An Efficient Way of a Non-Hermitian Wavelet-based Signal
Decomposition
BORIS M. SHUMILOV
Department of Applied Mathematics
Tomsk State University of Architecture and Building
2 Solyanaya, Tomsk, 634003
RUSSIA
Abstract: - The article investigates an implicit method of decomposition of the 7th degree non-Hermitian splines
into a series of wavelets with two zero moments. The system of linear algebraic equations connecting the co-
efficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the
embedded scale is obtained. The even-odd splitting of the wavelet decomposition algorithm into a solution of
the half-size five-diagonal system of linear equations and some local averaging formulas are substantiated. The
results of numerical experiments on accuracy on polynomials and compression of spline-wavelet decomposition
are presented.
Key-Words: -B-splines, wavelets, implicit decomposition relations, sweep method, data processing
Received: March 20, 2021. Revised: January 11, 2022. Accepted: February 5, 2022. Published: March 2, 2022.
1 Introduction
Laser scanning is a new direction in high-precision
3D measurements [1]. As to mobile scanning, scopes
of application include positioning of the automobile
and railroads, bridges, overpasses, city streets, the
coastline, etc. The main advantage of laser scanning
is the possibility to work at objects with heavy traf-
fic, at industrial facilities without stopping production
and in hard-to-reach sites, and also at objects having
a complex configuration [2]. As the principle of a
mobile scanning system allows working on the road
without traffic overlapping, the cloud of data points
from laser scanning devices contains a roadside land-
scape and hindrances on the carriageway (reflection
from people who are on the object, technique, vegeta-
tion, etc.). The main goals of preliminary processing
of laser scanned data are removal of a roadside and
surrounding landscape, filling of gaps in the scanned
data, created by cars passing by on the carriageway,
and the elaboration of the planned axial line of the
road [3]. From the mathematical point of view, the
production of the axis of the road allows transform-
ing bends of the highway to some rectangular area,
for which it is possible to apply standard methods of
two-dimensional spline interpolation [4] on a rectan-
gular grid, keeping structural lines of the road (edges,
brows), unlike the popular method of restoration of
a surface of the highway by triangulation of chaotic
points [5]. Very important is that at such approach
high precision of detection of cracks and damages of
road pavement in the places demanding repair is guar-
anteed and construction and application of wavelet-
transformation of interpolation splines for compres-
sion of the scanned information in the places of high-
ways not demanding repair are significantly facili-
tated also. Cubic splines of smoothness of C2be-
came popular for road engineers as an adequate way
of mathematical representation of picket method of
tracing of the reconstructing highways. As for Her-
mite splines they allow in an explicit form, using the
values of spline coefficients, to consider geometrical
restrictions on control points (pickets of the automo-
bile route), and on route tangents (the directions of
tangent lines at the entrance on bridge construction or
at adjunction) and radiuses of transition curves (val-
ues of curvature on accelerating and brake sections of
the route).
In the theory of a multi-scale analysis, the basis
of the space that fills the difference between approx-
imating spaces on a dense grid and a thinned grid
are called wavelets [6, p. 41]. Unlike Fourier trans-
form, which provides only amplitude-frequency in-
formation but loses time information, discrete wavelet
transform analyses the signal in a time-frequency do-
main [6-9]. For example, effective denoising methods
are required for processing measured signals, because
conventional denoising methods may not be practical
due to the noisy nature of the measuring as well as
its spectral overlap with different noise sources [10,
11]. The unique properties of wavelets allow design-
ing a basis in which the data representation can be
expressed with a small number of non-zero coeffi-
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Volume 18, 2022
cients. This property makes wavelets attractive for
data compression, including video and audio infor-
mation. Wavelet transform can be viewed as one of
the methods of primary signal processing to improve
the efficiency of its compression. In this case, direct
compression is performed by classical methods only
for significant coefficients of the wavelet decomposi-
tion of the signal, and its reconstruction according to
these coefficients is performed at the stage of restora-
tion (decompression) [12]. With multimedia becom-
ing widely popular, the conflict between massive data
and finite memory devices has been continuously in-
tensified; so, it requires more convenient, efficient,
and high-quality transmission and storage technology,
and the fast wavelet analysis is what people want to
use for highly efficient compression technology [13-
16].
