Poisson Noise Model Research Articles

Total Variation Regularization Algorithms for Images Corrupted with Different Noise Models: A Review

Total Variation (TV) regularization has evolved from an image denoising method for images corrupted with Gaussian noise into a more general technique for inverse problems such as deblurring, blind deconvolution, and inpainting, which also encompasses the Impulse, Poisson, Speckle, and mixed noise models. This paper focuses on giving a summary of the most relevant TV numerical algorithms for solving the restoration problem for grayscale/color images corrupted with several noise models, that is, Gaussian, Salt & Pepper, Poisson, and Speckle (Gamma) noise models as well as for the mixed noise scenarios, such the mixed Gaussian and impulse model. We also include the description of the maximum a posteriori (MAP) estimator for each model as well as a summary of general optimization procedures that are typically used to solve the TV problem.

Gaussian Markov random field priors for inverse problems

In this paper, our focus is on the connections between the methods of (quadratic) regularization for inverse problems and Gaussian Markov random field (GMRF) priors for problems in spatial statistics. We begin with the most standard GMRFs defined on a uniform computational grid, which correspond to the oft-used discrete negative-Laplacian regularization matrix. Next, we present a class of GMRFs that allow for the formation of edges in reconstructed images, and then draw concrete connections between these GMRFs and numerical discretizations of more general diffusion operators. The benefit of the GMRF interpretation of quadratic regularization is that a GMRF is built-up from concrete statistical assumptions about the values of the unknown at each pixel given the values of its neighbors. Thus the regularization term corresponds to a concrete spatial statistical model for the unknown, encapsulated in the prior. Throughout our discussion, strong ties between specific GMRFs, numerical discretizations of diffusion operators, and corresponding regularization matrices, are established. We then show how such GMRF priors can be used for edge-preserving reconstruction of images, in both image deblurring and medical imaging test cases. Moreover, we demonstrate the effectiveness of GMRF priors for data arising from both Gaussian and Poisson noise models.

An Approach based on Run Length Count for Denoising the Kannada Characters

Optical Character Recognition (OCR) is one of the important fields in image processing and pattern recognition domain. OCR with high accuracy finds application in offices, banks, healthcare etc. The accuracy of the OCR is primarily dependent on the quality of the input image. So, to achieve high accuracy OCR we should provide a high quality image, which is free from different types of noises, degradation, skews etc. In this paper, we have made an attempt to remove the noise, which is present in the input image. A novel method based on run length count is proposed to denoise the images. In this approach first the noisy image is binarized. Based on the horizontal and vertical run length count, the noise in the image will be identified and eliminated. The algorithm is tested with noisy epigraphical document images, noisy printed document images. The effectiveness of the algorithm is verified with images having synthetic noise derived from Gaussian, Speckle and Poisson noise models. The experimental results show that the proposed method is efficient for noise elimination.

Noise reduction with low dose CT data based on a modified ROF model

In order to reduce the radiation exposure caused by Computed Tomography (CT) scanning, low dose CT has gained much interest in research as well as in industry. One fundamental difficulty for low dose CT lies in its heavy noise pollution in the raw data which leads to quality deterioration for reconstructed images. In this paper, we propose a modified ROF model to denoise low dose CT measurement data in light of Poisson noise model. Experimental results indicate that the reconstructed CT images based on measurement data processed by our model are in better quality, compared to the original ROF model or bilateral filtering.

Benchmarking Quickpalm and Other Molecule Localization Software for Super-Resolution Microscopy

Image processing software is used to estimate the positions of fluorescent molecules in super-resolution microscopy images. The rejection methods (for distinguishing single-molecule and multi-molecule images) and fitting algorithms (for estimating molecular positions) used determine the quality and reliability of the reconstructed images. We benchmarked and analyzed the ImageJ plug-in called QuickPALM. To do this, we used simulations of two-molecule images (with realistic point spread functions and Poisson noise model) as input. These images were generated with known molecule positions, to test the predictions of QuickPALM against the actual images. Our analysis showed that the accuracy dropped significantly when molecules were less than one wavelength apart, as there was rejection of multiple molecules in close proximity. With our results, we were also able to show that localization bias occurred towards the center of each pixel, distorting the overall image. The result was confirmed by simulating a PALM experiment on the US Air Force Test Pattern. We also wrote our own molecule localization plug-in (for ImageJ) utilizing second-moments (as a rejection criterion) and a fast maximum likelihood estimator (for estimating molecular positions), to compare with QuickPALM. We were able to show that the new plug-in had improved precision and accuracy when using the same single-molecule and two-molecule images. We also showed that the maximum likelihood estimator had no bias towards the center of each pixel when processing the US Air Force Test Pattern.

