The optimal transport problem for measures supported on non-Euclidean spaces has recently gained ample interest in diverse applications involving representation learning. In this paper, we focus on circular probability measures, i.e., probability measures supported on the unit circle, and introduce a new computationally efficient metric for these measures, denoted as Linear Circular Optimal Transport (LCOT). The proposed metric comes with an explicit linear embedding that allows one to apply Machine Learning (ML) algorithms to the embedded measures and seamlessly modify the underlying metric for the ML algorithm to LCOT. We show that the proposed metric is rooted in the Circular Optimal Transport (COT) and can be considered the linearization of the COT metric with respect to a fixed reference measure. We provide a theoretical analysis of the proposed metric and derive the computational complexities for pairwise comparison of circular probability measures. Lastly, through a set of numerical experiments, we demonstrate the benefits of LCOT in learning representations from circular measures.

This paper presents a new end-to-end signal classification method using the signed cumulative distribution transform (SCDT). We adopt a transport generative model to define the classification problem. We then make use of mathematical properties of the SCDT to render the problem easier in transform domain, and solve for the class of an unknown sample using a nearest local subspace (NLS) search algorithm in SCDT domain. Experiments show that the proposed method provides high accuracy classification results while being computationally cheap, data efficient, and robust to out-of-distribution samples with respect to the existing end-to-end classification methods. The implementation of the proposed method in Python language is integrated as a part of the software package PyTransKit [1].

The ability to perform quantitative and automated neurological assessment could enhance diagnosis and treatment in the pre-hospital setting, such as during telemedicine or emergency medical services (EMS) encounters. Such a tool could be developed by adapting clinically significant information such as symmetry of eye movement or conjugate eye movement. Here we describe a digital camera-based eye tracking method “NeuroGaze” to capture the symmetry of eye movement while performing neurological eye examination. The proposed method was developed based on detecting the center of the pupil for both eyes from a given video and measuring eye conjugacy by transforming the pupil center coordinates to relative gaze. The method was tested on healthy volunteers while performing three neurological eye examinations1. We also compared our proposed approach to state-of-the-art digital camera-based eye-tracking methods and commercial off-the-shelf (COTS) eye trackers. NeuroGaze outperformed digital camera-based eye tracking methods by reporting a mean Spearman rank-order correlation coefficient of 0.86 for the H-test, 0.87 for the Dot-test, and 0.56 for the OKN-Test, and shows similarity in trends for the relative gaze trajectories with a noticeable offset in the scale of the relative gaze angle compared to COTS eye tracker (see Fig.1). The study demonstrates that by using a pupil-center-based eye-tracking method, a digital camera can measure clinically relevant information regarding eye movement.

This paper presents a new method to classify 1D signals using the signed cumulative distribution transform (SCDT). The proposed method exploits certain linearization properties of the SCDT to render the problem easier to solve in the SCDT space. The method uses the nearest subspace search technique in the SCDT domain to provide a non-iterative, effective, and simple to implement classification algorithm. Experiments show that the proposed technique outperforms the state-of-the-art neural networks using a very low number of training samples and is also robust to out-of-distribution examples on simulated data. We also demonstrate the efficacy of the proposed technique in real-world applications by applying it to an ECG classification problem.

Abstract Building on past work showing that the Hilbert transform can be used for edge feature enhancement, a new edge feature enhancement method is developed using a two‐dimensional (2D) isotropic Hilbert transform of the Cauchy distribution. First, both the shape of the Hilbert kernel and the Hilbert transform of edge feature models (step and delta) and various simulated signals result in edge feature enhancement, the properties of which are derived and confirmed. Second, Cauchy distribution as low‐pass filter is introduced in Hilbert transform to get new edge feature enhancement operator, and its efficiency under various criteria is comprehensively discussed. Third, a 2D isotropic extension is presented using a circularly symmetric window function. Finally, two experiments, including edge detection and image segmentation, are performed to validate the proposed edge feature enhancement method. The experimental results of the method applied to the Berkeley Segmentation Dataset and remote sensing images demonstrate that the new method is effective for edge detection and image segmentation.

