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Image coding methods based on adaptive wavelet transform and those employing zerotree quantization have been shown to be successful in recent years. In this paper, we present a general zerotree structure for an arbitrary wavelet packet geometry in an image coding framework. A fast basis selection algorithm which uses a Markov chain based cost estimate of encoding the image using this structure is developed. As a result, our adaptive wavelet zerotree image coder has a relatively low computational complexity, performs comparably to the state-of-the-art image coders and is capable of progressively encoding images. [ABSTRACT FROM AUTHOR]

An amendment to this paper has been published and can be accessed via a link at the top of the paper. [ABSTRACT FROM AUTHOR]

Experimental work across species has demonstrated that spontaneously generated behaviors are robustly coupled to variations in neural activity within the cerebral cortex. Functional magnetic resonance imaging data suggest that temporal correlations in cortical networks vary across distinct behavioral states, providing for the dynamic reorganization of patterned activity. However, these data generally lack the temporal resolution to establish links between cortical signals and the continuously varying fluctuations in spontaneous behavior observed in awake animals. Here, we used wide-field mesoscopic calcium imaging to monitor cortical dynamics in awake mice and developed an approach to quantify rapidly time-varying functional connectivity. We show that spontaneous behaviors are represented by fast changes in both the magnitude and correlational structure of cortical network activity. Combining mesoscopic imaging with simultaneous cellular-resolution two-photon microscopy demonstrated that correlations among neighboring neurons and between local and large-scale networks also encode behavior. Finally, the dynamic functional connectivity of mesoscale signals revealed subnetworks not predicted by traditional anatomical atlas-based parcellation of the cortex. These results provide new insights into how behavioral information is represented across the neocortex and demonstrate an analytical framework for investigating time-varying functional connectivity in neural networks., (© 2023. The Author(s), under exclusive licence to Springer Nature America, Inc.)

Confinement breaks translational and rotational symmetry in materials and makes all physical properties functions of position. Such spatial variations are key to modulating material properties at the nanoscale, and characterizing them accurately is therefore an intense area of research in the molecular simulations community. This is relatively easy to accomplish for basic mechanical observables. Determining spatial profiles of transport properties, such as diffusivity, is, however, much more challenging, as it requires calculating position-dependent autocorrelations of mechanical observables. In our previous paper (Domingues, T.S.; Coifman, R.; Haji-Akbari, A. J. Phys. Chem. B 2023 , 127 , 5273 10.1021/acs.jpcb.3c00670), we analytically derive and numerically validate a set of filtered covariance estimators (FCEs) for quantifying spatial variations of the diffusivity tensor from stochastic trajectories. In this work, we adapt these estimators to extract diffusivity profiles from MD trajectories and validate them by applying them to a Lennard-Jones fluid within a slit pore. We find our MD-adapted estimator to exhibit the same qualitative features as its stochastic counterpart, as it accurately estimates the lateral diffusivity across the pore while systematically underestimating the normal diffusivity close to hard boundaries. We introduce a conceptually simple and numerically efficient correction scheme based on simulated annealing and diffusion maps to resolve the latter artifact and obtain normal diffusivity profiles that are consistent with the self-part of the van Hove correlation functions. Our findings demonstrate the potential of this MD-adapted estimator in accurately characterizing spatial variations of diffusivity in confined materials.

Materials under confinement can possess properties that deviate considerably from their bulk counterparts. Indeed, confinement makes all physical properties position-dependent and possibly anisotropic, and characterizing such spatial variations and directionality has been an intense area of focus in experimental and computational studies of confined matter. While this task is fairly straightforward for simple mechanical observables, it is far more daunting for transport properties such as diffusivity that can only be estimated from autocorrelations of mechanical observables. For instance, there are well established methods for estimating diffusivity from experimentally observed or computationally generated trajectories in bulk systems. No rigorous generalizations of such methods, however, exist for confined systems. In this work, we present two filtered covariance estimators for computing anisotropic and position-dependent diffusivity tensors and validate them by applying them to stochastic trajectories generated according to known diffusivity profiles. These estimators can accurately capture spatial variations that span over several orders of magnitude and that assume different functional forms. Our kernel-based approach is also very robust to implementation details such as the localization function and time discretization and performs significantly better than estimators that are solely based on local covariance. Moreover, the kernel function does not have to be localized and can instead belong to a dictionary of orthogonal functions. Therefore, the proposed estimator can be readily used to obtain functional estimates of diffusivity rather than a tabulated collection of pointwise estimates. Nonetheless, the susceptibility of the proposed estimators to time discretization is higher at the immediate vicinity of hard boundaries. We demonstrate this heightened susceptibility to be common among all covariance-based estimators.

