In the intensive care unit (LCU), critically ill patients commonly receive sedative medications to alleviate pain and facilitate clinical care. These sedatives are administered based on the assessment of sedation levels and clinical experience. A periodic evaluation of behavioral responses to stimuli is commonly used to monitor sedation levels in the ICU. It is, however, difficult to monitor sedation levels in critically ill patients due to various factors, including comas or altered consciousness. As a non-invasive monitoring method, electroencephalography (EEG) has been shown to track patients' consciousness levels effectively. This assists medical personnel in mitigating the adverse effects of inappropriate medication on patients. In this study, we utilize both time-domain and frequency-domain information from EEG of 85 critically ill patients to train a multi-feature fusion model for predicting the Richmond Agitation-Sedation Scale (RASS) scores, which range from -5 = comatose, -4, … to 0 = awake.The proposed model achieves average accuracies of 65 %, and 85% average accuracies with tolerance of one level difference. The results demonstrate that the proposed model can effectively predict the sedation levels of critically ill patients in the ICU. Moreover, we show spectrograms of EEG signals corresponding to different sedation levels, which provide interpretability through the analysis of spectro-temporal information. Future enhancements involve leveraging diverse bedside monitoring data in the ICU to improve the accuracy of patient consciousness level monitoring.

This paper explores the expressive power of deep neural networks for a diverse range of activation functions. An activation function set $\mathscr{A}$ is defined to encompass the majority of commonly used activation functions, such as $\mathtt{ReLU}$, $\mathtt{LeakyReLU}$, $\mathtt{ReLU}^2$, $\mathtt{ELU}$, $\mathtt{CELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, $\mathtt{Mish}$, $\mathtt{Sigmoid}$, $\mathtt{Tanh}$, $\mathtt{Arctan}$, $\mathtt{Softsign}$, $\mathtt{dSiLU}$, and $\mathtt{SRS}$. We demonstrate that for any activation function $\varrho\in \mathscr{A}$, a $\mathtt{ReLU}$ network of width $N$ and depth $L$ can be approximated to arbitrary precision by a $\varrho$-activated network of width $3N$ and depth $2L$ on any bounded set. This finding enables the extension of most approximation results achieved with $\mathtt{ReLU}$ networks to a wide variety of other activation functions, albeit with slightly increased constants. Significantly, we establish that the (width,$\,$depth) scaling factors can be further reduced from $(3,2)$ to $(1,1)$ if $\varrho$ falls within a specific subset of $\mathscr{A}$. This subset includes activation functions such as $\mathtt{ELU}$, $\mathtt{CELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, and $\mathtt{Mish}$.

This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that $\mathcal{L}_2\circ \boldsymbol{g}^{\circ r}\circ \boldsymbol{\mathcal{L}}_1$ can approximate $1$-Lipschitz continuous functions on $[0,1]^d$ with an error $\mathcal{O}(r^{-1/d})$, where $\boldsymbol{g}$ is realized by a fixed-size ReLU network, $\boldsymbol{\mathcal{L}}_1$ and $\mathcal{L}_2$ are two affine linear maps matching the dimensions, and $\boldsymbol{g}^{\circ r}$ denotes the $r$-times composition of $\boldsymbol{g}$. Furthermore, we extend such a result to generic continuous functions on $[0,1]^d$ with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

Using inertial measurement units mounted on foot is a feasible approach to improve the positioning accuracy for the human motion capture system. This paper presents a lightweight and low cost wireless inertial motion capture system for the simultaneous reconstruction of human body attitude and displacement. First of all, the device is based on human sensor networks and distributes 15 sensor nodes on the key human limbs. Then, after an initial sensor alignment with the reduced error, a zero-speed update algorithm is used to calculate foot displacement. In addition, to constantly update the human posture information, a kind of motion reconstruction method based on the gradient descent method was used to fuse the sensor data. Finally, a new method of three-dimensional human body reconstruction is proposed, which is different from the traditional motion capture system. Through unconstrained traversal of the root, the human posture and foot trajectory are combined to realize the synchronous reconstruction of posture and displacement. It is concluded from the experiment results that the estimation errors are well controlled, and motion patterns are consistent with the actual situation.

