=Paper=
{{Paper
|id=Vol-2391/paper4
|storemode=property
|title=Interpolation of multidimensional signals using the reduction of the dimension of parametric spaces of decision rules
|pdfUrl=https://ceur-ws.org/Vol-2391/paper4.pdf
|volume=Vol-2391
|authors=Mikhail Gashnikov
}}
==Interpolation of multidimensional signals using the reduction of the dimension of parametric spaces of decision rules ==
https://ceur-ws.org/Vol-2391/paper4.pdf
Interpolation of multidimensional signals using the reduction
of the dimension of parametric spaces of decision rules
M V Gashnikov1,2
1
Samara National Research University, Moskovskoe Shosse 34А, Samara, Russia, 443086
2
Image Processing Systems Institute of RAS - Branch of the FSRC "Crystallography and
Photonics" RAS, Molodogvardejskaya street 151, Samara, Russia, 443001
e-mail: mgash@smr.ru
Abstract. In this paper, we consider the interpolation of multidimensional signals problem. We
develop adaptive interpolators that select the most appropriate interpolating function at each
signal point. Parameterized decision rule selects the interpolating function based on local
features at each signal point. We optimize the adaptive interpolator in the parameter space of
this decision rule. For solving this optimization problem, we reduce the dimension of the
parametric space of the decision rule. Dimension reduction is based on the parameterization of
the ratio between local differences at each signal point. Then we optimize the adaptive
interpolator in parametric space of reduced dimension. Computational experiments to
investigate the effectiveness of an adaptive interpolator are conducted using real-world
multidimensional signals. The proposed adaptive interpolator used as a part of the hierarchical
compression method showed a gain of up to 51% in the size of the archive file compared to the
smoothing interpolator.
1. Introduction
Currently, the need to use multidimensional digital signals is becoming more acute [1]. It is primarily
about such areas as remote sensing [2-3], processing of multispectral and hyperspectral signals [4], as
well as video processing.
Now we know a large number of interpolation and approximation algorithms for such signals [1-
16], and the high-performance requirements often do not allow us to use trivial linear, bilinear or
bicubic interpolators [1]. In other words, there is a tendency to the more and more widespread use of
more complex interpolation methods, such as the support vector method [5], locally optimal well-
adapted basis functions [6], approximation by multidimensional orthogonal polynomials [7],
multidimensional approximation and interpolation [8] etc.
However, the presence of a large number of well-developed solutions did not stop research in the
development and modification of interpolation and approximation algorithms for multidimensional
signals. The approximation based on Kronecker bases [9], splines [10], and tensors [11] continues to
be improved. Artificial neural networks [7,12] are also increasingly used for interpolating signals.
Even the well-known least squares method (OLS) continues to be modified [13] in recent years. In the
foreign literature, special attention is paid to the sparse approximation method [14], which is the basis
for the “compressed sensing” approach [15–16].
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All the above algorithms have a sufficiently high accuracy in solving the corresponding applied
problems. However, these algorithms have high computational complexity. In this paper, we propose
fast interpolation algorithms for multidimensional signals that are adaptive due to automatic switching
between interpolating functions at each point of the signal. This adaptability makes it possible to
ensure high interpolation accuracy with low computational complexity, due to the simplicity of the
interpolating functions used.
The proposed interpolators are parameterized. Therefore, we can optimize them according to
various criteria, the choice of which is determined by the specifics of the applied problem being
solved. Optimization of adaptive interpolators is carried out in the space of their parameters. The
complexity of this optimization is substantial if resources are limited. In this paper, we propose an
algorithm for reducing the complexity of optimization due to the dimension reduction of the
interpolator parametric space.
