Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
Matrix inverse operation convolution: Three Models description.
JESUS MEDEL
Research Center of Applied
Science and Advance Technology and
Research Center for Computing,
National Polytechnic Institute IPN,
Av. Juan de Dios Batiz s/n, Edificio CIC
Col. Nueva Industrial Vallejo, 07738
Mexico City
MEXICO
[email protected]
CONSUELO V. GARCIA
Research Center of Applied
Science and Advance Technology,
National Polytechnic Institute IPN,
Legaria No. 694
Col. Irrigación, 11500
Mexico City
MEXICO
[email protected]
Abstract: This paper gives a short introduction and a description that defines the basic inverse convolution operation
structure and explains with short comments some of its applications in astronomy, microscopy and analysis of seismic
measures. The second part presents formal convolution and inverse convolution operation concepts about various
issues concerned: The first correspond to convolution as a matrix expands expression, which provides the necessary
preamble to define the three models respect to inverse convolution operation Matrix suggested in this work. Requiring
in probability manner the MSE (Mean Square Error), either the diagrams Decibels description into the proposed
inverse convolution models operation in simulation graphs.
Finally, it shows in which of the three inverse convolutions operation methods simulated was the best convergence
rate. Graphically speaking compared the inverse convolution operation signals respect to original signal before
convolutioned. The convergence error functional was obtained respect to MSE, using the second probability moment
and Decibels expression in recursive form. Analyzing the simulation results, could be concluded that the optimal
inverse convolution operation was the good enough estimation methods, respect to the last two methods considered.
Key-Words: Inverse convolution operation, error functional, estimation, filtering, identification, recursive decibels.
1 Introduction
reconstruction process 16, or the blind inverse
convolution operation in astronomy, which in turn
improved an initial identification of the real object
reaching a certain predetermined criteria 1, 16 and
[18]. The analysis of the earthquake required using a
division spectral method [2] and [14], or in digitalized
systems. In the last ten years was considered the
discrete wavelet packet transform as good enough
tools into inverse convolution operation 11, [12] and
[13]; but didn`t compare with other previous
techniques.
The problem of recovering an original signal, after
being convolution process was treated with different
methods 1,2, 6, 7, 12, 16 and [17]; for
example: inverse convolution operation, homomorphic
inverse convolution operation, online or iterative
inverse convolution operation 2, with intelligent
computational structures 1, and identification
systems techniques 16 , in the others. All of its
minimizing the MSE respect to original signal.
The inverse convolution operation is either a
mathematical operation as a signal restoration for to
recover data degraded by someone physical process
interacting 2. This operation has a lot of applications
into the earthquake analysis; microscopy and
astronomy, reserving in each case the resolution of
inverse convolution operation, treating as a complex
problem 6 and 7. For example, used in iterative
algorithms, obtaining the maximum likelihood
estimation in microscopy, in the others sciences, which
concerns itself with the identification in the
ISSN: 1790-5117
2 Convolution Operations
Convolutions in discrete form could be described as a
basic inner point operation respect two signals, each of
its delayed (basic signal) respect to the others,
generally the last of its perturbing the original natural
data signal evolution 8 and [13].
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Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
Theorem 1: The discrete convolution operation after
all delayed could be seeing as an expanded matrix
results:
(1)
𝐶𝑚𝑥 1 = 𝐺𝑚𝑥𝑛 𝐹𝑛𝑥 1 .
operation as a unique signal 𝐶𝑚 ×1 compounded by a
matrix with rows enlarged 𝐺𝑚 ×𝑛 ; and unknowing the
fix expanded vector 𝐹𝑚 ×1 .
Theorem 2: The inverse convolution operation
required for to find the known fix expanded
vector 𝐹𝑚 ×1 , considering that 𝐺𝑚 ×𝑚 is a simple
square matrix and not singular, is given:
(7)
𝐹𝑚 ×1 = 𝐺𝑚 ×𝑚 −1 𝐶𝑚×1 .
