On Adapting Randomized Nyström Preconditioners to Accelerate Variational Image Reconstruction

Model-based iterative reconstruction plays a key role in solving inverse problems. However, the associated minimization problems are generally large-scale, ill-posed, nonsmooth, and sometimes even nonconvex, which present challenges in designing efficient iterative solvers and often prevent their practical use. Preconditioning methods can significantly accelerate the convergence of iterative methods. In some applications, computing preconditioners on-the-fly is beneficial. Moreover, forward models in image reconstruction are typically represented as operators, and the corresponding explicit matrices are often unavailable, which brings additional challenges in designing preconditioners. Therefore, for practical use, computing and applying preconditioners should be computationally inexpensive. This paper adapts the randomized Nystr\"{o}m approximation to compute effective preconditioners that accelerate image reconstruction without requiring an explicit matrix for the forward model. We leverage modern GPU computational platforms to compute the preconditioner on-the-fly. Moreover, we propose efficient approaches for applying the preconditioner to problems with nonsmooth regularizers. Our numerical results on image deblurring, super-resolution with impulsive noise, and computed tomography reconstruction demonstrate the efficiency and effectiveness of the proposed preconditioner.

Patch-Based Diffusion Models Beat Whole-Image Models for Mismatched Distribution Inverse Problems

Diffusion models have achieved excellent success in solving inverse problems due to their ability to learn strong image priors, but existing approaches require a large training dataset of images that should come from the same distribution as the test dataset. When the training and test distributions are mismatched, artifacts and hallucinations can occur in reconstructed images due to the incorrect priors. In this work, we systematically study out of distribution (OOD) problems where a known training distribution is first provided. We first study the setting where only a single measurement obtained from the unknown test distribution is available. Next we study the setting where a very small sample of data belonging to the test distribution is available, and our goal is still to reconstruct an image from a measurement that came from the test distribution. In both settings, we use a patch-based diffusion prior that learns the image distribution solely from patches. Furthermore, in the first setting, we include a self-supervised loss that helps the network output maintain consistency with the measurement. Extensive experiments show that in both settings, the patch-based method can obtain high quality image reconstructions that can outperform whole-image models and can compete with methods that have access to large in-distribution training datasets. Furthermore, we show how whole-image models are prone to memorization and overfitting, leading to artifacts in the reconstructions, while a patch-based model can resolve these issues.

Shorter SPECT Scans Using Self-supervised Coordinate Learning to Synthesize Skipped Projection Views

Purpose: This study addresses the challenge of extended SPECT imaging duration under low-count conditions, as encountered in Lu-177 SPECT imaging, by developing a self-supervised learning approach to synthesize skipped SPECT projection views, thus shortening scan times in clinical settings. Methods: We employed a self-supervised coordinate-based learning technique, adapting the neural radiance field (NeRF) concept in computer vision to synthesize under-sampled SPECT projection views. For each single scan, we used self-supervised coordinate learning to estimate skipped SPECT projection views. The method was tested with various down-sampling factors (DFs=2, 4, 8) on both Lu-177 phantom SPECT/CT measurements and clinical SPECT/CT datasets, from 11 patients undergoing Lu-177 DOTATATE and 6 patients undergoing Lu-177 PSMA-617 radiopharmaceutical therapy. Results: For SPECT reconstructions, our method outperformed the use of linearly interpolated projections and partial projection views in relative contrast-to-noise-ratios (RCNR) averaged across different downsampling factors: 1) DOTATATE: 83% vs. 65% vs. 67% for lesions and 86% vs. 70% vs. 67% for kidney, 2) PSMA: 76% vs. 69% vs. 68% for lesions and 75% vs. 55% vs. 66% for organs, including kidneys, lacrimal glands, parotid glands, and submandibular glands. Conclusion: The proposed method enables reduction in acquisition time (by factors of 2, 4, or 8) while maintaining quantitative accuracy in clinical SPECT protocols by allowing for the collection of fewer projections. Importantly, the self-supervised nature of this NeRF-based approach eliminates the need for extensive training data, instead learning from each patient's projection data alone. The reduction in acquisition time is particularly relevant for imaging under low-count conditions and for protocols that require multiple-bed positions such as whole-body imaging.

