Image smoothing with Savitzky-Golay filters

Noise in medical images is common. It occurs during the image formation, recording, transmission, and subsequent image processing. Image smoothing attempts to locally preprocess these images primarily to suppress image noise by making use of the redundancy in the image data. One-dimensional Savitzky-Golay filtering provides smoothing without loss of resolution by assuming that the distant points have significant redundancy. This redundancy is exploited to reduce the noise level. Using this assumed redundancy, the underlying function is locally fitted by a polynomial whose coefficients are data independent and hence can be calculated in advance. Geometric representations of data as patches and surfaces have been used in volumetric modeling and reconstruction. Similar representations could also be used in image smoothing. This paper shows the two and three-dimensional extensions of one-dimensional Savitzky-Golay filters. The idea is to fit a two/three-dimensional polynomial to a two/three-dimensional sub region of the image. As in the one-dimensional case, the coefficients of the polynomial are computed a priori with a linear filter. The filter coefficients preserve higher moments. The coefficients always have a central positive lobe with smaller outlying corrections of both positive and negative magnitudes. To show the efficacy of this smoothing, it is used in-line with volume rendering while computing the sampling points and the gradient.