Digital Camera Technologies for Scientific Bio-Imaging.
Part 3: Noise and Signal-to-Noise Ratios
James Joubert, 1 Yashvinder Sabharwal 2 and Deepak Sharma 1
1. Photometrics and QImaging, 3440 East Britannia Drive, Tucson, AZ 85706-5006, USA
Tel: +1 520 889 9933 Email email@example.com
2. Solexis Advisors LLC, 6103 Cherrylawn Circle, Austin, TX 78723, USA
Mob: +1 512 534 8340 Email: firstname.lastname@example.org
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Follow these links to see Parts 1, 2, 3 and 4 in this series:
Digital Camera Technologies for Scientific Bio-Imaging. Part 1: The Sensors
Digital Camera Technologies for Scientific Bio-Imaging. Part 2: Sampling and Signal
Digital Camera Technologies for Scientific Bio-Imaging. Part 3: Noise and Signal-to-Noise Ratios
Digital Camera Technologies for Scientific Bio-Imaging. Part 4: Signal-to-Noise Ratio and Image Comparison of Cameras
Parts 1 and 2 of this series [1,2] provided an introduction to the three most common camera technologies, CCD, EMCCD, and CMOS sensors, and compared their performance with metrics ranging from sampling requirements to light throughput at the detector. Throughout this series, the discussion has focused on utilizing optical parameters of an imaging system in the selection of a camera for particular bio-imaging applications.
Here, in Part 3, the focus shifts to noise parameters with an introduction and discussion of various noise sources. In this installment, the noise properties of different scientific cameras are combined with the photon throughput model developed in Part 2 to generate signal-to-noise ratio (SNR) profiles for the three sensor types as a function of exposure time. The SNR of an image is a well-accepted metric of image quality and it is used in this discussion to allow quantitative comparison of the sensors.
In Part 4, in an effort to identify which sensor would be best employed for particular applications, the SNR analysis will be expanded to compare sensor performance for applications with different levels of signal (e.g. low-light level applications versus high-light level applications).
The concept of noise is often misunderstood. This term is often used to represent almost any error in the measured number of photons φ at a pixel in an image. However, not all errors should be considered as noise. When an error is deterministic, it is not noise. Noise is a probabilistic phenomenon where there is uncertainty in the occurrence of the event being measured. Examples of such phenomena in microscopy and imaging include the emission of photons during fluorescence, the detection of photons by the sensor, and the reading out of photoelectrons during the digitization process.
The uncertainty leads to variations from one measurement to the next, and this variation is the noise inherent in the camera system. The noise can be numerically specified as the standard deviation σ of the distribution describing the probabilistic behavior of the system. The errors due to noise cannot be eliminated; they can only be reduced. A familiar technique for reducing noise is data averaging. For imaging applications, the noise at each pixel can theoretically be reduced by the square root of the number of measurements averaged together. While this is fairly true for shot noise and CCD read noise, there can be significant deviations from this for CMOS read noise averaging .
Sensors such as CCDs and scientific CMOS have numerous sources of noise. The major sources of noise are dark noise σd , readout noise σr, and shot noise σn, which is the noise inherent to the light signal itself. There are other minor sources of noise which are very small relative to these noise sources and are therefore negligible for the purposes of this article.
The noise sources discussed in Table 1 (below) are generally considered to be independent of each other. This assumption simplifies the calculation to arrive at the total noise based on their individual contributions. Equation 1 describes how the total noise can be calculated from the individual
σ2 = σd2 + σr2 + σn2 + ..... (1)
Photon shot noise is the fluctuation in the image photons themselves. Due to the probabilistic nature of the individual photons that comprise the light flux hitting the detector, the intensity of light follows a Poisson distribution, even with a perfectly stable light source. The width of this distribution is characterized by the standard deviation, which is equal to the square root of the average number of photons and scales as such with intensity. Of the three noise sources listed in Table 1, only shot noise increases with the signal incident on the sensor. Therefore, when the light levels are sufficiently high, shot noise outweighs the other noise sources and the image quality is negligibly affected by dark noise and read noise.
