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We propose a novel mortality improvement model with the difference of death counts follows the Skellam distribution. We extend Mitchell et al. (2013) by considering the difference in Poisson death counts instead of the ratio of subsequent mortality rate, which does not have a known distribution. We derive the iterative estimators of the model from the Skellam distribution. Our model can employ maximum likelihood estimation for estimation issues such as missing data and provides a better fit than Mitchell et al. (2013) . Using English and Wales mortality rate age 0-89 data during 1950-2016, the model estimate suggests that the age-dependent mortality improvement is slower than the benchmark, which coincides with a recent observation by Office for National Statistics (2018) . The forecasting performance outperforms the Poisson and M10 model. We make inferences on the price of longevity swaps and analyze how the volatility shock of mortality improvement affects the premium of longevity swaps.

Using European soccer data sets, which contain data related to common European soccer leagues, players basic information, and teams’ goals, etc., this paper analyzes the characteristics of European soccer and players, explores data visualization regarding European soccer, and makes predictions of results of matches. Based on Python 3 and some of the packages inside, such as numpy, the author improves the data set to make it clear and user-friendly. Visualizations of data and basic statistics, including Poisson Distribution, are then utilized to determine the results. Finally, this paper analyzes the attacking and defending abilities of different leagues and teams in Europe, ascertains distributions of players’ attributes, and predicts match results by using Poisson distribution and Skellam Distribution. Generally, this paper analyzes data from leagues to matches to players. All these analyses are meaningful for the public to understand the characteristics of European soccer and the world behind the numbers.

The degree that the environment determines what genes contribute towards adaptation is a fundamental question in microbial evolution. Microbial populations are often experimentally passaged in different environments and sequenced in order to identify candidates for adaptation in a particular environment. However, there remains the need to develop an appropriate statistical framework to identify genes that acquired more mutations in one environment over the other (i.e., divergent evolution). Here we demonstrate how the evolutionary outcomes among replicate populations in the same environment, known as parallel evolution, can be leveraged to construct an intuitive statistical test for identifying the genes that contribute towards divergent evolution. To accomplish this task, we examined publicly available evolve-and-resequence experiment datasets and found that the distribution of mutation counts among genes can be predicted using an ensemble of independent Poisson random variables. Building on this result, we propose that the degree of divergent evolution at a given gene between populations from two different environments can be modeled as the difference between two Poisson random variables, known as the Skellam distribution. We then propose and apply a statistical test to identify specific genes that contribute towards divergent evolution. IMPORTANCE: There is currently no existing framework that can be leveraged to identify genes that contribute towards divergent evolution in microbial evolution experiments. To correct for this absence, we investigated the distribution of mutation counts among genes in order to identify an appropriate null model. Our observations suggest that divergent evolution within a given gene can be modeled as the difference in the total number of mutations observed between two environments. This quantity is described by a probability distribution known as the Skellam distribution, providing an appropriate statistical test for researchers seeking to identify the set of genes that contribute towards divergent evolution in evolution experiments.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

Photon-limited Poisson image denoising is urgent demand in many application fields, but is particularly challenging because the image structures are often damaged seriously. The effective using of the image prior is very important for improving the quality of the denoised image. In this paper, we exploit the prior of inner statistical relationships among pixels in local spatial neighborhood of the Poisson noisy image and introduce a new Skellam distribution based inner interaction potential function as a distance measurement between the pixels. In the framework of conditional random field (CRF) modeling, we propose a novel Poisson-Skellam distribution based regularization CRF model for photon-limited Poisson noise removal. Using the alternating direction method of multipliers technique, we extend our method to a flexible plug-and-play scheme in which we can combine powerful Gaussian denoising method to improve the denoising performance better.

The Inter-Symbol Interference (ISI) in Diffusive Molecular Communication (DMC) is unavoidable due to the stochastic behavior of molecular Brownian motion in a biological environment. In this paper, we analyze the channel capacity of DMC model by considering the ISI from all the previous time slots and the channel transmission probability in each time slot. We first use the Poisson distribution to approximate the Binomial distribution in order to obtain the total number of molecules received in the current time slot. Then we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. On this basis, we formulate the mutual information and bit error probability. Our numerical results are presented to show the channel capacity can be maximized with optimal lower bit error probability by controlling the channel transmission probability in each time slot and using the Skellam distribution. More importantly, under the same physical parameters, a higher capacity is achieved with using fewer molecules in our work.