Where the convolution matrix 𝐶𝑚𝑥 1 , is the inner
product between the matrix with rows enlarged
knowing symbolically as 𝐺𝑚𝑥𝑛 (conformed its by a
vector set displaced each time intervals and
representing
within the enlarged single parent
displacement columns referring to each time period
considered within convolution matrix) and the fixed
signal 𝐹𝑛𝑥 1 .
Proof (Direct form). Observing that the convolution
theorem expression by square matrix condition
expressed in (1), and in agreement the concepts
considered in 4 and 8, the final results has the
essential form (7) ■ .
Proof (Inductive technique). Symbolically speaking 2
and 8, the convolution between two signals:
Theorem 3: The inverse convolution vector operation
for 𝐹𝑚 ×1 , with no square matrix 𝐺𝑚 ×𝑛 is:
(8)
𝐹𝑛𝑥 1 = 𝐺′𝑛×𝑚 𝐺𝑚×𝑛 + 𝐺′𝑛×𝑚 𝐶𝑚 ×1 .
+∞
𝑐 𝑡 =
𝑔 𝑡 − 𝑥 𝑓 𝑥 𝑑𝑥 .
(2)
−∞
As a discrete approximation using finite differences
3, for each time, the convolution by t1
𝐶1 = 𝐶(𝑡1 ) = 𝑛𝑥=1 𝐺 𝑡1 − 𝑥 𝐹 𝑥 .
So
that
𝐶1 = 𝐺 𝑡1 − 𝑥 , 𝐹 𝑥
Proof (Direct form). Given the convolution expression
(1) and, being a rectangular matrix 𝐺𝑚 ×𝑛 , right
multiplying both sides of initial equation by its
transposed, finding a square matrix:
(9)
𝐺′𝑛×𝑚 𝐶𝑚×1 = 𝐺′𝑛×𝑚 𝐺𝑚×𝑛 𝐹𝑛×1 .
(3)
So that det⁡
(𝐺′𝑛×𝑚 𝐺𝑚 ×𝑛 ) = 0, could be obtained its
pseudoinverse [18] and [21] as an approximation
describing the fix vector 𝐹𝑛×1 depicted in (8) ■ .
Now, the convolution operation by t2 𝐶2 = 𝐶 𝑡2 =
𝑛
𝑥=1 𝐺 𝑡2 − 𝑥 𝐹 𝑥 . So that
𝐶2 = 𝐺 𝑡2 − 𝑥 , 𝐹 𝑥
And so on to tm, 𝐶𝑚 =
𝑛
𝑥=1 𝐺
(4)
Theorem 4: Considering now the inverse convolution
operation respect to 𝐺𝑚 ×𝑛 beside of 𝐹𝑛×1 and
considering the pseudoinverse properties [18] and
[21]:
𝑡𝑚 − 𝑥 𝐹 𝑥 . So that
𝐶𝑚 = 𝐺 𝑡𝑚 − 𝑥 , 𝐹 𝑥
(5)
𝐺𝑚 ×𝑛 = 𝐶𝑚×1 𝐹 ′1×𝑛
Symbolically speaking, it means that the each time
interval convolution viewed as a whole respect to fixed
𝐹 𝑥
considering the inner product properties [20]
and [21]:
𝐶𝑚𝑥 1
𝐺 𝑡1 − 𝑥1
𝐺 𝑡2 − 𝑥1
=
⋮
𝐺 𝑡𝑚 − 𝑥2
𝐺 𝑡1 − 𝑥2
𝐺 𝑡2 − 𝑥2
⋮
𝐺 𝑡𝑚 − 𝑥2
…
…
⋮
…
𝐺 𝑡1 − 𝑥𝑛
𝐺 𝑡2 − 𝑥𝑛
⋮
𝐺 𝑡𝑚 − 𝑥𝑛
𝑚𝑥𝑛
𝐹 𝑥1
𝐹 𝑥2
⋮
𝐹 𝑥𝑛
.
𝐹𝑛×1 𝐹 ′1×𝑛
+
.