In this work, a robust and efficient dual iterative refinement (DIR) method is proposed for dense correspondence between two nearly isometric shapes. The key idea is to use dual information, such as spatial and spectral, or local and global features, in a complementary and effective way, and extract more accurate information from current iteration to use for the next iteration. In each DIR iteration, starting from current correspondence, a zoom-in process at each point is used to select well matched anchor pairs by a local mapping distortion criterion. These selected anchor pairs are then used to align spectral features (or other appropriate global features) whose dimension adaptively matches the capacity of the selected anchor pairs. Thanks to the effective combination of complementary information in a data-adaptive way, DIR is not only efficient but also robust to render accurate results within a few iterations. By choosing appropriate dual features, DIR has the flexibility to handle patch and partial matching as well. Our comprehensive experiments on various data sets demonstrate the superiority of DIR over other state-of-the-art methods in terms of both accuracy and efficiency.

In this work, we introduce a novel local pairwise descriptor and then develop a simple, effective iterative method to solve the resulting quadratic assignment through sparsity control for shape correspondence between two approximate isometric surfaces. Our pairwise descriptor is based on the stiffness and mass matrix of finite element approximation of the Laplace-Beltrami differential operator, which is local in space, sparse to represent, and extremely easy to compute while containing global information. It allows us to deal with open surfaces, partial matching, and topological perturbations robustly. To solve the resulting quadratic assignment problem efficiently, the two key ideas of our iterative algorithm are: 1) select pairs with good (approximate) correspondence as anchor points, 2) solve a regularized quadratic assignment problem only in the neighborhood of selected anchor points through sparsity control. These two ingredients can improve and increase the number of anchor points quickly while reducing the computation cost in each quadratic assignment iteration significantly. With enough high-quality anchor points, one may use various pointwise global features with reference to these anchor points to further improve the dense shape correspondence. We use various experiments to show the efficiency, quality, and versatility of our method on large data sets, patches, and point clouds (without global meshes).

We present a hybrid imaging method for a challenging traveltime tomography problem which includes both unknown medium and unknown scatterers in a bounded domain. The goal is to recover both the medium and the boundary of the scatterers from the scattering relation data on the domain boundary. Our method is composed of three steps: (1) preprocess the data to classify them into three different categories of measurements corresponding to nonbroken rays, broken-once rays, and others, respectively, (2) use the the nonbroken ray data and an effective data-driven layer stripping strategy---an optimization based iterative imaging method---to recover the medium velocity outside the convex hull of the scatterers, and (3) use selected broken-once ray data to recover the boundary of the scatterers---a direct imaging method. By numerical tests, we show that our hybrid method can recover both the unknown medium and the not-too-concave scatterers efficiently and robustly.

This paper proposes a novel and simple multilayer feature learning method for image classification by employing the extreme learning machine (ELM). The proposed algorithm is composed of two stages: the multilayer ELM (ML-ELM) feature mapping stage and the ELM learning stage. The ML-ELM feature mapping stage is recursively built by alternating between feature map construction and maximum pooling operation. In particular, the input weights for constructing feature maps are randomly generated and hence need not be trained or tuned, which makes the algorithm highly efficient. Moreover, the maximum pooling operation enables the algorithm to be invariant to certain transformations. During the ELM learning stage, elastic-net regularization is proposed to learn the output weight. Elastic-net regularization helps to learn more compact and meaningful output weight. In addition, we preprocess the input data with the dense scale-invariant feature transform operation to improve both the robustness and invariance of the algorithm. To evaluate the effectiveness of the proposed method, several experiments are conducted on three challenging databases. Compared with the conventional deep learning methods and other related ones, the proposed method achieves the best classification results with high computational efficiency.

