Using a piece of paper and a pen, make dots that
represent the random nature of x-ray photons exposing a surface like
in an image. This is an example of how visual noise in an image
is formed.
Think of some common events that might occur
randomly with time (telephone calls, rain, etc). This is an example of
how radiation events, such as the emission of radiation from a
radioactive source occurs.
Let's think of a radioactive source
that is emitting an average of 100 photons each second.
Describe the actual number of photons that you would expect to be
emitted in each one-second interval over a 10-second period. There is
no way to know the actual number in each but use your knowledge of the
statistical distribution to produce reasonable estimates.
Now, draw a simple graph showing the general
distribution of how many times you would expect to have the different
number of photons in many one-second intervals. This is to
demonstrate the frequency that the different numbers of photons, like
98, 86, 100, 55, etc would be expected to occur.
Describe the general characteristics of a Gaussian
statistical distribution and how the standard deviation
(SD) applies to it. Think of some things in medicine and biology
that probably follow a Gaussian distribution.
If we are counting radiation photons from a
radioactive source to measure its activity, it appears that we will
get a different number each time we count (measure) because of the
normal statistical distribution. Let's assume that for our
example being used here, the average or mean of 100 photons per second
is an indication of the true activity. Calculate the errors (%
of true value) if you observe each of the following for several
one-second measurements: 95 counts, 80 counts, 75 counts.
That was easy, but the problem in the real world is if we make just
one measurement (lets say we get 85 counts) we cannot calculate
the error because we do not know what the mean values is, that would
take many measurements to determine.
If we make just one measurement of radioactivity by
counting emitted photons we do not know what the actual error is.
However, we can develop some knowledge of the range of possible
errors and the probability that our one measurement and the
associated error is within a specific range. Recognizing that
the SD actually describes a range of measured values some questions to
answer are:
1. What is the probability (% chance) that our one measurement value
was within one (1) SD of the true value, or mean value if we made many
measurements?
2. What range of measurement values, expressed in SDs, would we expect
our one measured value to be included in 95% of the time?
3. What range of measurement values, expressed in SDs, would we expect
our one measured value to be included in most (over 99%) of the time?
Describe the general relationship that is being
observed here between error values and the probability
of errors falling within certain ranges.
Describe the Poisson distribution characteristics
applied to radiation events that relate the value of the SD to the
mean value of many measurements if they were to be made.
Calculate (actually it is an estimation) the
value of the SD both in number of counted photons and as a percentage
of the number for the following: Number of PhotonsSD (Number)SD (%) 100
?
?
3600
?
?
1,000,000
?
?
Now describe the observed relationship between the number of photons
in a measurement and the range in size of the expected error.
Determine the number of photons that must be
collected in a measurement to have an error of not more than 1% at a
95% confidence level.
Consider a situation where on digital image is to
be subtracted from another, as in DSA. Here we will consider one
pixel in the images. The SD of the number of photons captured by
a pixel is a general indication of the expected noise in the image.
Calculate the SD (noise) for each of the following: ImagePhotons/PixelSD (%) A
1600
?
B
1000
?
A-B
600
?
Describe your observation on the effect of image subtraction on noise.
