We determine the underlying shapes of spiral and elliptical galaxies in the Sloan Digital Sky Survey Data Release 6 from the observed distribution of projected galaxy shapes, taking into account the effects of dust extinction and reddening. We assume that the underlying shapes of spirals and ellipticals are well approximated by triaxial ellipsoids. The elliptical galaxy data are consistent with oblate spheroids, with a correlation between luminosity and ellipticity: the mean values of minor to middle axis ratios are 0.41+-0.03 for Mr ~ -18 ellipticals, and 0.76+-0.04 for Mr ~-22.5 ellipticals. Ellipticals show almost no dependence of axial ratio on galaxy colour, implying a negligible dust optical depth. There is a strong variation of spiral galaxy shapes with colour indicating the presence of dust. The intrinsic shapes of spiral galaxies in the SDSS-DR6 are consistent with flat disks with a mean and dispersion of thickness to diameter ratio of (21+-2)%, and a face-on ellipticity, e, of ln(e)=-2.33+-0.79. Not including the effects of dust in the model leads to disks that are systematically rounder by up to 60%. More luminous spiral galaxies tend to have thicker and rounder disks than lower-luminosity spirals. Both elliptical and spiral galaxies tend to be rounder for larger galaxies. The marginalised value of the edge-on r-band dust extinction E_0 in spiral galaxies is E_0 ~ 0.45 magnitudes for galaxies of median colours, increasing to E_0=1 magnitudes for g-r>0.9 and E_0=1.9 for the luminous and most compact galaxies, with half-light radii <2kpc/h.
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