Deep Learning Solution of the Eigenvalue Problem for Differential Operators.

Solving the eigenvalue problem for differential operators is a common problem in many scientific fields. Classical numerical methods rely on intricate domain discretization and yield nonanalytic or nonsmooth approximations. We introduce a novel neural network-based solver for the eigenvalue problem of differential self-adjoint operators, where the eigenpairs are learned in an unsupervised end-to-end fashion. We propose several training procedures for solving increasingly challenging tasks toward the general eigenvalue problem. The proposed solver is capable of finding the M smallest eigenpairs for a general differential operator. We demonstrate the method on the Laplacian operator, which is of particular interest in image processing, computer vision, and shape analysis among many other applications. In addition, we solve the Legendre differential equation. Our proposed method simultaneously solves several eigenpairs and can be easily used on free-form domains. We exemplify it on L-shape and circular cut domains. A significant contribution of this work is an analysis of the numerical error of this method. In particular an upper bound for the (unknown) solution error is given in terms of the (measured) truncation error of the partial differential equation and the network structure.

Cross-site harmonization of multi-shell diffusion MRI measures based on rotational invariant spherical harmonics (RISH).

Quantification methods based on the acquisition of diffusion magnetic resonance imaging (dMRI) with multiple diffusion weightings (e.g., multi-shell) are becoming increasingly applied to study the in-vivo brain. Compared to single-shell data for diffusion tensor imaging (DTI), multi-shell data allows to apply more complex models such as diffusion kurtosis imaging (DKI), which attempts to capture both diffusion hindrance and restriction effects, or biophysical models such as NODDI, which attempt to increase specificity by separating biophysical components. Because of the strong dependence of the dMRI signal on the measurement hardware, DKI and NODDI metrics show scanner and site differences, much like other dMRI metrics. These effects limit the implementation of multi-shell approaches in multicenter studies, which are needed to collect large sample sizes for robust analyses. Recently, a post-processing technique based on rotation invariant spherical harmonics (RISH) features was introduced to mitigate cross-scanner differences in DTI metrics. Unlike statistical harmonization methods, which require repeated application to every dMRI metric of choice, RISH harmonization is applied once on the raw data, and can be followed by any analysis. RISH features harmonization has been tested on DTI features but not its generalizability to harmonize multi-shell dMRI. In this work, we investigated whether performing the RISH features harmonization of multi-shell dMRI data removes cross-site differences in DKI and NODDI metrics while retaining longitudinal effects. To this end, 46 subjects underwent a longitudinal (up to 3 time points) two-shell dMRI protocol at 3 imaging sites. DKI and NODDI metrics were derived before and after harmonization and compared both at the whole brain level and at the voxel level. Then, the harmonization effects on cross-sectional and on longitudinal group differences were evaluated. RISH features averaged for each of the 3 sites exhibited prominent between-site differences in the frontal and posterior part of the brain. Statistically significant differences in fractional anisotropy, mean diffusivity and mean kurtosis were observed both at the whole brain and voxel level between all the acquisition sites before harmonization, but not after. The RISH method also proved effective to harmonize NODDI metrics, particularly in white matter. The RISH based harmonization maintained the magnitude and variance of longitudinal changes as compared to the non-harmonized data of all considered metrics. In conclusion, the application of RISH feature based harmonization to multi-shell dMRI data can be used to remove cross-site differences in DKI metrics and NODDI analyses, while retaining inherent relations between longitudinal acquisitions.

Improving the predictive potential of diffusion MRI in schizophrenia using normative models-Towards subject-level classification.

Diffusion MRI studies consistently report group differences in white matter between individuals diagnosed with schizophrenia and healthy controls. Nevertheless, the abnormalities found at the group-level are often not observed at the individual level. Among the different approaches aiming to study white matter abnormalities at the subject level, normative modeling analysis takes a step towards subject-level predictions by identifying affected brain locations in individual subjects based on extreme deviations from a normative range. Here, we leveraged a large harmonized diffusion MRI dataset from 512 healthy controls and 601 individuals diagnosed with schizophrenia, to study whether normative modeling can improve subject-level predictions from a binary classifier. To this aim, individual deviations from a normative model of standard (fractional anisotropy) and advanced (free-water) dMRI measures, were calculated by means of age and sex-adjusted z-scores relative to control data, in 18 white matter regions. Even though larger effect sizes are found when testing for group differences in z-scores than are found with raw values (p < .001), predictions based on summary z-score measures achieved low predictive power (AUC < 0.63). Instead, we find that combining information from the different white matter tracts, while using multiple imaging measures simultaneously, improves prediction performance (the best predictor achieved AUC = 0.726). Our findings suggest that extreme deviations from a normative model are not optimal features for prediction. However, including the complete distribution of deviations across multiple imaging measures improves prediction, and could aid in subject-level classification.

