Search

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.

A growing world population together with a decrease in water resources have resulted in a deficit in water sources for agriculture. Nations with limited water resources are already using recycled treated water as a vital water source for irrigation. In 2019, treated wastewater made up 45% of the overall agricultural water consumption in Israel. Since recycling systems work continuously and are directly connected to irrigation reservoirs, there is a constant risk of irrigation system malfunctions in the treatment process. In cases of fecal origin, this may result in pathogens entering the irrigation water, the soil, and eventually the food chain. Early warning systems for biological water contamination are important in large-scale processing, yet less studied. This work proposes such a system, with a functional prototype for irrigation-water monitoring by measuring fluorescence spectra in the range of 300 – 520 nm at excitation wavelengths of 280 nm and 340 nm, to capture both tryptophan-like fluorescence and humic-like fluorescence. The prototype was successfully tested under simulated field conditions with treated irrigation water as the base substance, and milk injections to simulate dissolved organic matter breaches. Near real-time operation enabled recording biological and chemical dynamics within the substance for the sake of monitoring drift in process control.

We introduce a novel neural network-based PDEs solver for forward and inverse problems. The solver is grid free, mesh free, and shape free, and the solution is approximated by a neural network. We ...

Increasing global water deficit and demand for yield improvement call for high-resolution monitoring of irrigation, crop water stress, and crops' general condition. To provide high spatial resolution with high-temperature accuracy, remote sensing is conducted at low altitudes using radiometric longwave thermal infrared cameras. However, the radiometric cameras' price, and the low altitude leading to low coverage in a given time, limit the use of radiometric aerial surveys for agricultural needs. This paper presents progress toward solving both limitations using algorithmic and computational imaging methods: stabilizing the readout of low-cost thermal cameras to obtain radiometric data, and improving the latter's low resolution by applying convolutional neural network-based super-resolution. The two methods were merged by an end-to-end algorithm pipeline, providing a large mosaicked image of the field. First, the potential capabilities of a joint estimation method to correct unknown offset and gain were simulated on remotely sensed agricultural data. Comparison to ground-truth measurements showed radiometric accuracy with a root mean square error (RMSE) of 1.3 °C to 1.8 °C. Then, the proposed super-resolution method was demonstrated on experimental and simulated remotely sensed agricultural data. Preliminary experimental results showed 50% improvement in image sharpness relative to bicubic interpolation. The performance of the algorithm was evaluated on 22 simulated cases at × 2 and × 4 magnification. Finally, image mosaicking using the proposed pipeline was demonstrated. A mosaicked image composed of sub-images pre-processed by the proposed computational methods resulted in a RMSE in temperature of 0.8 °C, as compared to 8.2 °C without the initial processing.

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, We use a large (n = 1113) harmonized diffusion MRI dataset of individuals diagnosed with schizophrenia and healthy controls, in order to study the predictive performance of a normative modeling approach. This approach compares each individual with distribution derived from healthy controls to derive measures of deviation from the normal distribution. Our results show that combining deviation information from different white matter tracts, while using multiple imaging measures simultaneously, improves prediction performance, and could therefore aid in subject‐level classification.

Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals., 15 pages, v2 fixed minor typos

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. ispartof: Brain Imaging And Behavior vol:16 issue:1 ispartof: location:United States status: published

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.

Extracting meaningful representations from geometric data has prime importance in the areas of computer vision, computer graphics, and image processing. Classical approaches use tools from differential geometry for modeling the problem and employed efficient and robust numerical techniques to engineer them for a particular application. Recent advances in learning methods, particularly in the areas of deep-learning and neural networks provide an alternative mechanism of extracting meaningful features and doing data-engineering. These techniques have proven very successful for various kinds of visual and semantic cognition tasks achieving state-of-the art results. In this chapter we explore the synergy between these two seemingly disparate computational methodologies. First, we provide a short treatise on geometric invariants of planar curves and a scheme to discover them from data in a learning framework, where the invariants are modelled using neural networks. Secondly, we also demonstrate the reverse, that is, imputing principled geometric invariants like geometric moments into standard learning architectures enables a significant boost in performance. Our goal would not only be to achieve better performance, but also to provide a geometric insight into the learning process thereby establishing strong links between the two fields.

In order to bridge microscopic molecular motion with macroscopic diffusion MR signal in complex structures, we propose a general stochastic model for molecular motion in a magnetic field. The Fokker-Planck equation of this model governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad hoc selection of a propagator. In particular, the boundary conditions that implicitly incorporate the microstructure into the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal.

