FWHM Equations


If it is desired to reduce high frequency 2D spatial noise, a LPF (Low Passs Filter) can be used by selecting a LPF choice. Then prior to the FFT, the fid image is multiplied by the specified 2D filter.

The K-space gaussian filter has a HWHM (Half Width - Half Maximum) equal to the radius specified in Radius field. The FWHM (Full Width - Half Maximum) is simply equal to twice the radius. The values, g(r), of the gaussian filter are given for one dimension in Equation 1 for a radius = h and an image width of N pixels.


The tex2html_wrap_inline220 for the HWHM radius, h, is given in Equation 2.


Multiplication in K-space is equivalent to convolution in Image-space.


Thus the relationship between gaussian filter FWHM in K-space to the FWHM in Image-space can be determined by taking the Fourier Transform (FT) of the K-space gaussian.


But a gaussian with tex2html_wrap_inline222 in the numerator is just another gaussian with tex2html_wrap_inline224 in the denominator.


By equating the exponents and replacing tex2html_wrap_inline226, the Image-space tex2html_wrap_inline224 can be determined.


From equation 2 the Image-space HWHM, tex2html_wrap_inline230, is


Substituting for tex2html_wrap_inline222 and noting that FWHM = 2h


John Paul Strupp
Wed Jan 29 11:44:13 CST 1997