2. Adaptive interpolation of multidimensional signals
Let C x be a multidimensional digital signal, and x be the vector of arguments. Let an arbitrary
count C ( x) be necessary to interpolate using the nearest reference samples Cˆ x . Let
k
P Cˆ x be the set of interpolation functions used. Thus, for the current sample C( x) several
i
k
interpolating values can be calculated:
Pi x P Cˆ k ( x )
i
. (1)
The parameterized decision rule R performs the choice of the interpolating value for each signal
sample:
P x P x , i R х , lim ,
i
(2)
Rule R uses a local feature vector х , which is calculated based on the nearest reference samples
Ck x . Let the decision rule be parameterized, i.e. depends on the parameter lim . The value of this
parameter is determined by optimizing a specific criterion, which depends on the applied task. This
criterion can be, for example, the criterion for minimizing the energy of post-interpolation residues:
lim f x min
, f x C x P x ,
lim
x (3)
where f x are post-interpolation residues.
The criterion for minimizing the energy of post-interpolation residues can be used, in particular,
when solving the problem of matching [17-18] of heterogeneous signals that differ in resolution,
number of components, etc.
In this paper, we propose to use the criterion for minimizing the entropy [19] of post-interpolation
residues. This criterion is more suitable for the problem of signal compression than the criterion
considered above. When using the criterion for minimizing entropy, one should take into account that
post-interpolation residues are quantized before statistical coding in many compression methods, for
example, in differential [20-21] and hierarchical [22-23] compression methods. In this paper, the
quantizer with a uniform [19] scale is used to calculate quantized post-interpolation residues q x :
q x f x max 2max 1 sign f x
, (4)
where .. means the selection of the integer part of the number, max is the maximum error [20] when
quantizing.
Thus, when using the specified "entropy" optimization criterion for the interpolator, it is necessary
to minimize the entropy H of the quantized post-interpolation residues q x :
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M 1
H lim N q lim ln N q lim min
,
lim
M max C x ,
x
(5)
q M 1
is the number of quantized post-interpolation residues equal to q, and M is the
where N q lim
maximum value of the original signal.
3. Reduction of the parametric space dimension
With significant constraints on computational complexity, interpolation algorithms are often used
[1,4], which use “smoothing” (averaging) over some a set of nearest reference samples:
P x
1 N ˆ
Ck x
N k 1
, (6)
where Cˆ k x are the nearest reference samples, N is the number of these samples.
The specific of the interpolator application determines the arrangement of these reference samples.
This arrangement can be quite non-trivial (see below) for some tasks related, for example, to the use of
some image compression methods.
As mentioned above, the use of such simple interpolation algorithms is typical in situations where
it is necessary to minimize the computational complexity. In particular, interpolators of this type are
used in differential [20-21] and hierarchical [22-23] compression methods for multidimensional
signals.
The “smoothing” interpolation algorithm is sufficiently accurate on smoothly varying signal
regions since averaging reduces noise. However, the “smoothing” interpolator is always characterized
by an increase in the interpolation error at the boundaries of the indicated smoothly varying regions
(i.e., at the boundaries). To interpolate such boundaries, we can use algorithms that use the so-called
interpolation "along the border". For a two-dimensional signal, in particular, Graham's nonlinear
interpolation algorithm [20] works in this way.
When using this algorithm, the Interpolated value is equal to the reference signal sample to which
the local boundary is directed. However, this algorithm, for obvious reasons, has less accuracy on
smoothly varying parts of the signal.
In this article, we propose an adaptive parameterized interpolation algorithm that combines the
advantages of both the described approaches: "smoothing" approach and "boundary" approach. The
proposed interpolation algorithm is based on the approach described in Section 2. The proposed
algorithm automatically switches between “smoothing” and “boundary” interpolators, depending on
how sharp the boundary is in the local neighborhood of the processed sample.
Next, we describe the proposed adaptive interpolator. We also specify the interpolating functions
(1) and decision rule (2). Denote by Nc the number of boundary directions taken into account. Let
x : 0 i N be the set of averaged absolute values of differences between the reference
i c
samples Cˆ k x in each of the directions under consideration:
i x Cˆ t x Cˆ x (7)
where t and are the indices of the reference samples.