DiffusionBlend: Learning 3D Image Prior through Position-aware Diffusion Score Blending for 3D Computed Tomography Reconstruction

Diffusion models face significant challenges when employed for large-scale medical image reconstruction in real practice such as 3D Computed Tomography (CT). Due to the demanding memory, time, and data requirements, it is difficult to train a diffusion model directly on the entire volume of high-dimensional data to obtain an efficient 3D diffusion prior. Existing works utilizing diffusion priors on single 2D image slice with hand-crafted cross-slice regularization would sacrifice the z-axis consistency, which results in severe artifacts along the z-axis. In this work, we propose a novel framework that enables learning the 3D image prior through position-aware 3D-patch diffusion score blending for reconstructing large-scale 3D medical images. To the best of our knowledge, we are the first to utilize a 3D-patch diffusion prior for 3D medical image reconstruction. Extensive experiments on sparse view and limited angle CT reconstruction show that our DiffusionBlend method significantly outperforms previous methods and achieves state-of-the-art performance on real-world CT reconstruction problems with high-dimensional 3D image (i.e., $256 \times 256 \times 500$). Our algorithm also comes with better or comparable computational efficiency than previous state-of-the-art methods.

Learning Image Priors through Patch-based Diffusion Models for Solving Inverse Problems

Diffusion models can learn strong image priors from underlying data distribution and use them to solve inverse problems, but the training process is computationally expensive and requires lots of data. Such bottlenecks prevent most existing works from being feasible for high-dimensional and high-resolution data such as 3D images. This paper proposes a method to learn an efficient data prior for the entire image by training diffusion models only on patches of images. Specifically, we propose a patch-based position-aware diffusion inverse solver, called PaDIS, where we obtain the score function of the whole image through scores of patches and their positional encoding and utilize this as the prior for solving inverse problems. First of all, we show that this diffusion model achieves an improved memory efficiency and data efficiency while still maintaining the capability to generate entire images via positional encoding. Additionally, the proposed PaDIS model is highly flexible and can be plugged in with different diffusion inverse solvers (DIS). We demonstrate that the proposed PaDIS approach enables solving various inverse problems in both natural and medical image domains, including CT reconstruction, deblurring, and superresolution, given only patch-based priors. Notably, PaDIS outperforms previous DIS methods trained on entire image priors in the case of limited training data, demonstrating the data efficiency of our proposed approach by learning patch-based prior.

Provable Preconditioned Plug-and-Play Approach for Compressed Sensing MRI Reconstruction

Model-based methods play a key role in the reconstruction of compressed sensing (CS) MRI. Finding an effective prior to describe the statistical distribution of the image family of interest is crucial for model-based methods. Plug-and-play (PnP) is a general framework that uses denoising algorithms as the prior or regularizer. Recent work showed that PnP methods with denoisers based on pretrained convolutional neural networks outperform other classical regularizers in CS MRI reconstruction. However, the numerical solvers for PnP can be slow for CS MRI reconstruction. This paper proposes a preconditioned PnP (P^2nP) method to accelerate the convergence speed. Moreover, we provide proofs of the fixed-point convergence of the P^2nP iterates. Numerical experiments on CS MRI reconstruction with non-Cartesian sampling trajectories illustrate the effectiveness and efficiency of the P^2nP approach.

