A lossless DEM compression for fast retrieval method using fuzzy clustering and MANFIS neural network tài liệu, giáo án,...
Trang 1A lossless DEM compression for fast retrieval method using fuzzy
clustering and MANFIS neural network
Le Hoang Sona,n, Nguyen Duy Linha, Hoang Viet Longb
a
VNU University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
b Faculty of Basic Sciences, University of Transport and Communications, Vietnam
a r t i c l e i n f o
Article history:
Received 2 July 2013
Received in revised form
27 November 2013
Accepted 3 December 2013
Available online 16 December 2013
Keywords:
Data compression
DCR problem
Dem
Fuzzy clustering
Manfis
a b s t r a c t
In this paper, we propose an integrated approach between fuzzy C-means (FCM) and multi-active neuro fuzzy inference system (MANFIS) for the lossless DEM compression for fast retrieval (DCR) problem, aiming to compress digital elevation model (DEM) data with the priority of fast retrieval from the client machine over the Internet environment Previous researches of this problem either used the float wavelet transforms integrated with the SPIHT coding or constructed a predictor model using statistical correlation of DEM data in local neighborhoods; thus giving large-sized compressed data and slow transferring time of data between the server and the client Based on the observation that different non-linear transforms for predictive values in the sliding windows may increase the compression ratio, we herein present a novel approach for DCR problem and validated it experimentally on the benchmark DEM datasets The comparative results show that our method produces better compression ratio than the relevant ones
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1 Introduction
Digital elevation model (DEM) (a.k.a terrain) is the most popular
source among all kinds of terrain data that reflect the vertical and
horizontal dimensions of land surface and are expressed in terms
of the elevation, slope, and orientation of terrain features It has
been being applied for various applications such as flood or
drainage modeling, land-use studies, geological applications,
ren-dering of 3D visualizations, rectification of aerial photography or
satellite imagery, line-of-sight analysis, etc According toLi et al
(2005) andSon et al (2012,2013), DEM is commonly built using
remote sensing techniques, e.g photogrammetry, LiDAR, IfSAR, or
from land surveying so that its sizes are often large, for instances,
approximately several gigabytes for a medium resolution terrain
Transferring DEM over the Internet environment is necessary not
only for sharing new knowledge about terrain features but also for
the storing and retrieving relevant information of that terrain
Nevertheless, the sizes of DEM are large, and the compression
of this dataset is a must before sending it to the Internet As such,
the lossless DEM compression for fast retrieval (DCR) problem was
designed to compress DEM with the priority of fast retrieval from
the client machine over the Internet environment
Several works concerning the DCR problem were presented Boucheron and Creusere (2003,2005) used some float wavelet transforms such as 5/3, 9/7, max and min wavelets to decompose a terrain into sub-bands, such that lower sub-bands correspond to higher terrain frequencies and higher sub-bands correspond to lower terrain frequencies, where most of the image energy is concentrated Then, coefficients are quantified before encoding After quantification, coefficients can be encoded by the SPIHT coding which transmits the most important image information first The searching process is fast, however, the compression one
is slow since much computation is required in SPIHT and Wavelet transforms Kidner and Smith (2003)presented an algorithm for the DCR problem using the statistical correlation of terrain data in local neighborhoods Elevation data are pre-processed by simple linear prediction algorithms such as 3-points, 8-points, 12-points
or 24-points Lagrange The differences between the predictions and the real elevations are then compressed by Arithmetic Coding This algorithm is simple and obtains fast compressed time More-over, the compressed files are less than half the size of encoded DEM by GZIP However, the algorithm does not support for retrieval process Inanc (2008) introduced ODETCOM which is a predictor model using a causal template of size eight (linear predictor) An over-determined system of linear equations that correspond to the prediction and consist of the eight elevations in the causal template and a constant term was used to find a compromise for the best set of the coefficients The compression ratio of ODETCOM is better than those of JPEG 2000 (Marcellin
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n Corresponding author Tel.: þ 84 904171284; fax: þ 84 0438623938.
E-mail addresses: sonlh@vnu.edu.vn , chinhson2002@gmail.com (L.H Son)
Trang 2et al., 2000) (lossless mode) and JPEG-LS (Rane and Sapiro, 2001).