The Haar wavelets and Daubechies wavelets were
the first orthonormal wavelets with compact supports
[6]. Their generalizations were constructed by Cohen
et al in the form of biorthogonal wavelets [17, 18].
The compactness means that there exist explicit fi-
nite formulas for the discrete wavelet decomposition.
But the disadvantage of these wavelets is that the ex-
pansion coefficients are calculated using local aver-
aging formulas, that is, the information at the edges
of the image is lost during data processing [19]. For
the indicated problem of roads construction, it is im-
portant to complement smoothly the discretely given
functions. Not all types of wavelets meet these goals,
but only some of them, for example, wavelets based
on B-splines and Hermitian splines [20].
Usually, the construction of basis functions in the
wavelet space begins with the use of basis functions
in the approximating space with centers at odd nodes
of a dense grid (”lazy” wavelets [8]). We proposed
to use approximating basis functions with centers at
even nodes [21, 22] as ”lazy” wavelets. Since the
wavelet space must be the complement of the approx-
imating spaces on dense and thinned grids, the di-
mensions of these spaces must satisfy the so-called
dimension relation. Sometimes, to fulfill this condi-
tion, it is sufficient to restrict ourselves to consider-
ing the periodic case when the values of the coeffi-
cients at the ends of the approximation segment coin-
cide. This corresponds to the approximation of closed
curves and surfaces. For open curves, a mismatch oc-
curs. So we considered previously unknown types of
Hermitian wavelets of the third [21] and seventh [22]
degrees, proved the existence of finite implicit expan-
sion relations, and substantiated effective algorithms
for wavelet analysis based on them. Namely, for the
case of the third degree, we proposed to subtract from
the initial coordinates the equation of the straight line
connecting the first and last points. Then the two basis
functions along the edges of the segment are removed
from the bases, and the dimensions of the resulting
spaces provide the dimension relation. For the case of
the seventh degree, a variant was investigated in de-
tail that leads to the most compact solution when the
equation of a cubic parabola interpolating the function
and the second derivative at the first and last points is
subtracted from the initial coordinates. As a result,
the corresponding basis functions are removed from
the bases, and the dimension relation is also fulfilled.
For the cases of the first and fifth degrees, such sym-
metric solutions do not exist.
Wavelet transforms based on Hermite splines have
their drawbacks. First, in the problem of process-
ing measurement information, one must calculate the
approximate values of the derivatives at the nodes
of the densest grid with acceptable accuracy [23],
and only then can wavelet transform algorithms be
applied. Secondly, from the point of view of data
compression, the number of wavelet coefficients, in
this case, is greater than in methods based on B-
splines. Meanwhile, in the work of the author [24],
non-orthogonal wavelets of the third degree with the
first six zero moments, that is, orthogonal to all poly-
nomials of the five-degree, were considered; the ex-
istence of finite implicit decomposition relations was
proved, and an effective algorithm for wavelet analy-
sis based on them was justified. The importance of the
new algorithm in wavelets theory is its stability and
the simplicity of realization because a matrix possess-
ing strict diagonal prevalence is solved at each step of
resolution.
This article is a first attempt to find out if there
is an even-odd splitting algorithm for non-orthogonal
wavelets of the seventh degree with the two first zero
moments. Section 2 discusses the properties of the
splines of degree mof smoothness Cm−1on a uni-
form infinite grid of nodes and the non-orthogonal
spline-wavelets with n+ 1 vanishing moments. Sec-
tion 3 considers the use of the matrix notation to
wavelet transformation of hierarchical spline bases
and suggests the main idea of preconditioning the
wavelet transform system of equations. As a result,
in section 4 the case of spline-wavelets of seventh-
degree with 2 zero moments is fully resolved and in
section 5 the wavelet interpolation algorithm on the fi-
nite segment is implemented. In the sixth section, the
algorithm of wavelet analysis is presented as a whole,
and section 7 contains an example of the application.