Facial weakness is a symptom commonly associated to lack of facial muscle control due to neurological injury. Several diseases are associated with facial weakness such as stroke and Bell's palsy. The use of digital imaging through mobile phones, tablets, personal computers and other devices could provide timely opportunity for detection, which if accurate enough can improve treatment by enabling faster patient triage and recovery progress monitoring. Most of the existing facial weakness detection approaches from static images are based on facial landmarks from which geometric features can be calculated. Landmark-based methods, however, can suffer from inaccuracies in face landmarks localization. In this study, We also experimentally evaluate the performance of several feature extraction methods for measuring facial weakness, including the landmark-based features, as well as intensity-based features on a neurologist-certified dataset that comprises 186 images of normal, 125 images of left facial weakness, and 126 images of right facial weakness. We demonstrate that, for the application of facial weakness detection from single (static) images, approaches that incorporate the Histogram of Oriented Gradients (HoG) features tend to be more accurate.

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We further show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed proper metrics. Finally, we compare the numerical performance of the proposed distances on the generative modeling task of SW flows and report favorable results.

Motivation: Colocalization of structures in biomedical images can lead to insights into biological behaviors. One class of colocalization problems is examining an annular structure (disk-shaped such as a cell, vesicle or molecule) interacting with a network structure (vascular, neuronal, cytoskeletal, organellar). Examining colocalization events across conditions is often complicated by changes in density of both structure types, confounding traditional statistical approaches since colocalization cannot be normalized to the density of both structure types simultaneously. We have developed a technique to measure colocalization independent of structure density and applied it to characterizing intercellular colocation with blood vessel networks. This technique could be used to analyze colocalization of any annular structure with an arbitrarily shaped network structure. Results: We present the circular colocalization affinity with network structures test (CIRCOAST), a novel statistical hypothesis test to probe for enriched network colocalization in 2D z-projected multichannel images by using agent-based Monte Carlo modeling and image processing to generate the pseudo-null distribution of random cell placement unique to each image. This hypothesis test was validated by confirming that adipose-derived stem cells (ASCs) exhibit enriched colocalization with endothelial cells forming arborized networks in culture and then applied to show that locally delivered ASCs have enriched colocalization with murine retinal microvasculature in a model of diabetic retinopathy. We demonstrate that the CIRCOAST test provides superior power and type I error rates in characterizing intercellular colocalization compared to generic approaches that are confounded by changes in cell or vessel density. Availability and implementation: CIRCOAST source code available at: https://github.com/uva-peirce-cottler-lab/ARCAS. Supplementary information: Supplementary data are available at Bioinformatics online.

Bioinformatics (2019) https://doi.org/10.1093/bioinformatics/bty638, 35(3): 506–514. The publisher wishes to inform the reader that Figures 2, 5 and 6 appeared incorrectly in the above manuscript. Network area fraction dilated by cell radius determines the random cell colocation fraction. The mean ICF was calculated with the MCMRP over 10 000 trials with randomly selected parameters and displayed as a function of (A) cell diameter, (B) cell number, (C) network radius, (D) network fraction, (E) network length density and (F) cell-dilated network fraction (CDNF, N=2 500 images). Pearson correlation coefficient and associated 95% confidence interval and P-values are provided at the top of each scatterplot ASCs exhibit enriched colocalization with HUVECS network, while fluorescent microspheres (μSpheres) do not. (A) ASCs (red) co-cultured with HUVECS (green). (B) Distribution of simulated mean CIRCOAST P-values (blue) of random colocalization of ASC group compared to observed mean CIRCOAST P-value (red). (C) Fluorescent μSpheres seeded on a culture of HUVECs (scale bar 250 um). (D) Distribution of simulated mean CIRCOAST P-values (blue) of random colocalization from fluorescent μSpheres compared to actual mean P-value (red). (E) Circular cell density and (F) endothelial network density between groups. (G) Distribution of P-values (blue) derived from permuting CIRCOAST P-values in a Wilcox sum rank test between ASCs and μSpheres, with observed P-value (red) (N=6 wells, 3 images/well) Injected live and dead ASCs both exhibit enriched intercellular colocalization affinity with the vasculature. (A) Confocal image of retinal vasculature (green, preprocessed and thresholded) and injected with live DiI-labeled circular ASCs (red). (B) Distribution of simulated mean CIRCOAST P-values (blue) of random colocalization of ASC group compared to observed mean binomial P-value (red). (C) Dead DiI-labeled circular ASCs in the retinal vasculature (scale bars 150 um). (D) Distribution of simulated mean CIRCOAST P-values (blue) of random colocalization from dead cell group, compared to actual mean P-value (red). (E) Injected circular cell and (F) endothelial network density between study groups. (G) Distribution of permuted P-vales of Wilcox sum rank test of CIRCOAST P-values between study groups, with observed P-value (red) (N=6 mice, 3 images/mouse) The paper has been corrected online.