(10)
𝑤𝑖𝑡𝑕 𝑚 ≠ 𝑛
Proof (Direct form). Respect to (1) multiplying both
sides by the transpose of fix vector 𝐹𝑛×1 :
(11)
𝐶𝑚×1 𝐹′1×𝑛 = 𝐺𝑚 ×𝑛 𝐹𝑛×1 𝐹′1×𝑛
(6)
𝑛𝑥 1
Representing the matrix inner product operation
described in (1) ■ .
And considering the singularity matrix properties,
requires the pseudoinverse [18] and [21] matrix tools,
obtaining (10) ■ .
3 Matrix Inverse Convolution Models
Operation
4 Recursive Mean Square Error
Theorem 5: The (RMSE) Recursive Mean Square
Error between the trace of the original signal 𝑕𝑖 ∶=
Generally in the signal processing respect to 12, 13
and 16, only observe two signals in convolution
ISSN: 1790-5117
247
ISBN: 978-960-474-054-3
Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
𝑡𝑟𝑎𝑐𝑒 ( 𝐺𝑚 ×𝑛 ) and its estimation trace
𝑡𝑟𝑎𝑐𝑒 (𝐺𝑒𝑚 ×𝑛 ) for 𝑖 = 1, 𝑛, described as:
1
2
𝑀𝑆𝐸𝑛 = 2 𝑕𝑛 − 𝑕𝑛 + 𝑛 − 1 2 𝑀𝑆𝐸𝑛−1
𝑛
𝑕𝑖 ≔
𝐷 𝑛 = 10 𝑙𝑜𝑔10
𝑛−1
𝐷 𝑛 − 1 = 10 𝑙𝑜𝑔10
𝑖=1
𝑛−1
(𝑕𝑖 − 𝑕𝑖 )2
+
Where 10 𝑙𝑜𝑔10
(14)
𝑛−1
𝑖=1
10 𝑙𝑜𝑔10
𝑖=1
The MSE for 𝑖 = 1, 𝑛 − 1, delayed for stationary
conditions 8 , [18], [19] and [21]:
1
=
𝑛−1
(15)
(𝑕𝑖 − 𝑕𝑖 )2 = 𝑛 − 1 2 𝑀𝑆𝐸𝑛−1
(16)
2
𝑕𝑖2
(22)
1
𝑛−1
𝑛−1
𝑖=1
𝑕𝑖2
𝑕𝑖2
(23)
𝑕𝑖2
𝑕𝑖2
𝑕𝑖2
𝑕𝑖2
2
𝑛−1 𝑕 𝑖
𝑖=1 𝑕 2
− 10 𝑙𝑜𝑔10 𝑛 − 1
(24)
in terms of 𝐷 𝑛 − 1 :
𝑖
= 10 𝑙𝑜𝑔10 𝑛 − 1 + 𝐷 𝑛 − 1
(25)
Substituting (25) in (24):
𝑕𝑛2
𝐷 𝑛 = 10 𝑙𝑜𝑔10 2 − 10 𝑙𝑜𝑔10 2𝑛
𝑕𝑛
(26)
+ 10 𝑙𝑜𝑔10 𝑛 − 1
+𝐷 𝑛−1
Rewriting, using the logarithm properties, obtaining
(17) ■.
𝑛−1
(𝑕𝑖 − 𝑕𝑖 )2
𝑀𝑆𝐸𝑛−1
𝑕𝑖2
And logarithms properties in it:
For i = n, the last term for (13), in the same
expression:
𝑕𝑛 − 𝑕𝑛
𝑖=1
𝐷 𝑛 − 1 = 10 𝑙𝑜𝑔10
𝑖=1
2
𝑕𝑛2
𝑛−1
− 20 𝑙𝑜𝑔10 𝑛 + 10 𝑙𝑜𝑔10
In the same sense of the previous concepts, with
stationary properties, the decibel description for 𝑖 =
1, 𝑛 − 1:
(12)
Proof (Direct form). The MSE respect to the trace
description for Gi , 𝑖 = 1, 𝑛, it is defined as:
𝑛
1
(13)
𝑀𝑆𝐸𝑛 = 2
(𝑕𝑖 − 𝑕𝑖 )2
𝑛
1
𝑀𝑆𝐸𝑛 = 2
𝑛
𝑕𝑛2
𝑖=1
Where respect to (15):
𝑛−1
𝑖=1
6 Bode Recursive
Writing the MSE considering (16) in (14), obtaining
(12) ■.
