Resting-state functional MRI (rsfMRI) analysis relies on complex mathematical operations whose properties and pitfalls are often poorly understood, leading to interpretational errors and suboptimal processing choices. This work presents novel mathematical insights for rsfMRI analysis through three key contributions: (1) a unified geometric framework showing that all common preprocessing and analysis operations can be understood as rotations in time-series vector space, (2) identification of rotationally invariant properties that remain stable across different processing choices, and (3) mathematical equivalences between seemingly different connectivity measures. We demonstrate how the hemodynamic response function acts as a rotation operator in frequency space, derive closed-form expressions for the impact of preprocessing on effective degrees of freedom, and show that correlation and coherence measures can be unified through frequency-weighted integration. Common mathematical errors in the literature are identified and corrected with worked examples. This framework provides practical guidance for choosing connectivity measures, ordering preprocessing steps, and understanding the mathematical constraints imposed by operations such as global signal regression. By connecting abstract mathematical concepts to concrete rsfMRI applications, this work serves as both a theoretical foundation and a practical guide for researchers using functional connectivity methods. Practical applications include detecting network disruption in neuropsychiatric disorders (e.g., schizophrenia) with dramatically improved sensitivity, harmonizing multisite data without complex corrections, optimizing scan protocols for specific effect sizes, and providing robust quality control metrics that outperform traditional approaches.
{"title":"Rotational Invariance in Resting-State fMRI: A Geometric Framework for Understanding Signal Processing and Connectivity","authors":"Chisondi S. Warioba","doi":"10.1155/cmr/8852818","DOIUrl":"https://doi.org/10.1155/cmr/8852818","url":null,"abstract":"<p>Resting-state functional MRI (rsfMRI) analysis relies on complex mathematical operations whose properties and pitfalls are often poorly understood, leading to interpretational errors and suboptimal processing choices. This work presents novel mathematical insights for rsfMRI analysis through three key contributions: (1) a unified geometric framework showing that all common preprocessing and analysis operations can be understood as rotations in time-series vector space, (2) identification of rotationally invariant properties that remain stable across different processing choices, and (3) mathematical equivalences between seemingly different connectivity measures. We demonstrate how the hemodynamic response function acts as a rotation operator in frequency space, derive closed-form expressions for the impact of preprocessing on effective degrees of freedom, and show that correlation and coherence measures can be unified through frequency-weighted integration. Common mathematical errors in the literature are identified and corrected with worked examples. This framework provides practical guidance for choosing connectivity measures, ordering preprocessing steps, and understanding the mathematical constraints imposed by operations such as global signal regression. By connecting abstract mathematical concepts to concrete rsfMRI applications, this work serves as both a theoretical foundation and a practical guide for researchers using functional connectivity methods. Practical applications include detecting network disruption in neuropsychiatric disorders (e.g., schizophrenia) with dramatically improved sensitivity, harmonizing multisite data without complex corrections, optimizing scan protocols for specific effect sizes, and providing robust quality control metrics that outperform traditional approaches.</p>","PeriodicalId":55216,"journal":{"name":"Concepts in Magnetic Resonance Part A","volume":"2026 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/cmr/8852818","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146130397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Krithika Balaji, Zimu Huo, Michael Mendoza, Michael Hoff, Anil A. Bharath, Peter J. Lally, Neal K. Bangerter
MR images are often acquired using a phased array of radiofrequency (RF) channels, with each RF channel being sensitive to only part of the object being scanned. The images collected from each RF channel are combined to generate a single image with higher SNR and more uniform sensitivity than can be obtained with a single channel alone. It is generally desirable to combine the images before performing any analysis in a quantitative imaging (QI) experiment—this way, the voxel-level signals input into the fitting model have high SNR. Computationally, it is often also more efficient than performing a quantitative fitting process on each channel image individually. Although fitting is typically performed on a voxel-level signal magnitude, certain pulse sequences like phase-cycled balanced steady-state free precession (pc-bSSFP) encode important information about tissue properties in the signal phase as well. Therefore, it is desirable to preserve the complex signal during the coil combination process in order for QI analyses to be reliable. While a variety of different coil combination techniques exist, there is little information on which ones best preserve phase for pc-bSSFP. Pc-bSSFP is of particular interest as the complex-valued images are used for relaxometry. This study compared the phase preservation performance of various coil combination techniques: Eigenvalue-based approach for iterative self-consistent parallel imaging reconstruction (ESPIRiT), simple phase robust coil combination (SRCC), full phase robust coil combination (FRCC), adaptive reconstruction (AR), and intrinsic multi-channel phase alignment (IMPA). These techniques were tested on pc-bSSFP data in both a simulated phantom and in vivo knee cartilage. The comparisons were conducted across a range of SNR levels to reflect realistic scenarios. Results showed that ESPIRiT, AR, and IMPA best preserved phase across the range of SNR levels tested.