Nonetheless, ODETCOM has high computational complexity
depending on the time solving the over-determined linear
equa-tion systems.Zheng et al (2009)focused on the DEM multi-scale
representation, progressive compression and transmission based
on the integer lifting wavelet Through a series of experiments on
different wavelet transforms, the 2/6 integer wavelet was found to
be the most suitable transform for the DEM multi-scale
progres-sive compression among the 14 reversible integer wavelet
trans-forms compared (Adams and Kossentini, 2000) Our previous work
inSon et al (2011) presented an algorithm named as DCRA for the
DCR problem using the sliding windows and the modifiedKidner
and Smith (2003)method with 1-point Lagrange being applied for
all sliding windows Motivated by the multi-resolution
mechan-ism, the client machine can access some parts of DEM in
equiva-lent to a specific level of resolutions; thus accelerating the retrieval
process from the client Summary of the relevant works for the
DCR problem is highlighted below:
The relevant researches either used the float wavelet
trans-forms integrated with the SPIHT coding or constructed a
predictor model using statistical correlation of DEM in local
neighborhoods; thus giving large-sized compressed data (or
low compression ratio) and slow transferring time of data
between the server and the client
The comparison in the paper (Son et al., 2011) showed that
DCRA obtains faster compressed and retrieval time than the
Kidner and Smith (2003) method and other relevant ones
whilst the compression ratio is approximate to those of other
methods
From those remarks, we clearly recognize that low compression
ratio is the major problem of all algorithms that affect the total
processing time including the compressed, the transferring and
the retrieval time, and DCRA is the best method for the DCR
problem Thus, our objective in this study aims to enhance the
compression ratio of DCRA by using an integrated approach
between fuzzy C-means (FCM) and multi-active neuro fuzzy
infer-ence system (MANFIS) It is motivated by the fact that different
non-linear transforms for predictive values in the sliding windows
of DCRA may increase the compression ratio Our contribution in
this work is the introduction of a novel method named as
F-MANFIS that incorporates FCM (Bezdek et al., 1984) with MANFIS
to determine similar sliding windows andfind out the coefficients
of non-linear transforms for those sliding windows F-MANFIS
will be compared with DCRA and other methods in terms of
the compression ratio on the benchmark DEM datasets to verify
the effectiveness of the proposed method
The rest of the paper is organized as follows.Section 2analyzes
the DCR problem and the DCRA method The ideas and details of
F-MANFIS are introduced in Section 3 Section 4 validates the
proposed approach through a set of experiments involving
benchmark DEM data Finally,Section 5 draws the conclusions and delineates the future research directions
2 Analyses of the DCR problem and the DCRA method
In this section, we clearly describe the DCR problem and the DCRA method Now, let us briefly present some basic notations in Son et al (2011) as follows (Figs 1–3):
Definition 1 A sliding window (SW) with an original point ðx0; y0Þ and sizes ðw; hÞ is defined as
SWðx0; y0Þ ¼ fðx0þu; y0þvÞjuA½0; w; vA½0; hg: ð1Þ
Definition 2 A set of moving steps for the point ðx; yÞ with arguments ði; jÞ is denoted as
Δi;jðx; yÞ ¼ fðxþδ1; yþδ2Þjδ1¼ i; 0; i; δ2¼ j; 0; jg: ð2Þ
Definition 3 The set of original points and their sizes at the level k
of resolutions is specified below
ðx0i; y0jÞ ¼ W0
2k i;H0
2k j
;
i; j ¼ 1; 2k
ðw; hÞ ¼ W0
2k;H0
2k
where ðW0; H0Þ are the sizes of a DEM From now on, we denote
SWkðx0i; y0jÞ as a sliding window with the original point ðx0i; y0jÞ
at the level k
From those definitions, the DCR problem can be interpreted as the compression of DEM at the server machine that allows the fast display of a sliding window at a specific level of resolutions –
SWkðx0i; y0jÞ from the client over the Internet environment with-out extracting the whole compressed terrain
According toSection 1, DCRA, which is influenced by the multi-resolution mechanism, is the best method for the DCR problem This method consists of two phases: compression and retrieval, and the effectiveness of the compression process decides the fast retrieval from the client machine The mechanism of the compres-sion in DCRA is depicted inFig 4
It is obvious that sliding windows, generated from a DEM by the splitting mechanism at a specific level of resolutions, are transformed by the modified Lagrange 1-point The results of that procedure, including the template point and the prediction errors,
Fig 2 Two levels of resolutions.
Fig 3 The DCR problem.