The differences i x determine the presence and intensity of the boundary in the local
neighborhood of the current signal sample. We can use several thresholds ilim to decide on the
presence and direction of the boundary. We compare the described differences i with these threshold
values ilim . If there is no boundary in the neighborhood of the current signal sample, then we use a
“smoothing” interpolating function of the form (6):
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1 N ˆ
P x P x Ck x , if i ilim , i 0, N c
1
N k 1 (8) .
If there is a boundary in the local neighborhood, then for interpolation we use the average value
C j ( x) of the two nearest reference samples located in the direction of the local boundary:
P x P x C j x , if i ilim , i 0, N c
2
. (9)
Thus, we need to solve an optimization problem in a N c -dimensional parametric space to find the
best thresholds ilim .
The application determines the dimension N c of the parameter space ilim . As we show below, in
the problem of hierarchical compression [22-23] Nc 2 for a two-dimensional signal, Nc 4 for a
three-dimensional signal (in the simplest case), and then N c grows rapidly with increasing signal
dimension. However, the search for parameters can be an overly time-consuming task during
compression even when Nc 2 .
In this paper, we propose to reduce the dimension of the parameter space ilim to reduce the
computational complexity of finding these parameters. For this, we propose to use not the absolute
values of the differences i , but their ratio during interpolation. If there is no boundary (or the
boundary is weak) in the neighborhood of the current sample, all differences have close values. If
there is a clear boundary in this neighborhood, then the smallest difference j corresponds to the
direction of this boundary:
j x arg min i x
i
.
Moreover, if there is a boundary in the neighborhood, this difference is significantly different from
all other differences, including the nearest difference r :
r x arg min i x
i: i j
.
Based on this reasoning, in this article the feature of the boundary direction is defined as the
difference between the two smallest differences i :
.
x r x j x
(10)
When interpolating each sample, the feature x is compared with a threshold . If the feature lim
x is small enough (less than the threshold ), then it is considered that there is no boundary in
lim
the neighborhood of the current sample. Therefore, a “smoothing” interpolation of the form (8) is
used:
1 N
P x P x Cˆ xk , if x lim
1
N k 1 . (11)
The difference from (8) is that now this interpolation is one-parameter. If the feature x is larger
than the threshold lim , then interpolation of the form (9) “along the boundary” is performed:
P x P x Cˆ x , j arg min , if x lim .
2
j i
i (12)
Thus, by reducing the dimension of the parameter space, we reduce the multiparameter
optimization problem to a one-parameter one, in which the only parameter is lim . An automatic
optimization procedure similar to [23] is used to calculate this parameter.
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4. Multidimensional adaptive interpolator in the problem of hierarchical compression
The proposed multidimensional adaptive interpolator can be used in various signal processing tasks. In
particular, in this paper, we consider the use of this interpolator in the problem of compression. As an
example of the compression method, we consider the hierarchical compression method [22-23].
This method uses a unique hierarchical non-redundant representation (see Figures 1-2) of the
original multidimensional signal C C x as a set of L scale levels Cl :
L 1
C Cl
l 0
, Cl Cl x C x : x Il , (17)
where Il is the set of sample indices of the corresponding scale level Xl :
I L1 2 L1 x , I 2 x \ 2 x, 0 l L .
l
l l 1
(18)
(L – 1)
Thus, the most resampled scale level CL1 is a “grid” of signal samples with step of 2 , and all
other scale levels with the numbers l = (L-1),(L-2),..,1,0 are grids of signal samples with the step of 2l,
of which samples are removed with the step of 2l+1.
With hierarchical compression, the scale levels of the signal are compressed sequentially, from the
most resampled level CL1 to the least resampled level C0 . In this case, samples of more resampled
levels are used for interpolation of samples of less resampled levels.
3 1 2 1 3
1 1 1 1 1
2 1 2 1 2
1 1 1 1 1
3 1 2 1 3
Figure 1. Hierarchical representation of a Figure 2. Level numbers in the hierarchical
two-dimensional signal by four scale levels. representation of the signal (the level
number is zero in the empty cells).
