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QPCR: Application for real-time PCR data management and analysis

Scharfetter, Hermann

Issa, S.

 1 
Reduction of low-frequency noise in magnetic induction tomography systems  
H. Scharfetter, S. Issa 
Institute of Medical Engineering, Graz University of Technology, Kronesgasse 5, 8010 Graz, Austria 
Abstract—Magnetic induction tomography (MIT) aims at the 
contact-less imaging of the electrical conductivity and related 
pathophysiological processes. Systems operating at frequencies 
below 1 MHz are desirable but suffer from poor SNR which 
can be dramatically impaired by mechanical vibrations. We 
present an algorithm which suppresses vibration noise in the 
imaginary part of the data and improves the SNR up to 35 dB.  
Keywords— Magnetic induction tomography, conductivity 
imaging, noise suppression, signal to noise ratio. 
I. INTRODUCTION  
Magnetic induction tomography (MIT) [1],[2],[3] is a 
contact-less imaging modality which maps the electrical 
conductivity within a 3-D body. MIT requires an alternating 
magnetic excitation field to be coupled from an array of 
transmit coils (TXC) to the object. Changes of the conduc-
tivity cause a field perturbation which is sensed in form of 
induced voltages in an array of receiver coils (RXC). 
Images are usually reconstructed from voltage changes Δv 
between different states of the object (e. g. during move-
ments or when changing the excitation frequency), whereas 
the components of Δv stem from all possible independent 
TX-RX combinations. The image reconstruction problem 
has been described elsewhere and is out of the scope of this 
paper. Suffices to say that the reconstruction involves a 
highly ill-posed problem which is very sensitive to noise in 
the data, therefore high SNR is mandatory. 
 
Most MIT hardware so far developed for medical 
investigation uses frequencies of several MHz to provide 
high sensitivity in low conductivity tissues, i.e. 20 MHz, 
[2], 10 MHz [1,4]. However, measurements in the β-
dispersion range below 1 MHz are desirable because many 
pathological processes are reflected there [5]. Due to the 
poor SNR, however, this represents a real challenge. The 
lowest frequency so far tried was found in [6] (11.6 kHz). 
We have developed a fast parallel multi-frequency imaging 
system which operates from 50 kHz to 1 MHz [7]. It 
comprises 16 RXC and 16 TXC and allows frequency 
differential imaging [8], whereas 8+8 coils can be operated 
in single shot mode. However, due to the the low sensitivity 
vibrations and thermal drifts cause significant errors [7]. 
This paper describes a method for eliminating such errors. 
II. METHODS 
A. System description 
The Graz MIT Mk2 system consists of two ring shaped 
coil arrays with 8 solenoid TXC and 8 RXC on each ring.  
 
 
Fig. 1: Geometry of the coil array. The parameters RT, RR and h are 
variable in our mounting structure. D = 100 mm. 
The coils are arranged in a zig-zag pattern around the 
perimeter (fig. 1). Each RXC is a planar gradiometes placed 
in front of a TXC, facing the center of the ring and adjusted 
so as to cancel the net magnetic flux produced by the TXC.  
Fig. 2: Schematic block diagram of the Graz Mk2 MIT system 
In fig. 2 the block diagram of the complete hardware is 
shown. The excitation signals are generated as multisinus 
patterns by an 8-channel DAC board and fed to 8 power 
 2 
amplifiers (PAs). To drive all 16 TXCs a 8 Æ 16 
multiplexer is provided. The response voltages from the 16 
RXC are preamplified and demultiplexed 16 Æ 8 before 
being digitized by an 8-channel ADC board. As the 8 
TXCs are fed simultaneously they must be encoded by 
splitting the carrier frequencies into 8 closely neighboring 
sub-frequencies. The amplitude and phase of each 
TXC/RXC signal is then extracted by the FFT. 
B. Systematic errors 
To extract the complex conductivity information with the 
correct phase, the true EMFs in the RXCs should be de-
modulated with respect to a voltage induced in a reference 
coil by the primary field only. However, in the real system 
the digital demodulation is done with respect to the phase of 
the local oscillator (LO)  of the signal processing unit. As 
the signal travels through many stages to the TXC and 
through the receiver chain it suffers systematic phase 
changes so that the true phase relationship is lost. We ana-
lyze the problem according to fig. 3. The basic coordinate 
system is the physical frame (ReTX, ImTX) in which the EMF 
induced in an RXC by the primary magnetic flux of a TXC 
is purely real. Only changes of voltages are depicted.  
 
Assumption 1: In weak conductors (biological tissues) and 
below several MHz the real part σ of the conductivity 
causes only an imaginary change s in the frame (ReTX, 
ImTX). Movements of the TXC with respect to the RXC 
cause a modulation of the mutual inductance and hence of 
the induced voltage. The resulting  modulation signal m is 
purely real in (ReTX, ImTX). Given the need for resolving 
less than 1 ppm of the primary voltage mechanical vibra-
tions thus can heavily deteriorate the SNR of the real part 
while the imaginary part remains unaffected [10]. Aiming 
only at the reconstruction of σ the further analysis can be 
limited to the imaginary signal s. 
Assumption 2: The signal s is assumed to be much weaker 
modulated by the movement than m so that s can be 
considered as unmodulated in the further analysis. 
 