Trang 3are compressed by Arithmetic Coding and stored at the server
machine In the client machine, a sliding window and its
neigh-borhoods specified by the set of moving steps can be retrieved
from the equivalent compressed ones at the server Thus, this
enhances the compressed and retrieval time in comparison with
the Kidner and Smith method and other relevant ones since a
small number of compressed sliding windows is invoked and
transferred to the client instead of the whole DEM
Despite that the processing time is the advantage of DCRA, the
compression ratio is still approximate to those of other relevant
methods The reason for this fact is the using of the modified
Lagrange 1-point in the compression process described below
In Eq.(5), x is a template point in a sliding window and z is its
predictive value Beside the Lagrange 1-point, there exists some
other Lagrange transforms such as 3-points (JPEG encoder),
8-points, 12-points and 24-points that use
multiple-template-points in the sliding window to predict a value in a sliding
window Now, let us discuss some remarks of the Lagrange
transforms as follows:
The compression ratio can be enhanced by a
multiple-template-points Lagrange transform:Kidner and Smith (2003)argued that
Lagrange 1-point is often used since it requires less
computa-tion than other transforms Nevertheless, more template points
in the sliding window, e.g 8-points, 12-points and 24-points,
the transform uses to predict a value in a sliding window,
better accuracy the result obtains Thus, it should be a
multiple-template-points’ Lagrange transform for the consideration of
the improvement of compression ratio in DCRA
The non-linear Lagrange transform should be used instead of the
linear one: The above Lagrange transforms are linear and
expressed by Eq.(6) The advantages of linear transforms are
the simplicity and fast processing time However, linear
trans-forms do not minimize the prediction errors as expected
Indeed, it is better to use a non-linear Lagrange transform
instead of a linear one
z ¼ ∑n
It should be different transforms for different predictive values:
In the articles,Kidner and Smith (2003) andSon et al (2011),
the authors used a transform function for all predictive values
only Even though this reduces the computational time
of the algorithms, the prediction errors are not minimal since
each predictive value has different correlation with its
neighborhood As such, various transform functions should be used for predictive values in order to achieve high compre-ssion ratio
Based upon those remarks, we will design a novel compression algorithm so-called F-MANFIS that uses fuzzy C-means (FCM) (Bezdek et al., 1984) to determine similar sliding windows in a DEM and MANFIS neural network to find out the coefficients of non-linear transforms for those sliding windows
3 The F-MANFIS method 3.1 The algorithm ThroughSection 2, we clearly recognize that using various non-linear multiple-template-points’ Lagrange transforms for predic-tive values in a sliding window enhances the compression ratio of the algorithm Nonetheless, the number of predictive values in all sliding windows is large, and we cannot use the strategy above for those predictive values Instead of this, the same Lagrange trans-form can be applied to all predictive values in a sliding window and to those in neighbored sliding windows since nearby sliding windows in a DEM share similar characteristics This observation is
influenced by spatial interaction principle (Birkin and Clarke, 1991) that dominates the creation of DEM and other geographic sources The usage of Lagrange transforms in this work is different to that
in the DCRA method, which applies a transform to all sliding windows only Consequently, a fuzzy clustering method is required
to determine similar sliding windows
Fig 5 describes the flowchart of the F-MANFIS method According to thisfigure, based upon a desired level of resolution,
a DEM is divided into sliding windows by the splitting mechanism (Son et al., 2011) These sliding windows are then classified into groups by FCM (Bezdek et al., 1984), which is a widely known fuzzy clustering algorithm for a set of classification problems, e.g image segmentation,fingerprint and gesture recognition, etc
Fig 5 The F-MANFIS flowchart.
Fig 4 The compression process in DCRA.