Time delay estimation between signals propagating through nonlinear media is an important problem with application to radar, underwater acoustics, damage detection, and communications (to name a few). Here, we describe a simple approach for determining the time delay between two such signals via minimization of the one-dimensional Wasserstein distance. The solution can be computed efficiently digitally in O(N), or in linear time in circuitry. We demonstrate the approach in the estimation of time delay between acoustic (Lamb) waves generated in an aluminum plate, and compare to alternative approaches.

Due to the importance of nuclear structure in cancer diagnosis, several predictive models have been described for diagnosing a wide variety of cancers based on nuclear morphology. In many computer-aided diagnosis (CAD) systems, cancer detection tasks can be generally formulated as set classification problems, which can not be directly solved by classifying single instances. In this paper, we propose a novel set classification approach SetSVM to build a predictive model by considering any nuclei set as a whole without specific assumptions. SetSVM features highly discriminative power in cancer detection challenges in the sense that it not only optimizes the classifier decision boundary but also transfers discriminative information to set representation learning. During model training, these two processes are unified in the support vector machine (SVM) maximum separation margin problem. Experiment results show that SetSVM provides significant improvements compared with five commonly used approaches in cancer detection tasks utilizing 260 patients in total across three different cancer types, namely, thyroid cancer, liver cancer, and melanoma. In addition, we show that SetSVM enables visual interpretation of discriminative nuclear characteristics representing the nuclei set. These features make SetSVM a potentially practical tool in building accurate and interpretable CAD systems for cancer detection.

The automatic classification of breast cancer histopathological images has great significance in computer-aided diagnosis. Recently, deep learning via neural networks has enabled pattern detection and prediction using large, labeled datasets; whereas, collecting and annotating sufficient histological data using professional pathologists is time consuming, tedious, and extremely expensive. In the proposed paper, a deep active learning framework is designed and implemented for classification of breast cancer histopathological images, with the goal of maximizing the learning accuracy from very limited labeling. This method involves manual annotation of the most valuable unlabeled samples, which are then integrated into the training set. The model is then iteratively updated with an increasing training set. Here, two selection strategies are discussed for the proposed deep active learning framework: An entropy-based strategy and a confidence-boosting strategy. The proposed method has been validated using a publicly available breast cancer histopathological image dataset, wherein each image patch is binarily classified as benign or malignant. The experimental results demonstrate that, compared with a random selection, our proposed framework can reduce annotation costs up to 66.67%, with higher accuracy and less expensive annotation than standard query strategy.

Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm.

Epithelium-stroma classification is always considered as an important preprocessing step for morphological quantitative analysis in image-based histological researches of oncologic diseases. However, large-scale accurate ground-truth labeling is expensive in histopathological image analysis, thus the classification performances will still be limited with the insufficient labeled training samples. Considering that acquisition of public unlabeled histopathological images is much cheaper, an epithelium-stroma classification framework is developed, based on the deep convolutional neural network framework and the strategies of self-taught learning. The method has the ability of taking advantage of large-scale unlabeled public histopathological data as auxiliary data, and then transferring the knowledge to enhance the performances in epithelium-stroma classification with limited labeled training data. The experiments demonstrate that the proposed method outperforms traditional CNNs when the labeled training data size is decreasing dramatically.