Rapid super resolution for infrared imagery.

Infrared (IR) imagery is used in agriculture for irrigation monitoring and early detection of disease in plants. The common IR cameras in this field typically have low resolution. This work offers a method to obtain the super-resolution of IR images from low-power devices to enhance plant traits. The method is based on deep learning (DL). Most calculations are done in the low-resolution domain. The results of each layer are aggregated together to allow a better flow of information through the network. This work shows that good results can be achieved using depthwise separable convolution with roughly 300K multiply-accumulate computations (MACs), while state-of-the-art convolutional neural network-based super-resolution algorithms are performed with around 1500K MACs. MTF analysis of the proposed method shows a real ×4 improvement in the spatial resolution of the system, out-preforming the diffraction limit. The method is demonstrated on real agricultural images.

Single- and double-Diffusion encoding MRI for studying ex vivo apparent axon diameter distribution in spinal cord white matter.

Mapping average axon diameter (AAD) and axon diameter distribution (ADD) in neuronal tissues non-invasively is a challenging task that may have a tremendous effect on our understanding of the normal and diseased central nervous system (CNS). Water diffusion is used to probe microstructure in neuronal tissues, however, the different water populations and barriers that are present in these tissues turn this into a complex task. Therefore, it is not surprising that recently we have witnessed a burst in the development of new approaches and models that attempt to obtain, non-invasively, detailed microstructural information in the CNS. In this work, we aim at challenging and comparing the microstructural information obtained from single diffusion encoding (SDE) with double diffusion encoding (DDE) MRI. We first applied SDE and DDE MR spectroscopy (MRS) on microcapillary phantoms and then applied SDE and DDE MRI on an ex vivo porcine spinal cord (SC), using similar experimental conditions. The obtained diffusion MRI data were fitted by the same theoretical model, assuming that the signal in every voxel can be approximated as the superposition of a Gaussian-diffusing component and a series of restricted components having infinite cylindrical geometries. The diffusion MRI results were then compared with histological findings. We found a good agreement between the fittings and the experimental data in white matter (WM) voxels of the SC in both diffusion MRI methods. The microstructural information and apparent AADs extracted from SDE MRI were found to be similar or somewhat larger than those extracted from DDE MRI especially when the diffusion time was set to 40 ms. The apparent ADDs extracted from SDE and DDE MRI show reasonable agreement but somewhat weaker correspondence was observed between the diffusion MRI results and histology. The apparent subtle differences between the microstructural information obtained from SDE and DDE MRI are briefly discussed.

Joint estimation of unknown radiometric data, gain, and offset from thermal images.

Low cost, weight, and size microbolometer-based thermal focal plane arrays are attractive for thermal-imaging applications. Under environmental loads like those in agricultural remote sensing, these cameras tend to suffer from drift in gain and offset with time and thus require constant calibration. Our goal is to skip this step via computational imaging. In a previous work we estimated the unknown offset value and radiometric image of an object, given the calibrated gain, from a pair of successive images taken at two different blur levels, eliminating the need for offset calibration due to temperature variation. Here, we extend our model to a case with unknown gain and offset. We show that these values, as well as the objects' radiometric value, can be found jointly by minimizing a cost function relying on N pairs of blurred and sharp images. The method addresses both space-invariant and space-variant cases. Simulations show promising accuracy with error characterized by root mean squared error of less than 1.6°C.

The fiber-density-coreset for redundancy reduction in huge fiber-sets.

