In recent years,
there has been a transition in the Russian Federation to the provision of
public services from the traditional form to the electronic one. In the vast
majority of countries of the world, government initiatives are the main engine
of the development of informatization [1]. In this regard, the Ministry of Telecom
and Mass Communications of the Russian Federation is conducting systematic work
aimed at improving the quality and level of accessibility of state and
municipal services in electronic form.
One of the requirements involves the
possibility of operational work with a large amount of various data archives,
including the search for the desired element [2]. In the case of complex geographically
distributed systems for the implementation of mobile management and related
analysis, it becomes necessary to apply and develop cartographic services, the
functionality of which supports customized layers for each industry or consumer
group. Otherwise, potential consumers of the service will be disoriented, which
in turn generates lost profits for all market participants.
In recent years, cartographic services and
geographic information systems have gradually moved from using traditional
two-dimensional maps to three-dimensional visualization.
With the development of satellite
communications and the advent of open services for accessing globally
consistent digital elevation data with high spatial resolution, such as ASTER
Global Digital Elevation Map, Earth remote sensing data have gained the most
popularity.
All multiple regional information services,
most of which are built as cartographic services, should preferably be based on
the concept of a digital twin of the region's territory, the use of which
allows you to reduce financial and time costs.
Digital twins in the management of the
region are promising systems that allow you to create an interface between
cities and infrastructure and various information systems, make the necessary
variable changes and analyze their effectiveness.
For example, the
scheme of territorial planning of capital construction of gas supply facilities
in the Volgograd region is shown in Figure 1. It should be noted that this is
far from the only working urban web service.
Fig.
1. The scheme of the territorial planning of capital construction of gas supply
facilities in the Volgograd region.
The above scheme of territorial planning of
gas supply facilities is narrow-profile but contains a large number of existing
and under construction objects of various types, such as gas pipelines, gas
distribution stations, inter-settlement gas pipelines, underground
gas storages. All these objects have many
parameters to describe them in databases.
The main problem
in the design and development of cartographic services based on the use of a
digital twin of the territory of the region is the availability of technologies
for their implementation, including high-resolution visualization of the relief
of the Earth's surface [3].
One of the tasks
of visualizing digital elevation data is to create effective methods for coding
them for subsequent visualization for its full interpretation. In conditions of
extreme redundancy by nature of high resolution
digital
elevation data,
this problem is one of the most urgent, and its solution
directly affects the computational efficiency and, consequently, the
achievement of the Service Level Agreements.
The key role in
the process of interpreting and analyzing data is played by the visualization
of the elements of the digital twin of the territory of the region, this is a
technique for converting analytical abstraction about objects into geometric
views using computer graphics core as part of the visualization software.
Digital elevation
data visualization problem is shown in Figure 2. Digital elevation data is
characterized by redundant data representation. For example, an only array of
32-bit digital elevation data elements with a size of 10x10 km with a detail of
1 m will occupy 3200 MB.
Fig. 2.
Digital
elevation data visualization process.
As a consequence,
the transmission of digital elevation data via communication channels with
limited bandwidth
[4],
and subsequent loading into the RAM of a personal computer or
smartphone is expensive, and therefore the task of compact representation of
the relief of the Earth's surface is especially relevant.
A typical two-dimensional map, for example,
generated during route planning by navigation software, does not contain
additional information about the terrain, such as passing one road over another
without crossing them, leading to errors for drivers trying to navigate such a
map.
A modern alternative to such a display is
the development of a digital twin of the territory of the region [5], which is capable of displaying the environment
familiar to a person in the form of a three-dimensional image with objects
proportional to each other: buildings, shops, bus stops, green spaces, car
washes, gas stations, pharmacies, restaurants, hairdressers, etc.
In addition, three-dimensional maps allow
you to orientate the sky around the visual features of the area: hills,
forests, fields, mountainous terrain on the horizon.
When solving such problems as the
territorial planning of capital construction of gas supply facilities in the
region, or the modernization of the mobile communication network, it is also
necessary to take into account the features of the terrain.
A digital elevation model is a digital
model or digital cartographic dataset that has a three-dimensional
representation of the terrain surface, created from elevation data [6].
There are several methods that can be used
to generate digital elevation models: ground surveying, Light Detection and
Ranging (LiDAR), radar interferometry, multibeam echo sounder [7].
Digital elevation models are a core spatial dataset
required for many environmental, planning, and scientific applications: urban
and environmental planning, topographic maps, emergency responses, hydrological
functions, geological survey and analyses, transportation planning, military
applications, maritime transportation, etc.
In the field of representation of signals
and images, presentation methods are distinguished that are focused on
stationary and non-stationary signals [8].