2 Construction of spline wavelets
with vanishing moments
The basis for constructing wavelets is the presence
of a set of approximating spaces . . . VL−1⊂VL⊂
VL+1 . . . such that each basis function in VLcan be
expressed as a linear combination of basis functions in
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VL+1. In particular, splines – smooth functions glued
from pieces of polynomials of degree mon an em-
bedded sequence of grids, have this property. The
essence of the wavelet transform can be formulated
as follows: it allows one to decompose a given func-
tion VL+1 into a rough approximate representation VL
and local refinements WL=VL+1 −VL. The proce-
dure for dividing into a rough version and clarifying
details can be applied recursively to this part of VL.
Hence, the original function can be represented as a
hierarchy of rough versions of VL, VL−1, ... and re-
finements WL, WL−1, .... Such a recursive process is
called direct wavelet transformation (decomposition
or analysis) [9, p. 46]. Conversely, the function VL+1
can be reconstructed from the most compact repre-
sentation (reconstruction). Moreover, the values of
the coefficients of the wavelet decomposition can be
used to judge the significance of the corresponding
details of the refinement. Insignificant details can be
removed to compress the information. The main thing
here is to find a suitable basis for the space WLand
construct for it fast one-to-one formulas for the direct
and inverse wavelet transforms.
Let the space VLbe the space of splines of degree
mof smoothness Cm−1on a uniform grid of nodes
∆L:xi+1 =xi+ 1/2L, infinitely extended in both
directions for all i. It is well known that the basis
in this space is generated by functions ϕm(v−i)∀i,
where v= 2Lx, generated by contractions and shifts
of a function of the form [9, p. 89]:
ϕm(t) = 1
m!
m+1
X
j=0
(−1)j m+ 1
j!(t−j)m
+,
where tm
+= (max{t, 0})m.
They have the following supports,
supp ϕm= [0, m + 1],
and they satisfy the calibration relation [9, p. 91]:
ϕm(t) = 2−m
m+1
X
k=0 m+ 1
k!ϕm(2t−k).(1)
As a result, any spline on the mesh ∆Lcan be rep-
resented as
sL(x) =
∞
X
−∞
cL
iϕm(2Lx−i),(2)
where the coefficients cL
i∀iare the solution, for ex-
ample, of the cardinal interpolation problem:
sL(xi) = f(xi),−∞ < i < ∞.
If the grid ∆L−1is obtained from the grid ∆Lby
removing every second node, then the correspond-
ing space VL−1with basic functions ϕm(v/2 −i)∀i,
whose supports are twice as wide in width, is em-
bedded in VL. The difference between the spaces
VLand VL−1constitutes the wavelet space WL−1=
VL−VL−1[9].
In what follows we will use non-orthogonal
wavelets with n+ 1 vanishing moments, that is, or-
thogonal to all polynomials of degree n[25],
wm,n(t) = 2−m
n+1
X
k=0
(−1)k n+ 1
k!ϕm(2t−k).(3)
These functions have the following supports
supp wm,n = [0,m+n
2+ 1]
and, accordingly, n+ 1 zero moments
Z∞
−∞
xkwm,n(x)dx = 0, k = 0,1, . . . , n.
For the case, m= 7, the basis spline function takes
on a form (Fig. 1).
Figure 1: The graph of the scaling function ϕ7(x)
The graph of wavelet of the 7th degree, orthogonal
to all polynomials of the 1st degree, is shown in figure
2.
3 Construction of the defining system
of wavelet transform equations
We write the basic spline functions in the form of a
one-row matrix of infinite length,
ϕL(x) =
=h. . . , ϕm(2Lx−i), ϕm(2Lx−i−1), . . .i.
Introducing the notation
cL=h. . . , cL
i, cL
i+1, . . .iT
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Figure 2: The graph of the function 128 ·w7,1(x)
for a vector consisting of the spline coefficients, we
write the formula (2) in a vector form
sL(x) = ϕL(x)cL.
In the same way, we can write basic wavelet func-
tions at the decomposition level Lin the form of an
infinite row matrix as
ψL−1(x) =
=h. . . , wm,n(2Lx−i), wm,n(2Lx−i−1), . . .i.
