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The AAPM/RSNA Physics Tutorial for Residents¹

Counting Statistics

¹ From the Departments of Radiology, Case Western Reserve University and MetroHealth Medical Center, 2500 MetroHealth Dr, Cleveland, OH 44109-1998. From the AAPM/RSNA Physics Tutorial at the 1997 RSNA scientific assembly. Received January 13, 1999; revision requested February 22 and received March 2; accepted March 5. Address reprint requests to the author.

View larger version (8K): [in a new window]   Figure 1.  Measurement errors. Graphs show the results of taking 20 hypothetical measurements from the same sample in several different well counters. Four scenarios are shown. Each square represents one measurement. Two properties are demonstrated: randomness and bias. In the upper left graph, the data are random (spread out), and on average the measurements underestimate the true value. The upper right graph shows an equally random set of measurements, but now, on average, the true value is represented, and the bias has been eliminated. The lower left graph demonstrates low variability (good precision), but again a systematic error is introduced, which causes all measurements to be higher than expected, perhaps due to excessive background counts. The lower right graph shows a set of measurements with good precision and without bias, qualities that are our goal. Not shown is a blunder, which was outside the range of the graphs, whereupon it was recognized as an incorrect recording and the measurement was repeated.

View larger version (128K): [in a new window]   Figure 2a.  Frequency distribution. (a) Gamma camera uniformity flood image shows noise due to the natural fluctuations caused by radioactive decay. The mean pixel count for the 128 x 128 matrix image is 100 (1.6 million total counts). (b) Graph shows the pixel values observed along a single row of the image near the center. The x axis represents the position across the row from left to right. The y axis represents the recorded pixel values across the image. On average, the data are centered around the mean value of 100 but fluctuate in a random manner above and below this value. (c) Graph shows the distribution of pixel values for the entire image. The x axis represents the range of possible pixel values. The y axis represents the frequencies with which particular pixel values occur. Each black bar represents a grouping of data for several possible pixel values. Although the commonest pixel values are near the mean of 100, many pixels have values substantially higher or lower than the mean, a distribution characteristic of radioactive decay.

View larger version (11K): [in a new window]   Figure 2b.  Frequency distribution. (a) Gamma camera uniformity flood image shows noise due to the natural fluctuations caused by radioactive decay. The mean pixel count for the 128 x 128 matrix image is 100 (1.6 million total counts). (b) Graph shows the pixel values observed along a single row of the image near the center. The x axis represents the position across the row from left to right. The y axis represents the recorded pixel values across the image. On average, the data are centered around the mean value of 100 but fluctuate in a random manner above and below this value. (c) Graph shows the distribution of pixel values for the entire image. The x axis represents the range of possible pixel values. The y axis represents the frequencies with which particular pixel values occur. Each black bar represents a grouping of data for several possible pixel values. Although the commonest pixel values are near the mean of 100, many pixels have values substantially higher or lower than the mean, a distribution characteristic of radioactive decay.

View larger version (10K): [in a new window]   Figure 3.  Measures of central tendency. The mean is the average of all measurements. The median is the middle value when data are sorted in ascending numeric order. If there is an even number of measurements, the median is the average of the two middle values. The mode is the most frequent value.

View larger version (9K): [in a new window]   Figure 4.  Measures of variability. The range is the maximum value minus the minimum value. The variance is the sum of the squared deviations from the mean, divided by one less than the number of measurements used in the calculation. The standard deviation is the square root of the variance.

View larger version (12K): [in a new window]   Figure 5.  Probability density function. Graph shows Poisson probability density functions for three values of µ (mu). The y axis reflects the probability of obtaining a measurement that has the specific value listed on the x axis. For µ = 2, the probability distribution is asymmetric (skewed) with maxima at k = 1 and k = 2 yielding the same probability of 28%. As µ increases, the distributions become broader, more symmetric, and bell shaped and the probability of observing one specific number of events is reduced because the data are more widely dispersed.

