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302      DIGITAL IMAGE PROCESSING

                                 • For microscope images where the visibility of an extended object is of interest, the
                                    S/N ratio can be calculated based on dozens or even hundreds of pixels and not on
                                    a per pixel basis. In this case, an object covering a large patch of pixels can have an
                                    S/N value that is hundreds of times larger than the S/N of an individual pixel. For-
                                    mulas for determining the S/N of extended objects are given in this chapter.

                                    The fundamental principles are described in a series of excellent articles by New-
                                berry (1991, 1994a,b, 1995a,b,c) and at the Web sites of CCD camera manufacturers.


                                S/N and the Precision of Numeric Values

                                One valuable benefit of S/N analysis is that you can state the confidence limits (in units
                                of standard deviations) that a signal stands out from the noise of the background. This
                                calculation is possible because S/N is the ratio of the object signal to the standard devi-
                                ation of the background signal, and because standard deviations are directly related to
                                 -values and percent confidence. Consider an image with an object signal of 100 elec-

                                trons and a noise of  100   10 e , giving an S/N ratio of 10. The inverse of this value,
                                N/S   0.1, is called the relative error—that is, the fractional component of the signal
                                that is noise. For our example, we would say that noise is 0.1 or 10% of the signal. Since
                                photon noise is the same as the standard deviation (SD) of the photon signal, we can say
                                that fluctuations in the signal greater than 10% can be detected at a confidence level of
                                68% (1 SD). Extending this further, by multiplying by 2 for 2 standard deviations, we
                                would say that fluctuations in the signal  20% are detected at the 95% confidence level
                                (2 SD), and so on. Another way of interpreting this is to say that given a relative error of
                                0.1, we can say we detect a signal at 10% precision, or that our ability to measure the
                                signal precisely is no better than 10%. The ability to set confidence limits has profound
                                implications, because it allows the investigator to calculate in advance how many elec-
                                trons (and therefore how many ADUs and what exposure time) are required to visualize
                                fluctuations of a certain percent at a specified confidence level.
                                    In summary, given S/N   10, taking (1/S/N)   2 SD   100   20%, we say that
                                fluctuations  20% are detected at the 95% confidence level.


                                Correlation of S/N with Image Quality

                                The relationships between S/N, % fluctuations detected at 95% confidence, and image
                                quality are given in Table 15-1. An object image with S/N   10–20 is poor and grainy-
                                looking, whereas the appearance of the same object at S/N   100 is good to excellent.



                                Effect of the Background Signal on S/N

                                In microscopy, the background signal can be large, sometimes  90% of the total signal
                                representing an object. Figure 11-10 demonstrates the significance of the background
                                signal in an immunofluorescence image. In most cases, photon noise from the back-
                                ground is the major source of noise, not the read noise of the camera. Thus, the S/N
                                equation includes a term for the background noise as will be recognized in the follow-
                                ing sections. As seen in Table 15-2, a high background signal reduces S/N considerably.
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