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Multi-scale Representation and Automatic Generalization of Relief Based on Wavelet Analysis

WU Fan 1 ZHU Guorui 1 (1 School of Resource and Environment Science,Wuhan University,129 Luoyu Road,Wuhan,China,430079)  
With the development of GIS application ceaselessly,a mass of multi_scale geospatial data need to be analyzed and represented because users require different detailed spatial data to deal with different problems and output maps at different scales.It has become one of the key problems to applied GIS.The logic relations have to be established between spatial data sets at different scales so that one representation of spatial data can be transferred to another completely.The completeness refers that spatial precision and characteristics and a high information density that adapts to relevant abstract detail must be preserved,and the consistency of spatial semantics and spatial relations have to be maintained simultaneously.In addition,the deriving of new spatial data sets should be bi_directional on some constraint in GIS,from fine_scale to broad_scale and vice versa.Automatic generalization of geographical information is the core content of multi_scale representation of spatial data,but the scale_dependent generalization methods are far from abundance because of its extreme complicacy.Most existing algorithms about automatic generalization do not relate to scale directly or accurately,not forecast and control the generalized effects,and cannot assess the holistic consistency of the generalized results.The rational and quantitative methods and criterions of measuring the extent of generalization have not still been sought out.Wavelet analysis is a new branch of mathematics burgeoning at the end of 1980s.It has double meanings simultaneously on profundity of theory and extent of application.Because it has good local character at both time or space and frequency field simultaneously,and sample interval of signal can be adjusted automatically with different frequency components,any details of function,such as a sign or image etc.,can be analyzed at any scales by using wavelet analysis.Therefore,wavelet analysis suggests a new solution to the problems mentioned above.The fundamental characteristics of multi_scale spatial data can be detected and extracted,and represented by a set of wavelet coefficients,then handled and reconstructed,then the optimal representation of the spatial data sets can be got.This paper studies the multi_scale representation and automatic generalization of relief and the quantitative method and criterion of investigating the extent of generalization based on the above idea.The paper formulates briefly the basic principle of multiresolution analysis (MRA) on wavelet transform at first,and describes a model for multi_scale handling of spatial data based on MRA of wavelet.We know that subspace at a higher resolution includes completely all information at a lower resolution from the model,so multiple data sets such as V 1,V 2,...,V J may be derived from a basic set of spatial data V 0 at multiple scale by using MRA of wavelet,and the reverse procedure can be implemented completely by reconstructing.The decomposition and reconstruction are very stable.Accordingly,the model not only meets the need of automatic generalization but also is scale_dependent completely.Handling of automatic generalization is reverse based on the model.Two sections,approximation A jf and detail D e jf ,can be produced automatically by MRA of wavelet.The approximation describes the gentle and trend component of the characteristics of data,and the detail describes the fast and local one.They represent low and high frequency of data respectively.When data sets at scale j are derived from scale j+1 ,the loss of the approximation is W j because [FK(W28?40ZQ] V j+1 =V jW j and V jV j+1 ,described by D e jf .Therefore,{ D e jf } represents the detail generalized at stepped down scale.DEM is an abstract model about relief in GIS.The key problem of multi_scale representation of relief is how to derive the DEM at multiple scales.We propose a scheme for a multi_scale representation and generalization of scale_dependent relief based on the above model,which can be represented by
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