State of the art Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) protocols of white matter followed by advanced tractography techniques produce impressive reconstructions of White Matter (WM) pathways. These pathways often contain millions of trajectories (fibers). While for several applications the high number of fibers is essential, other applications (visualization, registration, some types of across-subject comparison) can achieve satisfying results using much smaller sets and may be overburdened by the computational load of the large fiber sets. In this paper we propose a novel, highly efficient algorithm for extracting a meaningful subset of fibers, which we term the Fiber-Density-Coreset (FDC). The reduced set is optimized to represent the main structures of the brain. FDC is based on an efficient geometric approximation paradigm named coresets, an optimization scheme showing much success in tasks requiring large computation time and/or memory. FDC was compared to two commonly used methods for selecting a reduced set of fibers: fiber-clustering and downsampling. The reduced sets were evaluated by several methods, including a novel structural comparison to the full sets called 3D indicator structure comparison (3D-ISC). The comparison was applied to High Angular Resolution Diffusion Imaging (HARDI) scans of 15 healthy individuals obtained from the Human Connectome Project. FDC produced the most satisfying subsets, consistently in all 15 subjects. It also displayed low memory usage and significantly lower running time than conventional fiber reduction schemes.

Radiometric imaging by double exposure and gain calibration.

Due to low cost and small size, uncooled microbolometer-based thermal focal plane arrays are very attractive for radiometry. However, being non-cooled, they suffer from temporally and spatially dependent changes that require constant calibration. While the gain calibration can be reasonably realized by two-point correction, the offset due to internal radiation loads poses a complicated calibration scheme. We present a new computational optics approach that simplifies the essential calibration for temperature offset. Using two successive images of the object taken with different known blur levels, one can eliminate the object term from the image-formation equation, resulting in an equation for the unknown sensor offset. A general algebraic model is presented for the space-variant case followed by solutions using both direct inverse method and iterative solver. The new scheme allows restoration of the radiometric value within 1% error with the direct method, and 0.2% error with the iterative scheme. Account of the influence of realistic lens positioning error on restoration accuracy was given. Results using direct inverse methods for restoring the radiometric values yield restoration error with a good average error of 3.7% and less.

Microstructural information from angular double-pulsed-field-gradient NMR: From model systems to nerves.

PURPOSE: To evaluate the ability of angular double-pulsed-field gradient (d-PFG) MR to provide microstructural information in complex phantoms and fixed nerves. METHODS: We modeled the signal in angular d-PFG MR experiments performed on phantoms of increasing complexity where the ground truth is known a priori. After analyzing the microstructural features of such phantoms the same methodology was used to study microstructural features in fixed nerves. RESULTS: We found that our modeling is able to determine with high accuracy and with very little prior knowledge the sizes and relative fractions of the restricted components as well as the fraction of the free diffusing water molecules. The same approach was used to study nerve microstructure. We found the apparent averaged axonal diameter (AAD) to be 2.3 ± 0.2 µm. However, here the results depended, to some extent, on the parameters used to collect the data and were affected by the diffusion time. CONCLUSION: Modeling of the angular d-PFG MR signal provides a means to obtain accurate microstructural information in complex phantoms where the ground truth is known. This approach also seems to be suitable for obtaining microstructural features in fixed nerves. Magn Reson Med 74:25-32, 2015. © 2014 Wiley Periodicals, Inc.

Blurred and noisy image pairs in parallel optics.

In previous works we have shown that parallel optics (PO) architecture can be used to improve the system matrix condition, which results in improving its immunity to additive noise in the image restoration process. PO is composed of a "main" system and an "auxiliary" system. Previously, we suggested the "trajectories" method to realize PO. In that method, a required auxiliary system is composed from auxiliary optics with a pixel confined response, followed by signal processing. In this paper, we emphasize the important secondary effects of the trajectories method. We show that in such a system, where the postprocessing comes after the detection, the postprocessing acts as a noise filter, hence allowing us to work with noisy data in the auxiliary channel. Roughly speaking, the SNR of an imaging system depends on the numerical aperture (NA). It follows that the main system, which typically has a higher NA, also has a higher SNR. Hence in the PO system, the ratio between the NA values of the main and auxiliary systems is expected to dictate the gap between their SNR values. In this paper, we show that when the system is implemented by the trajectories method, this expectation is too conservative. It is shown that due to the noise filtering, the auxiliary system can be noisier than expected. This claim is proved analytically and verified and exemplified by using experimental measurements.

Image formation of thick three-dimensional objects in differential-interference-contrast microscopy.

The differential-interference-contrast (DIC) microscope is of widespread use in life sciences as it enables noninvasive visualization of transparent objects. The goal of this work is to model the image formation process of thick three-dimensional objects in DIC microscopy. The model is based on the principles of electromagnetic wave propagation and scattering. It simulates light propagation through the components of the DIC microscope to the image plane using a combined geometrical and physical optics approach and replicates the DIC image of the illuminated object. The model is evaluated by comparing simulated images of three-dimensional spherical objects with the recorded images of polystyrene microspheres. Our computer simulations confirm that the model captures the major DIC image characteristics of the simulated object, and it is sensitive to the defocusing effects.