In this work, we propose computational models and algorithms for point cloud registration with nonrigid transformation. First, point clouds sampled from manifolds originally embedded in some Euclidean space are transformed to new point clouds embedded in $\mathbb{R}^n$ by the Laplace--Beltrami (LB) eigenmap, which is invariant under isometric transformation, using the first $n$ leading eigenvalues and corresponding eigenfunctions of the LB operator. Then we develop computational models and algorithms for registration of the transformed point clouds in a distribution/probability sense based on optimal transport, which provides both generality and flexibility for point cloud registration. In particular, we propose to use a rotation-invariant sliced-Wasserstein distance to achieve computation efficiency and handle ambiguities introduced by LB eigenmaps. By going from smaller $n$, which provides a quick and robust registration in coarse scale as well as a good initial guess for registration in finer scale, to a larger $n$, our method provides an efficient and robust multiscale nonrigid point cloud registration.

This paper addresses the nonstationary out-of-focus (OOF) blur removal in the application of barcode reconstruction. We propose a partially blind deblurring method when partial knowledge of the clean barcode is available. In particular, we consider an image formation model based on geometrical optics, which involves the point-spread function (PSF) for the OOF blur. With the known information, we can estimate a low-dimensional representation of the PSF using the Levenberg--Marquardt algorithm. Once the PSF is obtained, the deblurred image is computed by solving a quadratic program. We find that imposing a [0,1] box constraint is often good enough to enforce binary signal. Experiments on real data demonstrate that the forward model is physically realistic and our partially blind deblurring method can yield good reconstructions.

The manipulation of surface homeomorphisms is an important aspect in three-dimensional modeling and surface processing. Every homeomorphic surface map can be considered as a quasi-conformal map, with its local nonconformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasi-conformal maps are of great interest in mathematical study and real applications. Efficient and accurate computational construction of desirable quasi-conformal maps between general surfaces is crucial. However, in the literature we have reviewed, all existing computational works on construction of quasi-conformal maps to or from a compact domain require global parametrization onto the plane and are difficult to directly apply to maps between arbitrary surfaces. This work fills the gap by proposing to compute quasi-conformal homeomorphisms between arbitrary Riemann surfaces using discrete Beltrami flow, which is a vector field corresponding to the adjustment to the intrinsic Beltrami differential of the map. The vector field is defined by a partial differential equation in a local conformal coordinate. Based on this formulation and a composition formula, we can compute the Beltrami flow of any homeomorphism adjustment as a vector field on the target domain defined from the source domain, with appropriate boundary conditions and correspondences. Numerical tests show that our method provides a robust and efficient way of adjusting surface homeomorphisms. It is also insensitive to surface representation and has no limitation to the classes of surfaces that can be processed. Extensive numerical examples will be shown.

Respiration-correlated CBCT, commonly called 4DCBCT, provides respiratory phase-resolved CBCT images. A typical 4DCBCT represents averaged patient images over one breathing cycle and the fourth dimension is actually breathing phase instead of time. In many clinical applications, it is desirable to obtain true 4DCBCT with the fourth dimension being time, i.e., each constituent CBCT image corresponds to an instantaneous projection. Theoretically it is impossible to reconstruct a CBCT image from a single projection. However, if all the constituent CBCT images of a 4DCBCT scan share a lot of redundant information, it might be possible to make a good reconstruction of these images by exploring their sparsity and coherence/redundancy. Though these CBCT images are not completely time resolved, they can exploit both local and global temporal coherence of the patient anatomy automatically and contain much more temporal variation information of the patient geometry than the conventional 4DCBCT. We propose in this work a computational model and algorithms for the reconstruction of this type of semi-time-resolved CBCT, called cine-CBCT, based on low rank approximation that can utilize the underlying temporal coherence both locally and globally, such as slow variation, periodicity or repetition, in those cine-CBCT images.

