New GRASS modules for multiresolution analysis with wavelets

The recognition and the study of several spatial phenomena strongly depends on the resolution at which the analysis is carried out. The availability of an efficient way to represent the same phenomena at different resolutions is therefore very important (from both a mathematical and a computational point of view). Wavelets, families of base functions generated by dilation and translation of a mother function, provide a good balance of localization in space and localization in the frequency domain, unlike the Fourier transform (where the localization in space is lost) and spline (where the localization in frequency is lost). Wavelets can be used in a Multiresolution Analysis (MRA), where the signal and the difference from the previous resolution level are orthogonal, to provide an efficient tool for investigating resolution dependent phenomena. Four new different GRASS modules have been created for the wavelets analysis and synthesis with orthogonal and bi-orthogonal wavelets bases. A bidimensional signal is processed into four sub-images, one showing the original image at coarser resolution (de-noised), while the other three represent the difference (noise) between the actual level of decomposition and the previous level (in x, y and diagonal direction). In a geomorphological framework, by thresholding the “difference coefficients”, this allows to remove objects at a certain scale from a digital earth’s surface representation. These techniques can be used in two different ways: measurement errors can be removed by thresholding the difference coefficients at the finer scale and objects over the terrain can be removed by thresholding the difference coefficients at the resolution that matches their dimension. The latter can be applied also to natural terrain features to analyze geomorphological phenomena at various scales. Several tests have been carried out by applying this techniques to high resolution laser scan elevation data, where vegetation and artifacts have been removed to obtain a digital terrain model. Moreover, in application to a river basin, a geomorphological analysis has been carried out, obtaining the automatic extraction of scale dependent features and shapes.