We denote the corresponding wavelet approxima-
tion coefficients by dL−1
i,−∞ <i<∞, and intro-
duce the column vector
dL−1=h. . . , dL−1
i, dL−1
i+1 , . . .iT.
Since the spaces VL−1and WL−1are by definition
subspaces of VL, the functions ϕL−1(x)and ψL−1(x)
can be represented as linear combinations of the func-
tions ϕL(x):
ϕL−1(x) = ϕL(x)PL,
ψL−1(x) = ϕL(x)QL,
where the columns of the matrix PLare composed
of the relation coefficients (1) since each wide basic
function can be constructed from m+ 2 narrow basic
functions, while the elements of the columns of the
matrix QLare composed of the relation coefficients
(3).
Therefore, there is a chain of equalities:
ϕL(x)cL=ϕL−1(x)cL−1+ψL−1(x)dL−1=
=ϕL(x)PLcL−1+ϕL(x)QLdL−1.
Let the coefficients cL−1and dL−1be known.
Then the coefficients cLcan be easily obtained from
cL−1and dL−1as follows
cL=PLcL−1+QLdL−1.(4)
Using the notation for block matrices, we can
rewrite equality (4) in the form
cL=hPL|QLi"cL−1
dL−1#.(5)
Formula (5) is nothing more than a recovery algo-
rithm [9, p. 248], for the implementation of which,
due to tape matrices PLand QLthe moving average
scheme is successfully used. For example, for the
case m= 7, n = 1, the infinite matrix hPL|QLi
takes the following form:
hPL|QLi=1
128·
·
.
.
.
...0
...1.
.
.
...8 0
...28 1 ....
.
.
...56 8 ......0
...70 28 ......1.
.
.
...56 56 ......−2 0 ...
...28 70 ......1 1 ...
...8 56 ......0−2...
...1 28 ....
.
.1...
...0 8 ...0...
.
.
.1....
.
.
0...
.
.
.
.
Unfortunately, for the reverse process of calculat-
ing from the coefficients cLthe coarser version cL−1
and the refining coefficients dL−1, following the de-
composition algorithm [9, p. 247]
"cL−1
dL−1#="AL
BL#cL,
we obtain for all cases, except for the case of the Haar
wavelets, that the rows of matrices ALand BLare in-
finite numerical sequences, and their truncation leads
to errors.
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4 The algorithm with splitting
As can be seen from the above example, the result-
ing system of equations has some sort of a step-wise
structure, which permits the application of the method
of even-odd splitting to the system solving [24]. We
choose for this purpose some preconditioning matrix
RLto receive the easy invertible matrix
GL=hPL|QLiRL
by the conditions:
a) a matrix GLis a tape matrix with the minimum
possible number of nonzero diagonals;
b) RLis a tape matrix, with the minimum possible
number of elements.
By placing as many zeros as possible in the upper
and lower parts of each column of the matrix RL, we
ensure the compactness of the computational scheme;
by zeroing the elements spaced by an odd number
of steps from the main diagonal of the matrix GL,
we provide the possibility of using an efficient sweep
method for solving it; and additionally zeroing the el-
ements outside the main diagonal of the matrix GL,
we provide the possibility of the subsequent splitting
of the system into even and odd rows.
Then, assuming that the matrix GLis nondegener-
ate, we multiply the left and right sides of equality (5)
by the matrix RLGL−1. As a result, we get
RLGL−1cL=
=RLhPL|QLiRL−1hPL|QLi"cL−1
dL−1#=
=RLRL−1"AL
BL#hPL|QLi"cL−1
dL−1#=
="cL−1
dL−1#.
Thus, instead of directly solving a system of the
form (5), we can solve the system
GLhL=cL(6)
with respect to some values of hLand then just cal-
culate the values of cL−1and dL−1using the linear
transformation
"cL−1
dL−1#=RLhL.(7)
For the matrix, hPL|QLi, the nine-diagonal ma-
trix
hPL|QLi0
=1
128·
·
....
.
..
.
.
...0 0 .
.
.
...010.
.
.
...0800 .
.
.
...1 28 0 1 0 ...