Most often, to reduce computational complexity with hierarchical compression, a smoothing
interpolator of the form (6) is used. In the three-dimensional case, this interpolator can be written as:
1 1 1 1
Pl(1) 2m 1,2n 1,2k 1 Cl 1 m m, n n, k k
8 m0 n0 k 0 . (19)
Differences i (7) in this case take the form:
l 2m 1,2n 1,2k 1 Cl 1 m 1, n 1, k 1 Cl 1 m, n, k
(0)
, (20)
(1)
l 2m 1,2n 1,2k 1 Cl 1 m, n 1, k 1 Cl 1 m 1, n, k , (21)
(2)
l 2m 1,2n 1,2k 1 Cl 1 m, n, k 1 Cl 1 m 1, n 1, k , (22)
(3)
l 2m 1,2n 1,2k 1 Cl 1 m 1, n, k 1 Cl 1 m, n 1, k .
(23)
Thus, the adaptive three-dimensional interpolator allows us to automatically switch between
smoothing interpolation (19) and interpolation along the boundary of one of the four directions shown
in Fig. 3. In other words, at each point of the signal we can use the interpolating value (19) or one of
the following four interpolating values:
Vl(0) 2m 1,2n 1,2k 1 Cl 1 m 1, n 1, k 1 Cl 1 m, n, k
1
2 , (24)
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Cl 1 m, n 1, k 1 Cl 1 m 1, n, k
1
Vl(1) 2m 1,2n 1,2k 1
2 , (25)
Vl(2) 2m 1,2n 1,2k 1 Cl 1 m, n, k 1 Cl 1 m 1, n 1, k
1
2 , (26)
Vl(3) 2m 1,2n 1,2k 1 Cl 1 m 1, n, k 1 Cl 1 m, n 1, k
1
2 . (27)
The “boundary” interpolating function (12) in this situation takes the form:
Pl(2) 2m 1,2n 1,2k 1 Vl( j ) 2m 1,2n 1,2k 1 , j arg min i 2m 1,2n 1,2k 1
i . (28)
Figure 3. Three-dimensional interpolating functions (24-27) of the adaptive interpolator.
Thus, in the introduced notation, the three-dimensional adaptive interpolator is described by the
expression:
Pl 2m 1,2n 1,2k 1 , if l 2m 1,2n 1,2k 1 l
(1) lim
Pl 2m 1,2n 1,2k 1
Pl 2m 1,2n 1,2k 1 , if l 2m 1,2n 1,2k 1 l (29)
(2) lim
where, for each scale level, the feature l 2m 1,2n 1,2k 1 is calculated according to expressions
(8-10), and compared with threshold lim for each level.
In the two-dimensional case, the smoothing interpolator (6) can be written in the form:
Pl(1) 2m 1,2n 1 Cl 1 m, n Cl 1 m 1, n Cl 1 m, n 1 Cl 1 m 1, n 1
1
4 (30)
Differences i (7) in this case take the form:
l 2m 1,2n 1 Cl 1 m, n Cl 1 m 1, n 1
(0)
, (31)
(1)
l 2m 1,2n 1 Cl 1 m, n 1 Cl 1 m 1, n . (32)
The corresponding interpolating values are written as follows:
Vl(0) 2m 1,2n 1 Cl 1 m 1, n 1 Cl 1 m, n
1
2 , (33)
Vl(1) 2m 1,2n 1 Cl 1 m, n 1 Cl 1 m 1, n
1
2 . (34)
For the interpolation itself in the two-dimensional case, it is more appropriate to use a two-
parameter adaptive interpolation function:
Vl(0) 2m 1,2n 1 , if l 2m 1,2n 1 llim( )
Pl 2m 1,2n 1 Pl(1) 2m 1,2n 1 , if lim(
l
)
l 2m 1,2n 1 llim( )
(1)
Vl 2m 1,2n 1 , if l 2m 1,2n 1 l
lim( )
(35)
where l m, n is the feature of boundary direction.
l m, n l(0) m, n l(1) m, n
(36)
At each signal point, the feature l m, n is compared with two thresholds llim( ) , llim( ) , because
optimization by these parameters can be done separately.
