Fast HARDI Uncertainty Quantification and Visualization with Spherical Sampling

Tark Patel; Tushar M. Athawale; Timbwaoga A.J. Ouermi; Chris R. Johnson

doi:10.1111/cgf.70138

Fast HARDI Uncertainty Quantification and Visualization with Spherical Sampling

Tark Patel
, Tushar M. Athawale
, Timbwaoga A.J. Ouermi
, Chris R. Johnson

Visualization

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper, we study uncertainty quantification and visualization of orientation distribution functions (ODF), which corresponds to the diffusion profile of high angular resolution diffusion imaging (HARDI) data. The shape inclusion probability (SIP) function is the state-of-the-art method for capturing the uncertainty of ODF ensembles. The current method of computing the SIP function with a volumetric basis exhibits high computational and memory costs, which can be a bottleneck to integrating uncertainty into HARDI visualization techniques and tools. We propose a novel spherical sampling framework for faster computation of the SIP function with lower memory usage and increased accuracy. In particular, we propose direct extraction of SIP isosurfaces, which represent confidence intervals indicating spatial uncertainty of HARDI glyphs, by performing spherical sampling of ODFs. Our spherical sampling approach requires much less sampling than the state-of-the-art volume sampling method, thus providing significantly enhanced performance, scalability, and the ability to perform implicit ray tracing. Our experiments demonstrate that the SIP isosurfaces extracted with our spherical sampling approach can achieve up to 8164× speedup, 37282× memory reduction, and 50.2% less SIP isosurface error compared to the classical volume sampling approach. We demonstrate the efficacy of our methods through experiments on synthetic and human-brain HARDI datasets.

Original language	English
Article number	e70138
Journal	Computer Graphics Forum
Volume	44
Issue number	3
DOIs	https://doi.org/10.1111/cgf.70138
State	Published - Jun 2025

Funding

This work was supported in part by the Intel Center of Excellence and the U.S. Department of En- ergy (DOE) RAPIDS-2 SciDAC project under contract number DE-AC0500OR22725. Diffusion-weighted imaging (DWI) is a magnetic resonance imaging (MRI) technique for measuring water diffusion in fibrous tissue, such as muscle and nerve cells. Fibrous tissue has anisotropic diffusion, which can be measured along a set of gradient vectors. Diffusion is measured by acquiring multiple DWI scans, each with a gradient vector along which diffusion is measured. We refer to an ensemble of 3D DWI scans as a DWI volume. Diffusion imaging techniques typically construct approximate models per-voxel based on ensembles of diffusion-weighted images. Many models have been developed in the pursuit of creating a compact representation to design efficient visualization algorithms [ODWL19]. Diffusion tensor imaging (DTI) is a popular method for modeling diffusion with tensors, which are 3×3 symmetric matrices [BML94]. The DTI model can represent single fiber directions, but it fails to accurately capture multiple fiber populations with varying orientations, such as fiber crossings. High angular resolution diffusion imaging (HARDI) modeling methods overcome the limitation of a single prominent diffusion assumption [TRW*02]. HARDI methods typically construct orientation distribution functions (ODFs) of the diffusion or fiber orientation profiles. Though HARDI requires more scans, it can still be performed in a clinically feasible time [TCGC04]. Quantifying and visualizing uncertainty has been regarded as the top scientific challenge to mitigate data misrepresentation [Joh04, JS03, BOL12]. Because model fitting provides a best-fit estimate of the underlying DWI data, models discard the residual, which is the modeling error. Because residuals are influenced by noise, they provide insight into how well the model fits the measured signal. Bootstrapping is commonly used to investigate model uncertainty by generating samples based on residuals [CLH06, Jon08]. The shape inclusion probability (SIP) function is a state-of-the-art mathematical tool to capture and visualize the variation of an ensemble of ODFs or tensors, such as those generated from bootstrapping. We discuss SIP functions in more detail in Section 3.4. Jiao et al. [JPGJ12] compute the SIP function with a 3D structured grid, which we call the volume sampling method [JPGJ12]. We refer to the sampling grid as the sampling volume. A glyph is visualized by volume rendering the measured SIP function per voxel. Volume sampling has significant limitations concerning the performance and memory footprint for quantification and visualization. The computation of the SIP function exhibits quartic time computational complexity, O(R3N), where R is the sampling volume's resolution and N number of bootstrap simulations. This sampling volume is costly to store on disk and in memory. The SIP functions are typically visualized with volume rendering, which is limited by available memory. These limitations impede the integration of SIP function analysis into HARDI visualization tools and necessitate investigating more efficient sampling methods. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a non-exclusive, paid up, irrevocable, world-wide license to publish or reproduce the published form of the manuscript, or allow others to do so, for U.S. Government purposes. The DOE will provide public access to these results in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). To address volume sampling's storage and computational limitations, we propose spherical sampling, a novel method for computing the SIP distribution function with a spherical basis. Sampling exhibits cubic time complexity O(Θ2N) with a sorting cost of O(Θ2N logN), where Θ is the polar and azimuthal resolution and N is the number of bootstrap samples. In addition to the significant improvement in computation time, spherical sampling converges faster and has a smaller memory footprint than volume sampling. Additionally, we propose a novel upsampling method, which reduces sampling requirements and memory footprint. We propose another novel visualization method of volume rendering glyphs with implicit ray tracing, which enables continuous and data-efficient rendering without tessellation. Recent work by Peter et al. proposed an uncertainty glyph method. However, they lacked a method to construct real datasets [PPUJ23]. Our volumetric ray tracing glyphs are rendered using their framework. Our contributions are summarized below. We propose a fast, accurate, and memory-efficient method for evaluating ODF model uncertainty by computing the SIP function with spherical sampling. We extend the framework of ODF modeling to SIP isosurfaces, which provides a compact and continuous representation of the SIP isosurfaces that are efficiently tessellated and rendered. For SIP isosurfaces modeling with a spherical harmonics (SH) basis, we introduce an implicit uncertainty glyph with multiple volumes. Implicit ray tracing renders smooth, continuous surfaces without tessellation and has little memory overhead. This work was supported in part by the Intel Center of Excellence and the U.S. Department of En‐ ergy (DOE) RAPIDS‐2 SciDAC project under contract number DE‐AC0500OR22725.