Simultaneous Temperature Estimation and Nonuniformity Correction From Multiple Frames.

IR cameras are widely used for temperature measurements in various applications, including agriculture, medicine, and security. Low-cost IR cameras have the immense potential to replace expensive radiometric cameras in these applications; however, low-cost microbolometer-based IR cameras are prone to spatially variant nonuniformity and to drift in temperature measurements, which limit their usability in practical scenarios. To address these limitations, we propose a novel approach for simultaneous temperature estimation and nonuniformity correction (NUC) from multiple frames captured by low-cost microbolometer-based IR cameras. We leverage the camera's physical image-acquisition model and incorporate it into a deep-learning architecture termed kernel prediction network (KPN), which enables us to combine multiple frames despite imperfect registration between them. We also propose a novel offset block that incorporates the ambient temperature into the model and enables us to estimate the offset of the camera, which is a key factor in temperature estimation. Our findings demonstrate that the number of frames has a significant impact on the accuracy of the temperature estimation and NUC. Moreover, introduction of the offset block results in significantly improved performance compared to vanilla KPN. The method was tested on real data collected by a low-cost IR camera mounted on an unmanned aerial vehicle, showing only a small average error of 0.27-0.54° C relative to costly scientific-grade radiometric cameras. Real data collected horizontally resulted in similar errors of 0.48-0.68° C . Our method provides an accurate and efficient solution for simultaneous temperature estimation and NUC, which has important implications for a wide range of practical applications.

Modeling of the diffusion MR signal in calibrated model systems and nerves.

Diffusion NMR is a powerful tool for gleaning microstructural information on opaque systems. In this work, the signal decay in single-pulsed-field gradient diffusion NMR experiments performed on a series of phantoms of increasing complexity, where the ground truth is known a priori, was modeled and used to identify microstructural features of these complex phantoms. We were able to demonstrate that, without assuming the number of components or compartments, the modeling can identify the number of restricted components, detect their sizes with an accuracy of a fraction of a micrometer, determine their relative populations, and identify and characterize free diffusion when present in addition to the components exhibiting restricted diffusion. After the accuracy of the modeling had been demonstrated, this same approach was used to study fixed nerves under different experimental conditions. It seems that this approach is able to characterize both the averaged axon diameter and the relative population of the different diffusing components in the neuronal tissues examined.

Implications of fitting a two-compartment model in single-shell diffusion MRI.

It is becoming increasingly common for studies to fit single-shell diffusion MRI data to a two-compartment model, which comprises a hindered cellular compartment and a freely diffusing isotropic compartment. These studies consistently find that the fraction of the isotropic compartment (f) is sensitive to white matter (WM) conditions and pathologies, although the actual biological source of changes infhas not been validated. In this work we put aside the biological interpretation offand study the sensitivity implications of fitting single-shell data to a two-compartment model. We identify a nonlinear transformation between the one-compartment model (diffusion tensor imaging, DTI) and a two-compartment model in which the mean diffusivities of both compartments are effectively fixed. While the analytic relationship implies that fitting this two-compartment model does not offer any more information than DTI, it explains why metrics derived from a two-compartment model can exhibit enhanced sensitivity over DTI to certain types of WM processes, such as age-related WM differences. The sensitivity enhancement should not be viewed as a substitute for acquiring multi-shell data. Rather, the results of this study provide insight into the consequences of choosing a two-compartment model when only single-shell data is available.

Mapping apparent eccentricity and residual ensemble anisotropy in the gray matter using angular double-pulsed-field-gradient MRI.

Conventional diffusion MRI methods are mostly capable of portraying microarchitectural elements such as fiber orientation in white matter from detection of diffusion anisotropy, which arises from the coherent organization of anisotropic compartments. Double-pulsed-field-gradient MR methods provide a means for obtaining microstructural information such as compartment shape and microscopic anisotropies even in scenarios where macroscopic organization is absent. Here, we apply angular double-pulsed-gradient-spin-echo MRI in the rat brain both ex vivo and in vivo for the first time. Robust angular dependencies are detected in the brain at long mixing time (t(m) ). In many pixels, the oscillations seem to originate from residual directors in randomly oriented media, i.e., from residual ensemble anisotropy, as corroborated by quantitative simulations. We then developed an analysis scheme that enables one to map of structural indices such as apparent eccentricity (aE) and residual phase (φ) that enables characterization of the rat brain in general, and especially the rat gray matter. We conclude that double-pulsed-gradient-spin-echo MRI may in principle become important in characterizing gray matter morphological features and pathologies in both basic and applied neurosciences.

