Model-based two-object resolution from observations having counting statistics

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Abstract

This paper considers two-object resolution from the viewpoint of model fitting theory. The studied experiment consists in counting events, for example, an electron hitting a detector pixel. It is stated that the precision and the accuracy with which the locations of the objects can be estimated will determine the attainable resolution. Two different approaches are followed. For both, the special case of Gaussian peaks is further investigated. The first approach leads to the maximally attainable precision. It is shown that this precision is determined by a certain factor, which is a function of the distance of the peaks, their widths and the number of counts. This factor will be called the resolution factor. The influence of each of the quantities involved is determined by the way they enter this factor. The second approach defines a probability of resolution, i.e., the probability that the maximum likelihood estimates of the locations will be distinct. It is shown that the resolution factor, which resulted from the first approach, also determines the probability of resolution.

Introduction

The concept of resolution, the ability to distinguish details, is an important quality measure for imaging systems. Higher resolution often means that the interpretation of the image is easier. In the past, many resolution criteria have been proposed for two-point resolution, i.e., the ability to resolve two adjacent points. These criteria are mostly used in diffraction limited systems: systems where the image of a point is spread by a point-spread function. One of the earliest and most famous criteria is that of Rayleigh [1]. According to Rayleigh, resolution is limited by the width of the main lobe of the point spread function. However, this criterion is only based on the limitations of the human visual system, and does not take into account, for example, the presence of noise. Another classical criterion is that of Rose [2], which approaches resolution in terms of dose, i.e., the number of counts per area. An extensive survey on the concept of resolution can be found in [3]. In the present paper, two-object resolution is studied, i.e., the ability to resolve two objects of equal size and intensity. A global approach is followed: to investigate the attainable resolution, the size of the objects, the distance between them, as well as the dose are taken into account. It is assumed that a mathematical model for the objects exists and that this model is known, for example, the objects could be atoms, described by identical gaussian probability density functions. The locations of the objects are unknown and appear in the model as parameters. The ultimately attainable resolution is then achieved by using model-fitting techniques: the quality of the estimates of the location parameters will determine the attainable resolution. In this paper we follow two different approaches to investigate this quality.

The first approach is based on the available asymptotic parameter estimation theory. A survey of this theory can be found in handbooks on statistics and parameter estimation, for example, in [4], [5]. With the aid of this theory we deduce a lower limit on the attainable precision an unbiased estimator of the distance between the peaks can achieve. It is shown that this precision is determined by a factor that is a function of the distance of the peaks, their widths and the number of counts. This function will be called the resolution factor. The influence of each of the quantities involved is determined by the way they enter this factor. From this factor it can be seen, for instance, that the variance of an unbiased estimator will grow drastically if the distance between the peaks is decreased below a critical value. As a result, the estimated locations will no longer have meaningful values.

In the second approach, we investigate the possible estimates given by the maximum likelihood estimator, one of the most important estimators. It turns out that, for two closely located objects, collapse of the two objects can occur, i.e., for particular sets of observations, the estimated distance between the objects is exactly zero. This is due to a change of structure in the maximum likelihood criterion under the influence of the observations. This kind of structure change has been described in [6], [7], [8], [9] for the special case of a least-squares estimator. A rule is derived to calculate the probability that such a collapse will occur (this probability for resolution is similar to the one described in [10] for least-squares estimators).

In these two different approaches, the first related to precision (Standard deviation) and the second to accuracy (bias), the same expression, containing the total number of counts and the distance and width of the objects, appears. The paper is organized as follows. Section 2 gives a short introduction on the theory of parameter estimation and model fitting, which is used in this paper. In Section 3, the attainable precision of an unbiased estimator of the distance parameter is studied. In Section 4, the behavior of the maximum likelihood estimator is studied with the aid of catastrophe theory. In Section 5, a number of numerical examples are discussed.

Section snippets

Parameter estimation and model fitting

Consider an experiment that consists of counting events, for example, an electron hitting a pixel. The events are distributed over a number of intervals, described by {xi,i…,M}, by a probability density function (pdf). The observations are given by {ni,i=1,…,M}, where ni describes the number of counts or events in the interval xi. The total number of counts is defined by N, with N=∑Mi=1ni. The probability that an event occurs in the interval xi will be denoted by p(xi;θ), which is a pdf

Resolution in terms of statistical precision

We want to establish how close two objects in an experiment may get, before they can no longer be separated. If there were no noise present, one could resolve the objects even when they were extremely close together. However, in reality there is almost always a certain amount of noise present; consequently, the resolvability of the objects will be limited. The question to be answered then is: when are two objects resolved? One possible approach to this problem is looking at resolution in terms

Behavior of the ML-estimator

In this section, the behavior of the ML-estimator is investigated if the distance between the objects becomes small. In simulated experiments, it was noticed that there were two distinct possibilities: the ML-estimates for the locations of two objects coincided, meaning that their estimated distance was zero, or the ML-estimates were distinct, meaning that there were two equivalent maxima of the likelihood function as a function of the distance (equivalent because of the symmetry of the model).

Discussions and experiments

First, the CRLB of the distance parameter, Eq. (16), and its approximation, Eq. (33), will be investigated for Gaussian peaks, by means of an example. The resolution factor R, i.e., the standard deviation of A2 divided by A2 itself (sometimes called the relative error), is the most interesting quantity to investigate. We define Rexact as the non-approximated factor R, i.e., the root of Eq. (16), divided by A2. Fig. 1 shows Rexact and R as a function of the ratio A2/σ0, Fig. 1(a), and as a

Conclusion

Whether two objects, especially Gaussian peaks, can be resolved or not depends on a resolution factor, which is a function of the total number of counts, the distance between the peaks, and the width of the peaks. The ultimate precision with which the locations of the peaks can be estimated and the probability that the maximum likelihood solutions will coincide can be calculated using these quantities. From this, it is possible to deduce rules of thumb for the attainable resolution. For

Acknowledgments

E.B. was supported by the following projects of the Antwerp University: GOA Vision and GOA Generic Optimization.

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