Keywords

Bootstrapping
CCS Concepts
Ray tracing
• Computing methodologies → Uncertainty quantification
• Human-centered computing → Scientific visualization
• Mathematics of computing → Probabilistic algorithms

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10.1111/cgf.70138

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title = "Fast HARDI Uncertainty Quantification and Visualization with Spherical Sampling",

abstract = "In this paper, we study uncertainty quantification and visualization of orientation distribution functions (ODF), which corresponds to the diffusion profile of high angular resolution diffusion imaging (HARDI) data. The shape inclusion probability (SIP) function is the state-of-the-art method for capturing the uncertainty of ODF ensembles. The current method of computing the SIP function with a volumetric basis exhibits high computational and memory costs, which can be a bottleneck to integrating uncertainty into HARDI visualization techniques and tools. We propose a novel spherical sampling framework for faster computation of the SIP function with lower memory usage and increased accuracy. In particular, we propose direct extraction of SIP isosurfaces, which represent confidence intervals indicating spatial uncertainty of HARDI glyphs, by performing spherical sampling of ODFs. Our spherical sampling approach requires much less sampling than the state-of-the-art volume sampling method, thus providing significantly enhanced performance, scalability, and the ability to perform implicit ray tracing. Our experiments demonstrate that the SIP isosurfaces extracted with our spherical sampling approach can achieve up to 8164× speedup, 37282× memory reduction, and 50.2\% less SIP isosurface error compared to the classical volume sampling approach. We demonstrate the efficacy of our methods through experiments on synthetic and human-brain HARDI datasets.",

keywords = "Bootstrapping, CCS Concepts, Ray tracing, • Computing methodologies → Uncertainty quantification, • Human-centered computing → Scientific visualization, • Mathematics of computing → Probabilistic algorithms",

author = "Tark Patel and Athawale, \{Tushar M.\} and Ouermi, \{Timbwaoga A.J.\} and Johnson, \{Chris R.\}",

note = "Publisher Copyright: {\textcopyright} 2025 The Author(s). Computer Graphics Forum published by Eurographics - The European Association for Computer Graphics and John Wiley \& Sons Ltd.",

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AU - Johnson, Chris R.

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AB - In this paper, we study uncertainty quantification and visualization of orientation distribution functions (ODF), which corresponds to the diffusion profile of high angular resolution diffusion imaging (HARDI) data. The shape inclusion probability (SIP) function is the state-of-the-art method for capturing the uncertainty of ODF ensembles. The current method of computing the SIP function with a volumetric basis exhibits high computational and memory costs, which can be a bottleneck to integrating uncertainty into HARDI visualization techniques and tools. We propose a novel spherical sampling framework for faster computation of the SIP function with lower memory usage and increased accuracy. In particular, we propose direct extraction of SIP isosurfaces, which represent confidence intervals indicating spatial uncertainty of HARDI glyphs, by performing spherical sampling of ODFs. Our spherical sampling approach requires much less sampling than the state-of-the-art volume sampling method, thus providing significantly enhanced performance, scalability, and the ability to perform implicit ray tracing. Our experiments demonstrate that the SIP isosurfaces extracted with our spherical sampling approach can achieve up to 8164× speedup, 37282× memory reduction, and 50.2% less SIP isosurface error compared to the classical volume sampling approach. We demonstrate the efficacy of our methods through experiments on synthetic and human-brain HARDI datasets.

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KW - • Mathematics of computing → Probabilistic algorithms

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Fast HARDI Uncertainty Quantification and Visualization with Spherical Sampling

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