Study of Iterative Algorithms for Solving the Inverse
Problem of Electrocardiography
Yujing Lin
Department of Electrical & Systems Engineering
School of Engineering & Applied Science
Email: [email protected]
Tel: (314)255-3793
Supervisor: R. Martin Arthur
Department of Electrical & Systems Engineering
School of Engineering & Applied Science
Email: [email protected]
Tel: (314) 935-6167
Abstract
Changes in cardiac and torso geometry have both been investigated in their effects on the bodysurface electrocardiograms (ECGs). Inverse solution of ECG mappings is widely used to get the cardiac
electrophysiological information from the measured or simulated body-surface potentials (BSPs) so that
heart-surface potentials (HSPs) can be reconstructed [2]. Because of the ill-postedness of the ECG inverse
problem, regularization methods are usually used to obtain clinically reasonable solutions. In this project,
we present two algorithms: the iterative least-mean-square method and the L1-norm-based iterative
regularization technique in detail, and compare them with the traditional zero-order Tikhonov
regularization scheme. Furthermore, we reconstruct the 3D models of HSPs and BSPs according to these
three algorithms, and compare the spatial details of HSPs and BSPs that are generated from the different
algorithms. Based on these algorithms which are applied to the inverse problems of ECG mappings,
further studies regarding the spatial details of HSPs and BSPs, and the depolarization and repolarization
of action potential templates on HSPs can be conducted, which will have a significant impact on the study
of the identification of cardiac risks.
Keywords: inverse problem of ECG mappings; heart-surface potentials; body-surface potentials; zeroorder Tikhonov regularization; iterative least-mean-square method; L1-norm-based iterative
regularization; spatial details.
2
I.
In
ntroductio
on
Based
B
on previious clinical research,
r
we have
h
known tthat the changges in both caardiac and torso
geometry have effects on the body-ssurface electrrocardiogramss (ECGs). Forr example, thhe changes in
Gs
cardiac geeometry causeed by type II diabetes melllitus (T2DM) will affect paatients’ body--surface ECG
(Fig.1). The standard 12-lead
1
ECG set
s has been widely
w
applieed to identify cardiac risks [1]. This forw
ward
solution aims
a
in diagno
osing T2DM by
b identifying
g patients’ haabitus changes. However, aas shown in F
Fig. 1,
both diabeetes and obesity may causee cardiac dysffunction, so eeither habitus changes withh obesity or
cardiac so
ource changess with diabetees would affecct patients’ boody-surface E
ECGs. Thereffore, using thee
standard 12-lead
1
analysis alone prob
bably cannot provide enouugh spatial dettails for identtifying the
electrical phenotype off T2DM or so
ome other card
diac-associateed risks.
Fig.1. An
nterior body-surrface, iso-potentiial maps at the peak
p
of T wave. ((Left) Simulatedd map using boddy-surface potenttials
calculated
d from APs that combined
c
all of the
t regional incrreases. (Right) M
Measured map inn an obese diabettic subject. The w
white
dotss mark the locatio
ons of precordiaal electrodes in thhe standard 12-llead system [1].
n order to get more spatial information of
o patients’ heeart-surface ppotentials (HS
SPs), researchhers
In
are studyiing the inverse problem off ECG mappin
ng. The inversse ECG soluttion is used too get the cardiiac
electrophy
ysiological in
nformation fro
om the measu
ured or simulaated body-surrface potentials (BSPs) so tthat
the HSPs can be reconsstructed [2]. Because
B
of th
he ill-postedneess of the ECG inverse prooblems,
o
clinically reasonablee solutions. Z
Zero-order Tikkhonov (ZOT
T)
regularizaation methodss are used to obtain
regularizaation is one co
ommon techn
nique among the
t regularizaation schemess, which is oftten based on tthe
L2-norm data
d and the corresponding
c
g constraint teerms. Howevver, even thouugh L2-norm-bbased
regularizaation methodss can smooth the solution, the inverse soolution providded by L2-noorm-based meethods
is sensitiv
ve to measurem
ment errors. Additionally,
A
L2-norm-bassed methods ccannot localizze and distingguish
3
multiple proximal cardiac electrical sources [3]. Both of the above inadequacies of L2-norm-based
regularization methods would affect the accuracy of simulation result significantly. Thus, total variation
(TV) regularization method has been proposed to replace L2-norm-based regularization scheme in solving
the inverse problems of ECG mappings.
The TV regularization method is a L1-norm-based technique, which can overcome the
inadequacies of L2-norm-based regularization methods. Given the implementation of the L1-norm-based
regularization method, we can not only remove the error caused by L2-norm-based regularization scheme,
but also obtain more spatial information of cardiac electrical sources to reconstruct HSPs from measured
or simulated BSPs.
In this project, we will not only present the iterative algorithm of L1-norm-based regularization
scheme, but also come up with a completely different iterative algorithm which is derived by least-meansquare (LMS) method. Additionally, we will compare the reconstructions of HSPs that are generated from
the L1-norm-based regularization method and the LMS algorithm with the HSPs generated from the ZOT
regularization method. Moreover, we will apply these three different algorithms to reconstruct the
corresponding BSPs so that we can compare the spatial details provided by the different HSPs and BSPs.
Once proper algorithms can be developed to solve the inverse problems of ECG mappings, further clinical
and bioelectrical research regarding different cardiac risks associated with HSPs and BSPs may be
conducted, which will have a significant impact on the future study of cardiac diseases.
II.
Iterative Algorithms for Solving the ECG Inverse Problems
In this section, we will present three different algorithms for solving the ECG inverse problems:
the zero-order Tikhonov (ZOT) regularization method, the iterative least-mean-square (LMS) algorithm,
and the L1-norm-based iterative regularization scheme. Furthermore, we will compare the similarities and
differences among the reconstructions of the HSPs generated from these algorithms.

