Sarang Joshi

Haocheng Dai, Michael Penwarden, Robert M. Kirby et al.

Neural operator learning as a means of mapping between complex function spaces has garnered significant attention in the field of computational science and engineering (CS&E). In this paper, we apply Neural operator learning to the time-of-flight ultrasound computed tomography (USCT) problem. We learn the mapping between time-of-flight (TOF) data and the heterogeneous sound speed field using a full-wave solver to generate the training data. This novel application of operator learning circumnavigates the need to solve the computationally intensive iterative inverse problem. The operator learns the non-linear mapping offline and predicts the heterogeneous sound field with a single forward pass through the model. This is the first time operator learning has been used for ultrasound tomography and is the first step in potential real-time predictions of soft tissue distribution for tumor identification in beast imaging.

Mingzhen Shao, Tolga Tasdizen, Sarang Joshi

Homography estimation serves as a fundamental technique for image alignment in a wide array of applications. The advent of convolutional neural networks has introduced learning-based methodologies that have exhibited remarkable efficacy in this realm. Yet, the generalizability of these approaches across distinct domains remains underexplored. Unlike other conventional tasks, CNN-driven homography estimation models show a distinctive immunity to domain shifts, enabling seamless deployment from one dataset to another without the necessity of transfer learning. This study explores the resilience of a variety of deep homography estimation models to domain shifts, revealing that the network architecture itself is not a contributing factor to this remarkable adaptability. By closely examining the models' focal regions and subjecting input images to a variety of modifications, we confirm that the models heavily rely on local textures such as edges and corner points for homography estimation. Moreover, our analysis underscores that the domain shift immunity itself is intricately tied to the utilization of these local textures.

Haocheng Dai, Martin Bauer, P. Thomas Fletcher et al.

The goal of diffusion-weighted magnetic resonance imaging (DWI) is to infer the structural connectivity of an individual subject's brain in vivo. To statistically study the variability and differences between normal and abnormal brain connectomes, a mathematical model of the neural connections is required. In this paper, we represent the brain connectome as a Riemannian manifold, which allows us to model neural connections as geodesics. This leads to the challenging problem of estimating a Riemannian metric that is compatible with the DWI data, i.e., a metric such that the geodesic curves represent individual fiber tracts of the connectomics. We reduce this problem to that of solving a highly nonlinear set of partial differential equations (PDEs) and study the applicability of convolutional encoder-decoder neural networks (CEDNNs) for solving this geometrically motivated PDE. Our method achieves excellent performance in the alignment of geodesics with white matter pathways and tackles a long-standing issue in previous geodesic tractography methods: the inability to recover crossing fibers with high fidelity.

Mokshagna Sai Teja Karanam, Krithika Iyer, Sarang Joshi et al.

Spatial transformations that capture population-level morphological statistics are critical for medical image analysis. Commonly used smoothness regularizers for image registration fail to integrate population statistics, leading to anatomically inconsistent transformations. Inverse consistency regularizers promote geometric consistency but lack population morphometrics integration. Regularizers that constrain deformation to low-dimensional manifold methods address this. However, they prioritize reconstruction over interpretability and neglect diffeomorphic properties, such as group composition and inverse consistency. We introduce MORPH-LER, a Log-Euclidean regularization framework for population-aware unsupervised image registration. MORPH-LER learns population morphometrics from spatial transformations to guide and regularize registration networks, ensuring anatomically plausible deformations. It features a bottleneck autoencoder that computes the principal logarithm of deformation fields via iterative square-root predictions. It creates a linearized latent space that respects diffeomorphic properties and enforces inverse consistency. By integrating a registration network with a diffeomorphic autoencoder, MORPH-LER produces smooth, meaningful deformation fields. The framework offers two main contributions: (1) a data-driven regularization strategy that incorporates population-level anatomical statistics to enhance transformation validity and (2) a linearized latent space that enables compact and interpretable deformation fields for efficient population morphometrics analysis. We validate MORPH-LER across two families of deep learning-based registration networks, demonstrating its ability to produce anatomically accurate, computationally efficient, and statistically meaningful transformations on the OASIS-1 brain imaging dataset. https://github.com/iyerkrithika21/MORPH_LER

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.

Seunghoi Kim, Henry F. J. Tregidgo, Matteo Figini et al.

