WO2010003162A1

WO2010003162A1 - Correction of phase error in magnetic induction tomography

Info

Publication number: WO2010003162A1
Application number: PCT/AT2009/000266
Authority: WO
Inventors: Hermann Scharfetter
Original assignee: Technische Universität Graz; Forschungsholding Tu Graz Gmbh
Priority date: 2008-07-11
Filing date: 2009-07-08
Publication date: 2010-01-14

Abstract

The invention relates to correcting signals measured in magnetic induction tomography with regard to a phase error. In the tomography system, an object (OBJ) is exposed to alternating magnetic fields by transmitter coils (SP1, SP2, SP3) located at specific transmitter positions, alternating voltage signals are picked up through receiver coils (ES1, ES2, ES3) located at specific receiver positions, and an image of the spatial distribution of the electrical properties within the object is reconstructed from the voltage signals based on their amplitudes and phases. At least one of the receiver positions is modulated with regard to the transmitter positions, such as a linear vibration, causing an alternating modulation signal which superimposes the corresponding picked-up alternating voltage signal, and in a preprocessing unit (PRP) a phase of said modulation signal is determined and the alternating voltage signal is corrected by applying a phase shift compensating said modulation signal phase.

Description

CORRECTION OF PHASE ERROR IN MAGNEΉC INDUCTION TOMOGRAPHY

Field of the invention and description of prior art

The present invention relates to a method for correcting measured signals in magnetic induction tomography, wherein

- an object having inhomogeneous passive electrical properties is exposed to alternating magnetic fields by means of transmitter coils located at specific transmitter positions,

- alternating voltage signals are picked up through receiver coils located at specific receiver positions, which voltage signals contain information relating to the electric conductivity and its spatial distribution within the object, and

- an image of the spatial distribution of the electrical properties within the object is reconstructed from the alternating voltage signals based on their amplitudes and phases, wherein the correction relates to a compensation of a phase error of the voltage signals measured.

Likewise, the present invention relates to a magnetic induction tomography system for carrying out the mentioned method.

A method and apparatus of this kind is disclosed in WO 2008/011649 Al by the applicants. In that document, a method is proposed to determine the phase shift of the signal measured in magnetic induction tomography, namely by employing measurements at various frequency as well as a reference measurement.

Magnetic induction tomography (MIT) has been proposed for medical diagnostic as an imaging method which allows quick and non-expensive examination of a patient without the need to apply electrodes or ionising radiation. MIT may be particularly useful in applications such as the monitoring of lung ventilation, the early recognition of hydration disorders (e.g. edema) but also the early detection of cancers, for instance mammography with regard to breast cancer. The method is based on the application of a time-dependent (in particular alternating) magnetic field and measurement of the electric conductivity of the examined specimen (such as the examined body part of the patient) by means of non-vanishing imaginary part of the magnetic response, since the imaginary part corresponds to the energy loss response which is usually due to dissipation by electrical resistivity. By using several sets of transmitter and receiver coils, and employing image reconstruction algorithms, the measured signals are used to determine the location of regions of anomalous electrical conductivity (such as a lesion or a tumour) in the specimen. A description of the basics of magnetic induction lithography can be found in the article by H. Griffiths, "Magnetic induction tomography", Meas. Sci. Technol. 26, pp.1126-1131, 2001. A further description relating to the multifrequency modification of MIT (which is then called magnetic induction spectroscopy) is discussed by H. Scharfetter, R. Casanas and J. Rosell, "Biological Tissue Characterization by Magnetic Induction Spectroscopy (MIS): Requirements and Limitations", IEEE Trans. Biomed. Eng. 50, pp.870-880, 2003. Also documents WO 2004/026136 Al and RU 2129406 deal with the determination of the spatial variation of dielectrical properties, such as the electric conductivity, of an object.

One problem of magnetic induction tomography is the proper splitting of the signal into real and imaginary parts (reactive and dissipative signal components). This problem comes about because of the presence of complex impedances in the transmitter and receiver circuitries (cf . Fig. 5) which cause additional phase shifts between the excitation and measured signal (phase error). Therefore, correct determination of the phase error is a prerequisite for a correct analysis of the measured signal.

