Ross Whitaker

Surojit Saha, Sarang Joshi, Ross Whitaker

The variational autoencoder (VAE) is a well-studied, deep, latent-variable model (DLVM) that efficiently optimizes the variational lower bound of the log marginal data likelihood and has a strong theoretical foundation. However, the VAE's known failure to match the aggregate posterior often results in \emph{pockets/holes} in the latent distribution (i.e., a failure to match the prior) and/or \emph{posterior collapse}, which is associated with a loss of information in the latent space. This paper addresses these shortcomings in VAEs by reformulating the objective function associated with VAEs in order to match the aggregate/marginal posterior distribution to the prior. We use kernel density estimate (KDE) to model the aggregate posterior in high dimensions. The proposed method is named the \emph{aggregate variational autoencoder} (AVAE) and is built on the theoretical framework of the VAE. Empirical evaluation of the proposed method on multiple benchmark data sets demonstrates the effectiveness of the AVAE relative to state-of-the-art (SOTA) methods.

Riddhish Bhalodia, Shireen Elhabian, Ladislav Kavan et al.

Deep networks are an integral part of the current machine learning paradigm. Their inherent ability to learn complex functional mappings between data and various target variables, while discovering hidden, task-driven features, makes them a powerful technology in a wide variety of applications. Nonetheless, the success of these networks typically relies on the availability of sufficient training data to optimize a large number of free parameters while avoiding overfitting, especially for networks with large capacity. In scenarios with limited training budgets, e.g., supervised tasks with limited labeled samples, several generic and/or task-specific regularization techniques, including data augmentation, have been applied to improve the generalization of deep networks.Typically such regularizations are introduced independently of that data or training scenario, and must therefore be tuned, tested, and modified to meet the needs of a particular network. In this paper, we propose a novel regularization framework that is driven by the population-level statistics of the feature space to be learned. The regularization is in the form of a \textbf{cooperating subnetwork}, which is an auto-encoder architecture attached to the feature space and trained in conjunction with the primary network. We introduce the architecture and training methodology and demonstrate the effectiveness of the proposed cooperative network-based regularization in a variety of tasks and architectures from the literature. Our code is freely available at \url{https://github.com/riddhishb/CoopSubNet

Anupama Goparaju, Ibolya Csecs, Alan Morris et al.

Statistical shape modeling (SSM) has proven useful in many areas of biology and medicine as a new generation of morphometric approaches for the quantitative analysis of anatomical shapes. Recently, the increased availability of high-resolution in vivo images of anatomy has led to the development and distribution of open-source computational tools to model anatomical shapes and their variability within populations with unprecedented detail and statistical power. Nonetheless, there is little work on the evaluation and validation of such tools as related to clinical applications that rely on morphometric quantifications for treatment planning. To address this lack of validation, we systematically assess the outcome of widely used off-the-shelf SSM tools, namely ShapeWorks, SPHARM-PDM, and Deformetrica, in the context of designing closure devices for left atrium appendage (LAA) in atrial fibrillation (AF) patients to prevent stroke, where an incomplete LAA closure may be worse than no closure. This study is motivated by the potential role of SSM in the geometric design of closure devices, which could be informed by population-level statistics, and patient-specific device selection, which is driven by anatomical measurements that could be automated by relating patient-level anatomy to population-level morphometrics. Hence, understanding the consequences of different SSM tools for the final analysis is critical for the careful choice of the tool to be deployed in real clinical scenarios. Results demonstrate that estimated measurements from ShapeWorks model are more consistent compared to models from Deformetrica and SPHARM-PDM. Furthermore, ShapeWorks and Deformetrica shape models capture clinically relevant population-level variability compared to SPHARM-PDM models.

Xiwen Li, Rehman Mohammed, Tristalee Mangin et al.

Idling vehicle detection (IVD) can be helpful in monitoring and reducing unnecessary idling and can be integrated into real-time systems to address the resulting pollution and harmful products. The previous approach [13], a non-end-to-end model, requires extra user clicks to specify a part of the input, making system deployment more error-prone or even not feasible. In contrast, we introduce an end-to-end joint audio-visual IVD task designed to detect vehicles visually under three states: moving, idling and engine off. Unlike feature co-occurrence task such as audio-visual vehicle tracking, our IVD task addresses complementary features, where labels cannot be determined by a single modality alone. To this end, we propose AVIVD-Net, a novel network that integrates audio and visual features through a bidirectional attention mechanism. AVIVD-Net streamlines the input process by learning a joint feature space, reducing the deployment complexity of previous methods. Additionally, we introduce the AVIVD dataset, which is seven times larger than previous datasets, offering significantly more annotated samples to study the IVD problem. Our model achieves performance comparable to prior approaches, making it suitable for automated deployment. Furthermore, by evaluating AVIVDNet on the feature co-occurrence public dataset MAVD [23], we demonstrate its potential for extension to self-driving vehicle video-camera setups.