Dark current is the signal that electrons generate by thermal excitation instead of by photoexcitation. Just as the amount of signal generated increases with longer exposures to light, dark current also increases with long exposures. Additionally, larger pixels increase the likelihood of dark current electrons being generated.
Being thermally generated, dark current can be reduced by cooling. From an image processing perspective, dark current can be subtracted out using a dark image. However, the dark noise, the inherent random fluctuation in this dark current cannot, and is approximately equal to the square root of the dark current . The SNR analysis provided later in this article includes the exposure time for the very reason that dark noise will have an increased effect for longer exposure times.
READ NOISE: The Difference Between CCD and CMOS
The third type of noise associated with a camera is read noise, and this is the primary source of noise at very low light levels. In CCD imaging sensors, datasheets typically list a single read noise for the CCD. In fact, this is an average value since read noise typically follows a Gaussian distribution across all the pixels in the sensor. This is characteristic of a single data digitizer with a single Gaussian noise distribution associated with it (Figure 1). To obtain this specification, the read noise is usually characterized by measuring the standard deviation of a bias image, an image with zero exposure time and no incident light. This value is in intensity units, also called grey levels or analog-to-digital units (ADUs), but is typically converted to a standard unit of electrons by multiplying by the camera’s gain factor. The gain can be measured with a standard mean-variance test. For a CCD sensor this average read noise obtained from a bias image and a mean variance falls at the center of the left curve in Figure 1, which is also at the peak of the symmetric Gaussian distribution.
Figure 1: Plot of read noise distribution for standard CCDs (left) and for scientific CMOS (right).
Pixels in recently developed scientific grade CMOS cameras produce a different type of noise distribution, one that follows a non-Gaussian skewed distribution of read noise values across a chip. The blue diamond curves in Figure 1 represent the best-fit Gaussian curve for the data set and demonstrate how closely the read noise distribution of a CCD sensor on the left matches a normal Gaussian distribution while the skewed distribution of a scientific CMOS sensor on the right does not.
Because of the scientific grade CMOS camera’s skewed distribution, comparisons of scientific grade CMOS cameras can be difficult, and so a choice of which read noise value to use must be made. Two possibilities are the average (mean) read noise measured with a mean variance and a bias or the modal (peak) value of the skewed distribution. The standard average value takes the noisier pixels in the skewed end of the distribution into account somewhat and thus is higher than the modal value, which does not. For example, for the scientific CMOS shown in Figure 1, the standard average read noise value for this camera is 33% higher than the modal value. Additionally, this tailing means a significant number of pixels will have a read noise higher than average.
To further complicate matters, the amount of tailing varies among different scientific CMOS cameras. To quantify this, the read noise data can be compared against a standard Gaussian distribution to determine the percentage of total pixels that fall outside this typical Gaussian distribution. As expected, the CCD read noise distribution follows a Gaussian distribution with less than 1.5% of the total pixels falling outside the Gaussian distribution (Figure 1, left). On the other hand, the tailing in the scientific CMOS data causes this percentage of pixels outside the Gaussian to be 42%, or more than a third of the pixels (Figure 1, right). This large number of tailing pixels is a complication that should be considered when comparing CMOS cameras to CCDs for low-to-medium light imaging, where read noise dominates. At high light levels, where photon shot noise dominates, the skewed read noise is much less of an issue.
The tailing in the skewed distribution manifests itself in scientific grade CMOS camera images in several ways, such as ‘salt-and-pepper noise’ (see Figure 2) which appears as random bright pixels and dark pixels superimposed on the image and a bar-code effect of vertical lines of different levels of dark gray intensity appearing at random spacing across the image.
Figure 2: Salt-and-pepper speckle noise in a scientific CMOS bias image. The higher concentration of bright and dark speckles is readily apparent. The high amount of speckling corresponds to the long tail in the read noise distribution.
