| Other Abstract | Spatial events and patterns only takes place in a certain scale, so the spatial
information after scientifically scale-abstraction is more valuable in practices. Multi-scale
geographical information system (GIS) is more useful for users than single-scale GIS.
Digital elevation model (DEM), as a significant approach to represent topography, is a
fundamental support dataset of the research of other areas. The applications of DEM are no
longer confined to the traditionally surface topographic representation and have expanded
to some higher levels, such as surface-process dynamically simulation and multi-scale
geographical modeling. These applications need the DEM database which can provide
representation of terrain with continuously-changing scale (or resolution) and consistent
spatial location base well. The idea of multi-scale resolution analysis (MRA) introduces an
effective way to represent and analyze the spatial information (e.g., DEM) with different
resolution. The MRA can adaptively control the sampling rate of different frequency
components of signals in spatial domain. For spatial information, this methodology can
reasonably analyze and process their low-frequency macro-scale characteristics and
high-frequency micro-scale details as well. This methodology thus shows desirable
advantages in representing the fundamental characteristics of the spatial information
database of multi-sources, multi-scale and mass-quantity. It is of great value in theory and
practice to use MRA methods to establish the scale-depended spatial information
representation model and to employ the multi-scale and automatic processing to spatial
information dataset. Researchers have carried out studies in several ways. However, some
key problems are still left behind.
Firstly, the present studies have not analyzed and revealed the fundamental principle
of 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 have
some drawbacks. Some improved MRA methods (e.g., Contourlet, Multiwavelet) have
been shown some significant advantages in many image processing fields, but haven’t been
introduced into the application field of spatial information scale-transformation. This work
focuses on the above issues and studies the application of MRA to DEM multi-scale
representation in the following aspects.
(1) The fundamental principle of the MRA based DEM multi-scale
representation is analyzed. According to the geophysical theory and signal processing
principle, this study describes the relationship among the scale, resolution and sampling
interval of DEM and analyzes the constraint relation between the highest frequency (in
frequency domain) of DEM and their resolution (in spatial domain). Furthermore, the work
reveals the fundamental principle of the MRA based DEM multi-scale representation,
which generates DEM with levelly/continuously coarsening resolution by
levelly/continuously lessening the highest frequency of DEM.
(2) This study reveals the fundamental principle of the traditional wavelet
transform (WT) based DEM multi-scale representation methods, improves the
structure of the WT used by the present WT based multi-scale representation
methods, constructs three families of high-order balanced M-band ( M>2 and
M ) multiwavelet system which is more desirable than the WT in structure, and
finally verify their advantages via practical application in DEM multi-scale
generalization. Firstly, we study the frequency-domain behavior of the traditional WT
(including 2-band and M-band), describe the whole process of the WT based DEM
generalization which generates DEM with levelly-coarsening resolution by levelly-filtering
(of low-pass) in frequency domain, explain the relationship between the resolution of
generalized DEM and decomposition levels of WT when using traditional WT based
generalization method, and clarify the resolution obtained by the methods. We then study
the existing construction theory of high-order balanced 2-band multiwavelet that constructs
the WT filter bank by solving polynomial equation systems with the help of Gröbner-base
technique. Inspired by this idea, we successfully construct three families of M-band
multiwavelet, including three-band symmetric family, three-band flipped family and
four-band symmetric family. Each family is with the multiplicity of two, indexed by an
increasingly balanced order ( {1,2,3} , i.e. the balanced order of the multiwavelets in
each family ranges from one to three), and supported with the minimal length according to
every balanced order. The coefficients of the filter bank of each wavelet system are obtained. Finally, we apply these constructed multiwavelet systems to the application of
DEM generalization. The results show when providing same generalization scale or
resolution, they can effectively reduce the generalized errors compared with 2-band
multiwavelets, M-band scalar wavelets and 2-band wavelets. Also, the effectiveness of
high-order balanced property in these multiwavelet systems are tested and verified.
(3) The study proposes a scale-continuously-changing transform of DEM which
is effective and signal-processing theory based. We study the principle of the proposed
transform that is based on sampling theory and generates DEM with assigned resolution by
quantitatively lessening the highest frequency of DEM. Then, we improve a kind of
rational-dilation wavelet transform (RWT) to meet the requirement of the proposed
transform, i.e., to make the Q-factor at each decomposition level tunable to realize freely
frequency-domain partition and to expand the one-dimensional RWT to its
two-dimensional counterpart by using tensor product operation. We also evaluate the
performance of the scale-continuously-changing transform in DEM generalization
application, and the experimental results show its effectiveness (including the satisfactory
of generalized results with the sampling theory, the desirable-consistency spatial location
base, and effectiveness of the generalization).
(4) We propose a new effective image representation approach (Tunable-Q
Contourlet Transform) to improve the RWT based scale-continuously-changing
transform of DEM, and to improve the ability in retaining the local terrain and
contours. We integrate the multi-scale transform scheme of the rational-dilation wavelet
transforms and the directional filter bank of original contourlets, and eliminate the
frequency-domain aliasing component in the contourlets by reasonably employing the
stop-edge frequency of the filter bank of the multi-scale transform. We consequently obtain
the anti-aliasing tunable-Q contourlet transform which provides finer frequency-domain
partition and more sensitive directional information representation. Experiments are
implemented to evaluate its performance in image processing and the results show its
advantages in image approximation and de-noising. By using the scale (or resolution)
continuously-changing generalization principle of the proposed rational-dilation wavelet
transform based DEM representation, we apply the tunable-Q contourlet transform to DEM
generalization field and our results show its applicability and advantages.
(5) We take a comparative analysis to different DEM multi-scale representation
methods, including the three proposed methods (the M-band multiwavelet transform
based method, the rational-dilation wavelet transform based method and the
tunable-Q contourlet based method) and the traditional interpolation approaches (bilinear and bicubic), in order to evaluate their practical performance and
applicability environment. The datasets used in the experiments include the DEM
datasets with different terrain characteristics and the DEM derived from ideal mathematic
surface which provides reference real value in comparison. The experimental results show
the DEM multi-scale transform based on the proposed M-band multiwavelet transform can
effectively reduce the generalization error compared with traditional interpolation methods.
Meanwhile, its algorithm is comparatively simple and the execution is efficient, so it only
requires 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. The
rational-dilation wavelet transform based DEM generalization method does this well and
its generalized results provide us with continuously-changing resolution. The method
results in same generalization precision compared with the M-band multiwavelet based
method, for same goal resolutions. Additionally, it has moderate-level computation
complexity 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 not
only better retains the integrity and continuity of the contours or edges of DEM, but also
more effectively reduces the ‘artificial’ terrain features. On the other side, it incurs a more
complicate computational complexity and requires more computational time. So this
method is suitable for application environment with high-level execution equipment.
(6) According the main idea of the proposed DEM representation based on
rational-dilation wavelet transform and sampling theory, we design application program for
scale continuously-changing generalization of DEM. The program is established in
MATLAB software and supported by its Graphical User Interface (GUI) and its
interpreting and publishing packages. Our program provides a friendly interactive interface
and stable and effective execution process.
Key words: Multi-scale representation; DEM generalization; Multi-resolution analysis;
Sampling theory |
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