Surojit Saha, Sarang Joshi, Ross Whitaker

The variational autoencoder (VAE) is a popular, deep, latent-variable model (DLVM) due to its simple yet effective formulation for modeling the data distribution. Moreover, optimizing the VAE objective function is more manageable than other DLVMs. The bottleneck dimension of the VAE is a crucial design choice, and it has strong ramifications for the model's performance, such as finding the hidden explanatory factors of a dataset using the representations learned by the VAE. However, the size of the latent dimension of the VAE is often treated as a hyperparameter estimated empirically through trial and error. To this end, we propose a statistical formulation to discover the relevant latent factors required for modeling a dataset. In this work, we use a hierarchical prior in the latent space that estimates the variance of the latent axes using the encoded data, which identifies the relevant latent dimensions. For this, we replace the fixed prior in the VAE objective function with a hierarchical prior, keeping the remainder of the formulation unchanged. We call the proposed method the automatic relevancy detection in the variational autoencoder (ARD-VAE). We demonstrate the efficacy of the ARD-VAE on multiple benchmark datasets in finding the relevant latent dimensions and their effect on different evaluation metrics, such as FID score and disentanglement analysis.

Surojit Saha, Sarang Joshi, Ross Whitaker

Deep latent variable models (DLVMs) are designed to learn meaningful representations in an unsupervised manner, such that the hidden explanatory factors are interpretable by independent latent variables (aka disentanglement). The variational autoencoder (VAE) is a popular DLVM widely studied in disentanglement analysis due to the modeling of the posterior distribution using a factorized Gaussian distribution that encourages the alignment of the latent factors with the latent axes. Several metrics have been proposed recently, assuming that the latent variables explaining the variation in data are aligned with the latent axes (cardinal directions). However, there are other DLVMs, such as the AAE and WAE-MMD (matching the aggregate posterior to the prior), where the latent variables might not be aligned with the latent axes. In this work, we propose a statistical method to evaluate disentanglement for any DLVMs in general. The proposed technique discovers the latent vectors representing the generative factors of a dataset that can be different from the cardinal latent axes. We empirically demonstrate the advantage of the method on two datasets.

Siddhant Ranade, Gonçalo Dias Pais, Ross Tyler Whitaker et al.

We propose a fast and accurate surface reconstruction algorithm for unorganized point clouds using an implicit representation. Recent learning methods are either single-object representations with small neural models that allow for high surface details but require per-object training or generalized representations that require larger models and generalize to newer shapes but lack details, and inference is slow. We propose a new implicit representation for general 3D shapes that is faster than all the baselines at their optimum resolution, with only a marginal loss in performance compared to the state-of-the-art. We achieve the best accuracy-speed trade-off using three key contributions. Many implicit methods extract features from the point cloud to classify whether a query point is inside or outside the object. First, to speed up the reconstruction, we show that this feature extraction does not need to use the query point at an early stage (lazy query). Second, we use a parallel multi-scale grid representation to develop robust features for different noise levels and input resolutions. Finally, we show that attention across scales can provide improved reconstruction results.

Surojit Saha, Ross Whitaker

Automated interpretation of seismic images using deep learning methods is challenging because of the limited availability of training data. Few-shot learning is a suitable learning paradigm in such scenarios due to its ability to adapt to a new task with limited supervision (small training budget). Existing few-shot semantic segmentation (FSSS) methods fix the number of target classes. Therefore, they do not support joint training on multiple datasets varying in the number of classes. In the context of the interpretation of seismic facies, fixing the number of target classes inhibits the generalization capability of a model trained on one facies dataset to another, which is likely to have a different number of facies. To address this shortcoming, we propose a few-shot semantic segmentation method for interpreting seismic facies that can adapt to the varying number of facies across the dataset, dubbed the AdaSemSeg. In general, the backbone network of FSSS methods is initialized with the statistics learned from the ImageNet dataset for better performance. The lack of such a huge annotated dataset for seismic images motivates using a self-supervised algorithm on seismic datasets to initialize the backbone network. We have trained the AdaSemSeg on three public seismic facies datasets with different numbers of facies and evaluated the proposed method on multiple metrics. The performance of the AdaSemSeg on unseen datasets (not used in training) is better than the prototype-based few-shot method and baselines.

