Gaussian function
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✎ This article/section is a stub — some half-sorted notes, not necessarily checked, not necessarily correct. Feel free to ignore, or tell me about it.
Gaussian function
Either expressed as a probability density function, defined in terms of normal distribution's mean and standard deviation:
....or as as the slightly more generalized-and-flexible:
Where:
- a is the height of the peak
- b the (x-axis) position of the peak (often written μ)
- c the standard deviation (σ), controlling the width
- d is an offset - often unused so would fall away
in 2D
Mostly just makes the exponent part more interesting because you deal with more dimensions.
When centered at zero (μ=0), it simplifies somewhat, to:
Related properties
c (is σ) is related to the Full Width at Half Maximum of the peak like:
- FWHM = 2*sqrt(2*ln(2)) c)
- so point at which the value is half the peak is at approximately 2.35482*c
The peak's height is that one-over-sigma-sqrt-2pi factor in front, approximately 0.399/sigma
At b+c and b-c (also the inflection points), the amount of falloff is 0.60653 (verify)
The product of two Gaussian functions is a Gaussian, with a different mean and sigma:
μf*g = (μf σg2 + μg σf2 ) / (σf2 + σg2)
σf*g = sqrt( σf2 * σg2 / (σf2 + σg2) )