A doubly-stochastic model to analyze neuronal activity in the visual cortex

Event Abstract Back to Event A doubly-stochastic model to analyze neuronal activity in the visual cortex Robbe L. Goris1*, Eero P. Simoncelli1 and J. Anthony Movshon1 1 NYU, United States Sensory neurons encode information probabilistically: repeated stimulus presentations elicit variable firing. This variability is often described using a cascade model, in which spikes arise from a Poisson process whose rate is a deterministic function of the stimulus. However, it has recently been shown that time-dependent rate variability is a wide-spread phenomenon in cortex (Churchland et al 2010, 2011). Consequently, the Poisson noise model commonly underestimates the variability of visual cortical recordings, which can lead to systematic errors in inferring neuronal characteristics. We measured responses to a variety of stimuli in visual cortex in anesthetized monkeys. The fluctuations in neural responsiveness that typically occur over the timescale of these experiments are significantly greater than predicted by a Poisson model – estimated Fano factors as high as 10 occur in the acute preparation. We propose a doubly stochastic model, in which the stimulus-driven firing rate is modulated according to a stimulus-independent gamma-distributed random variable. This fluctuating rate generates spikes according to a Poisson process. Fitting the resulting mixture of Poisson processes to neural data reveals that the model is statistically superior for all neurons, and therefore provides an improved framework for analyzing neuronal tuning. The framework offers two further advantages over existing methods. First, it provides a natural means of estimating and tracking fluctuations in responsiveness (state changes) that occur during the course of an experiment. Second, it offers an efficient and accurate estimate of the upper bound on discrimination performance that can be supported by each neuron. Application of this new method can substantially improve the analysis of neuronal data, both in fitting explicit models and in assessing the limits of neuronal performance. References Churchland AK et al. Neuron 69: 818-831, 2011. Churchland MM et al. Nat Neurosci. 13: 369-378, 2010. Keywords: Neural coding, spike count distributions, Visual Cortex Conference: Bernstein Conference 2012, Munich, Germany, 12 Sep - 14 Sep, 2012. Presentation Type: Poster Topic: Data analysis, machine learning, neuroinformatics Citation: Goris RL, Simoncelli EP and Movshon J (2012). A doubly-stochastic model to analyze neuronal activity in the visual cortex. Front. Comput. Neurosci. Conference Abstract: Bernstein Conference 2012. doi: 10.3389/conf.fncom.2012.55.00222 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 11 May 2012; Published Online: 12 Sep 2012. * Correspondence: Dr. Robbe L Goris, NYU, New York City, United States, robbe.goris@nyu.edu Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Robbe L Goris Eero P Simoncelli J. Anthony Movshon Google Robbe L Goris Eero P Simoncelli J. Anthony Movshon Google Scholar Robbe L Goris Eero P Simoncelli J. Anthony Movshon PubMed Robbe L Goris Eero P Simoncelli J. Anthony Movshon Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Dual Norm Based Iterative Methods for Image Restoration