The aim of this paper is to study and develop Bayesian models for the analysis of volleyball match outcomes as recorded by the set-difference. Due to the peculiarity of the outcome variable (set-difference) which takes discrete values from $-3$ to $3$, we cannot consider standard models based on the usual Poisson or binomial assumptions used for other sports such as football/soccer. Hence, the first and foremost challenge was to build models appropriate for the set-differences of each volleyball match. Here we consider two major approaches: a) an ordered multinomial logistic regression model and b) a model based on a truncated version of the Skellam distribution. For the first model, we consider the set-difference as an ordinal response variable within the framework of multinomial logistic regression models. Concerning the second model, we adjust the Skellam distribution in order to account for the volleyball rules. We fit and compare both models with the same covariate structure as in Karlis & Ntzoufras (2003). Both models are fitted, illustrated and compared within Bayesian framework using data from both the regular season and the play-offs of the season 2016/17 of the Greek national men's volleyball league A1., IMA Journal of Management Mathematics, 2021

This paper considers the modeling of nonstationary integer valued time series with conditional heteroskedasticity using Skellam distribution. Two approaches of estimation of the model’s parameters are treated and discussed. The obtained results are verified through some numerical simulation. In addition, the proposed model is applied to real time series.

The pixel values of images taken by an image sensor are said to be corrupted by Poisson noise. To date, multiscale Poisson image denoising techniques have processed Haar frame and wavelet coefficients—the modeling of coefficients is enabled by the Skellam distribution analysis. We extend these results by solving for shrinkage operators for Skellam that minimizes the risk functional in the multiscale Poisson image denoising setting. The minimum risk shrinkage operator of this kind effectively produces denoised wavelet coefficients with minimum attainable $L^{2}$ error.

In this work we investigate the problem of private statistical analysis of time-series data in the distributed and semi-honest setting. In particular, we study some properties of Private Stream Aggregation (PSA), first introduced by Shi et al. 2011. This is a computationally secure protocol for the collection and aggregation of data in a distributed network and has a very small communication cost. In the non-adaptive query model, a secure PSA scheme can be built upon any key-homomorphic weak pseudo-random function as shown by Valovich 2017, yielding security guarantees in the standard model which is in contrast to Shi et al. We show that every mechanism which preserves \((\epsilon ,\delta )\)-differential privacy in effect preserves computational \((\epsilon ,\delta )\)-differential privacy when it is executed through a secure PSA scheme. Furthermore, we introduce a novel perturbation mechanism based on the symmetric Skellam distribution that is suited for preserving differential privacy in the distributed setting, and find that its performances in terms of privacy and accuracy are comparable to those of previous solutions. On the other hand, we leverage its specific properties to construct a computationally efficient prospective post-quantum protocol for differentially private time-series data analysis in the distributed model. The security of this protocol is based on the hardness of a new variant of the Decisional Learning with Errors (DLWE) problem. In this variant the errors are taken from the symmetric Skellam distribution. We show that this new variant is hard based on the hardness of the standard Learning with Errors (LWE) problem where the errors are taken from the discrete Gaussian distribution. Thus, we provide a variant of the LWE problem that is hard based on conjecturally hard lattice problems and uses a discrete error distribution that is similar to the continuous Gaussian distribution in that it is closed under convolution. A consequent feature of the constructed prospective post-quantum protocol is the use of the same noise for security and for differential privacy.

Two-stage non-local methods represented by the Poisson non-local means (PNLM) method [1] and the non-local principal component analysis (NLPCA) [2] method perform well for photon-limited Poisson image reconstruction, where patch-similarity computation in the non-local reconstruction stage is guided and affected by a pre-reconstructed image obtained at the first stage. In this paper, we propose a new method to provide a better pre-reconstructed image with low computational cost. Firstly, we propose an adaptive method for fitting the linear relationship between image intensity and Poisson parameter; Secondly, we obtain an initial estimated image according to this relationship. Lastly, we obtain new pre-reconstructed image by adjusting the initial estimation according to Skellam distribution. Numerical experiments show that our method can provide a better pre-reconstructed image, and therefore can improve the performance of the PNLM method and reduce the computational cost of the NLPCA method efficiently.

What are the most surprising, most impressive and most exciting matches in the history of the highest division of English football? Fans, clubs, critics and broadcasters often attempt to interpret and compare the outcomes of football matches. This can be useful as a means of building a reputation for and interest in a particular club, match or competition. However, such analysis is often subjective with fan bias, personal preference and the prize on offer influencing expectation and interpretation of the results. In this paper statistical measures are developed to identify the most surprising, most impressive and most exciting outcomes from an objective ex post point of view. To achieve this it is assumed that goal scoring in soccer can approximately be considered as a Poisson process. First, the most surprising outcomes are estimated as those with the lowest Poisson probability to occur. Second, the most impressive outcomes are identified by their so-called Skellam percentile position. This measure varie...

