Poisson Noise Model Research Articles

2D signal estimation for sparse distributed target photon counting data

In this study, we explore the utilization of penalized likelihood estimation for the analysis of sparse photon counting data obtained from distributed target lidar systems. Specifically, we adapt the Poisson Total Variation processing technique to cater to this application. By assuming a Poisson noise model for the photon count observations, our approach yields denoised estimates of backscatter photon flux and related parameters. This facilitates the processing of raw photon counting signals with exceptionally high temporal and range resolutions (demonstrated here to 50 Hz and 75 cm resolutions), including data acquired through time-correlated single photon counting, without significant sacrifice of resolution. Through examination involving both simulated and real-world 2D atmospheric data, our method consistently demonstrates superior accuracy in signal recovery compared to the conventional histogram-based approach commonly employed in distributed target lidar applications.

Tutorial: Guide to error propagation for particle counting measurements

Forward error propagation is an established technique for uncertainty quantification (UQ). This article covers practical applications of forward error propagation in the context of particle counting measurements. We begin by presenting pertinent error models, including the Poisson noise model, and assess their role in UQ. Next, we describe several basic techniques for UQ, including Gauss’s formula, its generalization to the Law of Propagation of Uncertainty (LPU), and the use of Monte Carlo (MC) sampling. We conclude with demonstrations of increasing complexity, including total number concentration, total mass concentration, penetration, and mass-based filtration efficiency scenarios. These examples serve two functions: (1) providing examples in which theoretical concepts are practically applied to interpret particle counting data and (2) presenting expressions that can be used to compute uncertainties for specific problems in particle counting measurement.

Ratio-Based Pulse Shape Discrimination: Analytic Results for Gaussian and Poisson Noise Models.

In experiments in a range of fields including fast neutron spectroscopy and astroparticle physics, one can discriminate events of interest from background events based on the shapes of electronic pulses produced by energy deposits in a detector. Here, I focus on a well-known pulse shape discrimination method based on the ratio of the temporal integral of the pulse over an early interval Xp and the temporal integral over the entire pulse Xt. For both event classes, for both a Gaussian noise model and a Poisson noise model, I present analytic expressions for the conditional distribution of Xp given knowledge of the observed value of Xt and a scaled energy deposit corresponding to the product of the full energy deposit and a relative yield factor. I assume that the energy-dependent theoretical prompt fraction for both classes are known exactly. With a Bayesian approach that accounts for imperfect knowledge of the scaled energy deposit, I determine the posterior mean background acceptance probability given the target signal acceptance probability as a function of the observed value of Xt. My method enables one to determine receiver-operating-characteristic curves by numerical integration rather than by Monte Carlo simulation for these two noise models.

Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography.

In this paper, we address the problem of activity estimation in passive gamma emission tomography (PGET) of spent nuclear fuel. Two different noise models are considered and compared, namely, the isotropic Gaussian and the Poisson noise models. The problem is formulated within a Bayesian framework as a linear inverse problem and prior distributions are assigned to the unknown model parameters. In particular, a Bernoulli-truncated Gaussian prior model is considered to promote sparse pin configurations. A Markov chain Monte Carlo (MCMC) method, based on a split and augmented Gibbs sampler, is then used to sample the posterior distribution of the unknown parameters. The proposed algorithm is first validated by simulations conducted using synthetic data, generated using the nominal models. We then consider more realistic data simulated using a bespoke simulator, whose forward model is non-linear and not available analytically. In that case, the linear models used are mis-specified and we analyse their robustness for activity estimation. The results demonstrate superior performance of the proposed approach in estimating the pin activities in different assembly patterns, in addition to being able to quantify their uncertainty measures, in comparison with existing methods.

Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm.

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained "blind" ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.

Fast Roughness Minimizing Image Restoration Under Mixed Poisson-Gaussian Noise.