Zero-order Tikhonov Regularization Method
ZOT regularization is a common regularization scheme, which is based on the L2-norm data and
the corresponding constraints. Consider the following cost function [4]:
‖
where
and
‖
‖
are HSPs and BSPs, respectively, is a regularization parameter,
coefficient matrix relating HSPs to BSPs, and
is a transfer-
is a regularization matrix. Based on the previous research,
we choose = 3.8983e-005. The notation ‖∙‖ in the expression of
the cost function
‖ , 1
represents L2-norm data, so
can be rewritten as:
. 2
4
.T
Thus, taking thhe derivative of
The
T ZOT regularization method is based
d on minimizinng
with
w respect to
o
and settiing the gradieent function eequal to
minimize the cost funcction
givves the estimate
that caan
:
2
2
, 3
. 4
The
T regularizaation matrix
depends on the type of reegularization ttechnique.
could be eithher ,
the identitty matrix; , the
t Laplacian
n operator; or , the gradieent operator [44]. To analyze the ZOT
regularizaation, we shou
uld choose
. The estiimate
thatt can minimizze the cost funnction
is
shown bellow.
Fig.2: Estim
mate

generatted by the ZOT rregularization m
method.
Itterative Lea
ast Mean Sq
quare Algorrithm
The
T iterative LMS
L
algorithm
m is an adaptiive algorithm
m, which uses a gradient-baased method oof
steepest descent.
d
The general
g
idea of
o the LMS alg
gorithm is to use the estim
mates of the grradient vectorr
from the available
a
dataa. Meanwhile,, it incorporattes an iterativve procedure tthat makes suuccessive
correction
ns to the weig
ght vector, in the
t direction of the negativve of the graddient vector, w
which eventuaally
leads to th
he minimum mean-square
m
ms, the LMS
error. Compaared with otheer regularizatiion algorithm
5
algorithm is relatively simple because it does not require the calculations of correlation function and
matrix inversions.
Before we apply the LMS algorithm to the inverse problem of ECG mappings, let us consider the
following signal channel model [5]:
, 5
,
where
1 ,…,
1
Our objective is to find the estimate
,
0 ,
1 ,…,
1
.
from the measurement by minimize the following cost
function:
, 6
which is the norm of the error between the measurement
To determine the estimate
.
and the estimate
that can minimize
, take the derivative of
with respect to
so that we can get the gradient function:
2
2
. 7
Using gradient descent method to update , we can obtain:
1
2
1
1
2
2
2
, 8
where the parameter should satisfy0
,
Now let us define
.
,
, and apply the LMS algorithm (8) to the
inverse problem of ECG mappings. We can get the following iterative solution:
, 9
where the superscript
and
1 indicate the iteration numbers. The iterative procedure depends on the initial value of
algorithm, we define the initial value
Observe Eq. (9), the initial value
, and choose
, where
.
will result in the maximum error between estimate BSPs
6
and the parameter . To simplify the
and the meassurement BSP
Ps
increases,, the estimate BSPs
; that is,
i
. As
increases so that the coorresponding error
deccreases.
Furthermo
ore, if the iterrative solution
ns generated by
b Eq. (9) aree convergent, the error
will eventuallly reach , which
w
indicatees that
as th
he notation sttanding for thhe mean of staandard deviation of the QR
RS
Denote
D
region of
in thhat iteration.
. We usee the
to determine whethher the solutioon
convergess or not. When the estimatee
proviided by Eq. (99)
updatees, the changees of the HSP
Ps in the QRS
S region are m
much
more obviious than the changes in otther regions. Therefore,
T
th e changes in the QRS regiion can be eassily
observed so that they are
a usually ch
hosen to represent the overaall changes inn the estimatee
determinee if
. To
conv
verges, we run
n 50000 iterattions and gennerate the curvve showing thhe changes off
as follo
ows:
QRS
Fig.3: (Leftt) Estimate HSPss generated from
m the ZOT regulaarization; the reggion bounded byy the two blue linnes represents thhe QRS
region. (Right)
generatted for each iteraation using the L
LMS algorithm.
Frrom the right figure shown
n above, we see that the
50000 iterrations, which
h means that the
t estimate
more iteraation, but the values of
keeps increassing within thhese
does nott converge in these 50000 iterations. W
We test
still
s keep increeasing. Thereefore, we can conclude thaat
using the iterative LMS
S algorithm, it
i is very diffiicult to find a convergent ssolution
in some amoount
of iteratio
ons.
Because
B
it is hard
h
to find a global
g
minim
mum
local miniimum
, wee need to conssider if we cann determine a
in
n these 50000
0 iterations insstead. We takke the differennce of
7
betw
ween
each two consecutive iterations; thaat is,
amount Δ
, to observe tthe increasingg
between eaach two conseecutive iteratiions.
Fig. 4: Δ
between each
e
two consecu
utive iterations: (Left) 50000 iteerations; (Right) 50 iterations.
Frrom the left figure
f
in Fig.4
4, we can find
d that the incrreasing amounnt Δ
decreasses to
zero quick
kly when we run 50000 iteerations, so it is very difficuult to observee the Δ
in thhe
very begin
nning of thesee 50000 iterattions clearly. Hence, we teest 50 iterationns instead to observe the
Δ
when
is small. Frrom the right figure, we seee that Δ
before thee first three iteerations, and then it starts to
t decrease affter
decreasing
g after
3,
Fig. 5: log
gΔ
3. A
Although Δ
is still increaasing as long as the value oof Δ
between each two consecutivee iterations usingg the LMS algorrithm.
8
inccreases quicklly
iis
is posittive.
cleearly, we tookk the logarithhm of Δ
To
T observe thee change of
the value of log Δ
(Fiig. 5). From Fig.
