1. Introduction

Recent ground-breaking advances in fluorescent biomarker microscopy [1] have enabled the localization and tracking of single molecules at scales of mere tens of nanometers, scales that are 10–20x finer than those achieved by the more traditional diffraction limited microscopy. As the need for rapid localization and tracking has grown, the question of disambiguation of multiple point sources in close vicinity of one another has also become important. At the most elementary level, one must characterize well the minimum SNR requirements on finely resolving a single pair of equally bright point sources with a given sub-diffractive separation under varying levels of background, sensor, and signal noise. Such a characterization is critical to the design and construction of ever more capable microscopes that must perform these tasks with exquisite spatio-temporal resolution in our effort to understand biological processes at the cellular level.

We have recently analyzed [2] the minimum probability of error (MPE) in the localization of a single molecule to sub-diffractive scales. Our statistical approach employs the Bayesian multi-hypothesis testing (MHT) protocol that for the problem of transverse (2D) localization assigns a priori equal probability, $1/M_{\perp }^2$, that the molecule is located in any one of M_⊥ × M_⊥ possible equal-area square subcells into which a chosen square base-resolution cell may be subdivided. The image data, essentially the point-spread function (PSF) of the imaging system corrupted by a variety of noise sources, including background, sensor, and photon noise, add information about the source location, helping thus to reduce the MPE from $1-1/M_{\perp }^2$, that for the equi-probable prior alone, to a value that can be bounded below by means of the maximum a posteriori probability (MAP) criterion for choosing the decision regions. If the data are of a sufficiently high quality, then the MPE can dip below a certain threshold value, say 5% corresponding to a statistical confidence limit (CL) of 95%, for localizing the molecule with M_⊥ times higher linear precision along each transverse dimension. This 2D localization error analysis was extended to include the axial, or depth, dimension as well. A complete analysis, including its asymptotics, of the minimum signal strengths needed for super-localizing a point source to ever smaller spatial scales in both transverse and axial dimensions under varying background and sensor noise conditions was presented in [2].

The present paper specializes our MHT-based MPE analysis to the related problem of super-resolution imaging of multiple point sources that are closely separated and thus subject to being confused as a single point source of the same intensity as the combined intensity of the individual sources. This is particularly true if the spacing is well within the Abbe diffraction limit, 0.5λ/NA, of a microscope operating with numerical aperture NA and imaging wavelength λ. However, as a number of recent experimental results have shown [3–5], the diffraction limit furnishes a nominal benchmark at best, and improvements well beyond it are eminently possible based on diffraction-limited far-field image data.

Here we ask and answer, in the Bayesian MPE framework, the elementary question: What source brightness is needed to reduce the MPE below a given threshold for discriminating a single pair of closely spaced, equally bright point sources of a given separation from a single point source of the same brightness located at the midpoint of the line joining the pair? This defines a simpler problem of binary hypothesis testing within the Bayesian detection framework, avoiding a more involved problem in which the matter of such binary discrimination is coupled with the additional requirement of estimating the spacing between the point-source pair. The minimum signal strengths needed for this single source vs. pair source discrimination, as we shall see, are rather modest and consistent with growing experimental evidence for high-precision optical superresolution (OSR) of point sources.

Similar questions of source-pair OSR in the framework of Bayesian inference have been asked previously by others [6–9]. These analyses are more limited, however, because they either employ simpifying assumptions such as one-dimensional PDFs or do not account for non-Gaussian sources of noise or consider the full range of noise sources from additive sensor noise to signal and background-based shot noise comprehensively. Considerations of the minimum requirements for superresolution of closely spaced point sources have recently gained prominence in view of highly developed experimental technqiues for superresolution fluorescence microscopy [1] for localizing single biolabel fluorophors that are now quite the rage in live-cell biological and biomedical imaging.

The limits on the spatial resolution of a closely spaced pair of point sources have been examined previously [10] in the context of biomolecular microscopy by means of a non-Bayesian estimation-theoretic analysis involving the Cramér-Rao lower bound (CRB) on an unbiased estimation of their mutual spacing. This analysis, while exhaustive in terms of the orientation and the location of the midpoint of the separation vector near the in-focus plane of a conventional clear-circular-aperture imager, was largely numerical, providing only limited insights into the minimum requirements, e.g., on the combined source strength, to resolve a specific subdiffractive separation in the presence of varying amounts of background and sensor noise levels. Even more importantly, this analysis suffers from the same weakness as any other CRB-based analysis in requiring certain regularity conditions for its validity, including a non-vanishing local first-order derivative of the statistical distribution of the data with respect to (w.r.t.) the parameter being estimated. This specific condition breaks down when the sources are located along the optical axis and their separation vector is centered at the in-focus plane where the imager PSF has a vanishing first-order derivative w.r.t. the axial coordinate and the CRB thus in fact diverges [11].