Image acquisition in many biomedical imaging modalities is corrupted by Poisson noise followed by additive Gaussian noise. While total variation and related regularization methods for solving biomedical inverse problems are known to yield high quality reconstructions in most situations, such methods mostly use log-likelihood of either Gaussian or Poisson noise models, and rarely use mixed Poisson-Gaussian (PG) noise model. There is a recent work which deals with exact PG likelihood and total variation regularization. This method adapts the primal-dual approach involving gradients steps on the PG log-likelihood, with step size limited by the inverse of its Lipschitz constant. This leads to limitations in the convergence speed. Although the alternating direction method of multipliers (ADMM) algorithm does not have such step size restrictions, ADMM has never been applied for this problem, for the possible reason that PG log-likelihood is quite complex. In this paper, we develop an ADMM based optimization for roughness minimizing image restoration under PG log-likelihood. We achieve this by first developing a novel iterative method for computing the proximal solution of PG log-likelihood, deriving the termination conditions for this iterative method, and then integrating into a provably convergent ADMM scheme. We experimentally demonstrate that the proposed method outperform primal-dual method in most of the cases.

Collocation Method for Multiplicative Noise Removal Model

Image denoising is a fundamental problem in both image processing and computer vision with numerous applications. It can be formulated as an inverse problem. Variational methods are commonly used to solve noise removal problems. The Total Variation (TV) regularization has evolved from an image denoising method for images corrupted with multiplicative noise into a more general technique for inverse problems such as denoising, deblurring, blind deconvolution, and inpainting, which also encompasses the Impulse, Poisson, Speckle, and mixed noise models. Multiplicative noise removal based on TV regularization has been widely researched in image science. In multiplicative noise problems, original image is multiplied by a noise rather than added to the original image. This article proposes a novel meshless collocation technique for the solution of a model having multiplicative noise. This technique includes TV and local collocation along with Multiquadric Radial Basis Function (MQ-RBF) for the solution of associated Euler-Lagrange equation for restoring multiplicative noise from digital images. Numerical examples demonstrate that the proposed algorithm is able to preserve small image details while the noise in the homogeneous regions is removed sufficiently. As a consequence, our method yields better denoised results than those of the current state of the art methods with respect to the Peak-Signal to Noise Ratio (PSNR) values.

Joint 3D localization and classification of space debris using a multispectral rotating point spread function.

We consider the problem of joint three-dimensional (3D) localization and material classification of unresolved space debris using a multispectral rotating point spread function (RPSF). The use of RPSF allows one to estimate the 3D locations of point sources from their rotated images acquired by a single 2D sensor array, since the amount of rotation of each source image about its x, y location depends on its axial distance z. Using multispectral images, with one RPSF per spectral band, we are able not only to localize the 3D positions of the space debris but also classify their material composition. We propose a three-stage method for achieving joint localization and classification. In stage 1, we adopt an optimization scheme for localization in which the spectral signature of each material is assumed to be uniform, which significantly improves efficiency and yields better localization results than possible with a single spectral band. In stage 2, we estimate the spectral signature and refine the localization result via an alternating approach. We process classification in the final stage. Both Poisson noise and Gaussian noise models are considered, and the implementation of each is discussed. Numerical tests using multispectral data from NASA show the efficiency of our three-stage approach and illustrate the improvement of point source localization and spectral classification from using multiple bands over a single band.

Non-negative Matrix Factorization and Its Extensions for Spectral Image Data Analysis

Scanning transmission electron microscopy combined with electron energy-loss spectroscopy and energy-dispersive X-ray spectroscopy is useful for analyzing chemical states and elemental components in a new material. Using these instruments, the spectra over the spatial grid points in a region of interest can be observed. This measurement technique is called spectral imaging (SI). Because of the large size of SI data, the analysis cost is a bottleneck in the evaluation process of the material. To reduce the analysis cost, machine learning techniques can be applied, which can automatically extract essential information from the data. This paper reviews our developed machine learning method, which is based on non-negative matrix factorization and its extensions. A spatial orthogonality constraint and a generalized noise model, which includes Gaussian and Poisson noise models, are introduced. Numerical experiments demonstrate the effectiveness and characteristics of our developed methods.