Xiwen Li, Tristalee Mangin, Surojit Saha et al.

Combustion vehicle emissions contribute to poor air quality and release greenhouse gases into the atmosphere, and vehicle pollution has been associated with numerous adverse health effects. Roadways with extensive waiting and/or passenger drop off, such as schools and hospital drop-off zones, can result in high incidence and density of idling vehicles. This can produce micro-climates of increased vehicle pollution. Thus, the detection of idling vehicles can be helpful in monitoring and responding to unnecessary idling and be integrated into real-time or off-line systems to address the resulting pollution. In this paper we present a real-time, dynamic vehicle idling detection algorithm. The proposed idle detection algorithm and notification rely on an algorithm to detect these idling vehicles. The proposed method relies on a multi-sensor, audio-visual, machine-learning workflow to detect idling vehicles visually under three conditions: moving, static with the engine on, and static with the engine off. The visual vehicle motion detector is built in the first stage, and then a contrastive-learning-based latent space is trained for classifying static vehicle engine sound. We test our system in real-time at a hospital drop-off point in Salt Lake City. This in-situ dataset was collected and annotated, and it includes vehicles of varying models and types. The experiments show that the method can detect engine switching on or off instantly and achieves 71.02 average precision (AP) for idle detections and 91.06 for engine off detections.

Riddhish Bhalodia, Shireen Elhabian, Ladislav Kavan et al.

In current biological and medical research, statistical shape modeling (SSM) provides an essential framework for the characterization of anatomy/morphology. Such analysis is often driven by the identification of a relatively small number of geometrically consistent features found across the samples of a population. These features can subsequently provide information about the population shape variation. Dense correspondence models can provide ease of computation and yield an interpretable low-dimensional shape descriptor when followed by dimensionality reduction. However, automatic methods for obtaining such correspondences usually require image segmentation followed by significant preprocessing, which is taxing in terms of both computation as well as human resources. In many cases, the segmentation and subsequent processing require manual guidance and anatomy specific domain expertise. This paper proposes a self-supervised deep learning approach for discovering landmarks from images that can directly be used as a shape descriptor for subsequent analysis. We use landmark-driven image registration as the primary task to force the neural network to discover landmarks that register the images well. We also propose a regularization term that allows for robust optimization of the neural network and ensures that the landmarks uniformly span the image domain. The proposed method circumvents segmentation and preprocessing and directly produces a usable shape descriptor using just 2D or 3D images. In addition, we also propose two variants on the training loss function that allows for prior shape information to be integrated into the model. We apply this framework on several 2D and 3D datasets to obtain their shape descriptors, and analyze their utility for various applications.

Riddhish Bhalodia, Shireen Elhabian, Jadie Adams et al.

Statistical shape modeling (SSM) characterizes anatomical variations in a population of shapes generated from medical images. SSM requires consistent shape representation across samples in shape cohort. Establishing this representation entails a processing pipeline that includes anatomy segmentation, re-sampling, registration, and non-linear optimization. These shape representations are then used to extract low-dimensional shape descriptors that facilitate subsequent analyses in different applications. However, the current process of obtaining these shape descriptors from imaging data relies on human and computational resources, requiring domain expertise for segmenting anatomies of interest. Moreover, this same taxing pipeline needs to be repeated to infer shape descriptors for new image data using a pre-trained/existing shape model. Here, we propose DeepSSM, a deep learning-based framework for learning the functional mapping from images to low-dimensional shape descriptors and their associated shape representations, thereby inferring statistical representation of anatomy directly from 3D images. Once trained using an existing shape model, DeepSSM circumvents the heavy and manual pre-processing and segmentation and significantly improves the computational time, making it a viable solution for fully end-to-end SSM applications. In addition, we introduce a model-based data-augmentation strategy to address data scarcity. Finally, this paper presents and analyzes two different architectural variants of DeepSSM with different loss functions using three medical datasets and their downstream clinical application. Experiments showcase that DeepSSM performs comparably or better to the state-of-the-art SSM both quantitatively and on application-driven downstream tasks. Therefore, DeepSSM aims to provide a comprehensive blueprint for deep learning-based image-to-shape models.

