Application Of Multi-resolution Analysis Methods To DEM Multi-scale Representation

Spatial events and patterns only takes place in a certain scale, so the spatialinformation after scientifically scale-abstraction is more valuable in practices. Multi-scalegeographical information system (GIS) is more useful for users than single-scale GIS.Digital elevation model (DEM), as a significant approach to represent topography, is afundamental support dataset of the research of other areas. The applications of DEM are nolonger confined to the traditionally surface topographic representation and have expandedto some higher levels, such as surface-process dynamically simulation and multi-scalegeographical modeling. These applications need the DEM database which can providerepresentation of terrain with continuously-changing scale (or resolution) and consistentspatial location base well. The idea of multi-scale resolution analysis (MRA) introduces aneffective way to represent and analyze the spatial information (e.g., DEM) with differentresolution. The MRA can adaptively control the sampling rate of different frequencycomponents of signals in spatial domain. For spatial information, this methodology canreasonably analyze and process their low-frequency macro-scale characteristics andhigh-frequency micro-scale details as well. This methodology thus shows desirableadvantages in representing the fundamental characteristics of the spatial informationdatabase of multi-sources, multi-scale and mass-quantity. It is of great value in theory andpractice to use MRA methods to establish the scale-depended spatial informationrepresentation model and to employ the multi-scale and automatic processing to spatialinformation dataset. Researchers have carried out studies in several ways. However, somekey problems are still left behind.Firstly, the present studies have not analyzed and revealed the fundamental principleof the MRA based DEM multi-scale representation and the resolution obtained. Secondly, the MRA methods used by the existing MRA based DEM multi-scale representation havesome drawbacks. Some improved MRA methods (e.g., Contourlet, Multiwavelet) havebeen shown some significant advantages in many image processing fields, but haven’t beenintroduced into the application field of spatial information scale-transformation. This workfocuses on the above issues and studies the application of MRA to DEM multi-scalerepresentation in the following aspects.(1) The fundamental principle of the MRA based DEM multi-scalerepresentation is analyzed. According to the geophysical theory and signal processingprinciple, this study describes the relationship among the scale, resolution and samplinginterval of DEM and analyzes the constraint relation between the highest frequency (infrequency domain) of DEM and their resolution (in spatial domain). Furthermore, the workreveals the fundamental principle of the MRA based DEM multi-scale representation,which generates DEM with levelly/continuously coarsening resolution bylevelly/continuously lessening the highest frequency of DEM.(2) This study reveals the fundamental principle of the traditional wavelettransform (WT) based DEM multi-scale representation methods, improves thestructure of the WT used by the present WT based multi-scale representationmethods, constructs three families of high-order balanced M-band (M>2andM) multiwavelet system which is more desirable than the WT in structure, andfinally verify their advantages via practical application in DEM multi-scalegeneralization. Firstly, we study the frequency-domain behavior of the traditional WT(including2-band and M-band), describe the whole process of the WT based DEMgeneralization which generates DEM with levelly-coarsening resolution by levelly-filtering(of low-pass) in frequency domain, explain the relationship between the resolution ofgeneralized DEM and decomposition levels of WT when using traditional WT basedgeneralization method, and clarify the resolution obtained by the methods. We then studythe existing construction theory of high-order balanced2-band multiwavelet that constructsthe WT filter bank by solving polynomial equation systems with the help of Gr bner-basetechnique. Inspired by this idea, we successfully construct three families of M-bandmultiwavelet, including three-band symmetric family, three-band flipped family andfour-band symmetric family. Each family is with the multiplicity of two, indexed by anincreasingly balanced order ({1,2,3}, i.e. the balanced order of the multiwavelets ineach family ranges from one to three), and supported with the minimal length according toevery balanced order. The coefficients of the filter bank of each wavelet system are obtained. Finally, we apply these constructed multiwavelet systems to the application ofDEM generalization. The results show when providing same generalization scale orresolution, they can effectively reduce the generalized errors compared with2-bandmultiwavelets, M-band scalar wavelets and2-band wavelets. Also, the effectiveness ofhigh-order balanced property in these multiwavelet systems are tested and verified.(3) The study proposes a scale-continuously-changing transform of DEM whichis effective and signal-processing theory based. We study the principle of the proposedtransform that is based on sampling theory and generates DEM with assigned resolution byquantitatively lessening the highest frequency of DEM. Then, we improve a kind ofrational-dilation wavelet transform (RWT) to meet the requirement of the proposedtransform, i.e., to make the Q-factor at each decomposition level tunable to realize freelyfrequency-domain partition and to expand the one-dimensional RWT to itstwo-dimensional counterpart by using tensor product operation. We also evaluate theperformance of the scale-continuously-changing transform in DEM generalizationapplication, and the experimental results show its effectiveness (including the satisfactoryof generalized results with the sampling theory, the desirable-consistency spatial locationbase, and effectiveness of the generalization).(4) We propose a new effective image representation approach (Tunable-QContourlet Transform) to improve the RWT based scale-continuously-changingtransform of DEM, and to improve the ability in retaining the local terrain andcontours. We integrate the multi-scale transform scheme of the rational-dilation wavelettransforms and the directional filter bank of original contourlets, and eliminate thefrequency-domain aliasing component in the contourlets by reasonably employing thestop-edge frequency of the filter bank of the multi-scale transform. We consequently obtainthe anti-aliasing tunable-Q contourlet transform which provides finer frequency-domainpartition and more sensitive directional information representation. Experiments areimplemented to evaluate its performance in image processing and the results show itsadvantages in image approximation and de-noising. By using the scale (or resolution)continuously-changing generalization principle of the proposed rational-dilation wavelettransform based DEM representation, we apply the tunable-Q contourlet transform to DEMgeneralization field and our results show its applicability and advantages.(5) We take a comparative analysis to different DEM multi-scale representationmethods, including the three proposed methods (the M-band multiwavelet transformbased method, the rational-dilation wavelet transform based method and thetunable-Q contourlet based method) and the traditional interpolation approaches (bilinear and bicubic), in order to evaluate their practical performance andapplicability environment. The datasets used in the experiments include the DEMdatasets with different terrain characteristics and the DEM derived from ideal mathematicsurface which provides reference real value in comparison. The experimental results showthe DEM multi-scale transform based on the proposed M-band multiwavelet transform caneffectively reduce the generalization error compared with traditional interpolation methods.Meanwhile, its algorithm is comparatively simple and the execution is efficient, so it onlyrequires low-level equipment and is suitable for most of application environments.However, the resolution cases provided by its generalized results is some discreet array,this mean the methods can not realize a scale continuously-changing representation. Therational-dilation wavelet transform based DEM generalization method does this well andits generalized results provide us with continuously-changing resolution. The methodresults in same generalization precision compared with the M-band multiwavelet basedmethod, for same goal resolutions. Additionally, it has moderate-level computationcomplexity and execution time, and is suitable for general users. For all the test methods,the tunable-Q contourlet based method obtains the highest generalized precision. It notonly better retains the integrity and continuity of the contours or edges of DEM, but alsomore effectively reduces the ‘artificial’ terrain features. On the other side, it incurs a morecomplicate computational complexity and requires more computational time. So thismethod is suitable for application environment with high-level execution equipment.(6) According the main idea of the proposed DEM representation based onrational-dilation wavelet transform and sampling theory, we design application program forscale continuously-changing generalization of DEM. The program is established inMATLAB software and supported by its Graphical User Interface (GUI) and itsinterpreting and publishing packages. Our program provides a friendly interactive interfaceand stable and effective execution process.