Trang 4The pseudo-code below shows how FCM groups the sliding
windows
Input: -A set of sliding windows
-Number of groups (P)
Output: -P groups of sliding windows including centers
FCM algorithm:
1: For each sliding window, calculate the mean value of
all data points
2: Use FCM algorithm (Bezdek et al., 1984) to classify the
set of mean values into P groups
3: For each group
4: Assign the data member that is nearest to the center
as the new center, and its sliding window is called a
basic sliding window
5: The sliding window holding other data members is
called related sliding windows (R-)
6: End For
Example 1 InFig 6, we illustrate the activities of FCM algorithm A
DEM is divided into 16 sliding windows at the level 2 of resolutions
These sliding windows are classified into three groups with the
centers being marked as red, green and blue (basic sliding
win-dows) Sliding windows marked as R-1 belong to a cluster whose
center is green Similarly, R-2 (R-3) relates to the blue (red) center
Once we have the groups of sliding windows, the MANFIS
neural network is applied to the basic sliding windows of those
groups MANFIS is a generalization of CANFIS (Jang, 1993), which is
a fuzzy-neural network designed to construct a multiple-output
model with non-linear fuzzy rules such as thefirst-order Sugeno
fuzzy rule By integrating a modular neural network, CANFIS is
able to rapidly and accurately approximate complex functions
According toParthiban and Subramanian (2007), CANFIS can solve
problems more efficiently than other types of neural networks
when the underlying function to a model is highly variable or
locally extreme Nonetheless, in order tofind a non-linear
trans-form function that is best-suited for a sliding window, an
exten-sion of CANFIS is required Thus, we propose a novel fuzzy-neural
network the so-called multi-active neuro fuzzy inference system
(MANFIS), which is able to solve the MIMO (multiple inputs–
multiple outputs) model by variousfirst-order Sugeno fuzzy rules
Since the basic sliding window of a group contains principal
elevations that are closely related to those of other related sliding
windows, we firstly use the MANFIS neural network to the basic
sliding window and receive the template points, the prediction errors
and the MANFIS parameters used to restore the original basic sliding
window from the compressed one This step consists of several
time-consuming training iterations of MANFIS parameters in order to get
the optimal ones that minimize the prediction errors Next, those
parameters (a.k.a coefficients) are applied to other related sliding windows, e.g R-1 This reduces the computational complexity of the algorithm since the basic sliding window and other related sliding windows of a group have strong spatial correlation The outputted results are the template points and the prediction errors of related sliding windows All outputs of the group are compressed by the Arithmetic Coding algorithm and stored at the server as in the paper (Son et al., 2011) Similar activities are applied to other groups of sliding windows The F-MANFIS algorithm stops when all groups are compressed and stored at the server
3.2 Using MANFIS for basic sliding windows
As we can recognize in Fig 5, the most important part of F-MAFIS is using MANFIS neural network for the basic sliding window that generates the optimal MANFIS parameters for other related sliding windows This sub-section describes in detail the structure, the working mechanism and the training phase of MANFIS The structure of MANFIS is depicted inFig 7
The working mechanism of MANFIS is described as follows Two inputted variables x and y are the template points in the basic sliding window They are modeled by the fuzzy sets having domain of values {“High”, “Low”} MANFIS is capable to process the MIMO model by the first-order Sugeno fuzzy rules in the following equation:
Rule 1: IF ðx isA1Þ and ðy is B1Þ THEN
ðu1is C1Þ and ðu2 is C1Þ and ðu3is C1…ðunis C1Þ: ð7Þ MANFIS consists of four layers as described below:
Layer 1 (fuzzification): Each node in this layer is the member-ship grade of a fuzzy set and specifies the degree to which the given input belongs to one of the fuzzy sets The membership function is
μiðxÞ ¼ 1=ð1þexpðaiðxciÞÞÞ;
where ðai; ciÞ, 8i ¼ 1; 4 are the premise parameters
Layer 2: Each node in this layer is product of all the incoming
signals
wj¼ μAjðxÞ μBjðyÞ;
Layer 3 (synthesis): Each node in this layer is sum of output
values from the previous layer
O3j ¼ ∑2
i ¼ 1
wi Cij;
where n is the number of predictive values except two template points, and CijA½0; 1; 8j ¼ 1; n; 8i ¼ 1; 2 are the consequence parameters The last node in the layer is calculated as the direct sum of outputted values:
Layer 4 (output): Outputted results are specified as follows:
uj¼ O3
j=O3
n þ 1;
Obviously, based on the premise and consequence parameters, the predictive values u (j ¼ 1; n) can be calculated from two
Fig 6 Activities of FCM (For interpretation of the references to color in this figure
caption, the reader is referred to the web version of this paper.)
