Poisson-noise case

Equation 48-105 is the closest we can come to the form of equation 42-37, so compare the functions describing the relative precision for the constant-noise case to that of the Poisson-noise case. 

In future improvements in technology may mean that that read noise no longer is the dominant noise source, and Poisson noise arising from the quantum nature of light is in fact the limiting factor. In this case the variance of the centroid noise is equal to. 

From Figure 47-17 we note several ways in which the behavior of the transmittance noise for the Poisson-distributed detector noise case differs from the behavior of the constant-noise case. First we note as we did above that at T = 0 the noise is zero, rather than unity. This justifies our earlier replacement of E0 by E0 for both the sample and the reference readings. 

Now that we have completed our expository interlude, we continue our derivation along the same lines we did previously. The next step, as it was for the constant-noise case, is to derive the absorbance noise for Poisson-distributed detector noise as we previously did for constant detector noise. As we did above in the derivation of transmittance noise, we start by repeating the definition and the previously derived expressions for absorbance [3],... 

And again our departure from the derivation for the constant detector noise case is to note and use the fact that for Poisson-distributed noise, Var(A ,r) = Er and Var(A s) = Es ... 

Thus, in the constant-noise case the absorbance noise is again proportional to the N/S ratio, although this is clearer now than it was in the earlier chapter there, however, we were interested in making a different comparison. The comparison of interest here, of course, is the way the noise varies as T varies, which is immediately seen by comparing the expressions in the radicals in equations 47-94 - for Poisson noise and 47-96. 

We present the variation of absorbance noise for the two cases (equations 47-94 - for Poisson noise and 47-96, corresponding to the Poisson noise and constant noise cases) in Figure 47-18. While both curves diverge to infinity as the transmittance —0 (and the absorbance - oo), the situation for constant detector noise clearly does so more rapidly, at all transmittance levels. 

Figure 47-18 Comparison between absorbance noise for the constant-detector noise case and the Poisson-distributed detector noise case. Note that we present the curves only down to 7 = 0.1, since they both asymptotically oo as r 0, as per equations 94 and 96.

Another characteristic of scintillation noise is that, since it represents the amount of energy in the optical beam, it can never attain a negative value. In this respect it is similar to the Poisson distribution, which also can never attain a negative value. On the other hand, since it is a continuous distribution it will behave the same way as the constant-noise case in regard to achieving an actual zero any given reading can become... 

So let us begin our analysis. As we did for the analysis of shot (Poisson) noise [8], we start with equation 52-17, wherein we had derived the expression for variance of the transmittance without having introduced any special assumptions except that the noise was small compared to the signal, and that is where we begin our analysis here as well. For the derivation of this equation, we refer the reader to [2], So, for the case of noise proportional to the signal level, but small compared to the signal level we have... 

Table V. Exponent of Dependence of SNR, MDC, and MAT on Three Instrumental Variables for Signal-Limited and Background-Limited Detection Cases in the Poisson Noise Limit...

In the case of a Poisson noise process, the smoothing function Q is set to zero and is minimized by minimizing the likeUhood fimction. The solution is straightforward and is computed by setting the derivative of the likelihood to zero, with respect to o , for fixed i, resulting in... 

In the case of microscopy, Poisson noise is signal-dependent. It is therefore important to take into account the additive background b, mainly produced by the dark-field current of the detector, giving... 

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

Fig. 7.7. Effects of Poisson photon noise on calculated SE and FRET values. (A) Statistical distribution of number of incoming photons for the mean fluorescence intensities of 5,10, 20, 50, and 100 photons/pixel, respectively. For n = 100 (rightmost curve), the SD is 10 thus the relative coefficient of variation (RCV this is SD/mean) is 10 %. In this case, 95% of observations are between 80 and 120. For example, n — 10 the RCY has increased to 33%. (B) To visualize the spread in s.e. caused by the Poisson distribution of pixel intensities that averaged 100 photons for each A, D, and S (right-most curve), s.e. was calculated repeatedly using a Monte Carlo simulation approach. Realistic correction factors were used (a = 0.0023,/ = 0.59, y = 0.15, <5 = 0.0015) that determine 25% FRET efficiency. Note that spread in s.e. based on a population of pixels with RCY = 10 % amounts to RCV = 60 % for these particular settings Other curves for photon counts decreasing as in (A), the uncertainty further grows and an increasing fraction of calculated s.e. values are actually below zero. (C) Spread in Ed values for photon counts as in (A). Note that whereas the value of the mean remains the same, the spread (RCV) increases to several hundred percent. (D) Spread depends not only on photon counts but also on values of the correction...

Tossing a mental coin, the decision was to analyze the case of noise proportional to the square root of the signal. This, as you will recall, is Poisson-distributed noise, characteristic of the noise encountered when the limiting noise source is the shot noise that occurs when individual photons are detected and represent the ultimate sensitivity of the measurement. This is a situation that is fairly commonly encountered, since it occurs, as mentioned previously, in UV-Vis instrumentation as well as in X-ray and gamma-ray measurements. This noise source may also enter into readings made in mass spectrometers, if the detection method includes counting individual ions. We have, in... 

From this point, up to and including equation 47-17, the derivation is identical to what we did previously. To save time, space, forests and our readers patience we forbear to repeat all that here and refer the interested reader to Chapter 41 referenced as [2] for the details of those intermediate steps, here we present only equation 47-17, which serves as the starting point for the departure to work out the noise behavior for case of Poisson-distributed detector noise ... 

Furthermore, one of the steps taken during the omitted sequence between equation 47-4 and equation 47-17 was to neglect AEt compared to ET. Clearly this step is also only valid for large values of Er, both for the case of constant detector noise and for the current case of Poisson-distributed detector noise. Therefore, from both of these considerations, it is clear that equation 47-88 and Figure 47-17 should be used only when Ex is sufficiently large for the approximation to apply. 

F(x), here, is (Es + AEs)/ Er + A )), as we just noted. In the previous case, the weighting function was the Normal distribution. Our current interest is the Poisson distribution, and this is the distribution we need to use for the weighting factor. The interest in our current development is to find out what happens when the noise is Poisson-distributed, rather than Normally distributed, since that is the distribution that applies to data whose noise is shot-noise-limited. Using P to represent the Poisson distribution, equation 49-59 now becomes... 

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