Streaming Probabilistic PCA for Missing Data with Heteroscedastic Noise

Streaming principal component analysis (PCA) is an integral tool in large-scale machine learning for rapidly estimating low-dimensional subspaces of very high dimensional and high arrival-rate data with missing entries and corrupting noise. However, modern trends increasingly combine data from a variety of sources, meaning they may exhibit heterogeneous quality across samples. Since standard streaming PCA algorithms do not account for non-uniform noise, their subspace estimates can quickly degrade. On the other hand, the recently proposed Heteroscedastic Probabilistic PCA Technique (HePPCAT) addresses this heterogeneity, but it was not designed to handle missing entries and streaming data, nor does it adapt to non-stationary behavior in time series data. This paper proposes the Streaming HeteroscedASTic Algorithm for PCA (SHASTA-PCA) to bridge this divide. SHASTA-PCA employs a stochastic alternating expectation maximization approach that jointly learns the low-rank latent factors and the unknown noise variances from streaming data that may have missing entries and heteroscedastic noise, all while maintaining a low memory and computational footprint. Numerical experiments validate the superior subspace estimation of our method compared to state-of-the-art streaming PCA algorithms in the heteroscedastic setting. Finally, we illustrate SHASTA-PCA applied to highly-heterogeneous real data from astronomy.

ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction that is useful for various data science problems. However, many applications involve heterogeneous data that varies in quality due to noise characteristics associated with different sources of the data. Methods that deal with this mixed dataset are known as heteroscedastic methods. Current methods like HePPCAT make Gaussian assumptions of the basis coefficients that may not hold in practice. Other methods such as Weighted PCA (WPCA) assume the noise variances are known, which may be difficult to know in practice. This paper develops a PCA method that can estimate the sample-wise noise variances and use this information in the model to improve the estimate of the subspace basis associated with the low-rank structure of the data. This is done without distributional assumptions of the low-rank component and without assuming the noise variances are known. Simulations show the effectiveness of accounting for such heteroscedasticity in the data, the benefits of using such a method with all of the data versus retaining only good data, and comparisons are made against other PCA methods established in the literature like PCA, Robust PCA (RPCA), and HePPCAT. Code available at https://github.com/javiersc1/ALPCAH

AWFSD: Accelerated Wirtinger Flow with Score-based Diffusion Image Prior for Poisson-Gaussian Holographic Phase Retrieval

Phase retrieval (PR) is an essential problem in a number of coherent imaging systems. This work aims at resolving the holographic phase retrieval problem in real world scenarios where the measurements are corrupted by a mixture of Poisson and Gaussian (PG) noise that stems from optical imaging systems. To solve this problem, we develop a novel algorithm based on Accelerated Wirtinger Flow that uses Score-based Diffusion models as the generative prior (AWFSD). In particular, we frame the PR problem as an optimization task that involves both a data fidelity term and a regularization term. We derive the gradient of the PG log-likelihood function along with its corresponding Lipschitz constant, ensuring a more accurate data consistency term for practical measurements. We introduce a generative prior as part of our regularization approach by using a score-based diffusion model to capture (the gradient of) the image prior distribution. We provide theoretical analysis that establishes a critical-point convergence guarantee for the proposed AWFSD algorithm. Our simulation experiments demonstrate that: 1) The proposed algorithm based on the PG likelihood model enhances reconstruction compared to that solely based on either Gaussian or Poisson likelihood. 2) The proposed AWFSD algorithm produces reconstructions with higher image quality both qualitatively and quantitatively, and is more robust to variations in noise levels when compared with state-of-the-art methods for phase retrieval.

Dynamic Subspace Estimation with Grassmannian Geodesics

Dynamic subspace estimation, or subspace tracking, is a fundamental problem in statistical signal processing and machine learning. This paper considers a geodesic model for time-varying subspaces. The natural objective function for this model is non-convex. We propose a novel algorithm for minimizing this objective and estimating the parameters of the model from data with Grassmannian-constrained optimization. We show that with this algorithm, the objective is monotonically non-increasing. We demonstrate the performance of this model and our algorithm on synthetic data, video data, and dynamic fMRI data.