The linewidth observable refers to the FWHM of the (assumed) Gaussian absorption line profile for the 6173.3 Å line on the surface of the sun.
HMI samples the Fe I absorption line at 6173.3 Å at six points. Assuming that the "pure" solar Fe I line profile is a Gaussian, and the HMI filter filter transmission profiles are delta functions, the first and second Fourier coefficients of the of the Fe I line can be calculated and an estimate of the Doppler Velocity can be made. An estimate (proxy) for the continuum intensity Ic is obtained by "reconstructing" the solar line from the estimates of the Doppler shift λo, the linewidth σ, and the linedepth Id. More details can be found at http://jsoc.stanford.edu/relevant_papers/observables.pdf.