The general OSR problem subsumes a large collection of well defined problems, a fact that merits some discussion here. The many claims and counter-claims of the possibility of OSR debated over the past several decades have as much to do with the complexity of proper analysis as with the nature of the specific OSR problem being considered. At the risk of over-simplification, we sharpen this debate with the mention of two specific OSR problems here. For one, the prohibitively difficult prospects of bandwidth extension beyond the diffraction limited cut-off by means of data inversion performed by imposing physical constraints [12–17] constitute a very different problem from that of superresolving point sources. The degree of bandwidth extension is typically logarithmic in the signal-to-noise ratio (SNR), a fact that is also well understood from Shannon’s channel capacity theorem [14]. But the knowledge that all sources in a scene are point-like, rather than being extended with arbitrary variations of brightness across them, is sufficiently information-bearing that superresolving them to well within the diffraction limit requires, as we hope to prove, rather more modest source-signal strengths. Independent confirmations of this claim are provided by a number of studies involving actual computational algorithms, including those presented in [18–22]. These algorithms make use, in essential ways, of the point-like extension of sources being reconstructed from noisy, filtered image data.

It is also worth noting that Lucy’s early analysis [23] of the point-source OSR problem in which he derived a prohibitive eighth-power scaling of the minimum source strength w.r.t. the pair separation was based on his asking a different question: What source strength is needed to disambiguate this pair from a single extended, spatially symmetric source of the same source strength and the same second-order spatial moment of the intensity distribution? Here the alternative (“null”) hypothesis involves an extended source of the same root-mean-squared (RMS) width as the separated source-pair in the main hypothesis. It is this additional constraint of equal RMS width placed on the two hypotheses that renders pair OSR so much harder to achieve in his analysis.

We begin this paper with a brief review of our earlier analysis [2] of the MPE as applied to the problem of super-localizing a single point source in three dimensions. The specialization of this multi-hypothesis testing (MHT) based analysis and its asymptotic considerations to the present problem of pair OSR in the single transverse plane of best focus are summarized in Sec. 3, where we further specialize the problem to a Gaussian model PSF. This restriction to the Gaussian PSF shape, rather than the Bessel-function-based Airy-disk PSF, is not essential but does enable a more transparent and complete analytical treatment. The biomolecular localization microscopy literature [1, 24] on the specific form of the in-focus PSF is divided, since the presence of any residual aberrations in the optics and the dipolar emission pattern [25] from a flourophore typically predispose a single-molecule image to deviate significantly from the Airy form, so the more generic bell-shaped form is as good, typically, as any other to assume. Our considerations may, however, be adapted to a completely general image form by means of a Taylor expansion to the second order along the lines of [9, 26]. For the Gaussian PSF, we derive detailed expressions for the MPE and also their asymptotic form for signal and background dominated regimes, including an important logarithmic correction to the quartic scaling law for the minimum photon strength needed for a sought pair OSR enhancement factor in the strong-signal regime. In Sec. 4, we present and discuss the results of a numerical evaluation of our analytical expressions, involving certain numerically ill-behaved logarithmic integrals derived in Sec. 3, for arbitrary relative strengths of the signal and background levels. We conclude the paper in Sec. 5 with a brief summary and outlook of the work.

