A class of nonlinear regression filters based on robust theory is introduced. The goal of the filtering is to restore the shape and preserve the details of the original noise-free signal, while effectively attenuating both impulsive and nonimpulsive noise. The proposed filters are based on robust Least Trimmed Squares estimation, where very deviating samples do not contribute to the final output. Furthermore, if there is more than one statistical population present in the processing window the filter is very likely to select adaptively the samples that represent the majority and uses them for computing the output. We apply the regression filters on geometric signal shapes which can be found, for example, in range images. The proposed methods are also useful for extracting the trend of the signal without losing important amplitude information. We show experimental results on restoration of the original signal shape using real and synthetic data and both impulsive and nonimpulsive noise. In addition, we apply the robust approach for describing local image structure. We use the method for estimating spatial properties of the image in a local neighborhood. Such properties can be used for example, as a uniformity predicate in the segmentation phase of an image understanding task. The emphasis is on producing reliable results even if the assumptions on noise, data and model are not completely valid. The experimental results provide information about the validity of those assumptions. Image description results are shown using synthetic and real data, various signal shapes and impulsive and nonimpulsive noise.