F 5, we see that after
keeps deccreasing, and then
t
it starts to
t increase around
local miniimum point between
b
minimum
m log Δ
log Δ
, andd plot
3000, log Δ
48
8000. Thus, w
we may assum
me there existss a
3000
3
and 480
000, which m
means that wee can find the location of thhe
to deetermine the stop
s
iteration. We can find that the miniimum value oof
ap
ppears at
37189 by ussing Matlab. A
Additionally, we need to ppay attention tto the
parameterr , because itt can affect th
he increasing rate
r of
the increaasing amount for
in eaach iteration. The larger
. Recall Eq. (9), the value of aff
ffects
is, the less iteeration we neeed to reach thhe
local miniimum point. However,
H
if becomes too
o large, the cuurve of Δ
will oscillatte,
that is wh
hy there existss an upper bou
und for the ch
hoice of . In this study, w
we choose the parameter
0.63
379.
We
W generate 37189 iteration
ns, and then compare
c
the eestimate
with thee
generateed
from the ZOT
Z
regularizzation techniq
que.
Fig. 6: (Left) Estimate
E
generated from
m the LMS algorrithm. (Right) Esstimate
geneerated from the Z
ZOT
regularizatiion. The cyan lin
nes in both of the figures indicatte the standard ddeviation at eachh node.
de value of th e HSPs show
wn in Fig. 6, w
we see that thee
Frrom the overaall shape and the magnitud
estimate
generated
d from the iterrative LMS allgorithm and the ZOT reguularization meethod are veryy
close to eaach other. Caalculating the relative errorr (RE) and thee correlation ccoefficient (C
CC) between tthese
two estim
mate HSPs giv
ves: RE = 0.18
874 and CC = 0.9884. Thiss result showss that the iteraative LMS
algorithm
m is a good app
proximation to
t the ZOT reegularization m
method, and iit can generatte very close
9
without considering L2-norm data and the corresponding constraints, so the LMS algorithm can overcome
the inadequacies of the ZOT regularization technique. The only disadvantage of the LMS algorithm is that
it usually needs more than 30000 iterations to obtain a good estimate solution
, which takes too much
testing time. In the next part, we are going to introduce another iterative regularization algorithm.

L1-Norm-Based Iterative Regularization Scheme
The ZOT regularization is a very common L2-norm-based technique; however, although the L2-
norm-based algorithms can smooth the solutions, they may reduce the accuracy of localizing cardiac
sources. Additionally, they may affect the accuracy of resolving multiple sources in close proximity [2].
Since the total-variation (TV) regularization, which is also known as non-quadratic regularization method,
has been widely used in the image restoration, there also exists a development of L1-norm-based
approaches for magnetoencephalography (MEG) and electroencephalography (EEG). The L1-norm-based
technique has been established as superior to higher order norms. This technique penalizes the L1-norm
of the gradient function and yields less-smoothed solutions with more localized details. Hence, we are
going to apply a L1-norm-based regularization scheme here, which bases on the L1-norm of the normal
derivative of the HSPs. The cost function in Eq. 1 can be modified as:
‖ , 10
‖
where
is the L1-norm-based regularization parameter, the subscripts 1 and 2 indicate L1 norm and L2
norm, respectively. Define a normal derivative matrix
, which can be derived to relate
to
:
. 11
Then, the cost function can be expressed as:
‖
‖
‖ . 12
‖
To minimize the cost function shown in (12), the most important task is to identify the normal
derivative operator
. This operator
can be derived from the geometric relationship between the HSPs
and the BSPs [6]:
, 13
where
represents the angles on heart surface, and
is the gradient of HSPs on heart surface.
Eq. (12) shows a nonlinear optimization problem. Due to non-differentiability of the L1-norm
penalty function, an estimated solution can be obtained by [8]:
, 14
10
where
is the weight matrix of
, and
is equal to the normal derivative operator shown in Eq.
is obtained by:
(13). The diagonal weight matrix
1
2
where
1
, 15
is a small positive number, which can guarantee that the denominator of each element in the
10 .
is nonzero. In our problem, we choose
Consider L1-norm of a matrix A:
‖ ‖
max
|
| , 16
yields:
which is simply the maximum absolute column sum of the matrix. Substituting
max
′
elementlocatingin
| ′
| , 17
rowand
columnof
. As what we have mentioned in the introduction, regularization algorithms for the inverse ECG
mapping problems are often used to generate clinically reasonable solutions to ill-posed problems. For the
ZOT regularization method, the regularization parameter has been solved under previous related
research, but the L1-norm-based regularization parameter
is unknown. Thus, before we further
formularize the L1-norm-based iterative regularization algorithm, we first need to solve an appropriate
regularization parameter .
L-curve is a parametric plot of the size of regularized solution and the corresponding residual [10].
We know that a good method for choosing the regularization parameter for discrete ill-posed problems is
to incorporate information about the solution size in addition to using information about the residual size.
The corner of the L-curve corresponds to a good balance between minimization of the sizes, and the
corresponding regularization parameter
is a good one.
The L1-norm-based L-curve shown below reveals the relationship between the L1-norm of the
estimate HSPs and the residuals of the corresponding BSPs (BSPR). Each red dot in the plot represents a
pair of BSPR,
, and the blue asterisk represents the corner of the L-curve in our case, where it
corresponds to a good balance between minimization of the sizes. The corresponding regularization value
is 0.1836. Thus, we choose
0.1836 as our L1-norm-based regularization parameter.
11
Fig. 7: L-curve for the L1-norm-based
L
regularization
r
sccheme. The valuue of tvec at the bblue asterisk is tthe L1-norm-bassed
regularizzation parameterr .
Once
O
we obtain the regulariization param
meter , we cann formularizee the L1-norm
m-based iteratiive
algorithm
m as follows:
In
nitialization:
, 18
For step
1,2,
1 …
1
2
1
, 19
, 20
0
, 21
0.1836, 22
10 . 23
L
iterativee algorithm, we
w use the
Similar to the LMS
stop iterattion.
12
aas the criterioon to determinne the
Fig. 8: (Leeft)
for each iteratio
on using the L1-n
norm-based iteraative regularizattion. (Right) Δ
each two consecutive
c
iteraations.