...−2 56 0 8 0 0 · · ·
...1 70 1 28 0 1 0 · · ·
...0 56 −2 56 0 8 0 · · ·
...0 28 1 70 1 28 0 ...
· · · 0 8 0 56 −2 56 0 ...
· · · 0 1 0 28 1 70 1 ...
...0 0 8 0 56 −2...
.
.
.0 1 0 28 1 ...
.
.
.0 0 8 0 ...
.
.
..
.
..........
is obtained by permuting the columns of the matrix
hPL|QLiso that the columns of the matrices PLand
QLalternate. In practice, such a permutation is ac-
companied by a change in the order of the unknowns
in the system (5) and is often done to give the sys-
tem a tape-like form to facilitate the numerical solu-
tion of the system [8]. Let the matrix corresponding
to the indicated permutation of columns is denoted by
T. Then the representation is true [26]
hPL|QLi0
=hPL|QLiT. (8)
From the representation (8) we find
hPL|QLi−1·GL=T·hPL|QLi0−1·GL.(9)
Thus, the problem of finding the matrices RLand
GLis reduced to finding a solution of the system of
matrix equalities
hPL|QLi0
jRL0
j=GL
j,−∞ < j < ∞.(10)
Here, the subscripts in the notation of the matrices
indicate which elements of the columns of the matrix
RL0are calculated by the corresponding system (10).
Specifically, according to the assumed stepped struc-
ture of the matrices RL0and GL, system (10) splits
into blocks with matrices of the following form:
hPL|QLi0
j,j+1,...,j+6 =1
128·
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·
1 28 0 1
−2 56 0 8
1 70 1 28 0 1
0 56 −2 56 0 8
028170128
0 8 0 56 −2 56
1 0 28 1 70 1
0 0 8 0 56 −2
0 1 0 28 1
,
provided that the equations corresponding to the zero
rows of the matrix GLare removed from the system.
This system is solvable and underdetermined. There-
fore, we can choose non-trivial solutions that interest
us, namely:
1) r0=r6= 4; r1=r5= 0;
r2=r4= 28; r3= 1;
g0=g8= 5;
g1=g3=g5=g7= 0;
g2=g6= 60; g4= 126;
2) r0=r1=r3=r4=r5=r6= 0;
r2= 1;
g0=g1=g5=g6=g7=g8= 0;
g2=g4= 1; g3=−2;
3) r0=r1=r2=r3=r5=r6= 0;
r4= 1;
g0=g1=g2=g3=g7=g8= 0;
g4=g6= 1; g5=−2.
As a result, the matrix GLacquires a tape structure
with seven nonzero diagonals of the form
.........
...0 5 ...
...000...
...1 60 0 5 ...
...−2 0 0 0 ...
...1 126 1 60 0 ...
...0 0 −2 0 0 ...
...0 60 1 126 1 ...
...0000−2...
...0 5 0 60 1 ...
...0000...
...050...
...0 0 ...
.........
,
while the product of matrices hPL|QLi0−1·GLturns
out to have the following form:
.........
...0 0 ...
...040...
...0000...
...1 28 0 4 0 ...
...01000...
...0281280...
...00010...
...0 4 0 28 1 ...
...0000...
...040...
...0 0 ...
.........
.(11)
From equality (8) it follows that to find the matrix
RL=hPL|QLi−1·GL, it is required to apply to the
rows of the matrix (11) an inverse permutation, that
is, a permutation in which the records of the image
and the inverse image are interchanged. From this,
we can obtain the required in (7) representation of the
matrix RL.
5 The case of a finite segment
Recall that for the case of interpolation by splines on a
finite interval [0,2L], the most productive approach to
constructing basis functions is to set multiple nodes at
the ends of the interval, which corresponds to zeroing
of the approximating spline and some of its deriva-
tives at the ends of the interval [8]. Then the left basic
functions have the view forms (Fig. 3, Fig. 4).