Praful Agrawal, Ross T. Whitaker, Shireen Y. Elhabian

Correspondence-based shape models are key to various medical imaging applications that rely on a statistical analysis of anatomies. Such shape models are expected to represent consistent anatomical features across the population for population-specific shape statistics. Early approaches for correspondence placement rely on nearest neighbor search for simpler anatomies. Coordinate transformations for shape correspondence hold promise to address the increasing anatomical complexities. Nonetheless, due to the inherent shape-level geometric complexity and population-level shape variation, the coordinate-wise correspondence often does not translate to the anatomical correspondence. An alternative, group-wise approach for correspondence placement explicitly models the trade-off between geometric description and the population's statistical compactness. However, these models achieve limited success in resolving nonlinear shape correspondence. Recent works have addressed this limitation by adopting an application-specific notion of correspondence through lifting positional data to a higher dimensional feature space. However, they heavily rely on manual expertise to create domain-specific features and consistent landmarks. This paper proposes an automated feature learning approach, using deep convolutional neural networks to extract correspondence-friendly features from shape ensembles. Further, an unsupervised domain adaptation scheme is introduced to augment the pretrained geometric features with new anatomies. Results on anatomical datasets of human scapula, femur, and pelvis bones demonstrate that features learned in supervised fashion show improved performance for correspondence estimation compared to the manual features. Further, unsupervised learning is demonstrated to learn complex anatomy features using the supervised domain adaptation from features learned on simpler anatomy.

Surojit Saha, Shireen Elhabian, Ross T. Whitaker

Mapping data from and/or onto a known family of distributions has become an important topic in machine learning and data analysis. Deep generative models (e.g., generative adversarial networks ) have been used effectively to match known and unknown distributions. Nonetheless, when the form of the target distribution is known, analytical methods are advantageous in providing robust results with provable properties. In this paper, we propose and analyze the use of nonparametric density methods to estimate the Jensen-Shannon divergence for matching unknown data distributions to known target distributions, such Gaussian or mixtures of Gaussians, in latent spaces. This analytical method has several advantages: better behavior when training sample quantity is low, provable convergence properties, and relatively few parameters, which can be derived analytically. Using the proposed method, we enforce the latent representation of an autoencoder to match a target distribution in a learning framework that we call a {\em generative encoding network}. Here, we present the numerical methods; derive the expected distribution of the data in the latent space; evaluate the properties of the latent space, sample reconstruction, and generated samples; show the advantages over the adversarial counterpart; and demonstrate the application of the method in real world.

Wenzheng Tao, Riddhish Bhalodia, Erin Anstadt et al.

This work describes an unsupervised method to objectively quantify the abnormality of general anatomical shapes. The severity of an anatomical deformity often serves as a determinant in the clinical management of patients. However, experiential bias and distinctive random residuals among specialist individuals bring variability in diagnosis and patient management decisions, irrespective of the objective deformity degree. Therefore, supervised methods are prone to be misled given insufficient labeling of pathological samples that inevitably preserve human bias and inconsistency. Furthermore, subjects demonstrating a specific pathology are naturally rare relative to the normal population. To avoid relying on sufficient pathological samples by fully utilizing the power of normal samples, we propose the shape normality metric (SNM), which requires learning only from normal samples and zero knowledge about the pathology. We represent shapes by landmarks automatically inferred from the data and model the normal group by a multivariate Gaussian distribution. Extensive experiments on different anatomical datasets, including skulls, femurs, scapulae, and humeri, demonstrate that SNM can provide an effective normality measurement, which can significantly detect and indicate pathology. Therefore, SNM offers promising value in a variety of clinical applications.