Hallucinations are spurious structures not present in the ground truth, posing a critical challenge in medical image reconstruction, especially for data-driven conditional models. We hypothesize that combining an unconditional diffusion model with data consistency, trained on a diverse dataset, can reduce these hallucinations. Based on this, we propose DynamicDPS, a diffusion-based framework that integrates conditional and unconditional diffusion models to enhance low-quality medical images while systematically reducing hallucinations. Our approach first generates an initial reconstruction using a conditional model, then refines it with an adaptive diffusion-based inverse problem solver. DynamicDPS skips early stage in the reverse process by selecting an optimal starting time point per sample and applies Wolfe's line search for adaptive step sizes, improving both efficiency and image fidelity. Using diffusion priors and data consistency, our method effectively reduces hallucinations from any conditional model output. We validate its effectiveness in Image Quality Transfer for low-field MRI enhancement. Extensive evaluations on synthetic and real MR scans, including a downstream task for tissue volume estimation, show that DynamicDPS reduces hallucinations, improving relative volume estimation by over 15% for critical tissues while using only 5% of the sampling steps required by baseline diffusion models. As a model-agnostic and fine-tuning-free approach, DynamicDPS offers a robust solution for hallucination reduction in medical imaging. The code will be made publicly available upon publication.

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.

Krithika Iyer, Shireen Elhabian, Sarang Joshi

Image registration is a core task in computational anatomy that establishes correspondences between images. Invertible deformable registration, which computes a deformation field and handles complex, non-linear transformations, is essential for tracking anatomical variations, especially in neuroimaging applications where inter-subject differences and longitudinal changes are key. Analyzing the deformation fields is challenging due to their non-linearity, which limits statistical analysis. However, traditional approaches for analyzing deformation fields are computationally expensive, sensitive to initialization, and prone to numerical errors, especially when the deformation is far from the identity. To address these limitations, we propose the Log-Euclidean Diffeomorphism Autoencoder (LEDA), an innovative framework designed to compute the principal logarithm of deformation fields by efficiently predicting consecutive square roots. LEDA operates within a linearized latent space that adheres to the diffeomorphisms group action laws, enhancing our model's robustness and applicability. We also introduce a loss function to enforce inverse consistency, ensuring accurate latent representations of deformation fields. Extensive experiments with the OASIS-1 dataset demonstrate the effectiveness of LEDA in accurately modeling and analyzing complex non-linear deformations while maintaining inverse consistency. Additionally, we evaluate its ability to capture and incorporate clinical variables, enhancing its relevance for clinical applications.

Kristen M. Campbell, Haocheng Dai, Zhe Su et al.

The structural network of the brain, or structural connectome, can be represented by fiber bundles generated by a variety of tractography methods. While such methods give qualitative insights into brain structure, there is controversy over whether they can provide quantitative information, especially at the population level. In order to enable population-level statistical analysis of the structural connectome, we propose representing a connectome as a Riemannian metric, which is a point on an infinite-dimensional manifold. We equip this manifold with the Ebin metric, a natural metric structure for this space, to get a Riemannian manifold along with its associated geometric properties. We then use this Riemannian framework to apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. This formulation ties into the existing framework for diffeomorphic construction of image atlases, allowing us to construct a multimodal atlas by simultaneously integrating complementary white matter structure details from DWMRI and cortical details from T1-weighted MRI. We illustrate our framework with 2D data examples of connectome registration and atlas formation. Finally, we build an example 3D multimodal atlas using T1 images and connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.

Amanpreet Singh, Martin Bauer, Sarang Joshi

Optimal Mass Transport (OMT) is a well studied problem with a variety of applications in a diverse set of fields ranging from Physics to Computer Vision and in particular Statistics and Data Science. Since the original formulation of Monge in 1781 significant theoretical progress been made on the existence, uniqueness and properties of the optimal transport maps. The actual numerical computation of the transport maps, particularly in high dimensions, remains a challenging problem. By Brenier's theorem, the continuous OMT problem can be reduced to that of solving a non-linear PDE of Monge-Ampere type whose solution is a convex function. In this paper, building on recent developments of input convex neural networks and physics informed neural networks for solving PDE's, we propose a Deep Learning approach to solve the continuous OMT problem. To demonstrate the versatility of our framework we focus on the ubiquitous density estimation and generative modeling tasks in statistics and machine learning. Finally as an example we show how our framework can be incorporated with an autoencoder to estimate an effective probabilistic generative model.