A convergent iterative regularization procedure based on the square of a dual norm is introduced for image restoration models with general (quadratic or non-quadratic) convex fidelity terms. Iterative regularization methods have been previously employed for image deblurring or denoising in the presence of Gaussian noise, which use L 2 (Tadmor et al. in Multiscale Model. Simul. 2:554---579, 2004; Osher et al. in Multiscale Model. Simul. 4:460---489, 2005; Tadmor et al. in Commun. Math. Sci. 6(2):281---307, 2008), and L 1 (He et al. in J. Math. Imaging Vis. 26:167---184, 2005) data fidelity terms, with rigorous convergence results. Recently, Iusem and Resmerita (Set-Valued Var. Anal. 18(1):109---120, 2010) proposed a proximal point method using inexact Bregman distance for minimizing a convex function defined on a non-reflexive Banach space (e.g. BV(?)), which is the dual of a separable Banach space. Based on this method, we investigate several approaches for image restoration such as image deblurring in the presence of noise or image deblurring via (cartoon+texture) decomposition. We show that the resulting proximal point algorithms approximate stably a true image. For image denoising-deblurring we consider Gaussian, Laplace, and Poisson noise models with the corresponding convex fidelity terms as in the Bayesian approach. We test the behavior of proposed algorithms on synthetic and real images in several numerical experiments and compare the results with other state-of-the-art iterative procedures based on the total variation penalization as well as the corresponding existing one-step gradient descent implementations. The numerical experiments indicate that the iterative procedure yields high quality reconstructions and superior results to those obtained by one-step standard gradient descent, with faster computational time.

This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms—Theory and Practice

Observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f*) from Poisson data (y) cannot be effectively accomplished by minimizing a conventional penalized least-squares objective function. The problem addressed in this paper is the estimation of f* from y in an inverse problem setting, where the number of unknowns may potentially be larger than the number of observations and f* admits sparse approximation. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods.

Performance Bounds for Expander-Based Compressed Sensing in Poisson Noise

This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on the signal reconstruction error of the proposed algorithm. We support our results with experimental demonstrations of reconstructing average packet arrival rates and instantaneous packet counts at a router in a communication network, where the arrivals of packets in each flow follow a Poisson process.

POSITIVELY CONSTRAINED TOTAL VARIATION PENALIZED IMAGE RESTORATION

The total variation (TV) minimization models are widely used in image processing, mainly due to their remarkable ability in preserving edges. There are many methods for solving the TV model. These methods, however, seldom consider the positivity constraint one should impose on image-processing problems. In this paper we develop and implement a new approach for TV image restoration. Our method is based on the multiplicative iterative algorithm originally developed for tomographic image reconstruction. The advantages of our algorithm are that it is very easy to derive and implement under different image noise models and it respects the positivity constraint. Our method can be applied to various noise models commonly used in image restoration, such as the Gaussian noise model, the Poisson noise model, and the impulsive noise model. In the numerical tests, we apply our algorithm to deblur images corrupted by Gaussian noise. The results show that our method give better restored images than the forward–backward splitting algorithm.

An open‐source deconvolution software package for 3‐D quantitative fluorescence microscopy imaging

Deconvolution techniques have been widely used for restoring the 3-D quantitative information of an unknown specimen observed using a wide-field fluorescence microscope. Deconv, an open-source deconvolution software package, was developed for 3-D quantitative fluorescence microscopy imaging and was released under the GNU Public License. Deconv provides numerical routines for simulation of a 3-D point spread function and deconvolution routines implemented three constrained iterative deconvolution algorithms: one based on a Poisson noise model and two others based on a Gaussian noise model. These algorithms are presented and evaluated using synthetic images and experimentally obtained microscope images, and the use of the library is explained. Deconv allows users to assess the utility of these deconvolution algorithms and to determine which are suited for a particular imaging application. The design of Deconv makes it easy for deconvolution capabilities to be incorporated into existing imaging applications.

Fourier block noise reduction: an adaptive filter for reducing Poisson noise in scintigraphic images

Both qualitative and quantitative analysis in nuclear medicine can be undermined by Poisson noise in low-count clinical images. Whilst the conventional smoothing filters are typically used do reduce noise, they also degrade the image structure. Fourier block noise reduction (FBNR) is an adaptive filtering approach, which attempts to reduce image noise and maintain image resolution and structure. Although a degree of automated flexibility is possible using conventional stationary pre-filtering, e.g. using a total image count-dependent Metz filter, resolution and contrast is degraded across the image. Adaptive non-stationary filtering has been applied by others in an attempt to maintain structure whilst reducing noise: instead of analysing the whole image, only a subset is used to determine each pixel's correction. Whilst the new software algorithm FBNR shares some common components with other adaptive non-stationary filters, it expressly includes the Poisson noise model within a simple and robust algorithm that can be applied to a diverse range of clinical studies. No additional artefacts were seen post-application of FBNR during evaluation using simulated and clinical images. Mean normalised error values indicate FBNR processing is equivalent to obtaining an unprocessed image with at least 2.5 times the number of counts.

