Gaussian process regression can turn non-uniform and undersampled diffusion MRI data into diffusion spectrum imaging

We propose to use Gaussian process regression to accurately estimate the diffusion MRI signal at arbitrary locations in q-space. By estimating the signal on a grid, we can do synthetic diffusion spectrum imaging: reconstructing the ensemble averaged propagator (EAP) by an inverse Fourier transform. We also propose an alternative reconstruction method guaranteeing a nonnegative EAP that integrates to unity. The reconstruction is validated on data simulated from two Gaussians at various crossing angles. Moreover, we demonstrate on non-uniformly sampled in vivo data that the method is far superior to linear interpolation, and allows a drastic undersampling of the data with only a minor loss of accuracy. We envision the method as a potential replacement for standard diffusion spectrum imaging, in particular when acquistion time is limited.Comment: 5 page

Bayesian uncertainty quantification in linear models for diffusion MRI

Diffusion MRI (dMRI) is a valuable tool in the assessment of tissue microstructure. By fitting a model to the dMRI signal it is possible to derive various quantitative features. Several of the most popular dMRI signal models are expansions in an appropriately chosen basis, where the coefficients are determined using some variation of least-squares. However, such approaches lack any notion of uncertainty, which could be valuable in e.g. group analyses. In this work, we use a probabilistic interpretation of linear least-squares methods to recast popular dMRI models as Bayesian ones. This makes it possible to quantify the uncertainty of any derived quantity. In particular, for quantities that are affine functions of the coefficients, the posterior distribution can be expressed in closed-form. We simulated measurements from single- and double-tensor models where the correct values of several quantities are known, to validate that the theoretically derived quantiles agree with those observed empirically. We included results from residual bootstrap for comparison and found good agreement. The validation employed several different models: Diffusion Tensor Imaging (DTI), Mean Apparent Propagator MRI (MAP-MRI) and Constrained Spherical Deconvolution (CSD). We also used in vivo data to visualize maps of quantitative features and corresponding uncertainties, and to show how our approach can be used in a group analysis to downweight subjects with high uncertainty. In summary, we convert successful linear models for dMRI signal estimation to probabilistic models, capable of accurate uncertainty quantification.Comment: Added results from a group analysis and a comparison with residual bootstra

Privacy Amplification via Importance Sampling

We examine the privacy-enhancing properties of subsampling a data set via importance sampling as a pre-processing step for differentially private mechanisms. This extends the established privacy amplification by subsampling result to importance sampling where each data point is weighted by the reciprocal of its selection probability. The implications for privacy of weighting each point are not obvious. On the one hand, a lower selection probability leads to a stronger privacy amplification. On the other hand, the higher the weight, the stronger the influence of the point on the output of the mechanism in the event that the point does get selected. We provide a general result that quantifies the trade-off between these two effects. We show that heterogeneous sampling probabilities can lead to both stronger privacy and better utility than uniform subsampling while retaining the subsample size. In particular, we formulate and solve the problem of privacy-optimal sampling, that is, finding the importance weights that minimize the expected subset size subject to a given privacy budget. Empirically, we evaluate the privacy, efficiency, and accuracy of importance sampling-based privacy amplification on the example of k-means clustering.Comment: Under review for NeurIPS 202

Unsupervised dynamic modeling of medical image transformation

Spatiotemporal imaging has applications in e.g. cardiac diagnostics, surgical guidance, and radiotherapy monitoring, In this paper, we explain the temporal motion by identifying the underlying dynamics, only based on the sequential images. Our dynamical model maps the inputs of observed high-dimensional sequential images to a low-dimensional latent space wherein a linear relationship between a hidden state process and the lower-dimensional representation of the inputs holds. For this, we use a conditional variational auto-encoder (CVAE) to nonlinearly map the higher-dimensional image to a lower-dimensional space, wherein we model the dynamics with a linear Gaussian state-space model (LG-SSM). The model, a modified version of the Kalman variational auto-encoder, is end-to-end trainable, and the weights, both in the CVAE and LG-SSM, are simultaneously updated by maximizing the evidence lower bound of the marginal likelihood. In contrast to the original model, we explain the motion with a spatial transformation from one image to another. This results in sharper reconstructions and the possibility of transferring auxiliary information, such as segmentation, through the image sequence. Our experiments, on cardiac ultrasound time series, show that the dynamic model outperforms traditional image registration in execution time, to a similar performance. Further, our model offers the possibility to impute and extrapolate for missing samples.Comment: published in 2022 25th International Conference on Information Fusion (FUSION

On Feynman--Kac training of partial Bayesian neural networks

Recently, partial Bayesian neural networks (pBNNs), which only consider a subset of the parameters to be stochastic, were shown to perform competitively with full Bayesian neural networks. However, pBNNs are often multi-modal in the latent-variable space and thus challenging to approximate with parametric models. To address this problem, we propose an efficient sampling-based training strategy, wherein the training of a pBNN is formulated as simulating a Feynman--Kac model. We then describe variations of sequential Monte Carlo samplers that allow us to simultaneously estimate the parameters and the latent posterior distribution of this model at a tractable computational cost. We show on various synthetic and real-world datasets that our proposed training scheme outperforms the state of the art in terms of predictive performance.Comment: Under revie

Probabilistic Estimation of Chirp Instantaneous Frequency Using Gaussian Processes

We present a probabilistic approach for estimating chirp signal and its instantaneous frequency function when the true forms of the chirp and instantaneous frequency are unknown. To do so, we represent them by joint cascading Gaussian processes governed by a non-linear stochastic differential equation, and estimate their posterior distribution by using stochastic filters and smoothers. The model parameters are determined via maximum likelihood estimation. Theoretical results show that the estimation method has a bounded mean squared error. Experiments show that the method outperforms a number of baseline methods on a synthetic model, and we also apply the method to analyse a gravitational wave data.Comment: Submitted to IEEE Transactions on Signal Processin