The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

Skellam's name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson's correlation between the components make it simple to estimate the para-meters via the method of moments. More complex formulas for Kendall's tau and Spearman's rho are also provided.

Financial prices are usually modelled as continuous, often involving geometric Brownian motion with drift, leverage and possibly jump components. An alternative modelling approach allows financial observations to take integer values that are multiples of a fixed quantity, the ticksize - the monetary value associated with a single change during the price evolution. In the case of high-frequency data, the sample exhibits diverse trading operations in a few seconds. In this context, the observables are assumed to be conditionally independent and identically distributed from either of two flexible likelihoods: the Skellam distribution - defined as the difference between two independent Poisson distributions - or a mixture of Geometric distributions. Posterior inference is obtained via adaptive Gibbs sampling algorithms. Comparisons of the models applied to high-frequency financial data is provided.

Rankings based on counts are often presented to identify popular items, such as baby names, English words, or Web sites. This article shows that, in some examples, the number of correctly identified items can be very small. We introduce a standard error versus rank plot to diagnose possible misrankings. Then to explain the slowly growing number of correct ranks, we model the entire set of count data via a Zipf–Poisson ensemble with independent Xi ∼ Poi(Ni − α) for α > 1 and N > 0 and integers i ⩾ 1. We show that as N → ∞, the first n′(N) random variables have their proper order relative to each other, with probability tending to 1 for n′ up to (AN/log (N))1/(α + 2) for A = α2(α + 2)/4. We also show that the rate N 1/(α + 2) cannot be achieved. The ordering of the first n′(N) entities does not preclude for some interloping m > n′. However, we show that the first n″ random variables are correctly ordered exclusive of any interlopers, with probability tending to 1 if n″ ⩽ (BN/log (N))1/(α + 2) for any B < A....

By the laws of quantum physics, pixel intensity does not have a true value, but should be a random variable. Contrary to the conventional assumptions, the distribution of intensity may not be an additive Gaussian. We propose to directly model the intensity difference and show its validity by an experimental comparison to the conventional additive model. As a model of the intensity difference, we present a Skellam distribution derived from the Poisson photon noise model. This modeling induces a linear relationship between intensity and Skellam parameters, while conventional variance computation methods do not yield any significant relationship between these parameters under natural illumination. The intensity-Skellam line is invariant to scene, illumination, and even most of camera parameters. We also propose practical methods to obtain the line using a color pattern and an arbitrary image under natural illumination. Because the Skellam parameters that can be obtained from this linearity determine a noise distribution for each intensity value, we can statistically determine whether any intensity difference is caused by an underlying signal difference or by noise. We demonstrate the effectiveness of this new noise model by applying it to practical applications of background subtraction and edge detection.

The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of Haar-Fisz variance-stabilized data, along with accompanying low-complexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam shrinkage estimators both for the standard univariate wavelet test functions as well as a variety of test images taken from the image processing literature, confirming that they offer substantial performance improvements over existing alternatives., 27 pages, 8 figures, slight formatting changes; submitted for publication

We propose a method that compares Poisson distributed outcomes. Our method uses the exact distribution of the difference between two Poisson variables to calculate the sample size required to detect a given difference with prespecified power. When the true difference between the two Poisson rates is more than 1.2 units, the number of subjects and events needed at the desired power and Type I error rate is 5–10% less than that computed by simulation based on the normal approximation method. The normal approximation method is more comparable to the exact sample size method when the difference between the rates is less than 1.2 units. The proposed method is more intuitive, efficient, and less subjective than the normal approximation method. A simple code is developed in R to estimate the sample size and critical values.

For its unprecedented level of spatial resolution, chromatin immunoprecipitation combined with λ exonuclease digestion followed by high-throughput sequencing (ChIP-exo) has the potential to replace ChIP-seq as the standard approach for genome-wide mapping of protein-DNA interactions. In this assay, the midpoint between the strand-specific paired peaks, formed in the forward and reverse strands, is typically delimited by the exonuclease stop-sites, within which the protein-binding events are located. Although numerous algorithms have been developed for peak-calling in ChIP-seq data, none of them is fully adjusted for the analysis of ChIP-exo. This is because those statistical models do not make use of ChIP-exo’s strand-specificity for the identification of protein-DNA binding sites. Here, we present the CexoR algorithm, which aims to ease the analysis of replicated ChIP-exo data in BAM alignment format. The detection algorithm relies on the Skellam distribution (cross-correlation of two Poisson distributions) to calculate probabilities of consecutive punctate-sources of read-enrichment located nearby at Watson-and-Crick strands. ChIP-exo peak-pairs are identified and ranked by their irreproducible discovery rate estimated across biological replicates, and finally reported in BED format files. CexoR can potentially be applied to other ChIP-exo-based protocols, such as ChIP-nexus. Availability and implementation: CexoR has been implemented in R, and is freely available at http://bioconductor.org.

