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

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 V₁,V₂,…,V_J may be derived from a basic set of spatial data V₀ 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_j^ef,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 V_j+1=V_j⊕W_j and V_j⊂V_j+1,described by D_j^ef.Therefore,D_j^ef 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 a four tuple:MultiGeomorph=φ(x), ψ(x),(V_j)_j∈Z, (W_j)_j∈Z. The tuple includes all information about relief representation at multiple scales. It is an analysis system based on M RA of wavelet, and describes the mechanism about deriving multi-scale DEM. Furthermore, it is a dynamic sy stem studying the rule about state of data changing with scale. Therefo re, We can get the multi-scale DEM from the MultiGeomorph, the size of the sequence DEM s derived is 2^-k times of its original, and the relevant scale is stepped down a half. With the scale decreased, the fine characteristics of relief are reduced and filtrated down step by step, but the main characteristics are represented. So the detail extent of relief represented changes with scale. The practical examples are demonstrated in Fig. 1. Model generalization is foundation for cartog raphical generalization based on DEM. As a result, it is more adapted to analysis and application of GIS, and avoids the harmonization between group contours, which is very difficult for the methods based on contour generalization. The precondition of generalization is derivative of multi-scale DEM. The generalization can be considered as a procedure of info rmation reducing on certain conditions. Using MRA of wavelet will erase the topographic points and details, which are minimum contribution for constructing topographical surface. For example, Figs. 2 and 3 demonstrate the original contour map and the generalized counterpart. Assuming C_-Jrepresents the coefficients of wavelet that include all information after MRA of wavelet, their energy is norm ‖C_-J‖. If CPrepresents the wavelet coefficients after generalized, norm ‖CP‖is their energy. Accordingly, the percentage of ‖CP‖ to ‖C_-J‖ can measure the detail extent between the original data sets and their counterparts derived.