Riddhish Bhalodia, Ladislav Kavan, Ross Whitaker

Statistical shape analysis is a very useful tool in a wide range of medical and biological applications. However, it typically relies on the ability to produce a relatively small number of features that can capture the relevant variability in a population. State-of-the-art methods for obtaining such anatomical features rely on either extensive preprocessing or segmentation and/or significant tuning and post-processing. These shortcomings limit the widespread use of shape statistics. We propose that effective shape representations should provide sufficient information to align/register images. Using this assumption we propose a self-supervised, neural network approach for automatically positioning and detecting landmarks in images that can be used for subsequent analysis. The network discovers the landmarks corresponding to anatomical shape features that promote good image registration in the context of a particular class of transformations. In addition, we also propose a regularization for the proposed network which allows for a uniform distribution of these discovered landmarks. In this paper, we present a complete framework, which only takes a set of input images and produces landmarks that are immediately usable for statistical shape analysis. We evaluate the performance on a phantom dataset as well as 2D and 3D images.

Riddhish Bhalodia, Shireen Y. Elhabian, Ladislav Kavan et al.

Spatial transformations are enablers in a variety of medical image analysis applications that entail aligning images to a common coordinate systems. Population analysis of such transformations is expected to capture the underlying image and shape variations, and hence these transformations are required to produce anatomically feasible correspondences. This is usually enforced through some smoothness-based generic regularization on deformation field. Alternatively, population-based regularization has been shown to produce anatomically accurate correspondences in cases where anatomically unaware (i.e., data independent) fail. Recently, deep networks have been for unsupervised image registration, these methods are computationally faster and maintains the accuracy of state of the art methods. However, these networks use smoothness penalty on deformation fields and ignores population-level statistics of the transformations. We propose a novel neural network architecture that simultaneously learns and uses the population-level statistics of the spatial transformations to regularize the neural networks for unsupervised image registration. This regularization is in the form of a bottleneck autoencoder, which encodes the population level information of the deformation fields in a low-dimensional manifold. The proposed architecture produces deformation fields that describe the population-level features and associated correspondences in an anatomically relevant manner and are statistically compact relative to the state-of-the-art approaches while maintaining computational efficiency. We demonstrate the efficacy of the proposed architecture on synthetic data sets, as well as 2D and 3D medical data.

Alessandro Ferrero, Shireen Elhabian, Ross Whitaker

Generative adversarial networks (GANs) have proven effective in modeling distributions of high-dimensional data. However, their training instability is a well-known hindrance to convergence, which results in practical challenges in their applications to novel data. Furthermore, even when convergence is reached, GANs can be affected by mode collapse, a phenomenon for which the generator learns to model only a small part of the target distribution, disregarding the vast majority of the data manifold or distribution. This paper addresses these challenges by introducing SetGAN, an adversarial architecture that processes sets of generated and real samples, and discriminates between the origins of these sets (i.e., training versus generated data) in a flexible, permutation invariant manner. We also propose a new metric to quantitatively evaluate GANs that does not require previous knowledge of the application, apart from the data itself. Using the new metric, in conjunction with the state-of-the-art evaluation methods, we show that the proposed architecture, when compared with GAN variants stemming from similar strategies, produces more accurate models of the input data in a way that is also less sensitive to hyperparameter settings.

Tim Sodergren, Riddhish Bhalodia, Ross Whitaker et al.

Difficult image segmentation problems, for instance left atrium MRI, can be addressed by incorporating shape priors to find solutions that are consistent with known objects. Nonetheless, a single multivariate Gaussian is not an adequate model in cases with significant nonlinear shape variation or where the prior distribution is multimodal. Nonparametric density estimation is more general, but has a ravenous appetite for training samples and poses serious challenges in optimization, especially in high dimensional spaces. Here, we propose a maximum-a-posteriori formulation that relies on a generative image model by incorporating both local intensity and global shape priors. We use deep autoencoders to capture the complex intensity distribution while avoiding the careful selection of hand-crafted features. We formulate the shape prior as a mixture of Gaussians and learn the corresponding parameters in a high-dimensional shape space rather than pre-projecting onto a low-dimensional subspace. In segmentation, we treat the identity of the mixture component as a latent variable and marginalize it within a generalized expectation-maximization framework. We present a conditional maximization-based scheme that alternates between a closed-form solution for component-specific shape parameters that provides a global update-based optimization strategy, and an intensity-based energy minimization that translates the global notion of a nonlinear shape prior into a set of local penalties. We demonstrate our approach on the left atrial segmentation from gadolinium-enhanced MRI, which is useful in quantifying the atrial geometry in patients with atrial fibrillation.

Riddhish Bhalodia, Anupama Goparaju, Tim Sodergren et al.