Paired count data usually arise in medicine when before and after treatment measurements are considered. In the present paper we assume that the correlated paired count data follow a bivariate Poisson distribution in order to derive the distribution of their difference. The derived distribution is shown to be the same as the one derived for the difference of the independent Poisson variables, thus recasting interest on the distribution introduced by Skellam. Using this distribution we remove correlation, which naturally exists in paired data, and we improve the quality of our inference by using exact distributions instead of normal approximations. The zero-inflated version is considered to account for an excess of zero counts. Bayesian estimation and hypothesis testing for the models considered are discussed. An example from dental epidemiology is used to illustrate the proposed methodology.

We explore the influence of deuteron formation in the late stage of nucleus-nucleus reactions on the fluctuations observed in the final net-proton yields around midrapidity. At each investigated energy, the produced (anti-)proton yield at chemical freeze-out is assumed to fluctuate according to a Poisson distribution and in each event the probability for deuteron formation by coalescence is proportional to $(dN_{\mathrm{proton}}/dy)^2$. The protons that are then clustered in deuterons are usually not included in the experimental measurement of the net-proton fluctuations, therefore, we subtract these clustered protons from the final state proton number for the calculation of the net-proton fluctuations (the same is done in the anti-proton sector). Due to the non-linear deuteron formation probability the resulting distribution is not a Skellam distribution, but shows the interesting feature of a decrease in the kurtosis $\kappa \sigma^2$ and a local maximum in the skewness $S \sigma$ observables as the collision energy decreases., Comment: 5 pages, 5 figures

We introduce a dynamic statistical model for Skellam distributed random variables. The Skellam distribution can be obtained by taking differences between two Poisson distributed random variables. We treat cases where observations are measured over time and where possible serial correlation is modeled via stochastically time-varying intensities of the underlying Poisson counts. The likelihood function for our model is analytically intractable and we evaluate it via a multivariate extension of numerically accelerated importance sampling techniques. We illustrate the new model by two empirical studies and verify whether our framework can adequately handle large data sets. First, we analyze long univariate high-frequency time series of U.S. stock price changes, which evolve as discrete multiples of a fixed tick size of one dollar cent. In a second illustration, we analyze the score differences between rival soccer teams using a large, unbalanced panel of seven seasons of weekly matches in the German Bundesliga.In both empirical studies, the new model provides interesting and non-trivial dynamics with a clear interpretation.

In this paper, we consider the Riemann–Liouville fractional integral N α , ν ( t ) = 1 Γ ( α ) ∫ 0 t ( t − s ) α − 1 N ν ( s ) d s , where N ν ( t ) , t ≥ 0 , is a fractional Poisson process of order ν ∈ ( 0 , 1 ] , and α > 0 . We give the explicit bivariate distribution Pr { N ν ( s ) = k , N ν ( t ) = r } , for t ≥ s , r ≥ k , the mean E N α , ν ( t ) and the variance V ar N α , ν ( t ) . We study the process N α , 1 ( t ) for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on N 1 , 1 ( t ) are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalized harmonic numbers are discussed.

The Poisson distribution is extended over the set of all integers. The motivation comes from the many reflected versions of the gamma distribution, the continuous analog of the Poisson distribution, defined over the entire real line. Various mathematical properties of the extended Poisson distribution are derived. Estimation procedures by the methods of moments and maximum likelihood are also derived with their performance assessed by simulation. Finally, a real data application is illustrated.

We study a minimal model of active transport in crowded single-file environments which generalizes the emblematic model of single-file diffusion to the case when the tracer particle (TP) performs either an autonomous directed motion or is biased by an external force, while all other particles of the environment (bath) perform unbiased diffusions. We derive explicit expressions, valid in the limit of high density of bath particles, of the full distribution P((n))(X) of the TP position and of all its cumulants, for arbitrary values of the bias f and for any time n. Our analysis reveals striking features, such as the anomalous scaling [proportionality] √[n] of all cumulants, the equality of cumulants of the same parity characteristic of a Skellam distribution and a convergence to a Gaussian distribution in spite of asymmetric density profiles of bath particles. Altogether, our results provide the full statistics of the TP position and set the basis for a refined analysis of real trajectories of active particles in crowded single-file environments.