In practice, the excitation waveform is generated by the 
LO which defines the LO frame (ReLO, ImLO) for 
demodulation. The waveform is sent through the TX chain 
(ADC, PA, TXC) and suffers a systematic phase shift 
θ1 before showing up in the physical frame (ReTX, ImTX). 
Then both s and m travel further through the parasitic LRC 
network of the RX coils and suffer a phase shift ϕ and 
finally the signals encounter a further shift θ2 caused by the 
following amplifiers and the DAC. When demodulating 
these rotated replicas s’ and m’ with respect to the LO frame 
(ReLO, ImLO) a strong spurious imaginary modulation ms 
appears although in reality there is none in the physical 
frame (assumption 2). The imaginary signal ss is the 
projection of s’ to the axis ImLO and does not reflect 
correctly the magnitude of the true signal s. Hence 
mechanical vibrations can heavily deteriorate the SNR of 
the measured imaginary signal [7]. Also other errors which 
affect mainly the true real part (moving metallic parts, 
thermal drifts of the RXC [5]) project into ImLO. 
 
ϕ
θ1 Re TX
ReLO
Im‘
ImLO
ψ
Re‘
Im TX
m
s
s‘
m‘
θ2ms
ss
 
Fig. 3:  phase rotations along the signal path 
 
 
C. Phase correction 
In order to avoid the above spurious projections we must 
back-rotate the signals by ψ = (θ1+θ2+ϕ) whereas θ1 in 
previous concepts was provided by a separate reference coil 
which picked up the primary magentic flows from all TXC 
at once [7]. In the new approach the correction is achieved 
by rotating the signals until the spurious modulation mS 
vanishes, i. e. Var(sS) is minimum. This rotation accounts 
for all error phases ϕ, θ2, θ1 without the need for an extra 
reference channel.  Setting: z:=ImLO, x:=m’; y=s’ the 
imaginary part in the LO frame is 
 
Ψ+Ψ−= cosysinxz     (1)  
 
The variance of this imaginary part is   
  
 
( ){ } ( ){ }
{ } { } ( )
( ) ( ) ( )y,xCovsincosxVarsinyVarcos
sincosxyEsinxEcosyE
sinxcosyEzE)z(Var
ΨΨ−Ψ+Ψ=
=ΨΨ−Ψ+Ψ=
=Ψ−Ψ==
2
2
22
2222
22
        (2) 
 
With 21 acos;asin −=Ψ=Ψ  and setting f:=Var(z), 
U:=Var(x), V:=Var(y), W:=Cov(x,y) eq. (2) becomes: 
 
 3 
( ) ( ) WaaUaVaaf 222 121 −−+−=   (3) 
 
Minimization of Var(z) with respect to ψ yields: 
 
)cos(f
da
d
d
da
da
dff
d
dfmin ψψψ ===→Ψ 0  (4) 
 
Excluding the trivial case |ψ| = π/2 (4) is equivalent to: 
 
0
1
1
2
2
=⎟⎟⎠
⎞
⎜⎜⎝
⎛
−−
−−−
a
a
a
aWVU    (5) 
 
1
22
2
12 +⎟⎠
⎞⎜⎝
⎛ −±⎟⎠
⎞⎜⎝
⎛ −−=→
W
VU
W
VU)cot(ψ  
 (6) 
 
from which we pick that ψ which minimizes Var(z).  
   
D. Experimental evaluation 
U, V, W were calculated from 15 calibration frames 
during which we introduced vibrations (sinusoidal, 20 
Hz) in the coil system with a loudspeaker so as to get 
sufficient variance in the data. During acquisition of 
the following 26 measurement frames the vibration 
was stopped. The correction angles were calculated 
only from the calibration data and then applied to the 
measurement data. DAQ parameters: 12 bit, 7.5 
MSamples/s, 20 ms window length. Injected current: 
Two superimposed sinusoids, 0.85 App @ 200 kHz and 
0.31 App @ 450 kHz, subfrequency spacing 300 Hz. 
RX2
RX3
RX4
RX5
RX6
RX7
RX8
RX1
 
Fig. 4:  Experiment for the determination of the SNR. The blue circles 
illustrate the saline bottle which was shifted after 13 frames to the 
opposite side of the upper RX ring. Top view. 
The equivalent spectral noise voltage density 
(ENVD) was first determined with the empty coil 
system and the PA switched off. Then we loaded the 
system with a plastic cylinder filled with saline (8 
S/m, Ø 7 cm, height = 10cm, orientation in z-axis) 
placed as shown in fig. 4. After 13 frames the object 
was shifted as indicated by the arrow.  
III. RESULTS  
With the PA switched off the ENVD was always 
less then 2 nV/sqrt(Hz) which is close to the value 
expected from the pre-amplifiers (1.5 nV/sqrt(Hz)). 
 