We
W can see thaat the
bettween
of estim
mate HSPs gennerated from tthe L1-norm--based iterativve
regularizaation increases very fast at the beginning
g, and the diffference of
bettween each tw
wo
consecutiv
ve iterations almost
a
reachees zero within
n 10 iterationss. To obverse the change oof Δ
more cleaarly, we plot th
he correspond
ding log Δ
Fig. 9: log Δ
aas follows:
between each
e
two consecutive iterations uusing the L1-norrm-based iterativve regularizationn.
Frrom Fig. 9, we
w notice that the log Δ
from 0 to 33. After
keeps decreaasing smoothly when
33, the diffference beginss to oscillate, which meanss that the iteraation number
cannot go
o beyond 33 iff we want to locate
l
the local minimum ppoint.
13
rannges
The
decreases very fast within the 33 iterations, for example, when
Δ
80
100
we define Δ
increases to 20,
, and when
10,
decreases to 150
. If
as the criterion to determine the stop iteration, we can stop at
13, and the relative error and correlation coefficient between
and the
generated from the
ZOT regularization method are RE = 0.3131 and CC = 0.9525, respectively. An interesting result is that
when
increases from 13 to 32, even though the Δ
decreases from 100
relative error and correlation do not change. Hence, we can say the estimate
to 250
, the
almost maintains
consistent within a specific range of iterations.
To further explicitly show the difference of
maintains consistent, we generate a table to record the relative
iterations, within which the estimate
error and correlation coefficient between the
regularization and the
in each iteration, and to determine the range of
generated from the L1-norm-based iterative
generated from the ZOT regularization.
Iteration Number
Relative Error (RE)
Correlation Coefficient (CC)
1
0.4359
0.9096
2
0.3218
0.9470
3
0.3093
0.9517
4
0.3106
0.9524
5
0.3119
0.9525
6
0.3126
0.9525
7
0.3129
0.9525
8-9
0.3130
0.9525
10-33
0.3131
0.9525
Table 1: Relative error and correlation coefficient of
generated from the L1-norm regularization and
generated from the
ZOT regularization. The red-marked 0.9525 shown in the CC column show that CC doesn’t change after the fifth iteration.
Recall that if we use the iterative LMS algorithm, we need more than 30000 iterations to get RE =
0.1874 and CC = 0.9884. However, only one iteration can result in RE = 0.4359 and CC = 0.9096 if we
apply the iterative L1-norm-based regularization method. Moreover, when utilizing the iterative L1-normbased regularization method, the CC starts to maintain at 0.9525 from the 5th iteration. Additionally, from
10th iteration, neither the relative error nor the correlation coefficient change. These features of the results
have the following significant meanings.
14
We
W have show
wn that the LM
MS algorithm is a good appproximation to the ZOT regularization, and
the idea of the LMS alg
gorithm is verry straightforrward. Howevver, it usuallyy takes thousaands of iteratioons to
reach our expected HSPs. As anotheer iterative alg
gorithm, the L
L1-norm-baseed regularizattion method oonly
needs
of the time that the LMS
S method wou
uld use, so appplying L1-noorm-based alggorithm can saave a
large amo
ount of testing
g time. But co
ompared with LMS algorithhm, the RE ggenerated by L
L1-norm-baseed
algorithm
m is much larg
ger, even thou
ugh the CC is high. This meeans that the spatial detailss of the estim
mate
HSPs gen
nerated from the
t iterative L1-norm-base
L
d regularizatiion are differeent from the sspatial detailss of
the HSPs obtained by the
t iterative LMS
L
algorithm
m or the ZOT
T method. Thuus, to further compare andd
hese algorithm
ms, we may neeed to study the
t reconstrucctions of correesponding BS
SPs.
discuss th
Frrom Table 1, we know
compariso
on between
almost maaintains consisstent from k = 10, and if w
we pick k = 300, the
and the
Fig.
F 10: (Left) Esstimate
generated
d from the ZO
OT regularizaation is shownn below:
geneerated from the L1-norm
L
iterativve regularizationn. (Right) Reconsstructed
gennerated
from th
he ZOT regularizzation. The cyan lines indicate thhe standard deviaation at each nodde.
A brief table given below iss to summarizze the differennces between the LMS iterrative algorithhm
and the L1-norm-based
d regularizatio
on method. Frrom the data, we can clearrly see that coompared withh the
od, the LMS algorithm is m
more approxiimate to ZOT
T regularizatioon,
L1-norm-based regularrization metho
but it takees much moree iterations to generate the expected HSP
Ps.
Sttop Iteration
Relative Error
E
Corrrelation Cooefficient
R
Regularization Parameterr
LMS Alg
gorithm
37189
0.187
74
0.9884
3.8983e-005
L1 Algo
orithm
10
0.313
31
0.9525
0.18836
Table 2: Co
omparison betweeen the LMS iterrative algorithm
m and the L1-norm
m-based regularrization method. The relative error and
correlation co
oefficient are com
mpared with ZOT
T regularizationn method.
15
III.
Reconstruc
R
ctions of Body-Surfa
B
face Potenttials
Before
B
we com
mpare the calcculated BSPs generated froom different iiterative algorrithms, let us
further co
ompare the esttimate
fro
om the perspeective of reco nstructed 3D hearts, not siimply from thhe
magnitudee of HSPs.
Fig
g. 11: Compariso
on of 3D HSPs generated
g
from (Left)
(
the LMS aalgorithm and (R
Right) the ZOT rregularization.
Frrom the abov
ve comparison
n, we see that the reconstruuctions of HSPs from the itterative LMS
S
algorithm
m or the ZOT regularization
r
n are pretty close to each o ther. This ressult shows thaat the LMS
algorithm
m can provide very similar spatial
s
detailss of heart surfface as the ZO
OT technique again, and thhis
result also
o correspondss to the small relative errorr (RE) and higgh correlationn coefficient ((CC) betweenn the
estimate
generated
d from the LM
MS iterative algorithm
a
andd the ZOT reggularization, rrespectively. A
As
for the L1
1-norm-based algorithm, th
he result is a little
l
bit differrent.