Left atrium shape has been shown to be an independent predictor of recurrence after atrial fibrillation (AF) ablation. Shape-based representation is imperative to such an estimation process, where correspondence-based representation offers the most flexibility and ease-of-computation for population-level shape statistics. Nonetheless, population-level shape representations in the form of image segmentation and correspondence models derived from cardiac MRI require significant human resources with sufficient anatomy-specific expertise. In this paper, we propose a machine learning approach that uses deep networks to estimate AF recurrence by predicting shape descriptors directly from MRI images, with NO image pre-processing involved. We also propose a novel data augmentation scheme to effectively train a deep network in a limited training data setting. We compare this new method of estimating shape descriptors from images with the state-of-the-art correspondence-based shape modeling that requires image segmentation and correspondence optimization. Results show that the proposed method and the current state-of-the-art produce statistically similar outcomes on AF recurrence, eliminating the need for expensive pre-processing pipelines and associated human labor.

Riddhish Bhalodia, Shireen Y. Elhabian, Ladislav Kavan et al.

Statistical shape modeling is an important tool to characterize variation in anatomical morphology. Typical shapes of interest are measured using 3D imaging and a subsequent pipeline of registration, segmentation, and some extraction of shape features or projections onto some lower-dimensional shape space, which facilitates subsequent statistical analysis. Many methods for constructing compact shape representations have been proposed, but are often impractical due to the sequence of image preprocessing operations, which involve significant parameter tuning, manual delineation, and/or quality control by the users. We propose DeepSSM: a deep learning approach to extract a low-dimensional shape representation directly from 3D images, requiring virtually no parameter tuning or user assistance. DeepSSM uses a convolutional neural network (CNN) that simultaneously localizes the biological structure of interest, establishes correspondences, and projects these points onto a low-dimensional shape representation in the form of PCA loadings within a point distribution model. To overcome the challenge of the limited availability of training images, we present a novel data augmentation procedure that uses existing correspondences on a relatively small set of processed images with shape statistics to create plausible training samples with known shape parameters. Hence, we leverage the limited CT/MRI scans (40-50) into thousands of images needed to train a CNN. After the training, the CNN automatically produces accurate low-dimensional shape representations for unseen images. We validate DeepSSM for three different applications pertaining to modeling pediatric cranial CT for characterization of metopic craniosynostosis, femur CT scans identifying morphologic deformities of the hip due to femoroacetabular impingement, and left atrium MRI scans for atrial fibrillation recurrence prediction.

Yen-Yun Yu, Shireen Y. Elhabian, Ross T. Whitaker

Semi-supervised learning (SSL) has become important in current data analysis applications, where the amount of unlabeled data is growing exponentially and user input remains limited by logistics and expense. Constrained clustering, as a subclass of SSL, makes use of user input in the form of relationships between data points (e.g., pairs of data points belonging to the same class or different classes) and can remarkably improve the performance of unsupervised clustering in order to reflect user-defined knowledge of the relationships between particular data points. Existing algorithms incorporate such user input, heuristically, as either hard constraints or soft penalties, which are separate from any generative or statistical aspect of the clustering model; this results in formulations that are suboptimal and not sufficiently general. In this paper, we propose a principled, generative approach to probabilistically model, without ad hoc penalties, the joint distribution given by user-defined pairwise relations. The proposed model accounts for general underlying distributions without assuming a specific form and relies on expectation-maximization for model fitting. For distributions in a standard form, the proposed approach results in a closed-form solution for updated parameters.

Mahsa Mirzargar, Ross T. Whitaker, Robert M. Kirby

Heterogeneous data pose serious challenges to data analysis tasks, including exploration and visualization. Current techniques often utilize dimensionality reductions, aggregation, or conversion to numerical values to analyze heterogeneous data. However, the effectiveness of such techniques to find subtle structures such as the presence of multiple modes or detection of outliers is hindered by the challenge to find the proper subspaces or prior knowledge to reveal the structures. In this paper, we propose a generic similarity-based exploration technique that is applicable to a wide variety of datatypes and their combinations, including heterogeneous ensembles. The proposed concept of similarity has a close connection to statistical analysis and can be deployed for summarization, revealing fine structures such as the presence of multiple modes, and detection of anomalies or outliers. We then propose a visual encoding framework that enables the exploration of a heterogeneous dataset in different levels of detail and provides insightful information about both global and local structures. We demonstrate the utility of the proposed technique using various real datasets, including ensemble data.