Kristen M. Campbell, Haocheng Dai, Zhe Su et al.

The structural connectome is often represented by fiber bundles generated from various types of tractography. We propose a method of analyzing connectomes by representing them as a Riemannian metric, thereby viewing them as points in an infinite-dimensional manifold. After equipping this space with a natural metric structure, the Ebin metric, we apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. We demonstrate connectome registration and atlas formation using connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.

Blake E. Zimmerman, Sara Johnson, Henrik Odéen et al.

Noninvasive MR-guided focused ultrasound (MRgFUS) treatments are promising alternatives to the surgical removal of malignant tumors. A significant challenge is assessing the viability of treated tissue during and immediately after MRgFUS procedures. Current clinical assessment uses the nonperfused volume (NPV) biomarker immediately after treatment from contrast-enhanced MRI. The NPV has variable accuracy, and the use of contrast agent prevents continuing MRgFUS treatment if tumor coverage is inadequate. This work presents a novel, noncontrast, learned multiparametric MR biomarker that can be used during treatment for intratreatment assessment, validated in a VX2 rabbit tumor model. A deep convolutional neural network was trained on noncontrast multiparametric MR images using the NPV biomarker from follow-up MR imaging (3-5 days after MRgFUS treatment) as the accurate label of nonviable tissue. A novel volume-conserving registration algorithm yielded a voxel-wise correlation between treatment and follow-up NPV, providing a rigorous validation of the biomarker. The learned noncontrast multiparametric MR biomarker predicted the follow-up NPV with an average DICE coefficient of 0.71, substantially outperforming the current clinical standard (DICE coefficient = 0.53). Noncontrast multiparametric MR imaging integrated with a deep convolutional neural network provides a more accurate prediction of MRgFUS treatment outcome than current contrast-based techniques.

Markus D. Foote, Blake E. Zimmerman, Amit Sawant et al.

Radiation therapy presents a need for dynamic tracking of a target tumor volume. Fiducial markers such as implanted gold seeds have been used to gate radiation delivery but the markers are invasive and gating significantly increases treatment time. Pretreatment acquisition of a respiratory correlated 4DCT allows for determination of accurate motion tracking which is useful in treatment planning. We design a patient-specific motion subspace and a deep convolutional neural network to recover anatomical positions from a single fluoroscopic projection in real-time. We use this deep network to approximate the nonlinear inverse of a diffeomorphic deformation composed with radiographic projection. This network recovers subspace coordinates to define the patient-specific deformation of the lungs from a baseline anatomic position. The geometric accuracy of the subspace deformations on real patient data is similar to accuracy attained by original image registration between individual respiratory-phase image volumes.

Line Kuhnel, Tom Fletcher, Sarang Joshi et al.

Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back to a nonlinear Riemannian geometry on the latent space. The latent space thus provides a low-dimensional nonlinear representation of data and classical linear statistical techniques are no longer applicable. In this paper we show how statistics of data in their latent space representation can be performed using techniques from the field of nonlinear manifold statistics. Nonlinear manifold statistics provide generalizations of Euclidean statistical notions including means, principal component analysis, and maximum likelihood fits of parametric probability distributions. We develop new techniques for maximum likelihood inference in latent space, and adress the computational complexity of using geometric algorithms with high-dimensional data by training a separate neural network to approximate the Riemannian metric and cometric tensor capturing the shape of the learned data manifold.

Stefan Sommer, Alexis Arnaudon, Line Kuhnel et al.

We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood.

Caleb Rottman, Martin Bauer, Klas Modin et al.

In this article we study the problem of thoracic image registration, in particular the estimation of complex anatomical deformations associated with the breathing cycle. Using the intimate link between the Riemannian geometry of the space of diffeomorphisms and the space of densities, we develop an image registration framework that incorporates both the fundamental law of conservation of mass as well as spatially varying tissue compressibility properties. By exploiting the geometrical structure, the resulting algorithm is computationally efficient, yet widely general.

Jacob Hinkle, Prasanna Muralidharan, P. Thomas Fletcher et al.

In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.