DCT and Hard Thresholding
Low-pass filtering is based on the assumption that "natural" images have more energy in the low-frequency coefficients than in the high frequency coefficients ; while noise will affect equally all coefficients. Thus, removing the high frequency coefficients will comparatively eliminate more noise than signal. The problem is that there are "legit" high-frequency coefficients in images, such as edges. Denoising by low-pass filtering will attenuate those coefficients and cause edge blurring.

The property used in magnitude thresholding is sparsity - "natural" images are very likely to have only a small set of high non-zero coefficients in the frequency domain. Adding uniform (and independent from pixel to pixel) noise is equivalent to adding a small random value to all the frequency coefficients. The result is that all the coefficients that were 0 in the original image now have a small value ; while the coefficients that were high in the original image are comparatively untouched. Thresholding the small magnitude coefficients will cancel the contribution on noise on those coefficients - though it will not recover the effect of noise on the high coefficients.

If you replace edges by transients/attacks, the same thing applies to sounds by the way.