 Fig. 5 shows the ENVD of all 8 RXC and receiving 
from TXC6 after switching on the PA. The noise is 
highest for RXC6 and the neighboring coils RXC5 and 
RXC7 which have high magnetic coupling with TXC6 
and thus are very sensitive to vibrations. After phase 
correction the ENVD decreased by a factor of 21 in 
RXC6. 
 
1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 x 10
-8
RX #
EN
VD
 im
ag
in
ar
y 
pa
rt
 [V
/s
qr
t(H
z)
]
STD compare uncorr/corr data17:25
 
 
TX6
 
Fig. 5:  Comparison of the equivalent noise voltage density between 
uncorrected (o) and corrected (+) data of TX channel 6 
Most channels improved significantly, some by up 
to a factor of 100 while only in 4 channels the noise 
increased slightly. Fig 6 quantifies the SNR (the ratio 
between the std of the signal change due to the object 
shift and the standard deviation of the noise voltage) 
before (A) and after (B) correction (please note the 
change of the colorbar scaling). In several channels (e. 
g. TXC5/RXC5) the improvement is more than 30 dB 
as shown in fig. 7 for both frequencies. Channels with 
still low SNR (e. g. TXC2/RXC3) are only those 
which inherently exhibit poor sensitivity to the chosen 
locations of the test object. Typically the SNR is high 
at the crossings of rows/columns 1,4,5 and 8, 
 4 
respectively, because these reflect those coil 
combinations which were most close to the object. 
TX
R
X
 
 
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
-10
-5
0
5
10
15
20
25
A 
TX
R
X
 
 
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
-10
0
10
20
30
40
50
B 
Fig. 6: SNR [dB] of the uncorrected (A) and corrected (B) system. 
Please note the different scales on the colorbar. 
TX
R
X
 
 
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
-15
-10
-5
0
5
10
15
20
25
30
A 
TX
R
X
g g p [ ]
 
 
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8 -5
0
5
10
15
20
25
30
35
B 
Fig. 7: gain of SNR [dB] due to correction at 200kHz (A) and 450 kHz 
(B). Please note the different scales on the colorbar. 
DISCUSSION 
The proposed algorithm provides an effective cor-
rection of the phase errors in the presented MIT sys-
tem without extra reference channels. Vibrational low 
frequency noise could be reduced and SNR increased 
by up to 35 dB. The gain of SNR between f1=200 and 
f2=450 kHz is about 5 dB due to the lower excitation 
current I at f2. Actually the expected gain of sensitivity 
is I(f2)/I(f1)*(f2/f1)2 which yields 5.3 dB. In an ideally 
symmetric system the SNR matrix (fig. 6) should be 
nearly symmetric but due to different vibration 
dynamics of different coils the symmetry is not 
perfect. The achieved system performance is sufficient 
for low contrast conductivity imaging such as in [8]. 
ACKNOWLEDGMENT 
This work was supported by the SFB project F32-
N18 granted by the Austrian Science Fund. 
REFERENCES  
1. Griffiths H, et al (1999) Magnetic induction tomography. A measuring 
system for biological tissues. Ann NY Acad Sci 873: 335-345 
2. Korjenevsky A, et al (2000) Magnetic induction tomography: 
Experimental realization. Physiol Meas  21: 89-94 
3. Peyton A J et al (1996) An overview of electromagnetic inductance 
tomography: Description of three different systems. Meas Sci 
Technol  7:  261-271 
4. Vauhkonen M, et al (2008) A measurement system and image 
reconstruction in magnetic induction tomography Physiol Meas 29: 
S445-S454  
5. Scharfetter H, Casañas R and Rosell J (2003) Biological Tissue 
Characterization by Magnetic Induction Spectroscopy (MIS): 
Requirements and Limitations  IEEE Trans Biomed Eng 50: 870-80 
6. Karbeyaz UB, Gencer NG (2003) Electrical conductivity imaging via 
contactless measurements: an experimental study. IEEE Trans Med 
Imaging 22: 627 – 635. 
7. H Scharfetter et al (2008) Hardware for quasi-single-shot 
multifrequency magnetic induction tomography (MIT): the Graz Mk2 
system. Physiol. Meas 29: S431-S443  
8. Brunner P, et al. (2006) Reconstruction of the shape of conductivity 
spectra using differential multi-frequency magnetic induction 
tomography. Physiol Meas 27: S237–S248 
 5 
Address of the corresponding author: 
• Author: Hermann Scharfetter 
• Institute: Inst. Of Medical Engineering 
• Street:  Kronesgasse 5 
• City:  Graz 
• Country: Austria 
• Email:  hermann.scharfetter@tugraz.at


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