They have the following supports,
supp ϕb1= [0,7],supp ϕb2= [0,6],
and they satisfy the calibration relations
ϕb1(t) = 1
64ϕb1(2t) + 19
134ϕ7(2t) +
+59
160ϕ7(2t−1) + 327
640ϕ7(2t−2) +
+41
96ϕ7(2t−3) + 167
768ϕ7(2t−4) +
+1
16ϕ7(2t−5) + 1
128ϕ7(2t−6),
ϕb2(t) = 1
32ϕb2(2t) + 147
640ϕb1(2t) + 12299
25600ϕ3(2t) +
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+371
800ϕ7(2t−1) + 399
1600ϕ7(2t−2) +
+7
96ϕ7(2t−3) + 7
768ϕ7(2t−4).
As to boundary basic wavelets, we use the
wavelets of seventh-degree that are orthogonal to all
first-degree polynomials (Fig. 5, Fig. 6),
wb1(t) = 7
48ϕb2(2t)−15
64ϕb1(2t) + 49
512ϕ7(2t),
wb2(t) = 1
16ϕb2(2t)−11
128ϕ7(2t) + 5
128ϕ7(2t−2).
Figure 3: The graph of the scaling function ϕb1(x)
Figure 4: The graph of the scaling function ϕb2(x)
They have the following supports
supp wb1= [0,5],supp wb2= [0,4],
and, accordingly, they have two zero moments
Z5
0
xkwb1(x)dx =Z4
0
xkwb2(x)dx = 0,
Figure 5: The graph of the function 128 ·wb1(x)
Figure 6: The graph of the function 128 ·wb2(x)
for k= 0,1.
At the right end of the segment, the basic functions
mirror the functions ϕb1,2(t), wb1,2(t). As a result, for
any grid ∆L,L≥3, a seven-degree spline with zero
boundary conditions
sL(r)(v) = 0, r = 0,1,2,3, v = 0,2L,
can be represented as
sL(v) = cL
−2ϕb2(v) + cL
−1ϕb1(v) +
+
2L−8
X
i=0
cL
iϕ7(v−i) + cL
2L−3ϕb1(2L−v) +
+cL
2L−2ϕb2(2L−v),0≤v≤2L,
where the coefficients cL
i∀iare the solution, for ex-
ample, of the interpolation problem:
sL(i) = f(i), i = 2,3, . . . , 2L−2.
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To prepare the system (5) for the odd-even split-
ting we need to solve the system (11) for indices
j=−2,−1, . . . , 5with the following matrix
hPL|QLi0
−2,−1,...,5=1
128·
·
856
34 0
0−30 147
50 2
−11 49
4
12299
200 11216
67 0 1
0 0 1484
25 −2236
50 8
5 0 798
25 1327
51 28 0
0 0 28
30164
3−2 56 0
07
60167
61 70 1
0 0 8 0 56 −2
0 1 0 28 1
,
provided that the equations corresponding to the zero
rows of the matrix GLare removed from the system.
This system is solvable and underdetermined. There-
fore, we can choose any non-trivial solutions that in-
terest us, for example:
1) r−2= 1;
r−1=r0=r1=r2=r3=r4=r5= 0;
g−2= 8; g−1=g1=g3=g4=g5= 0;
g0=−11; g2= 5;
2) r−2=r0=r1=r2=r3=r4=r5= 0;
r−1= 1;
g−2=56
3;g−1=−30; g0=49
4;
g1=g2=g3=g4=g5= 0;
3) r−2=−418
25 ;r−1=147
25 ;
r0= 6; r1=4452
25 ;
r2=r4=r5=r6=r7=r8= 0;
r3= 28;
g−2=g−1=g1=g3=g5=
=g7=g8=g9=g10 =g11 = 0;
g0= 803; g2= 314; g4= 35;
4) r−2=r−1=r0=r2=r3=r4=r5= 0;
r1= 1;
g−2=g−1=g4=g5= 0;
g0=g2= 1; g1=−2;
5) r−2=−78973
7875 ;r−1=1327
375 ;
r0=124
35 ;r1=16094
125 ;r2= 1;
r3=658
15 ;r5= 4;
r4=r6=r7=r8= 0;
g−2=g−1=g1=g3=g5=g7=
=g8=g9=g10 =g11 = 0;
g0=43765811
84420 ;g2=94804
315 ;
g4=479
6;g6= 5;
6) r−2=r−1=r0=
r1=r2=r4=r5=
=r6=r7=r8= 0;
r3= 1;
g−2=g−1=g0=g6=
=g7=g8=g9=g10 =g11 = 0;
g2=g4= 1; g3=−2.