Fig. 12: Comparison
n of 3D HSPs gen
nerated from (Leeft) the L1-norm
m algorithm and (Right) the ZOT
T regularization.
16
Basically,
B
the reconstruction
r
n of HSPs usiing the L1-noorm-based reggularization m
method is veryy
close to th
he one generaated by the ZO
OT regularizaation scheme. The only obvvious differennce appears inn the
T-wave reegion (the blu
ue area at the top
t surface off the heart). T
The T-wave aarea shown in left heart, whhose
is gen
nerated from the
t L1-norm-based regularrization, is a llittle smaller tthan the T-waave area in thhe
ZOT hearrt, and this L1
1-norm heart is
i different fro
om the LMS heart as well.. Thus, the L11-norm heart may
provide us with differeent spatial dettails of HSPs. Moreover, iff we observe the color-bars in Fig.11 annd Fig.
12, we can
n find that thee range of thee color-bars fo
or the LMS hheart and the L
L1-norm hearrt are both larger
than the one
o of the ZOT heart. Amo
ong these three hearts, the L
L1-norm hearrt has the larggest range of ccolorbar, which
h also shows that the L1-norm-based alg
gorithm can ggive differentt spatial inform
mation regardding
energy disstribution on the heart surfface.
We
W know theree exists the fo
ollowing relattionship betw
ween HSPs andd BSPs:
. 24
The transffer-coefficien
nt matrix relatting HSPs to BSPs,
B
HSPs
, iss known, andd we have gennerated estimaate
from both th
he LMS algoriithm and the L1-norm-bassed algorithm,, so we can geenerate the
estimate BSPs
B
usin
ng Eq. (24) fo
or the two iterrative algorithhms, respectivvely. The resuults for the LM
MS
algorithm
m and the ZOT
T regularizatio
on are given below:
b
Fig. 13: Comparison of reconstructed BSPs
B
generated from
fr
(Left) the L
LMS algorithm aand (Right) the Z
ZOT regularization.
Fig. 13 shows that not only the shapes off the reconstruucted torso, bbut also the ennergy distribuutions
dy surface in the two plotss are very closse to each othher. This is noot a surprisingg simulation rresult,
on the bod
because
is simply determined
d
by
y
in our assumption,
a
aand we have aalready shownn that the
generated
d from the LM
MS algorithm is a very good
d approximattion to the
17
generated frrom the ZOT
he correspond
ding BSPs
regularizaation. Thus, th
should be cllose to each oother as well. However, if we
reconstrucct the BSPs frrom the HSPss generated frrom the L1-noorm-based iteerative algoritthm, we find tthat
the reconsstructed L1-norm torso is also
a very simiilar to the ZO
OT torso (Fig. 14).
Fig. 14: Comparison
C
of reeconstructed BSP
Ps generated from (Left) the L1--norm algorithm
m and (Right) thee ZOT regularizaation.
Before
B
we disccuss the resultt generated frrom the L1-noorm-based iteerative regularrization algorrithm,
let us look
k at the follow
wing the tablee first:
LMS
vs. ZOT
L1--Norm
vss. ZOT
Relative errror
0.0540
0
0.05500
Co
orrelation coeefficient
0.9985
0
0.99855
Table 3: Relative error an
nd correlation co
oefficient betweeen LMS/L1-norm
m
and ZOT
Recall
R
the dataa shown in Taable 2, we obttain a smallerr RE and a higgher CC of
.
from the
iterative LMS
L
algorithm
m than the L1
1-norm-based
d iterative reguularization m
method. Howeever, if we
reconstrucct the
from
m the
, thee RE and CC between LM
MS/ZOT and L
L1-norm/ZOT
T are almost thhe
same. Hen
nce, we can say that the reconstructed BSPs
B
from thee LMS algorithm and the L
L1-norm
regularizaation method are both very
y close to the BSPs
B
generatted from the Z
ZOT method. The LMS
algorithm
m and the L1-n
norm-based reegularization can provide uus with very ssimilar spatiaal details of thhe
BSPs, eveen though the spatial detaills of HSPs giv
ven by the tw
wo algorithms are different.
IV.
Discussion
D
and Concclusion
In
n this project, we study and
d compare thee algorithms oof the ZOT reegularization method, the
iterative LMS
L
techniqu
ue, and the L1
1-norm-based
d iterative reguularization sccheme in detaail. Generally,, the
18
LMS iterative algorithm is simpler than both the ZOT regularization and the L1-norm-based iterative
regularization, because the LMS algorithm does not involve computation of matrix inversions.
Additionally, the parameter is simply based on the coefficient matrix
and the computation of
eigenvalues, but we need to generate the L-curve to solve the regularization parameter
for the L1-norm-
based iterative regularization. However, compared with the LMS iterative algorithm, the L1-norm-based
method takes much less iteration to achieve a local minimum solution
.
As for the reconstructions of HSPs and BSPs generated from the ZOT regularization, the LMS
algorithm, and the L1-norm-based method, respectively, we see that the ZOT regularization and the LMS
algorithm can result in very close HSPs (RE = 0.1874 & CC = 0.9884). Thus, we could say the iterative
LMS algorithm may be a good replacement for the ZOT regularization, since the ZOT regularization
scheme needs to deal with the computations of matrix inversion and the regularization parameter. More
importantly, the inverse solution provided by the ZOT regularization is sensitive to measurement errors,
and the ZOT regularization cannot localize and distinguish multiple proximal cardiac electrical sources
[3]. Using the LMS iterative algorithm instead can provide very similar spatial details while overcoming
the inadequacies of the ZOT regularization scheme at the same time.