These noise types specific to the scientific grade CMOS result in part to the fundamentally different way in which the pixels are read out compared to a CCD (Figure 3). In CCDs the photoelectrons produced on each pixel are passed through the same serial register and then read out using one readout amplifier. This leads to the reduced variability seen among the chip’s pixels. Scientific CMOS, however, has a separate readout amplifier for each individual pixel, and the pixels are output with a separate amplifier and analog-to-digital convertor for each column. Furthermore, some scientific CMOS cameras switch between two different amplifiers when reading out the pixels within a single image. The end result is increased variability from pixel to pixel and, more noticeably, from column to column, manifesting itself in the bar code column noise mentioned above. The trade-off is the increased speed achievable by scientific CMOS with multiple amplifiers.
Figure 3: Sampling architecture of CCD and CMOS cameras. CCDs (left) sample through one amplifier and thus have a single read noise source. CMOS sensors (right) sample using a different amplifier for each pixel and for each row, introducing a variety of read noise sources.
The salt-and-pepper noise, or random telegraph noise, also results in part from defects in individual pixels combined with the use of correlated double sampling (CDS) . Defects in the pixel semiconductor can trap electrons, creating more than one level of electron current such that double sampling between different levels can cause either bright pixels or dark pixels, depending on the how the two samples are taken.
Certain pixels with more defects create more current-level variation, and thus have larger swings in signal and larger standard deviations. These pixels are what skew the read noise distributions to high noise values, as seen in Figures 1 and 4. Because certain pixels are noisier than others, the location of noisy pixels on the chip is somewhat predictable. However, since these noisy pixels are spread throughout the chip, they are impossible to avoid and impossible to predict when a ‘salt or pepper’ state will occur.
Figure 4: CMOS noise distribution and individual pixel response due to random telegraph (salt-and-pepper) noise. Defects in pixels create two states with different signals. Double sampling and subtraction of the two samples creates a high, an average, and a low signal so a pixel will fluctuate rapidly among these three signals over time (i.e. frame-to-frame). Different pixels have different defect numbers and different amounts of variation, leading to a read noise distribution skewed to higher noise values.
SIGNAL-TO-NOISE RATIO (SNR)
Having an expression for the total number of detected photons from Part 2 of the series and the expression for the total noise, we arrive at the signal-to-noise ratio, which is an often used metric to describe the quality of the image being acquired. In typical situations, microscopists take exposure times ranging from 10 ms to 1 s. Therefore, the SNR calculations presented in this section use exposure times within this range. Figure 5 provides images characterized by increasing SNR for visual reference. It shows increasing SNRs taken from a CCD camera using a test card.
Figure 5: Comparison of SNRs of 3:1, 12:1 and 18:1.
Equation 2 outlines the signal-to-noise ratio φ/σ calculation. The sources of noise used in this calculation are read noise σr, shot noise σn, and dark current noise σd, which are the dominant sources of noise in CCD and CMOS cameras:
SNRCCD = φ / σ = φ / √(σd2 + σr2 + σn2 ) (2)
It was previously mentioned that the shot noise is a function of the signal level. Equation 2 shows that as the signal becomes smaller and smaller, the shot noise will correspondingly fall and approach the read noise. Eventually the read noise becomes the dominant noise source limiting the SNR. In extremely low-light microscopy applications, an alternate technology, the electron multiplication CCD (EMCCD) has been developed to overcome this problem. Electron multiplication technology provides the ability to amplify the detected photons by a known factor before digitization, allowing the signal to be brought up above the read noise. This electronic amplification is achieved through a process called impact ionization.
Impact ionization is also a probabilistic process where the exact factor of multiplication in the register of the CCD can vary. This variation adds another noise source known as the excess-noise factor Ψ. This excess-noise factor only affects the sources of noise which must be digitized and not the noise induced by the digitization process itself. In other words, the shot noise and dark current are multiplied by the excess noise factor as they are subject to the multiplication process. As a result the SNR equation is modified for an EMCCD, as shown in Equation 3:
SNREMCCD = φ / σ = φ / √(σr2 + Ψ2σn2 + Ψ2σd2 ) (3)
PUTTING IT ALL TOGETHER
Clearly there are a number of variables that ultimately affect the SNR of the image being acquired. With the model that we have developed in this series, we can incorporate all of these variables to provide a graphical interpretation of the SNR.