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5. Experimental study of the adaptive interpolator for real-world multidimensional signals
The proposed adaptive interpolator (30-36) was implemented programmatically in C++ and built into
the hierarchical compression method. This software implementation was used in this article to study
the effectiveness of the proposed interpolator. To this end, computational experiments were performed
in real-world multidimensional signals (see Figure 4-5) of two signal sets:
set no. 1 - TokyoTech hyperspectral dataset [24] (signal sizes 500x500x31 samples, 13 bits)
set no. 2 - AVIRIS hyper-spectrometer [25] (signal sizes 1086x614x224 samples, 16 bits).
A measure of the effectiveness of the proposed interpolator was the relative gain in the archive
size, which was achieved by replacing the smoothing interpolator with an adaptive interpolator in the
frame of hierarchical compression method:
Kadapt Ksmooth 1 100%
,
(37)
where K smooth , Kadapt are the compression ratios of the hierarchical compression method when using a
smoothing and adaptive interpolator, respectively. Typical results of computational experiments are
shown in Fig. 6-7 and Table 1. The adaptive interpolator has a gain of up to 51% in the size of the
archive file compared to the smoothing interpolator.
Figure 4. Fragments of bands 10, 86 of Figure 5. Bands 0, 30 of test multidimensional
test multidimensional signal «Cuprite-2». signal «Fan2».
, %
51,00
41,00
31,00
21,00
11,00
1,00 max
0 50 100 150 200 250
color Character cd fan2
Figure 6. Gain in %) the adaptive interpolator for the averaging interpolator in test signals
«Color», «Character», «CD», «Fan2».
6. Conclusion
We proposed interpolation algorithms for multidimensional signals based on automatic switching
between simple interpolating functions at each point of the signal. We described the parameterized
decision rules that perform this switch. We optimized the parameters of these decision rules on the
criteria for the minimum energy of post-interpolation residues and criteria of the minimum entropy of
quantized post-interpolation residues. We proposed a method for reducing the dimension of the
parametric space of decision rules. We performed computational experiments to study the proposed
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interpolators on real-world multidimensional signals. We experimentally proved that using the
adaptive interpolator instead of a smoothing one can significantly improve the efficiency of
hierarchical signal compression.
Table 1. The gain in %) of the adaptive interpolator for the averaging interpolator depending
on maximum error max in test signals «Butterfly2», « Butterfly3», ... ,«Low Altitude», «Moffett
Field».