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.

To use VRI systems, a field is divided into irrigation management zones (IMZs). While IMZs are dynamic in nature, most of IMZs prescription maps are static. High-resolution thermal images (TI) coupled with measured atmospheric conditions have been utilized to map the within-field water status variability and to delineate in-season IMZs. Unfortunately, spaceborne TIs have coarse spatial resolution and aerial platforms require substantial financial investments, which may inhibit their large-scale adoption. Three approaches are proposed to facilitate large-scale adoption of TI-based IMZs: 1) increase of the capacity of aerial TI by enhancing their spatial resolution; 2) sharpening the spatial resolution of satellite TI by fusing satellite multi-spectral images in the visible-near-infrared (VIS-NIR) range; 3) increase the capacity of aerial TI by fusing satellite multi-spectral images in the VIS-NIR range. The scientific and engineering basis of each of the approaches is described together with initial results.

We discuss the topic of selecting optimal orthonormal bases for representing classes of signals defined either through statistics or via some deterministic characterizations, or combinations of the two. In all cases, the best bases result from spectral analysis of a Hermitian matrix that summarizes the prior information we have on the signals we want to represent, achieving optimal progressive approximations. We also provide uniqueness proofs for the discrete cases.

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.

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.

Angular double-pulsed-field gradient (d-PFG) MR methodology is increasingly used to non-invasively obtain pore sizes in opaque chemical and biological systems. In such MR experiments, the angular dependency of the signal at zero mixing time, through modeling, can be used to extract the pore size. In many systems not only the pore sizes but also their directions are of importance. Before applying d-PFG NMR to complex systems, it is of value to challenge the ability of the methodology to extract these microstructural parameters in samples where the ground truth is known. In the present study we explored whether modeling of the signal in angular d-PFG NMR experiments at zero mixing time, can simultaneously provide the size and the direction of tilted compartments with little prior knowledge. We showed that the angular d-PFG MR methodology enables simultaneous extraction of the pore size and the direction of mono-dispersed phantoms and of phantoms where the restricted compartments have different pore sizes. However, we found that in phantoms with two or more pore sizes, only averaged pore sizes were extracted for large azimuthal and polar angles.

Image segmentation is often considered a deterministic process with a single ground truth. Nevertheless, in practice, and in particular, when medical imaging analysis is considered, the extraction of regions of interest (ROIs) is ill-posed and the concept of ‘most probable’ segmentation is model-dependent. In this paper, a measure for segmentation uncertainty in the form of segmentation error margins is introduced. This measure provides a goodness quantity and allows a ‘fully informed’ comparison between extracted boundaries of related ROIs as well as more meaningful statistical analysis. The tool we present is based on a novel technique for segmentation sampling in the Fourier domain and Markov Chain Monte Carlo (MCMC). The method was applied to cortical and sub-cortical structure segmentation in MRI. Since the accuracy of segmentation error margins cannot be validated, we use receiver operating characteristic (ROC) curves to support the proposed method. Precision and recall scores with respect to expert annotation suggest this method as a promising tool for a variety of medical imaging applications including user-interactive segmentation, patient follow-up, and cross-sectional analysis.

A novel algorithm combining sparse dictionary learning with variational method segmentation enables recognizing outbursts of macrophomina in cotton field images taken from a drone.

We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space -- the so called window-signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window-signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window-signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window-signal space. As one example application, we show how continuous wavelet multipliers (also called Calder\'{o}n-Toeplitz Operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.

By the multiple correlation function (MCF) formalism, the nuclear magnetic resonance magnetization of diffusing spins can be represented for simple pore geometries. It may be used to infer geometric structure at the scale of microns. This is to be compared to the diffusion tensor imaging, which provides geometric information at the scale of millimeters. The MCF formulation was derived to special cases in which the gradients of the magnetic field are oriented in specific directions. A generalized approach allowing an arbitrary magnetic field direction was introduced by Ozarslan. In this article, we present a complete account of the generalized MCF mathematical derivation starting from Bloch–Torrey equations. It is aimed to the experts and novices alike. We present two approaches—the indirect derivation is based on Ozarslan's work, where the standard MCF equations are adopted to arbitrary gradient directions. Our alternative approach is based on direct calculations of the MCF matrices for specified gradient directions. We prove that the two approaches lead to the same equations. Finally, we revise Mitra's microscopic approach and show the relation to the macroscopic MCF approach. We prove that in some limit conditions the two signal equations coincide © 2015 Wiley Periodicals, Inc. Concepts Magn Reson Part A 43A: 16–53, 2015.