Trang 5template points x and y The mean square error (MSE) is
calculated as the difference between the basic sliding window
and its predictive sliding window:
MSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
∑n
j ¼ 1
ðujxjÞ2
s
where xj ðj ¼ 1; nÞ is the original value of ujin the basic sliding
window
In order to minimize the MSE value, the training phase of
MANFIS is required In this part, we use the hybrid algorithm
between gradient descent (GD) and particle swarm optimization
(PSO) (Kennedy and Eberhart, 1995) to train the MANFIS network
The training consists of two sub-phases: forward and backward In
the forward, wefix the premise parameters and use PSO to train
the consequence ones Then, in the backward, we use the optimal
consequence parameters corresponding to the global best swarm
particle in PSO to train the premise ones by the GD method Those
premise parameters are again used in the forward phase of the
next iteration step The training phase is repeatedly performed
until MSE is smaller than a pre-defined threshold (ε)
The reasons of using the hybrid method between GD and PSO
for the training are elicited as follows Since we have two kinds of
parameters to be trained that are the premise and the consequence,
there is a need of using different training methods to adapt with their
own characteristics More specifically, the number of consequence
parameters is equivalent to the number of elements in a basic
sliding window, and they contribute greatly to the calculation of the
MSE value between the basic sliding window and its predictive one
Thus, a meta-heuristic method that can handle large number of
particles and converge quickly to the optima such as PSO should
be opted Lastly, because the number of premise parameters is small
and their values are often affected by the consequence parameters so
that the fast GD training method is used Another advantage of using
the hybrid method between GD and PSO is the balance between
computational time and the compression ratio Thus, this explains
why we combine the GD and the PSO methods to train the
parameters
The GD training algorithm is well-known and carefully described in Ooyen and Nienhuis (1992) Indeed, we ignore the descriptions of this algorithm and concentrate on the PSO algo-rithm PSO is a population based stochastic optimization technique developed by Dr Eberhart and Dr Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling Generally, it is based on the principle: “The best strategy to find the food is to follow the bird which is nearest to it” Thus, each single solution in PSO is a“bird” or “particle” in the search space All particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles The particlesfly through the problem space by following the current optimum particles The training of forward sub-phase using PSO is described below:
Encode/decode: The beginning population is initiated with Q
particles where Q is a designed parameter Each particle is a vector v ¼ ðC11; C21; C12; C22; ::; C1n; C2nÞ where Cij
ð8 j ¼ 1; n 8i ¼ 1; 2Þ is a consequence parameter These particles are randomly initiated in [0,1]
Thefitness function:
f ðvÞ ¼ ð1=nÞ ∑n
i ¼ 1
½ðw1C1iþw2C2iÞ=ðw1þw2Þxi2; ð14Þ
where xi(i ¼ 1; n) is the original value of uiin the basic sliding window
The criteria to choose the best values in PSO:
After a number of training iterations ðTrain_IterÞ, the optimal solution of {premise, consequence} parameters is found We then need to perform the backward sub-phase to converge to the global solution
Now, F-MANFIS is totally described But before we evaluate the proposed method, let us raise an important question In the
Fig 7 The structure of MANFIS.
Trang 6flowchart of the algorithm inFig 5, we have used FCM to classify
similar sliding windows into groups However, how many groups
of sliding windows are enough for F-MANFIS? In the other words,
we have to specify the suitable value of parameter P in order to
balance between the compression ratio and the computational
complexity of the algorithm The following theorem helps us to
answer the question
Theorem 1 The acceptable range of parameter P is ½2; N0=3 where
N0is the number of sliding windows
Proof Since the number of premise and consequence parameters
of MANFIS is ð2n þ 8Þ where ðn þ 2Þ is the number of values in a
sliding window Indeed, the upper bound of parameter P is
½N0ðnþ2Þ=½2ðnþ2Þþ4 N0=3: □ ð16Þ
Using this result, we can choose the suitable value of parameter