So, the first seven columns of the matrix GLare
856
30 0 0 0 0
0−30 0 0 0 0 0
−11 49
4803 1 43765811
84420 0 5
0 0 0 −2 0 0 0
5 0 314 1 94804
315 1 60
0 0 0 0 0 −2 0
0 0 35 0 479
61 126
0 0 0 0 0 0 0
0 0 0 0 5 0 60
0 0 0 0 0 0 0
0 0 0 0 0 0 5
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
.
From the structure of the matrix GLit immediately
follows, that the values of hLat odd nodes are calcu-
lated from the explicit equations
cL
i
hi
=30, i =−1, , 2L−3;
2, i = 1,3, . . . , 2L−5,,
while for the values of hLat even nodes a system of
linear equations is solved:
8h−2=r−2,
−11h−2+ 803h0+43765811
84420 h2+ 5h4=r0,
5h−2+ 314h0+94804
315 h2+ 60h4+ 5h6=r2,
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Boris M. Shumilov
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35h0+479
6h2+ 126h4+ 60h6+ 5h8=r4,
i= 6,8, . . . , 2L−10 :
5hi−4+ 60hi−2+ 126hi+ 60hi+2 + 5hi+4
i= 2L−8 :
5hi−4+ 60hi−2+ 126hi+479
6hi+2 + 35hi+6
i= 2L−6 :
5hi−4+ 60hi−2+94804
315 hi+ 314hi+2 + 5hi+4
i= 2L−4 :
5hi−6+43765811
84420 hi−4+ 803hi−11hi+2
i= 2L−2 :
8hi
=ri.(12)
Here the right-hand sides of the equations (12) are
calculated from the formulas
r−2=cL
−2+56
3h−1,
r0=cL
0+49
4h−1+h1,
ri=
cL
i+hi−1+hi+1,
i= 2,4, . . . , 2L−6,
cL
i+hi−1+49
4hi+1,
i= 2L−4,
cL
i+56
3hi−1,
i= 2L−2.
Note that the matrix of the system (12) has not
a strict diagonal dominance [27, p. 78] over the
columns of the system. So, although the system of
equations has a unique solution because of linear in-
dependence of basis functions, the stability of the cal-
culations by the sweep method is not guaranteed.
Multiplying the matrices hPL|QLi−1and GLwe
can recheck the analytical evaluation provided to ob-
tain that the matrix RLis composed of two blocks ac-
cording to 2L−1−3basic spline functions of VL−1
and 2L−1basic wavelets of WL−1:
0 0 6 0 124
35 0 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1 ...
.
.
..
.
..
.
..
.
.0 0 0 ...
.
.
..
.
.......
1 0 −418
25 0−78973
7875 0 0
0 1 147
25 01327
375 0 0
0 0 4452
25 116094
125 0 4 ...
0 0 28 0 658
15 1 28 ...
0 0 0 0 4 0 28 ...
0 0 0 0 0 0 4 ...
.
.
..
.
..
.
..
.
.0 0 0 ...
.
.
..
.
.......
.
Here diagonal points mean that the preceding two
columns are repeated the appropriate number of times
while going each time two positions right and mov-
ing one position down. The last six columns of both
blocks of the matrix RLmirror the first six columns;
the empty positions of matrices are equal to zero.
As a result, the values of the spline coefficients on
the thinned grid and the wavelet coefficients are cal-
culated by the formulas (7). Thus, the operation of
wavelet decomposition can be performed without ex-
plicit representation and the use of a filter block.
The mathematical complexity of the algorithm for
solving system (12) by the sweeping method is valued
by the number of required arithmetic operations [27,
p. 100]: 11 ·(2L−1−1) additions, 11 ·(2L−1−1)
multiplications, and 2·(2L−1−1) + 1 divisions.