One approach to solve this problem, represented by the mentioned WO 2008/011649 Al, is to employ two or more frequencies in order to determine a correction to the measured signal. The algorithm described there is based on the assumption that the phase error φ is small and obeys a certain condition, namely that

However, this condition is only fulfilled if φ is dominated by the mismatch of the electrical parameters between the two gradiometer coils. Any additional phase error ψ caused by other components of the system does not obey this relation and must therefore be removed by calibration before. Therefore, the WO 2008/011649 Al requires to pre-calibrate the system with a phantom and determine the phase angle ψ in a reference measurement cycle (with a phantom) and to subtract this value from all subsequent measurements so as to leave φ.

This step was found to be possibly problematic in practice because often the phase angle ψ can be determined with a limited accuracy only. Hence the remaining mismatch angle after subtracting a noisy ψ from the measured angle leaves a noisy value for φ. It is then difficult to reach the required accuracy for successfully applying the multifrequency algorithm proposed in WO 2008/011649 Al. In such cases, an alternative way is needed for either deter- rnining ψ more accurately or for determining φ. Moreover, the synchronous demodulation process described in WO 2008/011649 Al requires a reference channel to establish the reference frame for imaginary and real part of the detected signal. This means that one ADC channel cannot be used for the receiver signals. In typical ADC boards with 8 or 16 channels this reduces the effective number of channels to 7 or 15, respectively. It would therefore be advantageous to avoid the extra reference channel.

The WO 2007/072343 A2 proposes, in order to provide a MIT system and method with high resolution without the need of increasing the number of coils, to use means for providing a relative movement between one or more generator coils and/ or on or more sensor coils on the one hand and the object to be studied on the other hand.

The WO 2005/082247 Al describes a device and method for correction of the position of a field sensor in a magnetic localization device. External field distortions are determined with the help of a reference sensor placed at a known position.

Summary of the invention

In view of the above, it is an aim of the present invention to overcome the above-stated problems and find a way to determine the phase error without the need of additional references, even if only a single frequency is used in the measurement.

This aim is met by a method for correcting measured signals in MIT as described in the beginning wherein at least one of the receiver positions is subject to a modulation with regard to the transmitter positions, causing an alternating modulation signal superimposing the corresponding picked-up alternating voltage signal, a 'modulation signal phase' of this modulation signal is determined, and the alternating voltage signal is corrected by applying a phase shift compensating the modulation signal phase. In the context of the present invention, modulation is understood as a modification of the transmitter/ receiver coil positions (including location and orientation) with respect to time, i.e., a general movement.

In order to further eliminate the modulation signal, the imaginary part of the corrected voltage signal may be taken, in which case the imaginary part thus obtained is used for the reconstruction of the image of spatial distribution. Preferably, the receiver and transmitter coils are provided outside the object.

The solution according to the invention uses a modification of the relative location between the transmitter and receiver coils (which is otherwise considered as a source of spurious noise) for generating a voltage response that can be used to determine the phase angle and determine the amount of correction angle needed. The modulation signal can be used as a standard for determination of the phase in the measuring setup. The measured signals are then rotated back by applying the inverse phase. After that correction, the image reconstruction follows according to known procedures. The modulation signal according to the invention offsets the need of an extra reference channel/ and also renders a phase calibration dispensable.

The invention is based on the realisation that the modulation signal can be separated from the actual signal from the object, using its quite different time-dependent behaviour according to its very nature: The actual signal is virtually stationary with respect to the modulation, which in turn will have typical frequencies much lower than the frequency of the AC magnetic field. The contribution of the modulation to the measured signal is, therefore, easy to recognize and is used to determine the phase shift, using the fact that the modulation signal has a defined phase, namely, it is a real signal (phase angle 0°) with regard to a system with corrected phases.

In one advantageous aspect of the invention, the modulation includes a generally linear (specifically translatory) movement of the receiver position of a receiver coil, preferably along a direction within a plane of the receiver coil.

One type of modulation implemented is a periodic modulation introduced actively, wherein the modulation includes a periodic movement of the receiver position with respect to a corresponding transmitter position at a modulation frequency by an actuating means of at least one of the associated transmitter and receiver coils. In this case, the modulation signal phase may be determined by means of the Fourier coefficients of the frequency sidebands relating to the modulation frequency. The modulation signal phase may then be determined by taking the inverse tangent of the expression

(or performing any mathematically generally equivalent computation) wherein C_USB and C_LSB denote the Fourier coefficients of the upper and lower sidebands, respectively.