Development of a 4D Digital Phantom for Cone-Beam CT (CBCT) Imaging on the Varian On-Board Imager (OBI)

Mitigating effects of respiratory motion during image guided radiotherapy (IGRT) is important especially during thoracic and abdomen scanning protocols such as cone-beam CT (CBCT) imaging. However, the lack of ‘ground-truth’ in validating new algorithms has always been a challenge. The objective of this study is to outline the development of a novel 4D digital phantom for simulation of respiratory motion effects during CBCT image reconstruction based on Varian On-Board Imager (OBI): Half-Fan (HF) operating mode geometry. A set of actual 4D Magnetic Resonance (MR) data was used to develop the digital phantom. Firstly, the MR data sequencewasextendedto mimic a standard CBCT imaging acquisition protocol. Then, the images were segmented into several organs of interest and assigned with respective CT attenuation values. Subsequently, 2D projections of the developed digital phantom were simulated using the Varian OBI geometry. A Poisson noise model was also incorporated to the projection data to realistically simulate quantum noise that is present in an actual clinical environment. Three types of projections were then reconstructed using the standard 3D Feldkamp-Davis-Kress (FDK) algorithm, projections: without noise, with noise, and with noise and reconstructed with an additional Hann filter. As validation, the reconstructed images were compared against a single-frame of the developed phantom; quantitatively, using normalized root mean squared error (NRMSE) and qualitatively, using difference images. The results indicated that the phantom managed to display a consistent trend in modeling the effects of respiratory motion on the reconstructed images. On average, the NRMSE values for all three reconstructed images within the entire field-of-view (FOV) were evaluated to be approximately 29.07±0.22%. Nonetheless, the difference images indicated a large error in areas largely affected by respiratory motion. The NRMSE of a region-of-interest (ROI) near the affected area was evaluated as 51.26% that constitute to a significant +22.19% difference.

Compound Poisson Noise Sources in Diffusion-Based Molecular Communication

Diffusion-based molecular communication (DMC) is one of the most promising approaches for realizing nano-scale communications for healthcare applications. The DMC systems in in-vivo environments may encounter biological entities that release molecules identical to the molecules used for signaling as part of their functionality. Such entities in the environment act as external noise sources from the DMC system’s perspective. In this paper, the release of molecules by external bio-inspired noise sources is particularly modeled as a compound Poisson process. The impact of compound Poisson noise sources (CPNSs) on the performance of a point-to-point DMC system is investigated. To this end, the noise from the CPNS observed at the receiver is characterized. Considering a simple on-off keying modulation and formulating symbol-by-symbol maximum likelihood (ML) detector, the performance of the DMC system in the presence of the CPNS is analyzed. For the special case of CPNS in a high-rate regime, the noise received from the CPNS is approximated as a Poisson process whose rate is normally distributed. In this case, it is proved that a simple single-threshold detector is an optimal ML detector. Our results reveal that in general, adopting the conventional simple homogeneous Poisson noise model may lead to overly optimistic performance predictions, if a CPNS is present.

Bayesian Spectral Deconvolution Based on Poisson Distribution: Bayesian Measurement and Virtual Measurement Analytics (VMA)

In this paper, we propose a new method of Bayesian measurement for spectral deconvolution, which regresses spectral data into the sum of unimodal basis function such as Gaussian or Lorentzian functions. Bayesian measurement is a framework for considering not only the target physical model but also the measurement model as a probabilistic model, and enables us to estimate the parameter of a physical model with its confidence interval through a Bayesian posterior distribution given a measurement data set. The measurement with Poisson noise is one of the most effective system to apply our proposed method. Since the measurement time is strongly related to the signal-to-noise ratio for the Poisson noise model, Bayesian measurement with Poisson noise model enables us to clarify the relationship between the measurement time and the limit of estimation. In this study, we establish the probabilistic model with Poisson noise for spectral deconvolution. Bayesian measurement enables us to perform virtual and computer simulation for a certain measurement through the established probabilistic model. This property is called "Virtual Measurement Analytics(VMA)" in this paper. We also show that the relationship between the measurement time and the limit of estimation can be extracted by using the proposed method in a simulation of synthetic data and real data for XPS measurement of MoS$_2$.

Nonconvex Optimization for 3-Dimensional Point Source Localization Using a Rotating Point Spread Function

We consider the high-resolution imaging problem of 3D point source image recovery from 2D data using a method based on point spread function (PSF) engineering. The method involves a new technique, recently proposed by S.~Prasad, based on the use of a rotating PSF with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. Applications include high-resolution single molecule localization microscopy as well as the problem addressed in this paper on localization of space debris using a space-based telescope. The localization problem is discretized on a cubical lattice where the coordinates of nonzero entries represent the 3D locations and the values of these entries the fluxes of the point sources. Finding the locations and fluxes of the point sources is a large-scale sparse 3D inverse problem. A new nonconvex regularization method with a data-fitting term based on Kullback-Leibler (KL) divergence is proposed for 3D localization for the Poisson noise model. In addition, we propose a new scheme of estimation of the source fluxes from the KL data-fitting term. Numerical experiments illustrate the efficiency and stability of the algorithms that are trained on a random subset of image data before being applied to other images. Our 3D localization algorithms can be readily applied to other kinds of depth-encoding PSFs as well.

Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers

Ptychography has risen as a reference X-ray imaging technique: it achieves resolutions of one billionth of a meter, macroscopic field of view, or the capability to retrieve chemical or magnetic contrast, among other features. A ptychographyic reconstruction is normally formulated as a blind phase retrieval problem, where both the image (sample) and the probe (illumination) have to be recovered from phaseless measured data. In this article we address a nonlinear least squares model for the blind ptychography problem with constraints on the image and the probe by maximum likelihood estimation of the Poisson noise model. We formulate a variant model that incorporates the information of phaseless measurements of the probe to eliminate possible artifacts. Next, we propose a generalized alternating direction method of multipliers designed for the proposed nonconvex models with convergence guarantee under mild conditions, where their subproblems can be solved by fast element-wise operations. Numerically, the proposed algorithm outperforms state-of-the-art algorithms in both speed and image quality.

Measurement Extraction for a Point Target From an Optical Sensor

This paper considers the measurement extraction for a point target from an optical sensor's focal plane array (FPA) with a dead zone separating neighboring pixels. Assuming that the energy density of the target deposited in the FPA conforms to a Gaussian point spread function and that the noise mean and variance in each pixel are proportional to the pixel area (i.e., according to a Poisson noise model), we derive the Cramer–Rao lower bound (CRLB) for the covariance of the estimated target location. It is observed that that there is an optimal pixel size that minimizes the CRLB for a given dead-zone width, and the maximum likelihood estimator is shown to be efficient via Monte Carlo runs for moderate-to-large signal-to-noise ratios. The test statistic for target detection is derived and it is shown to be a matched filter at the estimated location. The distributions of the test statistic under both hypotheses are derived using some approximations. The detection probability is then obtained.

Blind Poissonian reconstruction algorithm via curvelet regularization for an FTIR spectrometer.

An FTIR spectrometer often suffers from common problems of band overlap and Poisson noises. In this paper, we show that the issue of infrared (IR) spectrum degradation can be considered as a maximum a posterior (MAP) problem and solved by minimized a cost function that includes a likelihood term and two prior terms. In the MAP framework, the likelihood probability density function (PDF) is constructed based on the observed Poisson noise model. A fitted distribution of curvelet transform coefficient is used as spectral prior PDF, and the instrument response function (IRF) prior is described based on a Gauss-Markov function. Moreover, the split Bregman iteration method is employed to solve the resulting minimization problem, which highly reduces the computational load. As a result, the Poisson noises are perfectly removed, while the spectral structure information is well preserved. The novelty of the proposed method lies in its ability to estimate the IRF and latent spectrum in a joint framework, thus eliminating the degradation effects to a large extent. The reconstructed IR spectrum is more convenient for extracting the spectral feature and interpreting the unknown chemical or biological materials.

Total Variation--Based Phase Retrieval for Poisson Noise Removal

Phase retrieval plays an important role in vast industrial and scientific applications. We consider a noisy phase retrieval problem in which the magnitudes of the Fourier transform (or a general linear transform) of an underling object are corrupted by Poisson noise, since any optical sensors detect photons, and the number of detected photons follows the Poisson distribution. We propose a variational model for phase retrieval based on a total variation regularization as an image prior and maximum a posteriori estimation of a Poisson noise model, which is referred to as “TV-PoiPR”. We also propose an efficient numerical algorithm based on an alternating direction method of multipliers and establish its convergence. Extensive experiments for coded diffraction, holographic, and ptychographic patterns are conducted using both real- and complex-valued images to demonstrate the effectiveness of our proposed methods.