Pulse-Doppler radar is suitable for target detection and parameter estimation such as range and radial velocity. The signal processing, in this type of radars, is based on matched filter and discrete Fourier transform (DFT). Detection, measurement accuracy and good resolution require the transmission of a train of pulses, where each pulse is usually frequency modulated. In some cases, even few trains of pulses are required. In this paper, we use statistical framework for detection and parameter estimation. By using the proposed algorithm, we show that targets can be detected and resolved using a transmission of an unmodulated single pulse where other classical methods based on matched filter may fail.

A method that takes into consideration the influence of imaging-system blur on the local-intensity histogram enables recognizing mealybug clusters covering only a few pixels in cotton field images taken from a drone.

Recent years have brought about impressive reconstructions of white matter architecture, due to the advance of increasingly sophisticated MRI based acquisition methods and modeling techniques. These result in extremely large sets of streamelines (fibers ) for each subject. The sets require large amount of storage and are often unwieldy and difficult to manipulate and analyze. We propose to use sparse representations for fibers to achieve a more compact representation. We also propose the means for calculating inter-fiber similarities in the compressed space using a measure, which we term: Cosine with Dictionary Similarity Weighting (CWDS). The performance of both sparse representations and CWDS is evaluated on full brain fiber-sets of 15 healthy subjects. The results show that a reconstruction error of slightly below 2 mm is achieved, and that CWDS is highly correlated with the cosine similarity in the original space.

We address the problem of blind piecewise space-variant image deblurring where only part of the image is sharp, assuming a shallow depth of field which imposes significant defocus blur.

In principle, a still image can be reconstructed from a turbulent video of a static scene using the classic ‘sum and deblur’ approach (with or without a preliminary registration step). However, the performance of such turbulence recovery algorithms has so far been limited. In the last decade, significant progress has been achieved in non-rigid registration and in image deblurring. We revisit the turbulence recovery problem, incorporating state of the art registration and deblurring algorithms as building blocks within the sum and deblur framework. Accurate pre-registration of the input video frames narrows the spatial support of the effective blur kernel affecting the sum of the turbulent video sequence. Powerful registration is therefore crucial for successful reconstruction. We employ a two-phase registration process, consisting of rigid registration followed by non-rigid refinement. For rigid registration, we adopt the recent algorithm of Lazaridis and Petrou (2006) [1]. Using real turbulent video data, we demonstrate excellent turbulence recovery.

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.

The motivation to this paper stems from signal/image processing where it is desired to measure various attributes or physical quantities such as position, scale, direction and frequency of a signal or an image. These physical quantities are measured via a signal transform, for example, the short time Fourier transform measures the content of a signal at different times and frequencies. There are well known obstructions for completely accurate measurements formulated as "uncertainty principles". It has been shown recently that "conventional" localization notions, based on variances associated with Lie-group generators and their corresponding uncertainty inequality might be misleading, if they are applied to transformation groups which differ from the Heisenberg group, the latter being prevailing in signal analysis and quantum mechanics. In this paper we describe a generic signal transform as a procedure of measuring the content of a signal at different values of a set of given physical quantities. This viewpoint sheds a light on the relationship between signal transforms and uncertainty principles. In particular we introduce the concepts of "adjoint translations" and "adjoint observables", respectively. We show that the fundamental issue of interest is the measurement of physical quantities via the appropriate localization operators termed "adjoint observables". It is shown how one can define, for each localization operator, a family of related "adjoint translation" operators that translate the spectrum of that localization operator. The adjoint translations in the examples of this paper correspond to well-known transformations in signal processing such as the short time Fourier transform (STFT), the continuous wavelet transform (CWT) and the shearlet transform. We show how the means and variances of states transform appropriately under the translation action and compute associated minimizers and equalizers for the uncertainty criterion. Finally, the concept of adjoint observables is used to estimate concentration properties of ambiguity functions, the latter being an alternative localization concept frequently used in signal analysis.

Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it. These geometries are typically defined using either the shortest path length (geodesic distance), or properties of heat diffusion (diffusion distance) on the surface. Both measures are implicitly derived from the metric induced by the ambient Euclidean space. In this paper, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We apply equi-affine geometry for analyzing arbitrary shapes with positive Gaussian curvature. The potential of the proposed framework is explored in a range of applications such as shape matching and retrieval, symmetry detection, and computation of Voroni tessellation. We show that in some shape analysis tasks, equi-affine-invariant intrinsic geometries often outperform their Euclidean-based counterparts. We further explore the potential of this metric in facial anthropometry of newborns. We show that intrinsic properties of this homogeneous group are better captured using the equi-affine metric.

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.

Signal and image processing have seen an explosion of interest in the last few years in a new form of signal/image characterization via the concept of sparsity with respect to a dictionary. An active field of research is dictionary learning: the representation of a given large set of vectors (e.g. signals or images) as linear combinations of only few vectors (patterns). To further reduce the size of the representation, the combinations are usually required to be sparse, i.e., each signal is a linear combination of only a small number of patterns. This paper suggests a new computational approach to the problem of dictionary learning, known in computational geometry as coresets. A coreset for dictionary learning is a small smart non-uniform sample from the input signals such that the quality of any given dictionary with respect to the input can be approximated via the coreset. In particular, the optimal dictionary for the input can be approximated by learning the coreset. Since the coreset is small, the learning is faster. Moreover, using merge-and-reduce, the coreset can be constructed for streaming signals that do not fit in memory and can also be computed in parallel. We apply our coresets for dictionary learning of images using the K-SVD algorithm and bound their size and approximation error analytically. Our simulations demonstrate gain factors of up to 60 in computational time with the same, and even better, performance. We also demonstrate our ability to perform computations on larger patches and high-definition images, where the traditional approach breaks down.

In the context of fibre bundles theory, there exist some differential operators of order 2, called generalized Laplacians, acting on sections of vector bundles over Riemannian manifolds, and generalizing the Laplace-Beltrami operator. Such operators are determined by covariant derivatives on vector bundles. In this paper, we construct a class of generalized Laplacians, devoted to multi-channel image processing, from the construction of optimal covariant derivatives. The key idea is to consider an image as a section of an associate bundle, that is a vector bundle related to a principal bundle through a group representation. In this context, covariant derivatives are determined by connection 1-forms on principal bundles. We construct optimal connection 1-forms by the minimization of a variational problem on principal bundles. From the heat equations of the generalized Laplacians induced by the corresponding optimal covariant derivatives, we obtain diffusions whose behaviors depend of the choice of the group representation. We provide experiments on color images.

Recent advances in Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) of white matter in conjunction with improved tractography produce impressive reconstructions of White Matter (WM) pathways. These pathways (fiber sets) often contain hundreds of thousands of fibers, or more. In order to make fiber based analysis more practical, the fiber set needs to be preprocessed to eliminate redundancies and to keep only essential representative fibers. In this paper we demonstrate and compare two distinctive frameworks for selecting this reduced set of fibers. The first framework entails pre-clustering the fibers using k-means, followed by Hierarchical Clustering and replacing each cluster with one representative. For the second clustering stage seven distance metrics were evaluated. The second framework is based on an efficient geometric approximation paradigm named coresets. Coresets present a new approach to optimization and have huge success especially in tasks requiring large computation time and/or memory. We propose a modified version of the coresets algorithm, Density Coreset. It is used for extracting the main fibers from dense datasets, leaving a small set that represents the main structures and connectivity of the brain. A novel approach, based on a 3D indicator structure, is used for comparing the frameworks. This comparison was applied to High Angular Resolution Diffusion Imaging (HARDI) scans of 4 healthy individuals. We show that among the clustering based methods, that cosine distance gives the best performance. In comparing the clustering schemes with coresets, Density Coreset method achieves the best performance.

Micro-bolometer based thermal focal plane array is very attractive for radiometric imaging. We introduce a new method, to perform on site calibration of unknown spatially dependent drift and gain thermal images for this goal.

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.