On the other hand, even though the estimate HSPs from the L1-norm-based regularization is not
very close to the estimate HSPs from the ZOT regularization, we cannot say the L1-norm-based method is
not a good regularization scheme. As what mentioned in the last paragraph, we know that there exist
some inadequacies in solving the inverse ECG problems by using the ZOT regularization, so we cannot
say the ZOT regularization is the most appropriate regularization technique for the inverse problem of
ECG mappings. Hence, the difference between the estimate HSPs generated from the L1-norm-based
regularization method and the ZOT regularization can provide us with some new spatial information,
which may be very useful for the further studies of the depolarization and repolarization of action
potential templates on HSPs. Similarly, the difference of the reconstructions of BSPs may also bring
different spatial information for us.
Therefore, through this project, we discuss and compare different algorithms for solving the
inverse problems of ECG mappings. We derive the iterative LMS algorithm, which could be a good
equivalent algorithm for the ZOT regularization technique. Additionally, we develop the iterative L1norm-based regularization algorithm, which may provide us with different spatial information of HSPs
compared with the ZOT regularization technique and the LMS algorithm. Based on these different
approaches to the ECG inverse problems, further clinical and bioengineering studies can be conducted to
identify cardiac-associated risks.
19
V.
Acknowledge
The author is grateful to Professor R. Martin Arthur of the Washington University in St. Louis for
providing all the data used in this study.
VI.
Reference
[1] R. Martin Arthur, Yujing Lin, Shuli Wang and Jason W. Trobaugh, “Effects of Changes in Action
Potential Duration on the Electrocardiogram in Type II Diabetes,” International Journal of
Bioelectromagnetism, Vol. 14, No.3, December 2012.
[2] Guofa Shou, Ling Xia, and Mingfeng Jiang, “Total Variation Regularization in Electrocardiographic
Mapping,” Life System Modeling and Intelligent Computing, Vol. 6330, pp 51-59, 2010.
[3] Guofa Shou, Ling Xia, Feng Liu, Mingfeng Jiang and Stuart Crozier, “On Epicardial Potential
Reconstruction Using Regularization Schemes with the L1-norm Data Term,” Physics in Medicine and
Biology, November 30, 2010.
[4] Daryl G. Beetner and R. Martin Arthur, “Estimation of Heart-Surface Potentials Using Regularized
Multipole Sources,” Transactions on Biomedical Engineering, IEEE, Vol. 51, No. 8, August 2004.
[5] Todd K. Moon and Wynn C. Stirling, “Mathematical Methods and Algorithms for Signal Processing,”
Ch. 14, pp 643 – 648, New Jersey, 2000. Print.
[6] Roger C. Barr, Maynard Ramsey. III, and Madison S. Spach, “Relating Epicardial to Body Surface
Potential Distributions by Means of Transfer Coefficients Based on Geometry Measurements,”
Transactions on Biomedical Engineering, IEEE, Vol. BME-24, No. 1, January 1977.
[7] Lorange, M.; Gulrajani, R.M., “The forward and inverse problems of electrocardiography,”
Engineering in Medicine and Biology Magazine, IEEE, Vol. 17, No. 5, Sep/Oct 2008.
[8] Ghosh, Subham, “Electrocardiographic Imaging : Development of a Non-smooth Regularization
Method and Clinical Application in Patients with Wolff-Parkinson-White Syndrome and Heart Failure,”
Ph.D. dissertation, Washington Univ., St. Louis, MO, 2009.
[9] Daryl G. Beetner, “Inference of Spectral and Temporal Characteristics of Pericardial Potentials Using
Individualized Human Heart-Torso Models and the Multipole-Equivalent Method,” Ph.D. dissertation,
Washington Univ., St. Louis, MO, 1997.
[10] Per Christian Hansen, and Dianne Prost O’Leary, “The Use of the L-curve in the Regularization of
Discrete Ill-posted Problems,” Society for Industrial and Applied Mathematics, Vol. 14, No. 6, pp. 14871503, Nov. 1993.
20
VII.
Appendices

Appendix A: Analysis on the Parameter
of the Iterative LMS Algorithm [5]
For optimizing a function, to minimize it – is to iterate in such a way that
general framework is to update
. The
by
, 1
is a scalar, denoting a step size, and
where
is a direction of motion, selected so that the successive
steps decrease .
→
We know that for a differentiable function :
in the direction of the maximum increase of
in some open set D, the gradient
points
at the point . Thus, Eq. (1) can be expressed as:
, 2
the parameter
determines how far we move at step . To simplify this algorithm, we usually use
for some constant .
To obtain the , consider the following example:
2
where
∈
is symmetric positive definite and
, and the initial value is
2
2
Let
∗
denote the solution to
, 3
.
2 , 4
2
. 5
2
. Shifting coordinates centered around
∗
2 , then
, and letting
we can rewrite Eq. (5) as:
∗
Because
∗
is the solution to
,
∗
∗
, so
∗
, 6
∗
0. Substituting
and
∗
0, we obtain:
∗
∗
, 7 , 8
…
, 9
. 10
∗
Convergence of this equation from any initial point
Let Λ
, where
requires that ‖
1.
is the orthogonal matrix composed of eigenvectors of , and Λ is the
diagonal matrix of eigenvalues. Let
, then the Eq. (7) can be written as:
, 11
which leads to the solution:
21
‖
Λ
Since the matrix
. 12
Λ is diagonal, Eq. (12) can be expressed as the set of decoupled equations:
1
,
1
,
⋮
1
.
Therefore, it is clear that if we hope the convergence can occur from any starting point
, we must
guarantee that:
|
|1
1,
2
0
1,2, … ,
,
1,2, … ,
.
.
Because a separate is not provided for each direction, we must take the
2
0
satisfying all constraints:
.
Now in our case, the objective function is:
13
2
The gradient function is
2
. Based on the proof shown above,
1
, 14
Eq. (14) satisfies the form of Eq. (5), thus, the
in Eq. (14) can also be obtained from:
2
0
where

,
is the maximum eigenvalue of .