Super-Resolution Image Reconstruction for High-Density Three-Dimensional Single-Molecule Microscopy

Single-molecule localization based super-resolution microscopy, by localizing a sparse subset of stochastically activated emitters in each frame, achieves subdiffraction-limit spatial resolution. Its temporal resolution, however, is constrained by the maximal density of activated emitters that can be successfully reconstructed. The state-of-the-art three-dimensional (3-D) reconstruction algorithm based on compressed sensing suffers from high computational complexity and gridding error due to model mismatch. In this paper, we propose a novel super-resolution algorithm for 3-D image reconstruction, dubbed TVSTORM, which promotes the sparsity of activated emitters without discretizing their locations. Several strategies are pursued to improve the reconstruction quality under the Poisson noise model, and reduce the computational time by an order-of-magnitude. Numerical results on both simulated and cell imaging data are provided to validate the favorable performance of the proposed algorithm.

Blind Detection for SPAD-Based Underwater VLC System Under P–G Mixed Noise Model

There is Poisson slot noise generated by the single photon avalanche diode (SPAD) and Gaussian thermal noise generated by the transimpedance amplifier, respectively, if an analog SPAD detector is applied in the visible light communication (VLC) system. In this letter, for the long distance SPAD-based underwater VLC system, a fast blind multiple-symbol detection (MSD) algorithm is developed under the pure Poisson noise model. And then, a blind MSD algorithm based on generalized maximum-likelihood rule under the Poisson-Gaussian (P-G) mixed noise model is proposed. A generalized Anscombe transform (GAT) is employed to make the P-G mixed noise model as an equivalent additive white Gaussian noise model, then an equivalent minimum Euclidean distance criterion is presented to formulate the optimization problem. Simulation results show that our blind MSD scheme can be identical to the performance under a GAT-based detector with perfect channel state information.

Sensitivity evaluation and selective plane imaging geometry for x-ray-induced luminescence imaging.

X-ray-induced luminescence (XIL) is a hybrid x-ray/optical imaging modality that employs nanophosphors that luminescence in response to x-ray irradiation. X-ray-activated phosphorescent nanoparticles have potential applications in radiation therapy as theranostics, nanodosimeters, or radiosensitizers. Extracting clinically relevant information from the luminescent signal requires the development of a robust imaging model that can determine nanophosphor distributions at depth in an optically scattering environment from surface radiance measurements. The applications of XIL in radiotherapy will be limited by the dose-dependent sensitivity at depth in tissue. We propose a novel geometry called selective plane XIL (SPXIL), and apply it to experimental measurements in optical gel phantoms and sensitivity simulations. An imaging model is presented based on the selective plane geometry which can determine the detected diffuse optical signal for a given x-ray dose and nanophosphor distribution at depth in a semi-infinite, optically homogenous material. The surface radiance in the model is calculated using an analytical solution to the extrapolated boundary condition. Y2 O3 :Eu3+ nanoparticles are synthesized and inserted into various optical phantom in order to measure the luminescent output per unit dose for a given concentration of nanophosphors and calibrate an imaging model for XIL sensitivity simulations. SPXIL imaging with a dual-source optical gel phantom is performed, and an iterative Richardson-Lucy deconvolution using a shifted Poisson noise model is applied to the measurements in order to reconstruct the nanophosphor distribution. Nanophosphor characterizations showed a peak emission at 611nm, a linear luminescent response to tube current and nanoparticle concentration, and a quadratic luminescent response to tube voltage. The luminescent efficiency calculation accomplished with calibrated bioluminescence mouse phantoms determines 1.06 photons were emitted per keV of x-ray radiation absorbed per g/mL of nanophosphor concentration. Sensitivity simulations determined that XIL could detect a concentration of 1mg/mL of nanophosphors with a dose of 1cGy at a depth ranging from 2 to 4cm, depending on the optical parameters of the homogeneous diffuse optical environment. The deconvolution applied to the SPXIL measurements could resolve two sources 1cm apart up to a depth of 1.75cm in the diffuse phantom. We present a novel imaging geometry for XIL in a homogenous, diffuse optical environment. Basic characterization of Y2 O3 :Eu3+ nanophosphors are presented along with XIL/SPXIL measurements in optical gel phantoms. The diffuse optical imaging model is validated using these measurements and then calibrated in order to execute initial sensitivity simulations for the dose-depth limitations of XIL imaging. The SPXIL imaging model is used to perform a deconvolution on a dual-source phantom, which successfully reconstructs the nanophosphor distributions.