Another important type of modulation is that caused be external vibrations ("shaking") which passively cause a distortion of the measurement setup. Such a modulation will then include a random movement of the receiver position with respect to a corresponding transmitter position caused by external vibrations. In this aspect, the modulation signal phase may be determined by taking a statistical sample of measurements of the corresponding voltage signals and evaluating measures of the statistic dispersion in the statistical sample of the imaginary and real parts of the voltage signals, in particular the variances of the imaginary and real parts of the voltage signals as well as the covariance of said imaginary and real parts. The modulation signal phase may then be determined by taking the inverse cotangent of C, where C is a solution of the equation

V - U - X ( C - VC) = O, wherein V and U represent the variances of the real and imaginary parts of the voltage signals, respectively, and X the covariance of said imaginary and real parts (such solution should be checked whether it represents a local mininαurn of the variances); or by performing any mathematically generally equivalent computation.

First experimental results performed by the inventors showed that the method according to the invention produced very good results and increased the signal to noise ratio by a factor of at least 10 : 1.

Brief description of the drawings

In the following, the present invention is described in more detail with reference to the drawings, which show:

Fig. 1 a schematic block diagram of a MIT system according to an embodiment of the invention;

Fig. 2 a schematic arrangement of transmitter and receiver coils surrounding an object which is investigated for a region of inhomogeneity;

Fig. 3 illustrates exemplary configurations of a transmitter coil as well as a receiving coil realized as gradiometer coil;

Fig. 4 shows the signals of a simple transmitter-receiver combination in the complex plane;

Fig. 5 shows an equivalent circuit for a gradiometer receiver coil network;

Fig. 6 illustrates the signals obtained with a gradiometer coil according to the invention;

Fig. 7 illustrates the profile of the magnetic flux density along an axis of displacement of a gradiometer coil;

Fig. 8 is a modified detail of Fig. 6 showing the phase angles of the modulation signal components and sum modulation; and Figs. 9 to 12 illustrate various possible implementations of how to introduce the modulation.

Detailed description of the invention

The schematic of Fig. 1 represents an overview of an embodiment of a MIT system in which the invention can be realized. Using a number of transmitter coils SPl, SP2, SP3 and receiver coils ESl, ES2, ES3, a specimen or object OBJ is examined for an inhomogeneity IHO. The in- homogeneity looked for in the object may, for instance, be a lesion within a body part such as the human brain or (female) breast, as a lesion is characterized by a different conductivity as compared to other regions of the object (human tissue).

The transmitter coils SPl, SP2, SP3 are arranged at various locations outside the object OBJ, preferably surrounding the object as shown in Fig. 2. Preferably the transmitter coils are located as close at possible to the space where the object OBJ resides. Three transmitter coils are shown, but the actual number may vary in accordance with the desired resolution and the type of the object to be examined, for instance eight or sixteen coils. Referring again to Fig. 1 the transmitter coils are supplied with alternating currents of a given frequency, provided by a signal generator SIG in connection with respective amplifiers AMP.

The receiver coils ESl, ES2, ES3 serve to pick up signals induced by the magnetic field from the transmitter coils SPl... SP3 and the object OBJ (including the inhomogeneity IHO). The receiver coils ESl... ES3 may be located near to corresponding transmitter coils SPl... SP3 as shown in Fig. 2, but it is also possible that they are arranged at locations well different from those of the transmitter coils. In the system shown in Fig. 1, the receiver coils are each connected with a pre-amplifier PRE; the pre-amplifier outputs are connected through shielded line connections LEI to the inputs of amplifiers EMP whose outputs are fed to a lock-in detector SYD. The lock-in detector receives a lock-in signal (for phase information) from the sinus generator SIG. In the same unit as with the lock-in detector, an image reconstruction BIR is done. The output of the image reconstruction is displayed on a monitor screen ANZ or other output device such as a printer. A control unit STE controls the lock-in detector SYD, the amplifiers AMP and the image reconstruction BIR.