View larger version (12K): [in a new window]   Figure 6.  Difference between Gaussian and Poisson distributions. Graph shows two Poisson and two Gaussian probability density functions for µ = 4 and µ = 36. The Poisson function is defined only for a discrete number of events, and there is zero probability for observing less than zero events. The Gaussian function is continuous and thus takes on all values, including values less than zero as shown for the µ = 4 case. In this example, the standard deviations for the Gaussian functions have been set equal to the square root of µ so the widths of the curves are similar. Once µ exceeds 20 or so, the Gaussian and Poisson curves are nearly identical, as demonstrated for the µ = 36 example.

View larger version (16K): [in a new window]   Figure 7.  Percent uncertainty. Graph shows three Gaussian probability density functions with mean values of 20, 100, and 1,000. The standard deviations have been set equal to the square root of the mean to match the Poisson distribution. The x axis has been normalized to represent the percent deviation from the mean value. As the mean increases, the relative dispersion of values about the mean decreases on a percentage basis.

View larger version (44K): [in a new window]   Figure 8.  Effect of total counts on perceived noise. Gamma camera images (128 x 128 matrix, 2-mm section thickness) of a brain phantom show that as the total number of counts increases, the effect of noise is less obvious. The perceived variability in the image decreases as the total counts increase because, on a percentage basis, the noise is less apparent.

View larger version (43K): [in a new window]   Figure 9.  Confidence intervals. Graph shows the probability density function for the Poisson distribution with a mean of 100. The Poisson distribution has a bell shape for µ > 20 or so and a dispersion or width that is constrained by the mean value. A measurement between the mean plus or minus one standard deviation is observed 68.3% of the time. Similar confidence intervals for plus or minus two or three standard deviations encompass 95.5% and more than 99.7% of the measurements, respectively.

View larger version (17K): [in a new window]   Figure 10.  Application of the confidence interval concept. This formula would be used if we want to be 95.5% sure that two measurements will be within 1% of each other on the basis of the count statistics of our data. The percent uncertainty we expect to observe is 1% with two standard deviations used as the measurement window for 95.5% confidence. The square root of N over the square root of N is 1; this factor is used in the third step to simplify the equation. In addition, the square root of N times the square root of N is N, which cancels with N in the denominator in the fourth step. The result is rearranged to obtain the final answer, 40,000 counts.

View larger version (15K): [in a new window]   Figure 11a.  Process of subtracting background from a sample. In this case, the true background value is 25 and the true sample activity is 75. Both the background measurement and the sample measurement represent Poisson processes, and each has an associated dispersion about the true value. (a) Graph shows three typical sample and background calculations. In these examples, the net counts vary from 42 to 97. (b) Graph shows the distribution of the net counts as a dotted line. This line represents the distribution for the true sample activity. The width of the net counts curve is greater than that of the sample or background curve because the variability of each curve must be taken into account during the subtraction process. As a result, the net counts probability density function is no longer Poisson because the standard deviation for this distribution is greater than the square root of the mean.

View larger version (14K): [in a new window]   Figure 11b.  Process of subtracting background from a sample. In this case, the true background value is 25 and the true sample activity is 75. Both the background measurement and the sample measurement represent Poisson processes, and each has an associated dispersion about the true value. (a) Graph shows three typical sample and background calculations. In these examples, the net counts vary from 42 to 97. (b) Graph shows the distribution of the net counts as a dotted line. This line represents the distribution for the true sample activity. The width of the net counts curve is greater than that of the sample or background curve because the variability of each curve must be taken into account during the subtraction process. As a result, the net counts probability density function is no longer Poisson because the standard deviation for this distribution is greater than the square root of the mean.

View larger version (15K): [in a new window]   Figure 12.  Example of propagation of errors. One can compute the background-corrected sample count by subtracting the background from the sample. The standard deviation for the net count is computed by taking the square root of the sum of the squares of the standard deviations of the net count and the background count. The standard deviation for the sample activity is higher than either individual standard deviation. Thus, the noise (standard deviation) that results from combining two noisy measurements is greater than the noise in either individual measurement.