2. A brief review of our asymptotic Bayesian error analysis

Let the conditional statistics of the data X be specified by a probability density function (PDF), P(x|m), conditioned on the validity of a specific hypothesis ℋ_m, labeled by an integer m, m = 1,...,M. In the Bayesian framework, the posterior distribution, P(m|x), quantifies the information carried by the data about the likelihood of the hypothesis ℋ_m having given rise to the observed data. The maximum a posteriori (MAP) estimator provides one fundamental metric of minimum error, namely the MPE, in correctly identifying the operative hypothesis under all possible observations,

When many observations are involved, as, e.g., for the typical image dataset consisting of N pixels with N >> 1, $P_e^{(\mathit{min})}$ may be evaluated approximately under conditions of moderate to high signal-to-noise ratio (SNR) by replacing the inner sum over m′ in Eq. (3) by a single term m̃ that labels the decision region “closest” to ℛ_m in the following sense:

For the case of image data acquired under combined signal photon-number fluctuations, background fluctuations, and sensor read-out noise, the following pseudo-Gaussian conditional data PDF accurately describes the statistics of data at sufficiently large photon numbers:

For this problem, we derived in [2] the following expression valid in the asymptotic limit of high photon numbers, many pixels, and hypotheses that are hard to discriminate from one another because the corresponding mean signal vectors that separate their statistics are very similar:

3. Bayesian MPE analysis for the point-source-pair superresolution problem

As we have indicated earlier, the problem of discriminating a pair of closely spaced point sources from a single point source is a binary hypothesis testing (BHT) problem in which the sum (9) is limited to two terms only, m = 1 and m = 2 corresponding to the cases of a single point source and a pair of point sources, respectively. Since the passage from the double sum (3) to the single sum (5) is exact for the BHT problem, any error involved in the expression (8) is only due to any asymptotic analysis of the full MPE contribution from each decision region. In the asymptotic limit and for only two terms in the sum (8), the two norms, ‖U₁‖₂ and ‖U₂‖ ₂, as defined by relation (9) are essentially the same, namely

In the following analysis we shall restrict our attention to a circular Gaussian-shaped PSF that is azimuthally invariant in the ξη image plane,

The two hypotheses in the present problem are characterized by the mean signal matrices, x₁ and x₂, with elements given in terms of the discrete PSF (14) by

 

Fig. 1 The image of a pair of point sources of equal brightness under the Gaussian-shaped PSF (14), for (a) d/w = 0.5; (b) d/w = 2; and (c) d/w = 3.

Download Full Size | PDF

With the elements of the data-mean matrices given by Eqs. (16), let us now expand each term in the numerator Q of the MPE expression, defined by relation (11), in a power series in K, interchange the order of the resulting infinite sum and the pixel sum, and then perform the pixel sum by means of its continuous version. We arrive in this way at the following expression for Q:

The quantity q_k may be evaluated recursively for increasing values of k by a simple mathematical trick. Note first that u²q_k(u) is simply the integral, from 0 to u, of the sum

These expressions for the various q’s when computed and substituted into expression (19) evaluate Q fully. Note that while the power series expansions on which expression (19) is based are valid only when |u| < 1, expressions (22)–(24) are valid for arbitrary values of u except where they fail to be analytic. This is guaranteed by the principle of analytic continuation [27].

Expression (11) for R, which determines the denominator of the MPE expression (10), may be evaluated quite similarly as Q. In the small-spacing limit, d/w << 1, one may again substitute KH_1ij for x_1ij and KH_2ij for x_2ij, in that sum expression for R, then expand each term in the resulting expression in powers of K, interchange the power series sum with the pixel sum, and evaluate the latter, which in view of expressions (16) for H_1ij and H_2ij involves only Gaussian functions, approximately by converting it into an integral over the full ξη plane. Term by term, the Gaussian integrals are the same ones we evaluated for Q, so their lowest-order limiting form in powers of d/w is again quartic, and the following expression for R results:

Like the q’s, the r’s too may be evaluated recursively by means of an analogous integral relation they obey, namely

In terms of the quantities q_k and r_k we have just calculated, we may now express the argument (10) of the erfc function in the MPE expression (12) via relations (19) and (26) as

3.1. Photon-signal-dominated regime, u >> 1

In this regime, we may use asymptotic forms of the various p’s and q’s occurring in expression (32). The following asymptotic forms, as we show in Appendix C, are obtained in the limit u → ∞:

Let us set the threshold on the MPE at p as the minimum requirement on the fidelity of Bayesian discrimination between the single-point-source and symmetric binary-source hypotheses. A typical value taken for p is 0.05, corresponding to a statistical CL of 95%, but one can adjust p according to the stringency of the application. Equating the RHS of relation (12) to p and solving for the argument of the erfc function in terms of its inverse function, which we may denote erfc⁻¹, we require that ${\Vert U_0\Vert }_2/\sqrt{2}$ have the minimum value