The idea of representing multidimensional
signals based on wavelet transforms, as well as other methods based on
orthogonal transforms, is quite simple. Initially, as a result of the
transform, some spectral components are removed from the resulting data set
according to the specified criteria. The remaining set of coefficients is
usually encoded. The dataset is recovered by decoding the coefficients, if
necessary, and applying an inverse transform to the result [9]. It is assumed that not too much
information is lost during the decimation of the transform coefficients.
However, it
should be noted that, in order to obtain a high value of accuracy and
compression ratio, it is necessary to choose the optimal wavelet basis for this
type of signals.
Currently, it
has become obvious that the Fourier transform, which is a traditional approach
for analyzing stationary signals, is ineffective for representing signals with
local features that have been widely used in recent years.
One of the main disadvantages of the Fourier
transform is the use of a sine wave as a basis, because it has an infinite
scope of definition. In the case of a limited number of terms of the Fourier
series, a basis of this type is fundamentally incapable to represent
non-stationary signals [10].
The set of all measurable
defined on the interval
denote by
,
then we have:
.
|
(1)
|
The domain of the piecewise continuous
function is
,
so
is a
-periodic function with
root mean square convergence. Any function
can be represented as a Fourier series [11]:
,
|
(2)
|
where
are the Fourier
coefficients, which are defined as:
.
|
(3)
|
The meaning of the Fourier series expansion
is that some function
ï
is represented as a set
of integer extensions of the basis function
.
The function
which is a sine wave does
not belong to the space
for which the expression
is valid:
.
Accordingly, any function
must decay at
and
,
and to generate the space
,
which is one of the cases
of a Hilbert space, it is necessary that
decay to zero as
as, for example, in case
of Le Gall basis functions[12].
Wavelet analysis also allows any function
can be represented in terms of the basis of functions
:
.
|
(4)
|
The wavelet
transform of function
is
a sum of the basis functions
,
weighted by the coefficients
[13,
14]. The basis functions
are
given, and only the coefficients
contain
information about the original function
.
The number of wavelet coefficients
is
a criterion of efficiency. A more efficient wavelet
transform allows to represent the original function
with fewer wavelet coefficients. This is
possible if the wavelet function
localizes
the features of the original function
more precisely, but real signals are
localized in both the time and frequency domains [15] so the wavelet transform is better than Fourier transform because the
original function
is
decomposed into different frequency bands.
The concept of multiresolution
decomposition is based on the representation of a signal as a combination of
its rough representation and detailed local representations of the signal, i.e.
a sequence
of closed subspaces
satisfying the following
properties [16]:
A Riesz basis is a Riesz sequence that is a
basis in the Hilbert space
[17]. According to this definition, it can be
argued that there exists a function
such that
forms a Riesz basis in
.
The
multiresolution
decomposition
describes a sequence of nested approximation spaces
in
,
so all subspaces
are
uniquely determined from the space
by
changing the scale of the function
It
is called scaling function if it is possible to perform a multiresolution
decomposition [18] . 2 is selected as the scaling factor, because it is based on a
computationally fast shift operation, i.e. the function
of the space
when
scaled by 2, becomes an element of the space
with
orthonormal bases:
,
and
.
Let's put
,
then it follows from (8)
and (9) that the sequence
is a Riesz basis in
for any
,
then there is a sequence
for scaling function:
.
|
(10)
|
The space
is the orthogonal
complement of
â
,
i.e.
and
.
Due to (6) and (7) there
is a representation of the multiresolution representation with decomposition of
the space
into the space
and orthogonal addition
:
.
|
(11)
|
A wavelet function is a function
,
that the sequence
,
as a consequence of the definition of the scaling function
from
(10), is a Riesz basis in
for
any
,
then there is a sequence of scaling ratios
:
,
|
(12)
|
where
are
the filter coefficients for the scaling function
,
are
the filter coefficients for the wavelet function
.
The filter coefficients
and
make it possible to calculate the values of the functions
and
,
as well as the spectral coefficients of
the discrete wavelet transform
.
By virtue of the relation
it
seems possible to define the function
,
written in terms of the basis functions of the space
,
in terms of the basis functions of the spaces
and
taking
into account the expressions for the scaling function
and
the wavelet function
,
then the discrete wavelet transform is defined as some mapping
,
that translates the sequences
in
sequences
.
The wavelet transform decompose the signal
in some system of functions that are shifted and scaled copies of the basis
function, then any function
can be represented on some
given resolution level
as:
,
|
(13)
|
where the first
term of the sum is a rough approximation of the original function
,
which is an element of the low-frequency bands of the spectrum. As
a result of the successive addition of other detail terms of the series to the
rough approximation of the original function
,
the resolution of the reconstructed function increases.
To represent multidimensional digital
elevation model, the multi-scale decomposition must be performed in the
space or higher order.
There are two main approaches that allow the approximation of multidimensional
digital elevation data.
The first approach is based on the
computation of a wavelet transform using multidimensional wavelet basis.
Unfortunately, it has a high computational complexity and is therefore rarely
used.