Appendix B: Analysis on the Normal Derivative Operator D for the L1-norm-based
Iterative Regularization [9]
Green’s second identity gives the following relationship:
A ∙
A ∙
where V is the volume inside the surface S,
,
(1)
is an outward pointing vector of unit magnitude normal to
surface element dS, and A and B are two scalar functions of position. By dividing either body surface
or heart surface
into inner and outer portions, and utilizing the fact that ϕ
0 on the outer
surface, we can obtain the potential at point o described by [1]:
ϕ
̅∙
∙
22
̅∙
,
(2)
where
and
are heart surface and body surface, respectively. Denote
that subtended at an observation of the ith location on surface
and
Specifically, choose two locations of observations,
̅∙
Ω , which means
by an area element on surface .
, then (2) can be written as:
ϕ
Ω
∙
Ω
,
(3a)
ϕ
Ω
∙
Ω
.
(3b)
Assume that each of the terms in (3a) and (3b) can be discretized as:
∑
Ω
ϕ
,
∑
Ω
∙
(4a)
,
∑
(4b)
Γ .
(4c)
Substituting (4a)-(4c) into (3a) and (3b) gives:
Φ
Φ
Γ
0,
(5a)
Φ
Φ
Γ
0.
(5b)
th
By choosing the i location successively at all
at all
locations on surface
locations on surface
and then successively
as well, (5a) and (5b) can be written as:
Φ
Φ
Γ
0,
(5a)
Φ
Φ
Γ
0,
(5b)
where P’s and G’s are matrices of coefficients depending entirely on geometry. Γ can be obtained either
from (5a) or (5b); however,
is invertible and
consists of coefficients of heart gradients as seen
by an observer at body locations. Therefore, (5b) can provide a better Γ with less numerical error. From
(5b), we can have:
Γ
Φ
Φ
.
(6)
If only one surface is considered, for example, the heart-surface is taken into account, (5b) and (6)
can be simplified as:
Φ
Γ
Γ
0,
(7a)
Φ .
(7b)
Applying the above analysis to our problem, Γ exactly represents the operation:
Γ
Φ
Φ . 8
.
Thus, the normal derivative operator
23

Appendix C: Matlab Code for the Iterative LMS Algorithm
% Function: Using least-mean-square iterative algorithm to generate HSPs from BSPs.
% Variables:
% (1) zbheln:transfer-coefficient matrix relating HSPs to BSPs.
% (2) gecg: body-surface potentials
% (3) Ve_old/Ve_new: hear-surface potentials
% Input: iternum = iteration number
% Yujing Lin
% Washington University in St.Louis, ESE Dept.
% Apr.7,2013
function [logdiff1]=iteration_LMS(iternum)
load ar1018bip2;
Ve_old=zeros(80,1162);% initialization of Phi_H = 0
Ve_new=zeros(80,1162);
tic;
for k=1:1:iternum
Ve_old=Ve_new;
mu=1./(max(eig(zbheln'*zbheln)));
Ve_new=Ve_old+mu*(zbheln'*gecg'-zbheln'*zbheln*Ve_old);%
use LMS iterative algorithm to generate new heart-surface potentials
mean_std(k)=mean(std(Ve_new(:,(290:450))'));% used as a criterion to tell if iteration results
converge
end
toc;
for m=1:1:(iternum-1)
diffmeanstd(m)=mean_std(m+1)-mean_std(m);
end
logdiff1=log(diffmeanstd);
% plot and compare the estimate Phi_H and ppot
figure;subplot(211);plot(Ve_new');grid on;hold on; set(gca,'FontSize',14);
plot(std(Ve_new),'c','LineWidth',2);
title(['\Phi_H at k = ' num2str(k),' Using LMS Algorithm']);
24
xlabel('n');ylabel('mV');
subplot(212);plot(ppot);grid on;hold on;set(gca,'FontSize',14);
plot(std(ppot'),'c','LineWidth',2);
title('\Phi_H Using ZOT Regularization');
xlabel('n');ylabel('mv');
figure;plot(Ve_new');grid on;hold on;set(gca,'FontSize',14);
plot(std(Ve_new),'c','LineWidth',2);
title(['\Phi_H at k = ' num2str(k),' Using LMS Algorithm']);
xlabel('n');ylabel('mV');
print -dmeta;
figure;plot(ppot);grid on;hold on;set(gca,'FontSize',14);
plot(std(ppot'),'c','LineWidth',2);
title('\Phi_H Using ZOT Regularization');
xlabel('n');ylabel('mv');
print -dmeta;
% plot mean(std(QRS))and log(diff(mean(std(QRS))))
figure;plot(mean_std,'LineWidth',2);grid on;set(gca,'FontSize',14);
title('\mu(\sigma(QRS))');
xlabel('Iteration Number');ylabel('\mu(\sigma(QRS))');
figure;plot(diffmeanstd,'LineWidth',2);grid on;set(gca,'FontSize',14);
title('\Delta(\mu(\sigma(QRS)))');
xlabel('Iteration Number');ylabel('\Delta(\mu(\sigma(QRS)))');
figure;plot(logdiff1,'LineWidth',2);grid on;set(gca,'FontSize',14);
title('log(\Delta(\mu(\sigma(QRS)))) between Each Iteration');
xlabel('Iteration Number');ylabel('log(\Delta(\mu(\sigma(QRS))))');
% calculate the relative error and correlation coefficient
[re1,cc1,act1,est1]=reccst(ppot,Ve_new');re1,cc1,
% plot 3-D heart based on estimate Phi_H
[ii,jj]=max(std(Ve_new));set(gca,'FontSize',14);
figure;trisurfv(inxhrt,ndfhrt,Ve_new(:,jj));set(gca,'FontSize',14);
25
title('\Phi_H Using LMS Algorithm');
xlabel('x');ylabel('y');zlabel('z');
shading interp;colorbar;
[ii,jj]=max(std(ppot'));
figure;trisurfv(inxhrt,ndfhrt,ppot(jj,:));set(gca,'FontSize',14);
title('\Phi_H Using ZOT Regularization');
xlabel('x');ylabel('y');zlabel('z');
shading interp;colorbar;
print -dmeta;
bsp=zbh*Ve_new; % calcuate body-surface-potential from estimate Phi_H
[mm,nn]=max(std(bsp));
bspe=zbheln*Ve_new;
[re2,cc2,act2,est2]=reccst(bspe,gecg');re2,cc2,
figure;trisurfv(inxtor,ndftor,bsp(:,nn));set(gca,'FontSize',14);
title('\Phi_B Using LMS Algorithm');
xlabel('x');ylabel('y');zlabel('z');
shading interp;
colorbar;
[mm,nn]=max(std(bspp'));
figure;trisurfv(inxtor,ndftor,bspp(nn,:));set(gca,'FontSize',14);
title('\Phi_B Using ZOT Regularization');
xlabel('x');ylabel('y');zlabel('z');
shading interp;
colorbar;
print -dmeta;
return

Appendix D: Matlab Code for the L1-norm-based Regularization Algorithm
% Function: Using L1-norm-based iterative algorithm to generate HSPs from BSPs.