The Laplacian raised to fractional powers can be used to generate scale spaces as was shown in recent literature by Duits, Felsberg, Florack, and Platel [$\alpha$ scale spaces on a bounded domain, in Scale Space Methods in Computer Vision, L. D. Griffin and M. Lillholm, eds., Lecture Notes in Comput. Sci. 2695, Springer, Berlin, Heidelberg, 2003, pp. 494-510] and Duits, Florack, de Graaf, and ter Haar Romeny [J. Math. Imaging Vision, 20 (2004), pp. 267-298]. In this paper, we study the anisotropic diffusion processes by defining new generators that are fractional powers of an anisotropic scale space generator. This is done in a general framework that allows us to explain the relation between a differential operator that generates the flow and the generators that are constructed from its fractional powers. We then generalize this to any other function of the operator. We discuss important issues involved in the numerical implementation of this framework and present several examples of fractional versions of the Perona-Malik and Beltrami flows along with their properties.

The uncertainty principle is a fundamental concept in quantum mechanics, harmonic analysis and signal and information theory. It is rooted in the framework of quantum mechanics, where it is known as the Heisenberg uncertainty principle. In general, the uncertainty principle gives a lower bound on the product of variances for any state f with respect to two self-adjoint operators: $$v_f(A)v_f(B)\ge\frac{1}{4}|e_f([A,B])|^2.$$ The functions that attain the lower bound of the inequality have been investigated extensively, and are known as uncertainty minimizers.

Relating brain tissue properties to diffusion tensor imaging (DTI) is limited when an image voxel contains partial volume of brain tissue with free water, such as cerebrospinal fluid or edema, rendering the DTI indices no longer useful for describing the underlying tissue properties. We propose here a method for separating diffusion properties of brain tissue from surrounding free water while mapping the free water volume. This is achieved by fitting a bi-tensor model for which a mathematical framework is introduced to stabilize the fitting. Applying the method on datasets from a healthy subject and a patient with edema yielded corrected DTI indices and a more complete tract reconstruction that passed next to the ventricles and through the edema. We were able to segment the edema into areas according to the condition of the underlying tissue. In addition, the volume of free water is suggested as a new quantitative contrast of diffusion MRI. The findings suggest that free water is not limited to the borders of the brain parenchyma; it therefore contributes to the architecture surrounding neuronal bundles and may indicate specific anatomical processes. The analysis requires a conventional DTI acquisition and can be easily merged with existing DTI pipelines.

In this paper we discuss regularization of images that take their value in matrix Lie groups. We describe an image as a section in a principal bundle which is a fibre bundle where the fiber (the feature space) is a Lie group. Via the scalar product on the Lie algebra, we define a bi-invariant metric on the Lie-group manifold. Thus, the fiber becomes a Riemannian manifold with respect to this metric. The induced metric from the principal bundle to the image manifold is obtained by means of the bi-invariant metric. A functional over the space of sections, i.e., the image manifolds, is defined. The resulting equations of motion generate a flow which evolves the sections in the spatial-Lie-group manifold. We suggest two different approaches to treat this functional and the corresponding PDEs. In the first approach we derive a set of coupled PDEs for the local coordinates of the Lie-group manifold. In the second approach a coordinate-free framework is proposed where the PDE is defined directly with respect to the Lie-group elements. This is a parameterization-free method. The differences between these two methods are discussed. We exemplify this framework on the well-known orientation diffusion problem, namely, the unit-circle S 1 which is identified with the group of rotations in two dimensions, SO(2). Regularization of the group of rotations in 3D and 4D, SO(3) and SO(4), respectively, is demonstrated as well.

We propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on a local differential geometric approach. The manifold of symmetric positive-definite (SPD) matrices, P n , is parameterized via the Iwasawa coordinate system. In this framework distances on P n are measured in terms of a natural GL(n)-invariant metric. Via the mathematical concept of fiber bundles, we describe the tensor-valued image as a section where the metric over the section is induced by the metric over P n . Then, a functional over the sections accompanied by a suitable data fitting term is defined. The variation of this functional with respect to the Iwasawa coordinates leads to a set of $\frac{1}{2}n(n+1)$ coupled equations of motion. By means of the gradient descent method, these equations of motion define a Beltrami flow over P n . It turns out that the local coordinate approach via the Iwasawa coordinate system results in very simple numerics that leads to fast convergence of the algorithm. Regularization results as well as results of fibers tractography for DTI are presented.

