Poisson distributed noise

Even in the nominal absence of laser fluctuations or other imagedegrading aberrations, the number of photons that hit the detector during the data collection period of the image (i.e., the exposure time for a CCD image or the pixel dwell time for a confocal image) will contain considerable noise. The photon count x follows a Poisson distribution (Fig. 7.7A) with mean value fi as... 

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 ... 

Poisson-distributed noise, however, has an interesting characteristic for Poisson-distributed noise, the expected standard deviation of the data is equal to the square root of the expected mean of the data ([11], p. 714), and therefore the variance of the data is equal (and note, that is equal, not merely proportional) to the mean of the data. Therefore we can replace Var(A s) with Es in equation 47-17 and Var(A r) with Et ... 

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. 

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. 

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 ... 

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.

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

Figure 48-19 Relative absorbance precision for Poisson-distributed detector noise.

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... 

Previously, in the case of constant detector noise, we then set Var(A s) and Var(A r) equal to the same value. This is the point at which must we now depart from the previous derivation, since in the case of Poisson-distributed noise the sample and reference noise levels will rarely, if ever, be the same. However, we are fortunate in this case that Poisson-distributed noise has a unique and very useful property that we have indeed previously made use of the variance of Poisson-distributed noise is equal to the mean signal value. Hence we can substitute Es for Var(A s) and Er for Var(A r) ... 

Figure 50-27 Transmittance noise for Poisson-distributed data as a function of Et at different values of parameter T, from equation 50-134.

In the previous chapter, we have examined the situation in regard to determining the effect of noise on the computed transmittance. Now we wish to examine the behavior of the absorbance for Poisson-distributed noise when the reference signal is small. Our starting point for this is equation 51-24, which we derived previously [3] for the case of constant detector noise, but at the point we take it up the equations have not yet had any approximations, or any special assumptions relating to the noise behavior ... 

Figure 51-28 Absorbance noise for Poisson-distributed data at low values of the reference signal.

In the case of Poisson-distributed noise, we can do a systematic numerical calculation. The reasons we can do this now, when we could not do it for the Normal distribution, are the ones we have discussed previously ... 

Figure 51-29 Relative absorbance noise for Poisson-distributed data, determined by numerical computation using equation 51-77. Figure 51-29b is an ordinate expansion of Figure 51-29a. (see Color Plate 18)...

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... 

In the derivation of the transmittance noise in the case of Poisson-distributed noise, at this point we noted that the variances of Er and Es were proportional to Er and Es respectively. In the current case, the corresponding relationship is that the standard... 

Following our usual sequence, our next step here then, as it was for the previous two cases we treated (constant detector noise and Poisson-distributed noise), is to ascertain the effect of this noise on the expected value of computed transmittance. To do this we start with equation 53-5 (reference [2]) ... 

As we did in the analysis of Poisson-distributed noise, we compute the expected value of T as the weighted sum of the transmittance described by equation 49-5 (reference [10]) ... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

We define as modulation theory an approach to a noncanonical distribution based on the modulation of processes that with no modulation would yield canonical distributions. For instance, a double-well potential under the influence of white noise yields a Poisson distribution of the time of sojourn in the two wells [150]. In the case of a symmetric double-well potential we have... 

In dispersive spectrometers, the Rayleigh radiation may produce stray radiation in the entire spectrum, the intensity of which may be higher than that of the Raman lines. Interferometers transform the Poisson distribution of the light quanta of the Rayleigh radiation into white noise, which overlays the entire Raman spectrum. Therefore, all types of spectrometers must have means to reduce the radiant power of the exciting radiation accompanying the Raman radiation. 

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