Expression (36) exhibits a nearly quartic scaling of the minimum photon number needed to discriminate a point-source pair from its single-point-source equivalent, as a function of the inverse spacing d⁻¹ between the source pair. Specifically, this scaling is given by the ratio (w/d)⁴, namely the fourth power of the ratio of the characteristic width of the PSF and the spacing between the source pair being resolved, that is modified by a logarithmic dependence on the SNR. The latter factor increases relatively slowly with increasing K_min, but tends to exacerbate somewhat a purely quartic scaling of K_min as higher and higher values of the pair-OSR factor, w/d, are sought. Such logarithmic factors have not been predicted by previous researchers [7,9,23,28], as their analyses have not been sufficiently comprehensive in treating the full range of combined noise statistics. Further, the analyses of [7,9] assumed essentially white Gaussian additive noise, rather than photon-number-dominated Poisson noise treated here, and their power SNR scales quadratically, rather than linearly, with the photon number. This difference of the noise statistics accounts for their quadratic, rather than our essentially quartic, scaling of K_min w.r.t. the degree of pair OSR sought.

In spite of its modification by a logarithmic correction, the quartic scaling of K_min is considerably more modest than the eighth-power scaling predicted by Lucy [23] for a similar problem. As we have argued earlier, the assumption of a single point source in our analysis, rather than a single equivalent but extended source in Lucy’s analysis, for the “null” hypothesis of an unresolved source potentially provides more constraining information that an algorithm can make essential use of to resolve the source pair at a more modest signal strength.

3.2. Background-noise-dominated regime, u << 1

In the limit u → 0, the various q and r functions tend to the same values, q_k = r_k = (1/2)^k, so the ratio involving them in expression (32) simplifies to (3/8)^1/2

4. Numerical results and discussion

In Figs. 2(a)–2(c), we plot the exact result (12) for the MPE, evaluated numerically by substituting expression (32) into its right-hand side, as a function of the photon number K for three different background variance levels, b, and illustrate the transition from the quadratic scaling of the background-dominated regime to the approximately quartic scaling of the signal-dominated regime of operation of our Gaussian-PSF-based imager. In each of these plots, the sensor noise variance σ² was set equal to 1, while the ratio of the characteristic area under the PSF to the pixel area, 2πw²/ΔA, was set at 100. For the lowest background level, b = 10², that considered in Fig. 2(a), u >> 1 over much of the plotted range of K, and the plot displays the decrease of the MPE with increasing signal strength appropriate to the signal-dominated regime in which the approximation (34) for the argument of the erfc function is quite accurate. The different curves in the plots refer to different values of the OSR ratio w/d, starting at 2 (labeled as 2x) and increasing to 20 (labeled as 20x), with the higher w/d values requiring higher source strengths to bring the error down to the statistical CL threshold. The bottom figure, Fig. 2(c), displays the MPE behavior in the opposite limit of extreme-background-dominated regime in which at the largest source strength, K = 10¹¹, the ratio u is 1, and for all others it is less than 1, being in fact considerably smaller than 1 over the range of source strengths over which the MPE decreases from its highest value of 0.5 toward the CL threshold of 0.05. The upper right figure, Fig. 2(b), displays the results for the case for which its left half represents the background-dominated regime of operation and the right half the signal-dominated regime. Note an appreciable but expected rightward shift in the MPE curves as the spatially uniform background level increases from one figure to the next. As the background level increases further, it becomes increasingly harder, requiring increasingly higher source strengths, to discriminate between the null and alternative hypotheses.

 

Fig. 2 A plot of MPE vs. signal strength for three different background levels. (a) b = 10²; (b) b = 10⁶; and (c) b = 10⁹.

Download Full Size | PDF

Experimentally, the high background levels considered in Figs. 2(b) and 2(c) are unrealistic. Nevertheless, they were needed here to illustrate the various asymptotic scaling laws in the signal and background dominated regimes even at the largest OSR factor, namely 20x, considered here and of much interest in single-molecule microscopy. In most practical situations, the background level is unlikely to exceed 10³ counts per pixel, for which achieving moderate to large OSR would mean inevitably the signal-dominated regime of operation.