Lemma 1: The Bode description as recursive
expectation expressed in logarithmic scale is a
function of Decibel recursive, as:
(27)
𝐵 𝑛 = 0.5 𝐷(𝑛)
Proof. The stochastic error considering the Bode
properties and the error description in it respect to the
second probability moment, for 𝑖 = 1, 𝑛, is:
5 Decibels Recursive
Theorem 6: Regarding the identification defined
as 𝑒𝑖 ≔ 𝑕𝑖 𝑕𝑖 . His behavior decibel recursively as:
𝑛 − 1 𝑕𝑛2
𝐷 𝑛 = 10 𝑙𝑜𝑔10
+ 𝐷 𝑛 − 1 . (17)
2𝑛 𝑕𝑛2
Proof. The identification error sequence defined as an
expanded vector array in mathematical expectation
as:
1/2
′
(18)
𝑒1𝑥𝑛 ≅ 𝐸 𝑒1𝑥𝑛 𝑒𝑛𝑥
1
The decibel error in stochastic way respect to second
probability moment for sequence 𝑖 = 1, 𝑛 , is:
′
𝐷 𝑛 = 20 𝑙𝑜𝑔10 𝐸 𝑒1𝑥𝑛 𝑒𝑛𝑥
1 𝑖=1,𝑛
In discrete description series:
𝑛
1
𝑕𝑖2
𝐷 𝑛 = 10 𝑙𝑜𝑔10
,
𝑛
𝑕𝑖2
1/2
1/2
′
𝐵 𝑛 = 10 𝑙𝑜𝑔10 𝐸 𝑒1𝑥𝑛 𝑒𝑛𝑥
1 𝑖=1,𝑛
Considering the logarithm properties:
𝑛 − 1 𝑕𝑛2
𝐵 𝑛 = 5 𝑙𝑜𝑔10
+𝐵 𝑛−1 .
2𝑛 𝑕𝑛2
The decibel corresponds to (27) ∎.
The simulation illustrates the properties considered in
the previous concepts, without lost the quality between
two spaces; i.e., the simulation only indicates the
evolution about its algorithms; depicting the
convergence rate of all of its [22].
The basics distribution function considered was the
Normal, with bounded variance and cero mean. In this
experiment was considered a laptop Toshiba Computer
(20)
𝑖=1
𝑛 −1
ISSN: 1790-5117
(29)
7 Numerical Results
(19)
Expanding respect to the last term:
1 𝑕𝑛2 1
𝑕𝑖2
𝐷 𝑛 = 10 𝑙𝑜𝑔10
+
𝑛 𝑕𝑛2 𝑛
𝑕2
𝑖=1 𝑖
Using the logarithms properties:
(28)
(21)
248
ISBN: 978-960-474-054-3
Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
with Core Duo at 1.73 GHz, 2GB of RAM, using as
software environment, the Matlab 7.0.
-25
x 10
1.5
7.1 Numerical Results Theorem 2
1
MSE
Fig. 1 represents in illustrative sense the function
evolution 𝐹 𝑥 = 0.006 ∗ cos 𝑥 + 0.02 ∗ 𝑟𝑎𝑛𝑑𝑛 ,
where a pseudorandom number obtained is a 𝑟𝑎𝑛𝑑𝑛
function.
0.5
0.08
0.06
0
0
50
100
0.04
0.02
F
Fig. 3
150
200
Numerical Evolution
250
300
MSE between F and its estimation Fe.
0
Fig. 4, illustrates the convergence rate between both
signals (F and its estimation) [20], [21], and [22],
using the decibel recursive expression (17).