REPIMPACT - a prospective longitudinal multisite study on the effects of repetitive head impacts in youth soccer.

Repetitive head impacts (RHI) are common in youth athletes participating in contact sports. RHI differ from concussions; they are considered hits to the head that usually do not result in acute symptoms and are therefore also referred to as "subconcussive" head impacts. RHI occur e.g., when heading the ball or during contact with another player. Evidence suggests that exposure to RHI may have cumulative effects on brain structure and function. However, little is known about brain alterations associated with RHI, or about the risk factors that may lead to clinical or behavioral sequelae. REPIMPACT is a prospective longitudinal study of competitive youth soccer players and non-contact sport controls aged 14 to 16 years. The study aims to characterize consequences of exposure to RHI with regard to behavior (i.e., cognition, and motor function), clinical sequelae (i.e., psychiatric and neurological symptoms), brain structure, function, diffusion and biochemistry, as well as blood- and saliva-derived measures of molecular processes associated with exposure to RHI (e.g., circulating microRNAs, neuroproteins and cytokines). Here we present the structure of the REPIMPACT Consortium which consists of six teams of clinicians and scientists in six countries. We further provide detailed information on the specific aims and the design of the REPIMPACT study. The manuscript also describes the progress made in the study thus far. Finally, we discuss important challenges and approaches taken to overcome these challenges.

Three dimension optical imaging in a high throughput layered expression system.

Layered peptide array (LPA) system enables multiplex screening of biomarkers [1-3]. One of the main problems of the LPA system is the screening of the layered-membranes stack. Nowadays, each membrane is imaged separately using conventional fluorescent imaging. This process is time consuming and requires extensive manual interaction. This paper describes a general solution for optical imaging of a layered grid medium using photogrammetric methods. The suggested method enables visualization of the LPA membranes stack by using only two images of the stack. This study is a proof of concept of the suggested solution using MATLAB simulation and phantom experiments.

Trajectories in parallel optics.

In our previous work we showed the ability to improve the optical system's matrix condition by optical design, thereby improving its robustness to noise. It was shown that by using singular value decomposition, a target point-spread function (PSF) matrix can be defined for an auxiliary optical system, which works parallel to the original system to achieve such an improvement. In this paper, after briefly introducing the all optics implementation of the auxiliary system, we show a method to decompose the target PSF matrix. This is done through a series of shifted responses of auxiliary optics (named trajectories), where a complicated hardware filter is replaced by postprocessing. This process manipulates the pixel confined PSF response of simple auxiliary optics, which in turn creates an auxiliary system with the required PSF matrix. This method is simulated on two space variant systems and reduces their system condition number from 18,598 to 197 and from 87,640 to 5.75, respectively. We perform a study of the latter result and show significant improvement in image restoration performance, in comparison to a system without auxiliary optics and to other previously suggested hybrid solutions. Image restoration results show that in a range of low signal-to-noise ratio values, the trajectories method gives a significant advantage over alternative approaches. A third space invariant study case is explored only briefly, and we present a significant improvement in the matrix condition number from 1.9160e+013 to 34,526.

The finite harmonic oscillator and its associated sequences.

A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are explained.

The effect of metric selection on the analysis of diffusion tensor MRI data.

The measurement of the distance between diffusion tensors is the foundation on which any subsequent analysis or processing of these quantities, such as registration, regularization, interpolation, or statistical inference is based. In recent years a family of Riemannian tensor metrics based on geometric considerations has been introduced for this purpose. In this work we examine the properties one would use to select metrics for diffusion tensors, diffusion coefficients, and diffusion weighted MR image data. We show that empirical evidence supports the use of a Euclidean metric for diffusion tensors, based upon Monte Carlo simulations. Our findings suggest that affine invariance is not a desirable property for a diffusion tensor metric because it leads to substantial biases in tensor data. Rather, the relationship between distribution and distance is suggested as a novel criterion for metric selection.