max Signal set 0 10 50 100 150 200 250 max()
Butterfly2 set no.1 1.17 4.84 8.00 9.17 9.57 9.19 8.62 9.57
Butterfly3 set no.1 1.87 7.24 13.81 17.07 17.74 16.74 14.98 17.74
Butterfly4 set no.1 1.87 6.37 11.45 13.17 13.81 13.51 12.98 13.81
Butterfly5 set no.1 1.30 6.36 11.35 12.63 12.47 11.00 9.41 12.63
Butterfly6 set no.1 1.23 5.20 9.01 10.90 11.58 11.28 11.10 11.58
Butterfly7 set no.1 1.48 5.24 9.81 11.55 12.16 11.55 10.39 12.16
Butterfly set no.1 1.68 5.49 9.02 9.89 9.49 8.64 7.64 9.89
Butterfly8 set no.1 1.88 7.03 11.46 12.37 12.36 12.14 10.72 12.37
cd set no.1 2.66 11.18 19.79 20.74 20.64 19.25 17.69 20.74
Character set no.1 6.21 16.26 24.34 27.73 28.19 27.02 24.64 28.19
Chart24 set no.1 5.57 18.18 36.27 40.18 36.60 30.42 25.57 40.18
ChartRes set no.1 7.80 18.76 28.57 36.31 39.30 39.49 38.91 39.49
Cloth2 set no.1 0.64 1.62 2.28 2.13 1.92 1.48 1.14 2.28
Cloth3 set no.1 1.16 2.88 4.11 4.28 4.18 2.98 1.88 4.28
Cloth4 set no.1 0.58 1.32 2.11 2.18 1.51 0.66 0.06 2.18
Cloth5 set no.1 2.10 4.60 6.42 5.91 5.66 5.11 4.07 6.42
Cloth set no.1 0.84 1.94 2.92 2.74 2.40 1.81 1.27 2.92
colorchart set no.1 4.10 15.40 37.31 44.07 39.92 34.67 30.61 44.07
color set no.1 5.24 15.63 30.38 43.53 50.40 51.00 47.56 51.00
doll set no.1 0.44 1.34 1.72 1.82 1.83 1.87 1.78 1.87
fan2 set no.1 3.15 8.50 12.19 12.49 11.51 10.68 10.11 12.49
fan3 set no.1 2.50 6.54 9.77 9.92 9.41 8.82 8.23 9.92
fan set no.1 1.76 4.12 6.29 6.25 5.51 4.70 3.99 6.29
flower2 set no.1 1.60 4.35 5.63 4.59 2.75 1.24 -0.71 5.63
flower3 set no.1 2.28 6.74 9.43 8.57 6.85 4.97 3.07 9.43
flower set no.1 2.79 8.42 11.50 9.18 6.54 4.00 2.34 11.50
party set no.1 3.18 9.51 15.22 16.70 17.03 16.46 16.45 17.03
tape2 set no.1 4.60 9.87 14.46 14.54 13.07 11.37 9.95 14.54
tape set no.1 1.61 3.99 5.63 5.61 5.22 4.64 4.11 5.63
Tshirts2 set no.1 0.76 2.32 3.96 4.77 4.65 4.48 4.52 4.77
Tshirts set no.1 0.75 2.26 3.82 4.59 4.88 5.11 5.28 5.28
Cuprite-1 set no.2 0.73 1.89 2.96 3.40 3.44 3.41 2.47 3.44
Cuprite-2 set no.2 1.26 3.45 5.63 6.84 7.11 6.92 5.95 7.11
Low Altitude set no.2 1.15 2.91 3.47 2.29 0.79 -0.69 -2.16 3.47
Moffett Field set no.2 0.71 1.74 1.87 1.64 1.34 1.06 0.76 1.64
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, %
8,00
6,00
4,00
2,00
0,00 max
0 50 100 150 200 250
Cuprite-2 Cuprite-1
Figure 7. Gain in %) the adaptive interpolator for the averaging interpolator in test signals
«Cuprite-1», «Cuprite-2», «Low Altitude», «Moffett Field».
7. References
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V International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 39
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M V Gashnikov
[19] Sayood K 2012 Introduction to Data Compression (The Morgan Kaufmann Series in
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[24] TokyoTech 31-band Hyperspectral Dataset URL: http://www.ok.sc.e.titech.ac.jp/res/
MSI/MSIdata31.html (03.11.2018)
[25] AVIRIS Data – Ordering Free AVIRIS Standard Data Products Jet Propulsion Laboratory
URL: http://aviris.jpl.nasa.gov/data/free_data.html (03.11.2018)
Acknowledgments
The work was partly funded by RFBR according to the research project 18-01-00667 in parts of «2
Adaptive interpolation of multidimensional signals» – «5 Experimental study of the adaptive
interpolator for real-world multidimensional signals» and by the Russian Federation Ministry of
Science and Higher Education within a state contract with the "Crystallography and Photonics"
Research Center of the RAS under agreement 007-ГЗ/Ч3363/26 in part of «1 Introduction».
V International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 40