-0.02
-0.04
-0.06
0
50
100
150
200
Numerical Evolution
250
DR
E
300
200
0
Fig.1 Original Signal F.
-200
Fig 2, describes the inverse convolution operation
signal Fe (the original signal estimation of F), respect
to the convolution operation signal F with delayed
signal G (delayed a step time), where 𝐺(𝑚, 𝑛) = 0.3 ∗
𝑐𝑜𝑠(𝑑) + 0.02 ∗ 𝑟𝑎𝑛𝑑𝑛; so that, the new matrix
expanded is a square matrix as Gm×m.
-400
-600
-800
-1000
0
0.08
50
100
150
200
Numerical Evolution
250
300
0.06
Fig. 4
0.04
DRE (Decibel Recursive Error) between F
and its estimation Fe.
Fe
0.02
7.2 Numerical Results Theorem 3
0
-0.02
Fig. 5 represents in illustrative sense the function
evolution 𝐹 𝑥 = 0.006 ∗ cos 𝑥 + 0.02 ∗ 𝑟𝑎𝑛𝑑𝑛.
-0.04
-0.06
0
50
100
150
200
Numerical Evolution
250
300
Fig.2 Inverse convolution operation signal Fe.
Fig. 3 shows the Mean Square Error (MSE) between F
and its estimations Fe with a rate convergence round
to 10-25 units.
ISSN: 1790-5117
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ISBN: 978-960-474-054-3
Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
-20
0.08
4
0.06
3.5
x 10
3
0.04
2.5
F
MSE
0.02
2
0
1.5
-0.02
1
-0.04
-0.06
0.5
0
Fig.5
50
100
150
200
Numerical Evolution
250
0
300
Original Signal F.
0
Fig. 7
Fig 6, describes the inverse convolution operation
obtained as signal Fe (the original signal estimation of
F), respect to have the convolutioned signal F with
delayed signal G (delayed a step time),
where 𝐺(𝑚, 𝑛) = 0.3 ∗ 𝑐𝑜𝑠(𝑑) + 0.02 ∗ 𝑟𝑎𝑛𝑑𝑛 ;
so
that, the new matrix expanded generated is not square
matrix Gm×n.
50
100
150
200
Numerical Evolution
250
300
MSE between F and its estimation Fe.
Fig. 8, illustrates the convergence rate between both
signals (F and its estimation), using the decibel
recursive expression (17).
DRE
200
0
0.08
-200
0.06
-400
0.04
-600
Fe
0.02
-800
0
-1000
0
-0.02
50
100
150
200
Numerical Evolution
250
300
-0.04
-0.06
Fig. 8
0
50
100
150
200
Numerical Evolution
250
300
7.3 Numerical Results Theorem 4
Fig.6 Inverse convolution operation signal Fe.
Fig. 9 represents in illustrative sense the function
evolution 𝐺(𝑚, 𝑛) = 0.3 ∗ 𝑐𝑜𝑠(𝑑) + 0.02 ∗ 𝑟𝑎𝑛𝑑𝑛 ,
where 𝑟𝑎𝑛𝑑𝑛 is a pseudorandom Matlab number.
Fig. 7 shows the MSE between F and its estimation Fe
has a rate convergence round to 10-25 units.
ISSN: 1790-5117
DRE (Decibel Recursive Error) between F
and its estimation Fe.
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ISBN: 978-960-474-054-3
Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
0.4
0.09
0.3
0.08
0.07
0.2
0.06
MSE
G
0.1
0
0.05
0.04
-0.1
0.03
-0.2
0.02
-0.3
-0.4
0.01
0
50
100
150
200
Numerical Evolution
250
0
300
0
50
100
150
200
Numerical Evolution
250
300
Fig.9 Original Signal F.
Fig. 11
Fig. 10, describes the inverse convolution operation
signal Ge (signal estimation of G), respect to the
convolution operation signal F with delayed signal G
(delayed a step time), where 𝐹 𝑥 = 0.006 ∗
cos 𝑥 + 0.02 ∗ 𝑟𝑎𝑛𝑑𝑛.