In fluoroscopy quantum noise is dominant with respect to other common noise sources, whose effects can be often neglected. Quantum noise is originated by the limited number of photons (low-dose X-ray) involved in fluoroscopic image formation; this noise is commonly modeled as Poisson distributed. Estimation of noise-parameters is required for evaluation of X-ray digital imaging sensors and in several image processing applications (e.g. denoising). The first aim of this work is to validate the analytically derived noise-parameter models by real fluoroscopic image sequences, also considering non-linear gray level mappings currently employed by fluoroscopic devices. A plain procedure for estimation of noise pixel-intensity variance as a function of mean pixel-intensity, which does not require specific test-objects but only some images screening a still scene, has been provided. Besides, a procedure for noise-parameter estimation by differencing fluoroscopic static images has been proposed. It computes image-pair differences, estimates concise parameters of the resulting Skellam distribution and, then, quotes Poisson noise-parameters. Image sequences of a simple step-phantom, acquired with a conventional fluoroscopic device, were utilized for performing the noise measurements. Experimental results confirmed a great agreement of the measured noise-parameters with those analytically derived and the possibility to use static images to estimate noise.

Owing to the stochastic nature of discrete processes such as photon counts in imaging, a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. Certain wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts, taking in the simplest case the so-called Skellam distribution. We show that a Skellam mean estimator provides a Poisson intensity estimation method based on shrinkage of filterbank coefficients, and a means of estimating the risk of any Skellam mean estimator is derived in closed form under a frequentist model.

Owing to the stochastic nature of discrete processes such as photon counts in imaging, real-world data measurements often exhibit heteroscedastic behavior. In particular, time series components and other measurements may frequently be assumed to be non-iid Poisson random variables, whose rate parameter is proportional to the underlying signal of interest-witness literature in digital communications, signal processing, astronomy, and magnetic resonance imaging applications. In this work, we show that certain wavelet and filterbank transform coefficients corresponding to vector-valued measurements of this type are distributed as sums and differences of independent Poisson counts, taking the so-called Skellam distribution. While exact estimates rarely admit analytical forms, we present Skellam mean estimators under both frequentist and Bayes models, as well as computationally efficient approximations and shrinkage rules, that may be interpreted as Poisson rate estimation method performed in certain wavelet/filterbank transform domains. This indicates a promising potential approach for denoising of Poisson counts in the above-mentioned applications.

Conventional edge detectors suffer from inherent image noise and threshold determination. In this paper, we propose a noble edge detector based on the noise distribution for CCD or CMOS cameras. By assuming the dominant photon noise, we model the distribution of intensity differences between two neighborhood pixels. Since it is well known that photon noise follows a Poisson distribution, we introduce a Skellam distribution, which is the difference of two Poisson random variables. We show experimentally that the Skellam distribution can be used to model the noise distribution of pixels that are captured from the same scene radiance. For estimating the noise distribution given a single pixel, we find the important property that the Skellam parameters are linearly related to the intensity value of pixels. This linearity enables us to determine noise parameters according to the intensity value. In addition, parameters of the line are preserved under illumination, scene and camera setting changes except for only a gain change. Based on the noise distributions, we calculate intensity allowances of three channels for each pixel given a confidence interval. We propose a noble edge detector by skipping a pre-processing step of conventional Gaussian smoothing which is the main obstacle for robust and accurate edge detection. If the difference of intensity exceeds the intensity allowance at least in a single channel, the in- between pixel is marked as an edge pixel. We demonstrate that without conventional Gaussian smoothing the noise-model based approach can automatically extract the fine details of image structures, such as edge and corners, independent of camera setting.

In this paper, we introduce the Skellam distribution as a sensor noise model for CCD or CMOS cameras. This is derived from the Poisson distribution of photons that determine the sensor response. We show that the Skellam distribution can be used to measure the intensity difference of pixels in the spatial domain, as well as in the temporal domain. In addition, we show that Skellam parameters are linearly related to the intensity of the pixels. This property means that the brighter pixels tolerate greater variation of intensity than the darker pixels. This enables us to decide automatically whether two pixels have different colors. We apply this modeling to detect the edges in color images. The resulting algorithm requires only a confidence interval for a hypothesis test, because it uses the distribution of image noise directly. More importantly, we demonstrate that without conventional Gaussian smoothing the noise model-based approach can automatically extract the fine details of image structures, such as edges and corners, independent of camera setting.