P for a given level of resolutions and a given terrain
4 Results
4.1 Experimental environment
In this part, we describe the experimental environments such
as the following:
Experimental tools: We have implemented the proposed
F-MANFIS method in addition toKidner and Smith (2003)and
DCRA (Son et al., 2011) in Java programming language and
executed it on a PC Intel Pentium 4, CPU 2.66 GHz, 1 GB RAM,
80 GB HDD
Parameter setting: According to Kennedy and Eberhart (1995),
some values below are assigned to the PSO’s parameters in
order to achieve the best results:
○ The number of particles in PSO is: Q ¼ 100
○ The maximal number of iterations in PSO is 100
○ The parameters of velocity and positions in PSO is:
ðc1; c2; c3Þ ¼ ð0:2; 0:3; 0:5Þ
○ The threshold: ε ¼ 0:01 and the number of groups: P ¼ 2
○ The total number of iterations in the training phase:
Train_Iter ¼ 50
Experimental datasets: a benchmark set of 24 USGS 1:250,000
scale DEMs on a 1201 1201 grid (see:http://dds.cr.usgs.gov/
pub/data/DEM/250/) Each DEM is thefirst in each higher level
USGS directory (ordered alphabetically) for which there is a
DEM entry (there are no DEMs for X or Y) Summary statistics of
these data can be found atKidner and Smith (2003), including
the elevation ranges, mean elevations, standard deviations, and
the entropies or information contents of the original data
Objective: We evaluate the compression ratios and compressed
time of those algorithms Some experiments to choose the
suitable value of parameter Pand the comparison of training
methods in F-MANFIS are also considered
4.2 The comparison of compression ratio
In this section, we use three algorithms: Kidner & Smith, DCRA
and F-MANFIS to compress 24 DEM datasets and record their
compressed sizes The comparison of compressed sizes of those
algorithms is described inTable 1
Obviously, the compressed sizes of F-MANFIS are mostly
smaller than those of DCRA and Kidner & Smith For example,
the original size of dataset no 4 (Dalhart) is 7045.44 kilobytes
(kB)–6.88 Megabytes (MB) The compressed size of this dataset using Kidner & Smith (DCRA) is 242.12 (252.94) kB This value in case of F-MANFIS is 200.7 kB, which is approximate to 82.9% (79.3%) of that of Kidner & Smith (DCRA) Similarly, the com-pressed sizes of F-MANFIS, DCRA and Kidner & Smith for the dataset no 24 (Yakima) are 143.36, 587.99 and 551.16 kB, respec-tively The compressed size of F-MANFIS in this case is approx-imate to 26% (24.4%) of that of Kidner & Smith (DCRA)
Nonetheless, there exist some cases that show the inferiority of F-MANFIS to other algorithms such as the dataset no 1 (Aberd-een), no 10 (Jacksonville), no 18 (Racine) and no 22 (Valdosta) In those cases, the compressed sizes of F-MANFIS are larger than those of DCRA and Kidner & Smith However, the differences of sizes between algorithms are not large, and the number of those cases is small in comparison with the remaining Therefore, F-MANFIS achieves better compressed sizes of data than other relevant algorithms
FromTable 1, we canfind out that the maximal differences of compressed sizes between F-MANFIS and other algorithms are recorded at the dataset no 19 (Sacramento) In this case, the compressed sizes of F-MANFIS, DCRA and Kidner & Smith are 217.09, 870.08 and 826.87 kB, respectively The difference of compressed sizes between F-MANFIS and DCRA (Kidner & Smith)
is 652.99 (609.78) The compressed size of F-MANFIS in this case is approximate to 26.2% (24.95%) of that of Kidner & Smith (DCRA) Similarly, the minimal differences of compressed sizes between F-MANFIS and other algorithms are found at the dataset no 9 (Idaho Falls) The difference of compressed sizes between F-MANFIS and DCRA (Kidner & Smith) in this case is 3.95 (1.65) The compressed size of F-MANFIS in this case is approximate to 99.3% (98.3%) of that of Kidner & Smith (DCRA)
Fig 8visualizes the compressed sizes of three algorithms on 24 DEM datasets As we can recognize from this figure, the com-pressed ratio line of F-MANFIS is mostly below those lines of DCRA and Kidner & Smith The compressed size of F-MANFIS is approx-imate to 60.9% (61.6%) of that of DCRA (Kidner & Smith) Thus, using the hybrid approach between FCM and MANFIS in the proposed F-MANFIS helps the compression sizes of data better
In Fig 9, we depict the compression ratios of all algorithms, calculated by the percentages of the compressed size over the
Table 1 The comparison of compressed sizes of algorithms (kB).