View larger version (17K): [in a new window]   Figure 13.  Calculation of ejection fraction by using propagation of errors concepts. The ejection fraction and its standard deviation can be computed by using the appropriate application of propagation of errors, where N is the number of counts in a region of interest. Let Ndiastole = 2,000, Nsystole = 1,200, and Nbackground = 400 counts. The ejection fraction can be calculated by the reader and found to be 0.50 (50%) with a standard deviation of 0.04 (4%). With 95.5% confidence, one can conclude that the true ejection fraction is between 42% and 58% (EF ± 2 EF) on the basis of the count statistics in this example. The high background counts in this example have increased the standard deviation significantly.

View larger version (44K): [in a new window]   Figure 14.  Addition and subtraction of images. Gamma camera images (128 x 128 matrix, 2-mm section thickness) of a brain phantom show the results of adding (top row) and subtracting (bottom row) two noisy images. The left and middle images each contain approximately 16,000 counts; the final images are on the right. In the subtraction image on the lower right, a medium gray value was used to represent a difference of zero so that negative deviations appear dark gray and positive deviations appear white.

View larger version (42K): [in a new window]   Figure 15.  Multiplication and division of images. Gamma camera images (128 x 128 matrix, 2-mm section thickness) of a brain phantom show the results of multiplying (top row) and dividing (bottom row) two noisy images. The left and middle images each contain approximately 16,000 counts; the final images are on the right. The final images were rescaled so that the minimum and maximum pixel values use the entire gray scale for display. This rescaling yielded a darker overall image for the multiplication result.

View larger version (97K): [in a new window]   Figure 16.  Effect of information density on perception. Four pairs of 128 x 128 images from an 18-frame multiple gated blood pool study show the effect of total counts per frame on the visual appearance of the blood pool at end diastole (top row) and end systole (bottom row). The total counts in the study varied from 9,000 counts to 9 million counts from left to right when the total counts from each of the 18 frames were taken into account.

View larger version (120K): [in a new window]   Figure 17.  Effect of image subtraction on perception. End-systolic images (middle) were subtracted from end-diastolic images (left) to obtain stroke volume functional images (right) from a 9 million-count study (500,000 total counts per frame) (top row) and a 900,000-count study (50,000 total counts per frame) (bottom row). The images demonstrate the principle that subtracting two noisy images results in an image that is even noisier. This principle is especially evident in the low-total-count images (bottom row). The final images were adjusted so that a medium gray value represents a difference of zero, negative deviations appear dark gray, and positive deviations appear white; the acquisition parameters were the same as in Figure 16.

View larger version (93K): [in a new window]   Figure 18.  Concept behind a contrast detail diagram. Four 256 x 256 simulated contrast detail images show disks with diameters of 2, 4, 8, 16, and 32 pixels. These disks represent objects superimposed on normal tissue. The contrast varies from 5% in the top row of each image to 75% in the bottom row. The left images are noise free, and the right images follow Poisson statistics with a mean information density of 100 counts per pixel (6.5 million total counts). In the top images, the disks have higher activity (ie, are "hotter") than their surroundings; in the bottom images, the disks have lower activity (ie, are "colder") than their surroundings. The trade-off between object size (disk area) and required contrast is clearly seen: If the size is large or the contrast is high, the object is more easily recognized. There is a diagonal line of minimum object detectability that follows the Rose model.

View larger version (107K): [in a new window]   Figure 19.  Effect of total counts on perception. Simulated contrast detail images show simulated disk-shaped objects with six different noise levels. was varied in each of the images so that the mean number of counts per pixel ranges from approximately 150 to 4. These images are examples of the noise and contrast commonly seen in gamma camera imaging. The line of demarcation for object detection is a diagonal line from the lower left to upper right of each image, as predicted by the Rose model. One can barely recognize objects above this line. The line of object detectability is also altered by noise; it shifts to the left as more counts are used to produce the image.

View larger version (22K): [in a new window]   Figure 20.  Contrast detail diagram. Graph shows the line of demarcation where an object is barely perceived in an image; the three curves correspond to images with 250,000, 1 million, and 10 million counts. For a given curve, objects to the right of or above the line are perceived in the image. Objects to the left of or below the line cannot be perceived.