{"title":"Comparison of Coil Combination Technique Performance for Phase Preservation in bSSFP","authors":"Krithika Balaji, Zimu Huo, Michael Mendoza, Michael Hoff, Anil A. Bharath, Peter J. Lally, Neal K. Bangerter","doi":"10.1155/cmr/8276955","DOIUrl":"https://doi.org/10.1155/cmr/8276955","url":null,"abstract":"<p>MR images are often acquired using a phased array of radiofrequency (RF) channels, with each RF channel being sensitive to only part of the object being scanned. The images collected from each RF channel are combined to generate a single image with higher SNR and more uniform sensitivity than can be obtained with a single channel alone. It is generally desirable to combine the images before performing any analysis in a quantitative imaging (QI) experiment—this way, the voxel-level signals input into the fitting model have high SNR. Computationally, it is often also more efficient than performing a quantitative fitting process on each channel image individually. Although fitting is typically performed on a voxel-level signal magnitude, certain pulse sequences like phase-cycled balanced steady-state free precession (pc-bSSFP) encode important information about tissue properties in the signal phase as well. Therefore, it is desirable to preserve the complex signal during the coil combination process in order for QI analyses to be reliable. While a variety of different coil combination techniques exist, there is little information on which ones best preserve phase for pc-bSSFP. Pc-bSSFP is of particular interest as the complex-valued images are used for relaxometry. This study compared the phase preservation performance of various coil combination techniques: Eigenvalue-based approach for iterative self-consistent parallel imaging reconstruction (ESPIRiT), simple phase robust coil combination (SRCC), full phase robust coil combination (FRCC), adaptive reconstruction (AR), and intrinsic multi-channel phase alignment (IMPA). These techniques were tested on pc-bSSFP data in both a simulated phantom and in vivo knee cartilage. The comparisons were conducted across a range of SNR levels to reflect realistic scenarios. Results showed that ESPIRiT, AR, and IMPA best preserved phase across the range of SNR levels tested.</p>","PeriodicalId":55216,"journal":{"name":"Concepts in Magnetic Resonance Part A","volume":"2025 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/cmr/8276955","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145686078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tyler Hecht, Griffin S. Hampton, Ryan Neff, Richard G. Spencer, Pak-Wing Fok
In this paper, we contrast frequentist and Bayesian approaches to parameter estimation for a magnetic resonance (MR) relaxometry signal model which takes the form of a two-dimensional biexponential decay. The signal consists of two terms, each parameterized by an amplitude and a transverse and longitudinal relaxation time constant. There are two user-selected parameters, defining the two-dimensional character of the signal; these are an inversion time TI and a set of echo times, TE. Of particular interest is the fact that for two values of TI, which we call the null points, the signal becomes a monoexponential function in TE. Extracting the two parameters—the amplitude and decay constant—from the signal observed at or near a null point is particularly ill-posed since the monoexponential signal is highly overparameterized by the four parameter biexponential models. We seek to estimate these null points, which directly provide values for the longitudinal relaxation time constants, using both frequentist and Bayesian techniques. The frequentist approach uses nonlinear least squares (NLLS), and the Bayesian approach uses the Metropolis–Hastings algorithm. In addition to point estimates, both methods generate point clouds of parameter estimates representing uncertainties. Due to the symmetry of the biexponential model, these point clouds consist of two clusters. The variance of a single cluster and the separation between the two clusters, both of which capture the size of the point clouds, may be used as metrics for ill-posedness. Increasing point cloud size, indicating an undesired greater flexibility in parameter choice, illustrates a greater degree of ill-posedness. We find that both the frequentist and Bayesian approaches can estimate the null points using the extrema of these metrics and yield qualitatively similar and consistent results.
{"title":"Parameter Estimation in Two-Dimensional Biexponential Magnetic Resonance Relaxometry: A Case-Study Comparison of Frequentist and Bayesian Approaches","authors":"Tyler Hecht, Griffin S. Hampton, Ryan Neff, Richard G. Spencer, Pak-Wing Fok","doi":"10.1155/cmr/6678358","DOIUrl":"https://doi.org/10.1155/cmr/6678358","url":null,"abstract":"<p>In this paper, we contrast frequentist and Bayesian approaches to parameter estimation for a magnetic resonance (MR) relaxometry signal model which takes the form of a two-dimensional biexponential decay. The signal consists of two terms, each parameterized by an amplitude and a transverse and longitudinal relaxation time constant. There are two user-selected parameters, defining the two-dimensional character of the signal; these are an inversion time <i>T</i><i>I</i> and a set of echo times, <i>T</i><i>E</i>. Of particular interest is the fact that for two values of <i>T</i><i>I</i>, which we call the null points, the signal becomes a monoexponential function in <i>T</i><i>E</i>. Extracting the two parameters—the amplitude and decay constant—from the signal observed at or near a null point is particularly ill-posed since the monoexponential signal is highly overparameterized by the four parameter biexponential models. We seek to estimate these null points, which directly provide values for the longitudinal relaxation time constants, using both frequentist and Bayesian techniques. The frequentist approach uses nonlinear least squares (NLLS), and the Bayesian approach uses the Metropolis–Hastings algorithm. In addition to point estimates, both methods generate point clouds of parameter estimates representing uncertainties. Due to the symmetry of the biexponential model, these point clouds consist of two clusters. The variance of a single cluster and the separation between the two clusters, both of which capture the size of the point clouds, may be used as metrics for ill-posedness. Increasing point cloud size, indicating an undesired greater flexibility in parameter choice, illustrates a greater degree of ill-posedness. We find that both the frequentist and Bayesian approaches can estimate the null points using the extrema of these metrics and yield qualitatively similar and consistent results.</p>","PeriodicalId":55216,"journal":{"name":"Concepts in Magnetic Resonance Part A","volume":"2025 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/cmr/6678358","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144910452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}