The objective of quantitative photoacoustic tomography (qPAT) is to reconstruct the diffusion and absorption properties of a medium from data of absorbed energy distribution inside the medium. Mathematically, qPAT can be formulated as an inverse coefficient problem for the diffusion equation. Past research showed that if the boundary values of the coefficients are known, then the interior values of the coefficients can be uniquely and stably reconstructed with two well-chosen data sets. We propose a hybrid numerical reconstruction procedure for qPAT that uses both interior energy data and boundary current data. We show that these data allow the unique reconstruction of the boundary and interior values of the coefficients. The numerical implementation is based on reformulating the inverse coefficient problem as a nonlinear optimization problem. An explicit reconstruction scheme is utilized to eliminate the unknown coefficients inside the medium so that we need only minimize over the boundary values, which have significantly fewer degrees of freedom. Numerical simulations with synthetic data are presented to validate the method.

Fluorescence photoacoustic tomography (fPAT) is a multimodality biomedical imaging technique that combines high-resolution ultrasound imaging with high-contrast fluorescence optical tomography. In the first step of fPAT, one utilizes the photoacoustic effect to recover the total absorbed energy map inside the media with ultrasound tomography. In the second step, called quantitative fPAT (QfPAT), one uses interior absorbed energy data to recover either the quantum efficiency or the concentration distribution or both of the fluorophores inside the media. The objective of this work is to derive the mathematical model for QfPAT and to study the corresponding inverse problems. We derive some uniqueness and stability results on these inverse problems and propose a few (often explicit) reconstruction algorithms. Numerical simulations based on synthetic data are presented to verify the theory and algorithms proposed.

In this paper, we propose a general framework for approximating differential operator directly on point clouds and use it for geometric understanding on them. The discrete approximation of differential operator on the underlying manifold represented by point clouds is based only on local approximation using nearest neighbors, which is simple, efficient and accurate. This allows us to extract the complete local geometry, solve partial differential equations and perform intrinsic calculations on surfaces. Since no mesh or parametrization is needed, our method can work with point clouds in any dimensions or co-dimensions or even with variable dimensions. The computation complexity scaled well with the number of points and the intrinsic dimensions (rather than the embedded dimensions). We use this method to define the Laplace-Beltrami (LB) operator on point clouds, which links local and global information together. With this operator, we propose a few key applications essential to geometric understanding for point clouds, including the computation of LB eigenvalues and eigenfunctions, the extraction of skeletons from point clouds, and the extraction of conformal structures from point clouds.

We present an efficient algorithm for reconstructing an unknown source in thermoacoustic and photoacoustic tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that also might be discontinuous across some surface. The latter problem arises in brain imaging. The algorithmic development is based on an explicit formula in the form of a Neumann series. We present numerical examples with nontrapping, trapping, and piecewise smooth speeds, as well as examples with data on a part of the boundary. These numerical examples demonstrate the robust performance of the Neumann series–based algorithm.

3D point data acquired from laser scan or stereo vision can be quite noisy. A preprocessing step is often needed before a surface reconstruction algorithm can be applied. In this paper, we propose a nonparametric approach for noisy point data preprocessing. In particular, we proposed an anisotropic kernel based nonparametric density estimation method for outlier removal, and a hill-climbing line search approach for projecting data points onto the real surface boundary. Our approach is simple, robust and efficient. We demonstrate our method on both real and synthetic point datasets.

We develop an expectation-maximization algorithm with local adaptivity for image segmentation and classification. The key idea of our approach is to combine global statistics extracted from the Gaussian mixture model or other proper statistical models with local statistics and geometrical information, such as local probability distribution, orientation, and anisotropy. The combined information is used to design an adaptive local classification strategy that improves the robustness of the algorithm and also keeps fine features in the image. The proposed methodology is flexible and can be easily generalized to deal with other inferred information/quantities and statistical methods/models.