ASTR , Majewski [SPRING ]. Lecture Notes
In some applications, however, the full width at half maximum (FWHM) is often used instead. Relation between the standard deviation a and the full width at. Signal and Graph Terminology · Mean and Standard Deviation · Signal vs. . For example, all the PSFs in (a) have an FWHM of 1 unit. system resolution is to quote the frequency where the MTF is reduced to either 3%, 5% or 10%. Since a line is the derivative (or first difference) of an edge, the LSF is the derivative (or . The Relationship Between the FWHM and 1/e-Squared Halfwidth of a Gaussian Beam For Gaussian beam size measurements, Zemax uses.
Figure a shows profiles from three circularly symmetric PSFs: These are representative of the PSFs commonly found in imaging systems. As described in the last chapter, the pillbox can result from an improperly focused lens system. Likewise, the Gaussian is formed when random errors are combined, such as viewing stars through a turbulent atmosphere. An exponential PSF is generated when electrons or x-rays strike a phosphor layer and are converted into light.
This is used in radiation detectors, night vision light amplifiers, and CRT displays. The exact shape of these three PSFs is not important for this discussion, only that they broadly represent the PSFs seen in real world applications. The PSF contains complete information about the spatial resolution.
To express the spatial resolution by a single number, we can ignore the shape of the PSF and simply measure its width. Unfortunately, this method has two significant drawbacks. First, it does not match other measures of spatial resolution, including the subjective judgement of observers viewing the images. Second, it is usually very difficult to directly measure the PSF. Imagine feeding an impulse into an imaging system; that is, taking an image of a very small white dot on a black background.
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By definition, the acquired image will be the PSF of the system. The problem is, the measured PSF will only contain a few pixels, and its contrast will be low. Unless you are very careful, random noise will swamp the measurement. In loose terms, the signal in the impulse image is abouttimes weaker than a normal image. No wonder the signal-to-noise ratio will be bad; there's hardly any signal! A basic theme throughout this book is that signals should be understood in the domain where the information is encoded.
For instance, audio signals should be dealt with in the frequency domain, while image signals should be handled in the spatial domain. In spite of this, one way to measure image resolution is by looking at the frequency response.
This goes against the fundamental philosophy of this book; however, it is a common method and you need to become familiar with it. Taking the two-dimensional Fourier transform of the PSF provides the two-dimensional frequency response.
If the PSF is circularly symmetric, its frequency response will also be circularly symmetric. In this case, complete information about the frequency response is contained in its profile. In cases where the PSF is not circularly symmetric, the entire two-dimensional frequency response contains information.
However, it is usually sufficient to know the MTF curves in the vertical and horizontal directions i. We will come back to this issue shortly.
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As shown in Fig. Figure shows a line pair gauge, a device used to measure image resolution via the MTF. Line pair gauges come in different forms depending on the particular application.
For example, the black and white pattern shown in this figure could be directly used to test video cameras. For an x-ray imaging system, the ribs might be made from lead, with an x-ray transparent material between. The key feature is that the black and white lines have a closer spacing toward one end. When an image is taken of a line pair gauge, the lines at the closely spaced end will be blurred together, while at the other end they will be distinct.
Somewhere in the middle the lines will be just barely separable. An observer looks at the image, identifies this location, and reads the corresponding resolution on the calibrated scale.
The way that the ribs blur together is important in understanding the limitations of this measurement. Imagine acquiring an image of the line pair gauge in Fig. Figures a and b show examples of the profiles at low and high spatial frequencies. At the low frequency, shown in bthe curve is flat on the top and bottom, but the edges are blurred, At the higher spatial frequency, athe amplitude of the modulation has been reduced.
This is exactly what the MTF curve in Fig. This is related to the eye's ability to distinguish the low contrast difference between the peaks and valleys in the presence of image noise. A strong advantage of the line pair gauge measurement is that it is simple and fast.
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The strongest disadvantage is that it relies on the human eye, and therefore has a certain subjective component. Unfortunately, you will not always be told which of these values is being used; product data sheets frequently use vague terms such as "limiting resolution.
A subtle point to notice is that the MTF is defined in terms of sine waves, while the line pair gauge uses square waves. That is, the ribs are uniformly dark regions separated by uniformly light regions.
This is done for manufacturing convenience; it is very difficult to make lines that have a sinusoidally varying darkness. What are the consequences of using a square wave to measure the MTF? At high spatial frequencies, all frequency components but the fundamental of the square wave have been removed. Image and caption from Wikipedia: Dark blue is less than one standard deviation from the mean.
For the normal distribution, this accounts for The height of the distribution at its maximum is. To see this, consult the following figure, which is a graph showing the fractional area under the Gaussian curve with where i. Central Limit Theorem A feature that helps to make the Gaussian distribution of such widespread relevance is the Central Limit Theorem. One statement of this is as follows. If we repeat this procedure many times, since the individual x i are random, then the calculated means will have some distribution.
Full width at half maximum
If the x are already Gaussian distributed, then the distribution of is also Gaussian for all values of n from 1 upwards. But even if the x i have some other distribution say, for example, a uniform distribution over a finite range then the distribution of the sum or average of a few x i will already look Gaussian. Thus, whatever the original distribution, a linear combination of a few representatives from the distribution almost always approximates to a Gaussian distribution.
Regardless of the shape of the parent population, the distribution of the means calculated from samples quickly approaches the normal distribution as shown below for four very different parent populations left to right and by doing averages of an increasing number of independent "draws" from the parent population. Regardless of the shape of the parent population, the distribution of the means calculated from samples drawn from the parent population quickly approaches the normal distribution as the number of averaged samples, n, increases.