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal (i.e., no self loops) is robust to heteroskedastic noise. That is, the doubly-stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly-stochastic) noisy affinity matrix converges to its clean counterpart with rate m -1/2 , where m is the ambient dimension. We demonstrate this result numerically, and show that in contrast, the popular row-stochastic and symmetric normalizations behave unfavorably under heteroskedastic noise. Furthermore, we provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity, where the advantage of the doubly-stochastic normalization for exploratory analysis is evident.

The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between [Formula: see text] data points and a set of [Formula: see text] reference points, where [Formula: see text] can be drastically smaller than [Formula: see text]. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as [Formula: see text], and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions., (© The Author(s) 2019. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.)

It is known that if ( p n ) n ∈ ℕ is a sequence of orthogonal polynomials in L 2 ([-1,1], w ( x ) dx ), then the roots are distributed according to an arcsine distribution π -1 (1 - x 2 ) -1 dx for a wide variety of weights w ( x ). We connect this to a result of the Hilbert transform due to Tricomi: if f ( x )(1 - x 2 ) 1 / 4 ∈ L 2 (-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple of the arcsine distribution f ( x ) = c 1 - x 2 χ ( - 1 , 1 ) where c ∈ ℝ . We also prove a localized Parseval-type identity that seems to be new: if f ( x )(1- x 2 ) 1 / 4 ∈ L 2 (-1, 1) and f ( x ) 1 - x 2 has mean value 0 on (-1, 1), then ∫ - 1 1 ( H f ) ( x ) 2 1 - x 2 d x = ∫ - 1 1 f ( x ) 2 1 - x 2 d x . .

In this paper, we present a method for time series analysis based on empirical intrinsic geometry (EIG). EIG enables one to reveal the low-dimensional parametric manifold as well as to infer the underlying dynamics of high-dimensional time series. By incorporating concepts of information geometry, this method extends existing geometric analysis tools to support stochastic settings and parametrizes the geometry of empirical distributions. However, the statistical models are not required as priors; hence, EIG may be applied to a wide range of real signals without existing definitive models. We show that the inferred model is noise-resilient and invariant under different observation and instrumental modalities. In addition, we show that it can be extended efficiently to newly acquired measurements in a sequential manner. These two advantages enable us to revisit the Bayesian approach and incorporate empirical dynamics and intrinsic geometry into a nonlinear filtering framework. We show applications to nonlinear and non-Gaussian tracking problems as well as to acoustic signal localization.

Recovering the three-dimensional structure of molecules is important for understanding their functionality. We describe a spectral graph algorithm for reconstructing the three-dimensional structure of molecules from their cryo-electron microscopy images taken at random unknown orientations.We first identify a one-to-one correspondence between radial lines in three-dimensional Fourier space of the molecule and points on the unit sphere. The problem is then reduced to determining the coordinates of points on the sphere given a subset of their pairwise geodesic distances. To recover those coordinates, we exploit the special geometry of the problem, as rendered by the Fourier projection-slice theorem, to construct a weighted graph whose vertices are the radial Fourier lines and whose edges are linked using the common line property. The graph organizes the radial lines on the sphere in a global manner that reveals the acquisition direction of each image. This organization is derived from a global computation of a few eigenvectors of the graph's sparse adjacency matrix. Once the directions are obtained, the molecule can be reconstructed using classical tomography methods.The presented algorithm is direct (as opposed to iterative refinement schemes), does not require any prior model for the reconstructed object, and is shown to have favorable computational and numerical properties. Moreover, the algorithm does not impose any assumption on the distribution of the projection orientations. Physically, this means that the algorithm is applicable to molecules that have unknown spatial preference.

We introduce a graph Laplacian-based algorithm for the tomographic reconstruction of a planar object from its projections taken at random unknown directions. A Laplace-type operator is constructed on the data set of projections, and the eigenvectors of this operator reveal the projection orientations. The algorithm is shown to successfully reconstruct the Shepp-Logan phantom from its noisy projections. Such a reconstruction algorithm is desirable for the structuring of certain biological proteins using cryo-electron microscopy.