Trang 7original size of a DEM dataset The results show that the
compres-sion ratio of F-MANFIS is better than those of DCRA and Kidner &
Smith The average compression ratio of F-MANFIS on 24 DEM
datasets is 3.09 that means the compressed size of dataset is
approximate to 3.09% of the original size Meanwhile, the average
compression ratios of DCRA and Kidner & Smith are 7.22 and 6.84,
respectively
InFig 10, the performance of all algorithms, calculated by the
subtraction of 100 percents to the compression ratio, is described
Thisfigure shows how many percents of data can be compressed
by a compression algorithm It is clear that F-MANFIS is quite
stable through various cases The performance of F-MANFIS is
96.9% whilst those of DCRA and Kidner & Smith are 92.8% and
93.2%, respectively This re-confirms that F-MANFIS is better than
DCRA and Kidner & Smith in terms of compression ratio
We also investigate the compressed time of all algorithms
The results inTable 2show that the F-MANFIS takes more time
to compress the datasets than other algorithms Nonetheless,
the compressed time of F-MANFIS is around 303 s ( 5 min) to
compress a DEM It is small and is acceptable to our context
Moreover, the experimental results also affirm a remark in the
paper (Son et al., 2011) that proves the advantage of DCRA over other algorithms in terms of processing time
Some remarks found fromSection 4.2are the following:
The compression ratio of F-MANFIS is better than those other relevant algorithms
The performance of F-MANFIS is 96.9%
The compressed time of F-MANFIS is slower than those of other algorithms
4.3 Choosing optimal value for the number of groups
In this section, we made some experiments to determine the optimal number of groups of sliding windows (P) for our experi-mental datasets Even though we have a theoretical analysis of this determination throughTheorem 1, an experimentation to specify the exact value of P is still necessary We have re-run the F-MANFIS algorithm with various numbers of groups and on different datasets, especially those who are the bad cases in Section 4.2such as the dataset no 1 (Aberdeen), no 10 (Jacksonville), no 18 (Racine) and no
Fig 8 The compressed sizes of algorithms on 24 benchmark DEM datasets (kB).
Fig 9 The compression ratios of algorithms.
Trang 822 (Valdosta) The level of resolutions used for the experiments both
in the current section and in the previous one is 2 Thus, we have
N0¼ 42¼ 16 sliding windows According toTheorem 1,
Pr½N0=3 ¼ 6:
Therefore, the possible value of P belongs to the interval [2, 6] The
experimental results are shown inTable 3
Fig 11below visualizes the results inTable 3
From those results, we recognize that using large number of
groups increases the compressed size of data in F-MANFIS For
example, the compressed sizes of Aberdeen with P from 2 to 6 are
143.36, 155.65, 172.03, 188.42 and 204.80 Each time a group is
added, the compressed size increases 9.3% of the previous size of
data Similarly, the incremental ratios of Baker, Jacksonville, Racine
and Valdosta are 5.1%, 7.8%, 10.9% and 12.3%, respectively The
reason for this fact is that more MANFIS parameters needed to be
stored at the server when the number of groups increases In most datasets, the value of parameter P ¼ 2 often minimizes the compressed sizes of data Thus, it is our recommended number
of groups of sliding windows when compressing a DEM dataset The remark found fromSection 4.3is the following:
The optimal number of groups of sliding windows that should
be used is two
4.4 The comparison of training methods
In Section 3.2, we have already known the reasons of using the hybrid method between GD and PSO for the training phase However, is the hybrid method more effective than the stand alone PSO or GD for training? This section aims to answer this question
by performing the comparison between the original F-MANFIS (a k.a GD-PSO) and the modified F-MANFIS with the training methods being the stand alone PSO (a.k.a PSO*) and GD (a.k.a GD*) In PSO*, we randomly generate the premise parameters and use them to create the consequence ones The new-born para-meters are trained by the PSO algorithm We continue the process from random generation of premise parameters to training of consequence parameters several times and choose thefinal result having minimal MSE value among all In GD*, we also randomly generate the premise parameters and use them to create the consequence ones However, contrary to PSO*, those parameters are not trained but used to generate the next premise ones by the
GD method This process is repeatedly performed until the total number of iterations in the training phase (Train_Iter) is reached
Fig 10 The performance of algorithms.
Table 2
The comparison of compressed time of algorithms (sec).
Table 3 The compressed size of F-MANFIS through various numbers of groups (kB).