The second approach is to combine
one-dimensional wavelet transforms. In this case, the functions of this wavelet
transform are tensor products of the functions of a one-dimensional wavelet
transform.
For example, for a wavelet transform in
,
four generating functions
must be specified:
|
(14)
|
The remaining
functions of the wavelet transform are determined by the relation:
|
(15)
|
The space
generated by shifts of the
scaling function
on the same scale
,
and the detail spaces have the form:
|
(16)
|
For the wavelet transform in
we define eight generating
functions:
|
(17)
|
Then the remaining
functions of the wavelet transform can be described by the relation:
|
(18)
|
The space
generated by shifts of the
scaling function
on the same scale
,
and the detail spaces will have the form:
|
(19)
|
When decomposing the digital elevation
model, it is important to choose the optimal basis in terms of decorrelation of
wavelet coefficients. This problem arises due to the fact that the samples of
the original digital elevation model are related to each other with a
correlation coefficient reaching values close to one. This fact means that any
count can be reconstructed from near samples due to the fact that there is
redundancy in digital elevation data.
As an objective criterion for estimating
the difference between the original digital elevation data
and the reconstructed
-th
basis for decomposition of digital elevation data
we derive a criterion based on the mean square error vector for the
digital elevation model with dimensions
.
Then the problem of choosing the optimal
wavelet basis for representing the digital elevation model for a specific given
compression ratio will be written as:
|
(20)
|
where
– number of wavelet bases.
Based on the
proposed approach, a library of digital elevation models coding methods was
developed. The proposed approach was tested on 56 digital elevation models,
which have different morphological and spectral characteristics. To approximate
the initial models of the 3-dimensional surface relief, 89 different wavelet
bases were used.
As an optimality
criterion, the minimum mean square error of the reconstructed digital elevation
model obtained as a result of compression from the original model is used at a
given constant compression ratio. Since this criterion is not always consistent
with the visual perception of the quality of the resulting image, it is
possible to visually check the results obtained. An additional analysis of the
digital elevation models spectrum obtained as a result of the wavelet transform
allows us to evaluate how effectively this transform localizes energy in the
low-frequency region.
An example of
approximation of the original model of the 3-dimensional surface relief using
the Le Gall basis [19] is shown in Figure 3.
Fig. 3. Approximation of the original model of the
3-dimensional
surface
relief with the Le Gall basis, compression ratio is 16:1.
Figure 4 shows an estimate of the
proportion of distortions during compression of one of the
3-dimensional surface relief
models, presented using various wavelet bases.
It can be seen that the approximations of
the original digital elevation models based on the Antonini and Brislawn
wavelets have higher quality indicators than the approximations of the original
elevation model based on the traditional Haar, Daubechies db2, or Villasenor
1810 transformations under the same conditions, which is explained by their lower
efficiency in representing original digital elevation models.
The values of the optimality of
representation and the proportion of introduced distortions using different
wavelet transforms for the same digital elevation models can differ
significantly, which allows us to conclude that it is important to choose the
optimal wavelet basis and the efficiency of representing digital elevation
models by the selected wavelet basis.
Fig. 4. Dependence of the root mean square error on the compression
ratio for various wavelet bases.
The selection of the optimal wavelet basis
for decomposition of digital elevation model in the general case is a complex
and difficult to formalize problem. There are many properties known for
constructing an optimal wavelet basis, among which the most important are:
orthogonality, smoothness, orthonormality, approximation accuracy,
normalization, symmetry, size of the definition area. However, the optimal
combination of these parameters for a specific digital elevation model is
unknown.
In addition, a basis that works well for
one digital elevation models class may not work at all for another class. To
select the optimal wavelet basis, modeling of the quantization process is used,
since the analytical solution of this problem is rather complicated and up to
now there is no mathematical model that effectively describes all types of
triangulation models.
A technique is proposed for choosing the
optimal wavelet basis in terms of decorrelation of the wavelet coefficients
when solving the problem of representing digital elevation models. It is shown
that the efficiency of the selection of basis significantly affects the error
in computing the inverse spectral transform to reconstruction of digital elevation
models. As a result of the analysis, it was found that stationary signals are
effectively represented using classical decomposition methods, and
non-stationary signals, which include digital elevation models, are more
efficiently represented using wavelet transforms with an optimal basis. A
technique for solving the problem of representation for subsequent processing
of digital elevation models, adapted to its main features, is given.
The proposed method for selection of
optimal wavelet basis for representation of digital elevation model based on
the wavelet transform has linear computational complexity.
The important features of the proposed
method include the use of a single approach for representing and processing
digital elevation models, which made it possible to increase the number of
possible operations, for example, to process a fragment of the model at a given
scale. It was possible to achieve high accuracy in the representation of
digital elevation models, as well as to reduce the computation time, which
significantly increased the quality of service when providing digital elevation
models in the implemented regional geoinformation system.
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