Epithelium-stroma classification is a necessary preprocessing step in histopathological image analysis. Current deep learning based recognition methods for histology data require collection of large volumes of labeled data in order to train a new neural network when there are changes to the image acquisition procedure. However, it is extremely expensive for pathologists to manually label sufficient volumes of data for each pathology study in a professional manner, which results in limitations in real-world applications. A very simple but effective deep learning method, that introduces the concept of unsupervised domain adaptation to a simple convolutional neural network (CNN), has been proposed in this paper. Inspired by transfer learning, our paper assumes that the training data and testing data follow different distributions, and there is an adaptation operation to more accurately estimate the kernels in CNN in feature extraction, in order to enhance performance by transferring knowledge from labeled data in source domain to unlabeled data in target domain. The model has been evaluated using three independent public epithelium-stroma datasets by cross-dataset validations. The experimental results demonstrate that for epithelium-stroma classification, the proposed framework outperforms the state-of-the-art deep neural network model, and it also achieves better performance than other existing deep domain adaptation methods. The proposed model can be considered to be a better option for real-world applications in histopathological image analysis, since there is no longer a requirement for large-scale labeled data in each specified domain.

Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as powerful discrepancy measures for probability distributions. The recent developments on alternative formulations of the optimal transport have allowed for faster solutions to the problem and have revamped their practical applications in machine learning. In this paper, we exploit the widely used kernel methods and provide a family of provably positive definite kernels based on the Sliced Wasserstein distance and demonstrate the benefits of these kernels in a variety of learning tasks. Our work provides a new perspective on the application of optimal transport flavored distances through kernel methods in machine learning tasks.

Invertible image representation methods (transforms) are routinely employed as low-level image processing operations based on which feature extraction and recognition algorithms are developed. Most transforms in current use (e.g., Fourier, wavelet, and so on) are linear transforms and, by themselves, are unable to substantially simplify the representation of image classes for classification. Here, we describe a nonlinear, invertible, low-level image processing transform based on combining the well-known Radon transform for image data, and the 1D cumulative distribution transform proposed earlier. We describe a few of the properties of this new transform, and with both theoretical and experimental results show that it can often render certain problems linearly separable in a transform space.

Extracting high-resolution information from highly degraded facial images is an important problem with several applications in science and technology. Here we describe a single frame super resolution technique that uses a transport-based formulation of the problem. The method consists of a training and a testing phase. In the training phase, a nonlinear Lagrangian model of high-resolution facial appearance is constructed fully automatically. In the testing phase, the resolution of a degraded image is enhanced by finding the model parameters that best fit the given low resolution data. We test the approach on two face datasets, namely the extended Yale Face Database B and the AR face datasets, and compare it to state of the art methods. The proposed method outperforms existing solutions in problems related to enhancing images of very low resolution.

Characterizing the spatial distribution of proteins directly from microscopy images is a difficult problem with numerous applications in cell biology (e.g. identifying motor-related proteins) and clinical research (e.g. identification of cancer biomarkers). Here we describe the design of a system that provides automated analysis of punctate protein patterns in microscope images, including quantification of their relationships to microtubules. We constructed the system using confocal immunofluorescence microscopy images from the Human Protein Atlas project for 11 punctate proteins in three cultured cell lines. These proteins have previously been characterized as being primarily located in punctate structures, but their images had all been annotated by visual examination as being simply "vesicular". We were able to show that these patterns could be distinguished from each other with high accuracy, and we were able to assign to one of these subclasses hundreds of proteins whose subcellular localization had not previously been well defined. In addition to providing these novel annotations, we built a generative approach to modeling of punctate distributions that captures the essential characteristics of the distinct patterns. Such models are expected to be valuable for representing and summarizing each pattern and for constructing systems biology simulations of cell behaviors.

A crucial step in understanding the architecture of cells and tissues from microscopy images, and consequently explain important biological events such as wound healing and cancer metastases, is the complete extraction and enumeration of individual filaments from the cellular cytoskeletal network. Current efforts at quantitative estimation of filament length distribution, architecture and orientation from microscopy images are predominantly limited to visual estimation and indirect experimental inference. Here we demonstrate the application of a new algorithm to reliably estimate centerlines of biological filament bundles and extract individual filaments from the centerlines by systematically disambiguating filament intersections. We utilize a filament enhancement step followed by reverse diffusion based filament localization and an integer programming based set combination to systematically extract accurate filaments automatically from microscopy images. Experiments on simulated and real confocal microscope images of flat cells (2D images) show efficacy of the new method.

