Globally Injective ReLU Networks

Publisher Copyright: © 2022 Michael Puthawala, Konik Kothari, Matti Lassas, Ivan Dokmanic, Maarten de Hoop.Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end-rather than layerwise-doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.Peer reviewe

The clinical utilities of multi-pinhole single photon emission computed tomography

Single photon emission computed tomography (SPECT) is an important imaging modality for various applications in nuclear medicine. The use of multi-pinhole (MPH) collimators can provide superior resolution-sensitivity trade-off when imaging small field-of-view compared to conventional parallel-hole and fan-beam collimators. Besides the very successful application in small animal imaging, there has been a resurgence of the use of MPH collimators for clinical cardiac and brain studies, as well as other small field-of-view applications. This article reviews the basic principles of MPH collimators and introduces currently available and proposed clinical MPH SPECT systems

Investigation of Axial and Angular Sampling in Multi-Detector Pinhole-SPECT Brain Imaging

We designed a dedicated multi-detector multi-pinhole brain SPECT scanner to generate images of higher quality compared to general-purpose systems. The system, AdaptiSPECT-C, is intended to adapt its sensitivity-resolution trade-off by varying its aperture configurations allowing both high-sensitivity dynamic and high-spatial-resolution static imaging. The current system design consists of 23 detector heads arranged in a truncated spherical geometry. In this work, we investigated the axial and angular sampling capability of the current stationary system design. Two data acquisition schemes using limited rotation of the gantry and two others using axial translation of the imaging bed were also evaluated concerning their impact on image quality through improved sampling. Increasing both angular and axial sampling in the current prototype system resulted in quantitative improvements in image quality metrics and qualitative appearance of the images as determined in studies with specifically selected phantoms. Visual improvements for the brain phantoms with clinical distributions were less pronounced but presented quantitative improvements in the fidelity (normalized root-mean-square error (NRMSE)) and striatal specific binding ratio (SBR) for a dopamine transporter (DAT) distribution, and in NRMSE and activity recovery for a brain perfusion distribution. More pronounced improvements with increased sampling were seen in contrast recovery coefficient, bias, and coefficient of variation for a lesion in the brain perfusion distribution. The negligible impact of the most cranial ring of detectors on axial sampling, but its significant impact on sensitivity and angular sampling in the cranial portion of the imaging volume-of-interest were also determined

Creation of 3D Digital Anthropomorphic Phantoms which Model Actual Patient Non-rigid Body Motion as Determined from MRI and Position Tracking Studies of Volunteers

Background: Patient motion during emission imaging can create artifacts in the reconstructed emission distributions, which may mislead the diagnosis. For example, in myocardial-perfusion imaging, these artifacts can be mistaken for defects. Various software and hardware approaches have been developed to detect and compensate for motion. There are various ways of testing the effectiveness of motion correction methods applied in emission tomography, including the use of realistic digital anthropomorphic phantoms. Purpose: The purpose of this study was to create 3D digital anthropomorphic phantoms based on MRI data of volunteers undergoing a series of clinically relevant motions. These phantoms with combined position tracking were used to investigate both imaging-data-driven and motion tracking strategies to estimate and correct for patient motion. Methods: MRI scans were obtained of volunteers undergoing a series of clinically relevant movements. During the MRI, the motions were recorded by near-infra-red cameras tracking using external markers on the chest and abdomen. Individual-specific extended cardiac-torso (XCAT) phantoms were created fit to our volunteer MRI imaging data representing pre- and post-motion states. These XCAT phantoms were then used to generate activity and attenuation distributions. Monte Carlo methods will then be performed to simulate SPECT acquisitions, which will be used to evaluate various motion estimation and correction strategies. Results: Three volunteers were scanned in the MRI with concurrent external motion tracking. Each volunteer performed five separate motions including an axial slide, roll, shoulder twist, spine bend, and arm motion. These MRI scans were then manually digitalized into 3D anthropomorphic XCAT phantoms. Activity and attenuation distributions were created for each XCAT phantom, representing fifteen individual-specific motions. Conclusions: Our results will be combined with the external motion tracking data to determine if external motion tracking accurately reflects heart position in patients undergoing cardiac SPECT imaging. This data will also be used to evaluate other motion correction methods in the future

Exact form factors of the SU(N) Gross-Neveu model and 1/N expansion

The general SU(N) form factor formula is constructed. Exact form factors for the field, the energy momentum and the current operators are derived and compared with the 1/N-expansion of the chiral Gross-Neveu model and full agreement is found. As an application of the form factor approach the equal time commutation rules of arbitrary local fields are derived and in general anyonic behavior is found.Comment: 35 pages Published version of the paper, which includes minor corrections and improved acknowledgement

Inclusion of quasi-vertex views in a brain-dedicated multi-pinhole SPECT system for improved imaging performance

With brain-dedicated multi-detector systems employing pinhole apertures the usage of detectors facing the top of the patient\u27s head (i.e., quasi-vertex views) can provide the advantage of additional viewing from close to the brain for improved detector coverage. In this paper, we report the results of simulation and reconstruction studies to investigate the impact of the quasi-vertex views on the imaging performance of AdaptiSPECT-C, a brain-dedicated stationary SPECT system under development. In this design, both primary and scatter photons from regions located inferior to the brain can contribute to SPECT projections acquired by the quasi-vertex views, and thus degrade AdaptiSPECT-C imaging performance. In this work, we determined the proportion, origin, and nature (i.e., primary, scatter, and multiple-scatter) of counts emitted from structures within the head and throughout the body contributing to projections from the different AdaptiSPECT-C detector rings, as well as from a true vertex view detector. We simulated phantoms used to assess different aspects of image quality (i.e., uniform sphere and Derenzo), as well as anthropomorphic phantoms with multiple count levels emulating clinical(123)I activity distributions (i.e., DaTscan and perfusion). We determined that attenuation and scatter in the patient\u27s body greatly diminish the probability of the photons emitted outside the volume of interest reaching to detectors and being recorded within the 15% photopeak energy window. In addition, we demonstrated that the inclusion of the residual of such counts in the system acquisition does not degrade visual interpretation or quantitative analysis. The addition of the quasi-vertex detectors increases volumetric sensitivity, angular sampling, and spatial resolution leading to significant enhancement in image quality, especially in the striato-thalamic and superior regions of the brain. Besides, the use of quasi-vertex detectors improves the recovery of clinically relevant metrics such as the striatal binding ratio and mean activity in selected cerebral structures

Fully Discrete Wavelet Galerkin Schemes

The present paper is intended to give a survey of the developments of the wavelet Galerkin boundary element method. Using appropriate wavelet bases for the discretization of boundary integral operators yields numerically sparse system matrices. These system matrices can be compressed to O(N_j) nonzero matrix entries without loss of accuracy of the underlying Galerkin scheme. Herein, O(N_j) denotes the number of unknowns. As we show in the present paper, the assembly of the compressed system matrix can be performed within optimal complexity. By numerical experiments we provide examples which corroborate the theory

Some Remarks on Quadrature Formulas for Refinable Functions and Wavelets

This paper is concerned with the efficient computation of integrals of (smooth) functions against refinable functions and wavelets, respectively. We derive quadrature formulas of Gauss-type using these functions as weight functions. The methods are tested for several model problems and possible practical applications are discussed