. Furthermore, this strategy results in a greater than 60% increase in measured SNR when compared with the same variable-density spiral data corrected with a conventional DCF (P < 0.01). Perfusion defects could be clearly visualized with minimal off-resonance and dark-rim artifacts. Variable-density spiral pulse sequences using an apodized DCF produce high-quality first-pass perfusion images with minimal dark-rim and off-resonance artifacts, high SNR and contrast-to-noise ratio, and good delineation of resting perfusion abnormalities.