Trang 9The evaluation criteria for this experiment are the compressed
sizes and time of algorithms
The results inTable 4show that the compressed sizes of
GD-PSO are smaller than those of GD* and GD-PSO* For example, the
compressed size of GD-PSO on the Aberdeen dataset is 143.36 kB
whilst those of GD* and PSO* are 481.09 and 250.18 kB,
respec-tively Similarly, the results on some datasets such as the Dalhart,
Eagle Pass, Fairmont, Macon, Tallahassee and Waco also confirm
the finding above even though the compressed sizes of three
algorithms are approximate to the others Thus, this means that
using the hybrid method between GD and PSO results in better
compressed sizes of algorithm than using the stand alone GD
and PSO
The explanation for this fact is that in PSO*, the consequence
parameters generated by PSO is just the best values for a given
premise parameters This does not guarantee the optimal value
since other premise parameters may obtain better values of consequence parameters In the other words, the solution found
by PSO* is the local optima Similarly, the solution of GD* is the local optima too By combining both GD and PSO methods, the final solution is optimal both on the premise and consequence parameters so that it can be considered as the global optima The experimental results stated that the achieved global optima is better than the local optima of GD* and PSO* in terms of compressed sizes of algorithms
Nonetheless, there are some exceptions inTable 4showing that the compressed sizes of GD-PSO are larger than those of PSO* and GD* For instance, the results on the Idaho Falls, Jacksonville and Racine datasets clearly confirm that the compressed sizes of PSO* are smaller than those of GD-PSO, especially on the Racine dataset both results of GD* and PSO* are better than that of GD-PSO This fact can be explained by the reverse impact of consequence to premise parameters In cases that the achieved consequence parameters is the global optima, they are still used to generate the premise parameters and the next consequence ones; thus increasing the MSE values since the next consequence parameters are not the global optima again Nevertheless, those cases are rare and most of times the GD-PSO method still produces better results than PSO* and GD* as stated inTable 4
Anotherfinding found fromTable 4is the superiority of PSO* over GD* The results show that the compressed sizes of PSO* are mostly smaller than those of GD* The differences of values are quite obvious, e.g 230 kB (Aberdeen) or 110 kB (Hailey) There are also some exceptions that reflect the larger values of PSO* to GD* such as on the Nashville and Yakima datasets However, those exceptions are not much, and PSO* is assumed to be better than GD* This clearly confirms the significant role of consequence parameters in the system, and using a meta-heuristic algorithm such as PSO to train this kind of parameters achieves better MSE values and the compressed sizes of algorithms than using the GD algorithm to train the premise parameters only
Fig 12 describes the compressed time of GD-PSO, PSO* and GD* The results show that using the GD method for the training of premise parameters in GD* takes less computational time than using the PSO method for consequence parameters in PSO* Both GD* and PSO* are faster than GD-PSO The average compressed time of GD-PSO is 303 s whilst those of GD* and PSO* are 150 and
256 s, respectively Even though the compressed time of GD-PSO is
Fig 11 The compressed size of F-MANFIS by the number of groups.
Table 4
The compressed sizes of algorithms using various training methods (kB).
Trang 10the largest among all, it is assumed to be acceptable to our context
and objectives
The remark found fromSection 4.4is the following:
The hybrid GD-PSO for the training in F-MANFIS obtains better
compressed sizes of algorithm than the stand alone GD and
PSO methods
PSO* is better than GD* in terms of compressed sizes.
GD-PSO has large compressed time but is acceptable.
5 Conclusions
In this paper, we concentrated on the DCR problem and
presented a hybrid approach between fuzzy clustering (FCM)
and MANFIS neural network to improve the compression ratio of
the state-of-the-art compression algorithm– DCRA The proposed
method (F-MANFIS) used FCM to determine similar sliding
win-dows in a DEM and used MANFIS tofind out the coefficients of
non-linear transforms for those sliding windows F-MANFIS was
compared with some best compression algorithms for DCR such as
DCRA and Kidner & Smith on the benchmark DEM datasets The
results showed that the compression ratio of F-MANFIS is better
than those of other algorithms Theoretical and experimental
analyses on the optimal number of groups of sliding windows
used in F-MANFIS and the efficiency of using the hybrid GD and
PSO in the training phase over stand alone methods were
conducted
Further works of this theme aim to answer the questions as
follows: (i) How to choose two template points in a sliding
window to minimize the objective function?; (ii) How many
template points is optimal?
Acknowledgment
The authors are greatly indebted to the editor-in-chief, Prof
B Grabot and anonymous reviewers for their comments and their
valuable suggestions that improved the quality and clarity of
paper Other thanks will be sent to Mr Tran Van Huong and
Ms Bui Thi Cuc for some experimental works This work is sponsored by the NAFOSTED under Contract no 102.01-2012.14 References
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