Multigrid tomographic inversion with variable resolution data and image spaces

A multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed. Since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements, the proposed algorithm further reduces the computation relative to conventional multigrid approaches, which change only the image space resolution at coarse scales. The advantage is particularly important for data-rich applications, where data resolutions may differ for different scales. Applications of the approach to Bayesian reconstruction algorithms in transmission and emission tomography with a generalized Gaussian Markov random field image prior are presented, both with a Poisson noise model and with a quadratic data term. Simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed-grid iterative coordinate descent method and a multigrid method with fixed-data resolution.

Information content of the near field: two-dimensional samples.

Limits on the effective resolution of many optical near-field experiments are investigated. The results are applicable to variants of total-internal-reflection microscopy (TIRM), photon-scanning-tunneling microscopy (PSTM), and near-field-scanning-optical microscopy (NSOM) in which the sample is weakly scattering and the direction of illumination may be controlled. Analytical expressions for the variance of the estimate of the complex susceptibility of an unknown two-dimensional object as a function of spatial frequency are obtained for Gaussian and Poisson noise models, and a model-independent measure is examined. The results are used to explore the transition from near-zone to far-zone detection. It is demonstrated that the information content of the measurements made at a distance of even one wavelength away from the sample is already not much different from the information content of the far field.

Generalized approach for accelerated maximum likelihood based image restoration applied to three-dimensional fluorescence microscopy.

For deconvolution applications in three-dimensional microscopy we derived and implemented a generic, accelerated maximum likelihood image restoration algorithm. A conjugate gradient iteration scheme was used considering either Gaussian or Poisson noise models. Poisson models are better suited to low intensity fluorescent image data; typically, they show smaller restoration errors and smoother results. For the regularization, we modified the standard Tikhonov method. However, the generic design of the algorithm allows for more regularization approaches. The Hessian matrix of the restoration functional was used to determine the step size. We compared restoration error and convergence behaviour between the classical line-search and the Hessian matrix method. Under typical working conditions, the restoration error did not increase over that of the line-search and the speed of convergence did not significantly decrease allowing for a twofold increase in processing speed. To determine the regularization parameter, we modified the generalized cross-validation method. Tests that were done on both simulated and experimental fluorescence wide-field data show reliable results.

Total variation regulated EM algorithm [SPECT reconstruction

An iterative Bayesian reconstruction algorithm based on the total variation (TV) norm constraint is proposed. The motivation for using TV regularization is that it is extremely effective for recovering edges of images. This paper extends the TV norm minimization constraint to the field of SPECT image reconstruction with a Poisson noise model. The regularization norm is included in the OSL-EM (one step late expectation maximization) algorithm. Unlike many other edge-preserving regularization techniques, the TV based method depends one parameter. Reconstructions of computer simulations and patient data show that the proposed algorithm has the capacity to smooth noise and maintain sharp edges without introducing over/under shoots and ripples around the edges.

Joint estimation of object and aberrations by using phase diversity

The joint estimation of an object and the aberrations of an incoherent imaging system from multiple images incorporating phase diversity is investigated. Maximum-likelihood estimation is considered under additive Gaussian and Poisson noise models. Expressions for an aberration-only objective function that accommodates an arbitrary number of diversity images and its gradient are derived for the case of a Gaussian noise model. Expressions for the log-likelihood function and its gradient are presented for the case of Poisson noise. An expectation-maximization algorithm that enforces a nonnegativity constraint in a natural fashion is constructed for use in the Poisson noise case.

PPM Performance for Reed-Solomon Decoding Over an Optical-RF Relay Link

Consideration is presently being given to an optical-RF relay deep space communication link that transmits optical PPM data from spacecraft to orbital relay, then retransmits the data via microwave to ground. It is generally advantageous to use Reed-Solomon (RS) encoding over the PPM optical link for improved error correction. In this paper several demodulating schemes are considered for generating the RS symbols at the relay photodetector output, and the effect of each on overall RS decoding performance is computed. Both Poisson and Gaussian optical noise models are used in the evaluation. The effect on performance of bit errors in the relay downlink is also examined.