The variation of the minimum photon number required to achieve the CL threshold as a function of the OSR ratio, w/d, is plotted in Fig. 3 for the three different background levels of Fig. 2. The values of this number were simply read off from the intersections of the MPE curves in Figs. 2 with the CL threshold line. The doubly logarithmic plots show the expected nearly quartic and quadratic scaling with w/d for the photon-dominated and background-dominated regimes, while for the intermediate background level, b = 10⁶, the scaling is nearly quadratic for the smaller d/w values for which K_min is smaller than or comparable to (2πw²/ΔA), or 100, times b, but for larger w/d, the slow change of slope consistent with the nearly quartic scaling of the signal-dominated regime can be discerned. The transition from slope-2 to slope-4 is rather gradual, taking several decades of increase in K_min to complete.

 

Fig. 3 Log-log plots of the minimum source strength, K_min, vs. the OSR ratio, w/d, for the same three different background levels as in Figs. 2.

Download Full Size | PDF

It is rather remarkable that one can achieve any OSR at all when the spatially uniform mean background level and its shot-noise fluctuations dominate, as in Fig. 2(c), the signal part of the data with its information-bearing spatial variations that are consistent with the background-free image of a separated source pair. The higher source strengths needed when even relatively small OSR ratios are sought, as seen in Fig. 3, confirm the difficulty of attaining OSR in the extreme-background-dominated regime.

We also note that increasing the background and sensor noise variances, specifically the sum (b + σ²), can in fact help, rather than hurt, the degree of OSR that is achievable at sufficiently high source flux. This is seen from Fig. 4 by comparing the solid and dashed plots of MPE vs. K, corresponding to the background-variance values b = 1 and b = 100, respectively, for each of four different values of the OSR factor considered here. We note that for sufficiently large values of K and the degree of OSR, a given value of the MPE is associated with a smaller K for the larger value of (b + σ²) (dashed curves), which for σ = 1 is approximately 50 times larger than its smaller value (solid curves). This seemingly paradoxical result supports the argument that because of its spatial non-uniformity across the pixel array, the signal-dependent part of the total noise variance per pixel, which is simply the mean signal count as given in (16), undermines the ability of the imaging system to superresolve a pair of closely spaced sources presumably more than the signal-independent part of that noise variance, namely (b + σ²), that is spatially uniform across the sensor array. This would imply that the degree of achievable OSR at a fixed source strength could in fact increase as the signal-dependent noise variance decreases even as the signal-independent noise variance increases. This result is also consistent with the fact that when the total noise variance is spatially uniform and signal independent, as for read-noise dominated detection, the minimum photon number, K_min, required for a specified degree of OSR has a far more modest quadratic, rather than quartic, scaling [9].

 

Fig. 4 Plots of MPE vs. signal strength for two different background levels, b = 1 (solid curves) and b = 100 (dashed curves), for four different OSR factors, 4x, 8x, 12x, and 20x, and sensor noise variance, σ² = 1.

Download Full Size | PDF

Effect of changing pixel size

With changing pixel size, two competing effects control the Bayesian fidelity of pair OSR as a function of the source strength. At low signal strengths, if the detector pixels are so small that the signal falling on each pixel is overwhelmed by the background and sensor read noise, assuming the two noise variances to be fixed independent of pixel size, then the MPE for deciding between the single-source and double-source hypotheses is expected to be large. The opposite is true when the pixel size is large. The error balance reverses at high enough source strengths for which the signal dominates the noise, pixel by pixel. In this limit, the smaller the pixel size, the better the maximum possible optical resolution between close-by point sources because of the higher native resolution of such a detector, with the result being a smaller MPE in discriminating between the two hypotheses.

We display this behavior in Figs. 5(a)–(b) where for five different values of the OSR factor, namely 4, 8, 12, 16, and 20, we plot the MPE as a function of the source strength K for three different pixel sizes, namely $\delta a\stackrel{\text{def}}{=}\mathrm{\Delta }A/(2\pi w^2)$ being equal to 1/100, 4/100, and 9/100, corresponding to pixels with widths in the ratio 1:2:3, respectively. The two figures display results for two different read noise and background noise levels. The cross-over (CO) of the MPE curves is easily seen in these plots to occur at a common source strength K_CO that is empirically given by the expression

 

Fig. 5 A plot of MPE vs. signal strength for two different noise levels, (a) b = 1, σ = 1 and (b) b = 100, σ = 10, for three different pixel sizes, ΔA/(2πw²) equal to 1/100, 4/100, and 9/100. The dashed (red), solid (blue), and dot-dashed (black) curves correspond to the plots for the smallest to the largest pixel sizes, respectively. Five different values of the OSR factor, namely 4x, 8x, 12x, 16x, and 20x, were considered in these plots. The vertical dashed line passes through the common intersection points of the curves grouped by the OSR factor.