% Variables:
% (1) zbheln:transfer-coefficient matrix relating HSPs to BSPs.
% (2)gecg: body-surface potentials
% (3) phi_old/phi_new: hear-surface potentials
26
% Input: iternum = iteration number
% Yujing Lin
% Washington University in St.Louis, ESE Dept.
% Apr.7,2013
function [D,W,bsp]=iteration_L1(iternum)
load ar1018bip2;
tau_new=0.1836;
D=-inv(ghhn)*phhn; % differentiation operator D
phi_old=inv(zbheln'*zbheln+tau_new*D'*D)*zbheln'*gecg'; % initialization of estimate heart surface
potentials
phi_new=inv(zbheln'*zbheln+tau_new*D'*D)*zbheln'*gecg';
beta=10^(-5);
% generate the weighting matrix W and the corresponding estimate Phi_H
for k=1:1:iternum
phi_old=phi_new;
D_phi=D*phi_old;
L1_D=zeros(1,1162);
for j=1:1:1162
L1_D(1,j)=sum(abs(D_phi(:,j))); % take the absolute sum of each column of D
end
L1_D_L1norm=max(L1_D);% updating D*Phi_H
W=diag(1/2./(sqrt((L1_D_L1norm)^2+beta))*ones(1,80));
D_new=D'*W*D;
phi_new=inv(zbheln'*zbheln+tau_new.*D_new)*zbheln'*gecg';
mean_std(k)=mean(std(phi_new(:,(290:450))));
end
% calculate the difference of mean(std(QRS)) between each iteration
for m=1:1:(iternum-1)
diffmeanstd(m)=mean_std(m+1)-mean_std(m);
end
27
% plot and compare the estimate Phi_H and ppot
figure;plot(phi_new');grid on;hold on;
set(gca,'FontSize',14);
plot(std(phi_new),'c','LineWidth',2);
title(['\Phi_H at k = ' num2str(k),' Using L1-norm Algorithm']);
xlabel('n');ylabel('mV');
print -dmeta;
figure;plot(ppot);grid on;hold on;set(gca,'FontSize',14);
plot(std(ppot'),'c','LineWidth',2);
title('\Phi_H Using ZOT Regularization');
xlabel('n');ylabel('mV');
print -dmeta;
% plot mean(std(QRS))
figure;plot(mean_std,'LineWidth',2);grid on;set(gca,'FontSize',14);
title('\mu(\sigma(QRS))');
xlabel('Iteration Number');ylabel('\mu(\sigma(QRS))');
print -dmeta;
% plot diff(mean(std(QRS)))
figure;plot(diffmeanstd,'LineWidth',2);grid on;set(gca,'FontSize',14);
title('\Delta(\mu(\sigma(QRS)))');
xlabel('Iteration Number');ylabel('\Delta(\mu(\sigma(QRS)))');
print -dmeta;
figure;plot(20*log10(abs(diffmeanstd)),'LineWidth',2);
set(gca,'FontSize',14);grid on;
xlabel('Iteration Number'); ylabel('dB');
title('log|\Delta(\mu[std(QRS)])|');
print -dmeta;
% calculate the relative error and correlation coefficient
[re1,cc1,act1,est1]=reccst(ppot,phi_new');re1,cc1,
28
% plot 3-D heart based on estimate Phi_H
[ii,jj]=max(std(phi_new));
figure;trisurfv(inxhrt,ndfhrt,phi_new(:,jj));set(gca,'FontSize',14);
title('\Phi_H Using L1-norm Algorithm');
xlabel('x');ylabel('y');zlabel('z');
shading interp;colorbar;
[ii,jj]=max(std(ppot'));
figure;trisurfv(inxhrt,ndfhrt,ppot(jj,:));set(gca,'FontSize',14);
title('\Phi_H Using ZOT Regularization');
xlabel('x');ylabel('y');zlabel('z');
shading interp;colorbar;
print -dmeta;
bsp=zbh*phi_new; % calcuate body-surface-potential from estimate Phi_H
[mm,nn]=max(std(bsp));
bspe=zbheln*phi_new;
[re2,cc2,act2,est2]=reccst(bspe,gecg');re2,cc2,
figure;trisurfv(inxtor,ndftor,bsp(:,nn));set(gca,'FontSize',14);
title('\Phi_B Using L1-norm Algorithm');
xlabel('x');ylabel('y');zlabel('z');
shading interp;
colorbar;
[mm,nn]=max(std(bspp'));
figure;trisurfv(inxtor,ndftor,bspp(nn,:));set(gca,'FontSize',14);
title('\Phi_B Using ZOT Regularization');
xlabel('x');ylabel('y');zlabel('z');
shading interp;
colorbar;
print -dmeta;
return;
29