Fig. 1, illustrates the convergence rate between both
signals (F and its estimation), using the decibel
recursive expression (17).
DRE
MSE between F and its estimation Fe.
4000
3500
0.08
3000
0.06
2500
0.04
2000
Ge
0.02
1500
0
1000
-0.02
500
-0.04
0
-0.06
-0.08
0
50
100
150
200
Numerical Evolution
250
0
Inverse convolution operation signal Fe.
60
80
100 120 140
Numerical Evolution
160
180
200
DRE (Decibel Recursive Error) between F
and its estimation Fe.
The first simulation model respect to inverse
convolution operation (7) showed a rate convergence
round to 10-25 (Fig. 3). The second model of inverse
convolution operation (8) showed a rate convergence
round to 10-20 (Fig. 7). Finally the third model (10)
showed a rate convergence round to 10-4 (Fig. 11).
Moreover the first and the second model in its decibel
diagram illustrated the convergence rate between both
signals: F and its estimation (Fig. 4 and Fig. 8,
respectively). But in the case of the third model
illustrated in decibel form didn`t convergence between
both signals: G and its estimation (Fig. 12).
Fig. 11 shows the MSE between G and its estimation
Ge with a rate convergence round to 10-25 Matlab
units.
ISSN: 1790-5117
40
300
Fig. 12
Fig.10
20
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ISBN: 978-960-474-054-3
Proceedings of the 8th WSEAS International Conference on SIGNAL PROCESSING, ROBOTICS and AUTOMATION
8 Conclusions
This paper gave a short introduction and description
defining the inverse convolution operation in basic
structure and explains with short comments some of its
applications in astronomy, microscopy and analysis of
seismic measures, in the others. The second part
presented formal convolution and inverse convolution
operation concepts about various issues concerned.
The first corresponded to convolution as a matrix
expands expression, which provided the necessary
preamble to define the three models respect to inverse
convolution operation matrix suggested in this work.
Requiring in probability manner the mean square
error, observing its convergence rate trough diagrams
Decibels description according to the proposed Inverse
convolution operation models in simulation graphs.
Showing this section observing the three inverse
convolution operation methods in simulation scheme
where the third was the best in convergence rate.
Graphically speaking compared the inverse
convolution operation signals respect to original signal
in trace operation sense. The convergence error
functional was obtained respect to mean square error
in vector description, using the second probability
moment and Decibels expression in recursive form. So
that, analyzing the results could be concluded that the
novel optimal inverse convolution operation was the
good enough estimation methods.
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W. Mendenhall., T. Sincich., Probability and
Statistics for Engineering and Science, Prentice
Hall, 1997.
[10] Walpole., Myers., Probability and Statistics,
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[11] M.J. Marques., Probability and Statistics for
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[12] W.H. Hayt., J.E. Kemmerly., Analysis Circuit
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The first simulation model respect to inverse
convolution operation (7) showed a rate convergence
round to 10-25 (Fig. 3). The second model of inverse
convolution operation (8) showed a rate convergence
round to 10-20 (Fig. 7). Finally the third model (10)
showed a rate convergence round to 10-4 (Fig. 11).
Moreover the first and the second model in its decibel
diagram illustrated the convergence rate between both
signals: F and its estimation (Fig. 4 and Fig. 8,
respectively). But in the case of the third model
illustrated in decibel form didn`t convergence between
both signals: G and its estimation (Fig. 12).
[14] R.L. Boylestad., Introductory Analysis Circuit,
Pretince Hall, 1997.
[15] D.C. Baird., Experimentation, An Introduction
to the Theory of Measurement and Design of
Experiments, Pearson, 1991.
[16] M.S. Grewal., A.P. Andrews., Kalman Filtering
Theory and Practice, Prentice Hall, 1993.
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The matrix inverse convolution operation model, in
discrete form, recovered the signal through the inverse
matrices and pseudoinverse schemes.
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