The authors apply the maximum empirical likelihood method to the problem of estimating the time delay of a measured digital signal when the signal can be seen as an instance of a stationary random process with additive independently and identically distributed (i.i.d.) noise. It is shown that, under these assumptions, an approximate log‐likelihood function can be estimated from the measured data itself, and therefore a maximum likelihood estimate can be obtained without the prior knowledge of the formula for the signal likelihood. The Cramer–Rao lower bounds (CRLB) for two additive noise models (mixed‐Gaussian and generalised normal distribution) are derived. The authors also show that the error produced by the maximum log‐likelihood estimates (when the likelihood function is estimated from the measured data) better approximates the CRLB than other estimators for noise models other than Gaussian or Laplacian (special case of the generalised normal).

Filter bank-based methods for pixel classification are attractive due to the potential of fast implementation with convolution operations. The design of optimal filter sets, however, is a challenging task given the nonlinear aspects of the problem. This letter extends the well known linear discriminant analysis method into a novel framework for local texture feature discrimination tasks. It proposes a mixture of linear models as a nonlinear classifier, where a number of filters are optimized locally by minimizing the prediction error. Through these filters, the `best separable' features are selected. Experiments performed on two standard texture databases show that our method produces results which are comparable to state-of-the-art techniques while at the same time maintaining low computational complexity.

Nuclear morphology and structure as visualized from histopathology microscopy images can yield important diagnostic clues in some benign and malignant tissue lesions. Precise quantitative information about nuclear structure and morphology, however, is currently not available for many diagnostic challenges. This is due, in part, to the lack of methods to quantify these differences from image data. We describe a method to characterize and contrast the distribution of nuclear structure in different tissue classes (normal, benign, cancer, etc.). The approach is based on quantifying chromatin morphology in different groups of cells using the optimal transportation (Kantorovich-Wasserstein) metric in combination with the Fisher discriminant analysis and multidimensional scaling techniques. We show that the optimal transportation metric is able to measure relevant biological information as it enables automatic determination of the class (e.g., normal versus cancer) of a set of nuclei. We show that the classification accuracies obtained using this metric are, on average, as good or better than those obtained utilizing a set of previously described numerical features. We apply our methods to two diagnostic challenges for surgical pathology: one in the liver and one in the thyroid. Results automatically computed using this technique show potentially biologically relevant differences in nuclear structure in liver and thyroid cancers.

We consider the problem of registering (aligning) two images to sub-pixel accuracy by optimization of objective functions constructed from the images' intensity values. We show that some widely used interpolation methods can introduce multiple local optima in the energy of the interpolated image which, if not counter-balanced by other terms, can cause local optima in registration objective functions including the sum of squared differences, cross correlation, and mutual information. We discuss different solutions to address the problem based on high degree B-spline interpolation, low pass filtering the images, and stochastic integration. Numerical examples using synthetic and real signals and images are shown.

We look at the problem of extracting geometric functions such as spatial transformations and curves delineating edges to subpixel accuracy from noisy, sampled data. We analyze the stochastic properties of continuous B-spline interpolation to show that, in general noisy circumstances, sub-pixel accuracy is not obtainable when using low degree B-splines. Results using magnetic resonance image (MRI) data are shown

Nonrigid registration of medical images is important for a number of applications such as the creation of population averages, atlas-based segmentation, or geometric correction of functional magnetic resonance imaging (fMRI) images to name a few. In recent years, a number of methods have been proposed to solve this problem, one class of which involves maximizing a mutual information (MI)-based objective function over a regular grid of splines. This approach has produced good results but its computational complexity is proportional to the compliance of the transformation required to register the smallest structures in the image. Here, we propose a method that permits the spatial adaptation of the transformation's compliance. This spatial adaptation allows us to reduce the number of degrees of freedom in the overall transformation, thus speeding up the process and improving its convergence properties. To develop this method, we introduce several novelties: 1) we rely on radially symmetric basis functions rather than B-splines traditionally used to model the deformation field; 2) we propose a metric to identify regions that are poorly registered and over which the transformation needs to be improved; 3) we partition the global registration problem into several smaller ones; and 4) we introduce a new constraint scheme that allows us to produce transformations that are topologically correct. We compare the approach we propose to more traditional ones and show that our new algorithm compares favorably to those in current use.