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Boris M. Shumilov
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33
Calculating the right-hand sides of the equations
requires 2·(2L−1−2) additions; obtaining the spline-
coefficients at the nodes of the sparse grid requires 2
additions and 4multiplications. The calculation of the
wavelet-coefficients: 4·(2L−1−4) + 16 multiplica-
tions and 4·(2L−1−4) + 14 additions. If we make
no differences between the arithmetic operations, then
the total number of such operations for one step of the
wavelet decomposition is approximately 34 ·2L−1.
6 The algorithm of wavelet analysis
consists of performing the following steps:
6.1 On the finite segment [a, b]
the values of the observation results are reset to zero
at the ends by subtracting values of the fifth degree
Hermitian polynomial [20]
2
X
i=0
(b−a)ihf(i)(a)ϕi(1 −v) + f(i)(b)ϕi(v)i,(13)
v=x−a
b−a,
where
ϕi(t) =
t3(6t2−15t+ 10), i = 0,
−t3(3t2−7t+ 4), i = 1,
t3
2(t2−2t+ 1), i = 2,
from the entire time series.
6.2 The wavelet interpolation algorithm
for a given Lis incorporated.
6.3 If L > 3,
then the value of Ldecreases by 1, and the algorithm
goes to step 2.
6.4 Otherwise, at each level L
of the decomposition, the rejection of insignificant
wavelet coefficients is performed according to some
criterion [8, 21], and the spline coefficients are se-
quentially restored according to the moving average
algorithm (4).
6.5 After wavelet analysis
of the differences obtained at the first stage of the
algorithm and reconstructing (of course, with some
approximation) the spline coefficients for the densest
mesh, the values of the polynomial (13) are added to
values of the approximating spline of the seventh de-
gree.
7 Numerical experiments
7.1 Checking accuracy on polynomials
For x∈[0,1], assuming L= 4 at the upper resolu-
tion level, we find the grid step length 2−4= 1/16.
When performing the wavelet transform, the values
of the function at the nodes of the grid ∆Lare used as
initial data, assuming zero values of the function and
the derivatives at the ends of the segment, a total of
13 numbers.
Because the seventh-degree polynomial with
eight zero boundary conditions does not exist we
will bound considering the nine-degree polynomial
f(x) = x4(x−1)4(2x−1) as a test function. We
find at the last stage of the recursive wavelet de-
composition algorithm, four values of the boundary
coefficients of the spline S3(x)at the ends of the seg-
ment, C3= [−9.483 ·10−6,−7.916 ·10−6,0,7.916 ·
10−6,9.483 ·10−6]T. In this case, the wavelet
coefficients are equal to D3= [−8.275·10−6,2.712 ·
10−7,−2.779 ·10−5,3.064 ·10−5,−3.064 ·
10−5,2.779 ·10−5,−2.712 ·10−7,8.275 ·10−6]T.
To demonstrate the accuracy property of the ap-
proximation scheme for polynomials of the ninth
degree, we will neglect all the wavelet coefficients,
providing a compression factor of 13/4=3.25, and we
will show here the graph of the difference between
the approximation spline and polynomial (Fig. 7),
which has the alternating character from the ends of
the interval.
7.2 The compression
coefficient in the considered example is small (only
8 wavelet coefficients are discarded). However, with
an increase in the length of the smoothness intervals,
the compression quality will undoubtedly be higher.
Figure 7: Graph of difference between the 7th-degree
approximation spline and the nine-degree polynomial
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8 Conclusion
The article discusses the further development of the
author’s procedure [22] for an even-odd partition of
the defining system of the Hermite wavelet expan-
sion for the practically important case of approximat-
ing that do not require specifying the values of the
derivatives of the functions, based on B-splines of the
seventh degree.
The advantage of the new algorithm is its simplic-
ity of realization because of a matrix possessing five
diagonals is solved at each step of decomposition.
The directions of our future research consist in the
extension of the proposed approach to obtaining the
seven-diagonal splitting method, possessing strict di-
agonal dominance, and to splines of a higher degree
and of a larger number of zero moments which can
provide new opportunities for the development of al-
gorithms for performing wavelet-based signal de- and
re-composition for Cartesian components of the geo-
data from laser scanning devices that issued in the
form of the array (”cloud”) of points, in which there
is no division into separate cross scans.
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