We evaluate the dense optical flow between two frames via variational approach. In this paper, a new framework for deriving the regularization term is introduced giving a geometric insight into the action of a smoothing term. The framework is based on the Beltrami paradigm in image denoising. It includes a general formulation that unifies several previous methods. Using the proposed framework we also derive two novel anisotropic regularizers incorporating a new criterion that requires co-linearity between the gradients of optical flow components and possibly the intensity gradient. We call this criterion "alignment" and reveal its existence also in the celebrated Nagel and Enkelmann's formulation. It is shown that the physical model of rotational motion of a rigid body, pure divergent/convergent flow and irrotational fluid flow, satisfy the alignment criterion in the flow field. Experimental tests in comparison to a recently published method show the capability of the new criterion in improving the optical flow estimations.

An affine-group-based design methodology of Gabor-type filter bank is presented for the purpose of image analysis and synthesis. Various tessellations of the combined spatial-feature space are considered. We combine ideas introduced by Daugman [J.G. Daugman, Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters, J. Opt. Soc. Am. 2 (7) (1985) 1160–1169], Lee [T.S. Lee, Image representation using 2D Gabor-wavelets, IEEE Trans. PAMI 18 (10) (1996) 959–971] and Manjunath and Ma [B.S. Manjunath, W.Y. Ma, Texture features browsing and retrieval of image data, IEEE Trans. PAMI 18 (8) (1996) 837–842], and extend them by applying the action of the full affine group on Gaborian-type mother wavelets. In this approach we adopt optimality criteria of minimal spatial-features combined uncertainty, as well as tightness of the frame tessellating this combined space. In this work, scalings in the x and y directions, allowing for independent dilations in these two directions, as well as rotations and translations are allowed. For each discrete set of scalings, rotations and translations the frame bounds are calculated. For frames where the frame operator is well approximated by a multiple of the identity, we use the same set of functions in the analysis and synthesis, as though the frame is equivalent to an orthogonal basis. Moreover, we show that in the case of independent scalings in the x and y directions, the number of dominant (characteristic) orientations of the filter bank may depend on scale. We further show that the orientation bandwidths thus obtained, resemble those attained under the constraint imposed by the uncertainty principle.

Every stereovision application must cope with the correspondence problem. The space of the matching variables, often consisting of spatial coordinates, intensity and disparity, is commonly referred as the data term (space). Since the data is often noisy a-priori, preference is required to result a smooth disparity (or piecewise smooth). To this end, each local method (e.g. window correlation techniques) performs a regularization of the data space. In this paper we propose a geometric framework for anisotropic regularization of the data space seeking to preserve the discontinuities in this space when filtering out the noise. On the other hand, the global methods consider a non-regularized data term with a smoothing constraint imposed directly on the disparity. This paper also proposes a new idea where the data space is regularized in a global method prior to the disparity evaluation. The idea is implemented on the state of the art variational method. Experimental results on the Middlebury real images demonstrate the advantages of the proposed approach.

We present a novel variational approach for simultaneous segmentation of two images of the same object taken from different viewpoints. Due to noise, clutter and occlusions, neither of the images contains sufficient information for correct object-background partitioning. The evolving object contour in each image provides a dynamic prior for the segmentation of the other object view. We call this process mutual segmentation. The foundation of the proposed method is a unified level-set framework for region and edge based segmentation, associated with a shape similarity term. The suggested shape term incorporates the semantic knowledge gained in the segmentation process of the image pair, accounting for excess or deficient parts in the estimated object shape. Transformations, including planar projectivities, between the object views are accommodated by a registration process held concurrently with the segmentation. The proposed segmentation algorithm is demonstrated on a variety of image pairs. The homography between each of the image pairs is estimated and its accuracy is evaluated.

We analyze the discrete maximum principle for the Beltrami color flow. The Beltrami flow can display linear as well as nonlinear behavior according to the values of a parameter β, which represents the ratio between spatial and color distances. In general, the standard schemes fail to satisfy the discrete maximum principle. In this work we show that a nonnegative second order difference scheme can be built for this flow only for small β, i.e. linear-like diffusion. Since this limitation is too severe, we construct a novel finite difference scheme, which is not nonnegative and satisfies the discrete maximum principle for all values of β. Numerical results support the analysis.

Image restoration, i.e., the recovery of images that have been degraded by blur and noise, is a challenging inverse problem. A unified variational approach to edge-preserving image deconvolution and impulsive noise removal has recently been suggested by the authors and shown to be effective. It leads to a minimization problem that is iteratively solved by alternate minimization for both the recovered image and the discontinuity set. The variational formulation yields a nonlinear integro-differential equation. This equation was linearized by fixed point iteration. In this paper, we analyze and prove the convergence of the iterative method.