Download Full Size | PDF

It is worth noting that the sensor read and background noise variances per pixel occur together in all our expressions as σ² + b. In the pseudo-Gaussian approximation for the PDF for combined sensor, background, and signal photon-number fluctuations, the various noise sources are equivalent. However, that is not so in a more accurate treatment needed to incorporate either background or signal photon-number fluctuations at much lower strengths of order 10 or lower, where their discrete Poisson statistics can no longer be approximated well by a continuous Gaussian PDF.

5. Conclusions

In this paper we have presented a detailed Bayesian error analysis based on binary-hypothesis testing (BHT) to derive expressions for the minimum combined source strength needed to superresolve a pair of closely spaced point sources located in the plane of best focus from their image formed under a Gaussian PSF approximation. The statistical metric we have used here to determine this minimum value of the source strength needed for pair OSR is the minimum probability of error (MPE) for a successful BHT protocol. Specifically, we considered the pair to be resolved if the MPE for the associated BHT problem falls below a small threshold value, taken here to be 0.05. We considered a variety of operating conditions characterized by arbitrary values of the source signal strength, background count, and sensor noise variance.

Our detailed calculations predict an approximately quartic dependence of the minimum source strength on the reciprocal of the pair spacing in the regime where the signal dominates the background. This scaling is in fact slightly steeper than quartic via a multiplicative correction that is logarithmic in the ratio of the signal and background levels. In the opposite limit of the background-dominated regime of operation, this scaling is more modestly quadratic.

We have noted an interesting, seemingly paradoxical competition between the spatially nonuniform, signal dependent noise variance and the spatially uniform sensor-background noise variance that seems to improve the fidelity of OSR when the latter noise variance is increased while reducing the former by reducing the source strength in the signal-dominated regime. This may have implications for the design of super-resolution imaging instruments.

We have also observed a sensitivity vs. resolution trade-off when the sensor-background noise level per pixel is held fixed but the size of the sensor pixel is changed. In the over-sampled limit considered here, the asymptotically valid MPE vs. K plots exhibit a cross-over (CO) behavior with increasing K in which at values of K less the CO threshold a larger-pixel detector has a higher OSR fidelity than a smaller-pixel detector, since larger pixels see a larger signal than smaller pixels. At K larger than the CO threshold, the behavior reverses, however, since smaller pixels allow for a higher sampled resolution without a commensurate sensitivity loss at such high source brightnesses.

The results of the paper can be used to rank different OSR algorithms if by extensive simulation one could compile reliable statistics on the probability of error in discriminating between a single point source and a source pair. By comparing the probability of error for each algorithm with the minimum probabilty of error calculated in the present paper, a user could then learn about how well different superresolution algorithms perform relative to one another.

With increasing demand for efficient super-localization of multiple point sources with overlapping PSFs, we must extend our asymptotic error analysis to treat the problem of super-resolution of more than two closely spaced sources. While challenging, such analysis is likely to be essentially predicated on the pair-OSR analysis presented here, as the N-source problem can be reduced, at least in the asymptotic limit, to N(N − 1)/2 subproblems, each treating a single pair OSR problem. The MPE for such a problem will be dominated in this limit by the MPE corresponding to the source pair having the smallest separation. An adaptation of the M-ary hypothesis testing approach presented in the companion paper [2] will be needed to perform a more accurate asymptotic analysis of the multi-source OSR problem.

The problem of pair OSR when the sources are along the optical axis and thus in a common line of sight (LOS) is expected to entail much steeper scaling of minimum source power with inverse spacing in both the signal-dominated and background-dominated regimes. The main reason for this expected difference of on-axis, or longitudinal, OSR from transverse OSR treated in the present paper is that in the former case, unlike the latter, the PSF has no first order sensitivity on the spacing, which must imply more stringent requirements on any LOS pair OSR. However, the mathematical divergences seen in local first-order analyses using the Cramér-Rao lower bound on unbiased estimation are expected to be moderated in a Bayesian analysis that includes in effect non-vanishing higher-order sensitivities of the data PDF on the parameter being estimated, namely the pair spacing. This problem will be treated in a subsequent paper.