Most quantitative light microscopy applications have used CCD and EMCCD sensors. However, the recent emergence of scientific grade CMOS sensor technology suggests that these sensors may be viable for such scientific applications as well. As a result, this analysis evaluates these three sensor categories. By plotting the SNR as a function of the exposure time for the acquired image, the impact of dark noise with increasing exposure time is included in the analysis.
Again, as mentioned in Part 2, the calculations assume a sample labeled with enhanced cyan fluorescent protein (ECFP, e = 3300 cpsm) having a concentration of 200 nM in the sample. These calculations further assume a 603, 1.4 NA oil-immersion objective. As a demonstration of how the different noise sources of each sensor category affect the SNR, Figure 6 plots the SNR for each category of sensor assuming that all sensors have the same pixel size and QE. Read noise values characteristic of each class of sensor (3.5 e- for scientific CMOS, 6 e- for CCD, and <0.5 e- for EMCCD) and other noise sources are included.
Figure 6: Simplified SNR comparison for different sensors with 14 μm pixels and the same QE typical of front-illumination.
For SNR levels below around 3, the EMCCD class of sensor will perform the best because it has the lowest read noise of all three and the excess noise factor is also still minimal. As the exposure time increases and the detected signal increases, the excess noise factor starts to increase the total noise and the scientific CMOS class of sensor demonstrates the best performance. For very long exposure times, the CCD and scientific grade CMOS are equivalent as the shot noise, which is independent of sensor, then dominates.
The previous discussion was purely an academic exercise intended to demonstrate how the different sources of noise make the different sensor classes better or worse in terms of SNR depending on the parameters of the acquisition. If the actual pixel sizes of the different sensors and their QEs are considered, then the resulting SNR profiles are shown in Figure 7. In this analysis, the 3.5 µm scientific CMOS with a 0.5X coupler was also included since it was shown in Part 2 that this configuration could be used without sacrificing resolution. Alternatively, a 40X, high NA objective could be used without a coupler with the 3.5 µm scientific CMOS sensor to increase the signal while still Nyquist sampling.
Figure 7: SNR comparison for different sensors and with 3.5 μm-pixel sensor with a 0.5X coupler.
To tie the pixel analysis together, Figure 7 combines the signal-to-noise ratios for the 14 µm-pixel EMCCD sensor, the 6.5 µm-pixel CCD sensor and the 3.5 µm-pixel CMOS sensor. These were selected due to their accurate representation of the majority of each respective sensor family. Since the small pixel of the CMOS sensor will sufficiently sample when using a 0.5X coupler in the optical system, it was also used in this graph to show the benefit in the signal-to-noise ratio.
Figure 7 provides a fascinating result. The EMCCD sensor maintains its superior SNR as exposure time is increased because the much larger pixel size allows it to capture significantly more photons. Similarly, the CCD peforms better than the scientific CMOS because of its relatively larger pixel size. The scientific grade CMOS with its very small pixel has the worst SNR. However, if the CMOS sensor is used with the 0.5X coupler, then its effective pixel area is increased by a factor of 4. This increase in effective area coupled with lower read noise provides better performance than the CCD and the scientific CMOS without the coupler. >
Utilizing imaging and radiometric analysis based on the parameters of magnification, immersion media, and numerical aperture, the number of detected incident photons per second on sensor can be determined. Combining this with the known properties of the different sensors such as quantum efficiency, read noise, pixel size, and dark current, we can extend the optical analysis into a camera performance analysis. With the incorporation of the noise properties of the sensor into the model, it is possible to calculate the signal-to-noise ratio and hence get an estimate of image quality which can be used to compare different sensors. These comparisons are intended to provide guidelines when looking to purchase a camera for microscopy applications.
In Part 4 of the series, the SNR analysis will be extended to consider how variations in the light level of the object drive camera selection. Images from the different sensors at these SNR values will be presented along with the numerical analysis.
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