Typical values of the exciting currents in the transmitter coils are from several 100 mA to several Amperes, e.g. 2 A (all values given for AC quantities are effective values), using solenoids of diameter 2 to 10 cm with 1 to 10 turns. Magnetic fields are as much as several 100 μT close to the coils, occasionally up to several mT. However, other dimensions and numbers of windings may be used in applications other than the mentioned ones. The signals which are actually received through the receiving coils are considerably smaller than the excitation signals of the transmitter coils, typically by several orders of magnitude. Therefore, in order to ensure that the effect of the fields of the transmitter coils does not directly enter the receiver coil signals, the receiver coils are realized as so-called gradiometer coils, as illustrated in Fig. 3. Moreover, the receiver coils may be oriented along a plane orthogonal to that of the respective transmitter coils. Gradiometer coils of this kind are largely insensitive to magnetic fields, inasmuch those magnetic fields are homogeneous (with respect to the spatial direction spanned by the two halves of the coil) since each coil half will, by way of magnetic induction, produce voltages of same size but opposite signs. However, since the geometry of the receiver coils will not be completely perfect and also inhomogene- ous stray fields and other field imperfections cannot be excluded, disturbing signals of potentially large size may occur, in particular from long-wave or short-wave radio transmissions. Signal processing by a lock-in detector can, as known in prior art, considerably reduce tile level of disturbances.

The signals picked up in the receiver coils ESl... ES3 depend, among others, on the spatial distribution of the electric conductivity within the object OBJ to be investigated. The experimental finding is that modified tissue (for instance, located within a breast) causes a deviation of the electric conductivity which are sufficiently large to enable efficient measurement and processing by the microprocessor of the image reconstruction BIR and to produce a tomographic representation. Details about such a procedure can be easily found in prior art, such as the references already mentioned.

As already mentioned, the actually interesting signal is quite small in comparison to the overall signal at the output of the receiver coils, namely, typically down to voltages in the range of nanovolts (nV). It will be obvious, therefore, that minute changes in the field geometry may cause substantial changes in the signals, which is usually an effect detrimental to the measurement performed and hence unwanted. Examples of usual error sources are systematic deviations of the mutual arrangement of the coils involved, which may be adversely affected already by minor temperature changes or mechanical strain distorting the frame of the measuring apparatus, or metallic parts that are passing outside though near to the investigated regions (such as coins or a bunch of keys in a pocket of a person passing the patient), or, simply, vibrations. Thus, for instance cars passing the building may realize a multiple source of disturbance.

The invention aims at using just those vibrations and the resulting signals for determination of the phase angle. As mentioned in the introductory part, the transmitter and receiver circuitries contain various impedances causing a phase shift the amount of which is generally not known, and which needs to be corrected for prior to image reconstruction. As also mentioned, the minute distortions between the transmitter and receiver coils, caused by the vibrations, will result in an additional signal superposing the measured signals. A preprocessing unit PRP, which is connected upstream of the image reconstruction BIR, uses this additional signal as "modulation signal" for a standard for determination of the phase in the measuring setup. The measured signals are then rotated back by applying an inverse phase. After that correction in the preprocessing unit PRP, the image reconstruction follows according to known procedures.

ha the following, the principle of the invention is explained first for a simple transmitter- receiver coil arrangement and then for a combination of transmitter coil and gradiometer, i.e. a differential transformer; then, the treatment of periodic movements to reconstruct of the correction angle is discussed, as well as the analogous treatment using random movements.

1. Simple transmitter-receiver combination:

Referring to Fig.4, an excitation signal having a frequency ooc is generated by a local oscillator (LO, corresponding to signal generator SIG of Fig. 1). This local oscillator shall define the reference frame ReLO and Im^o for demodulation, so the excitation signal is purely real in this frame. The excitation signal is sent through the transmitter chain (analog-digital converter [not shown], power amplifier, transmitter coil) and will undergo a system-inherent phase shift θi. The axes for the field actually transmitted are therefore Reix and Imτχ.

We now consider the effect of a weakly conducting object, which corresponds to the object to be investigated. It can be assumed that the real part of the object's conductivity, i. e. σ, causes only a change of the transmitted signal by a purely imaginary contribution S. This assumption is valid to sufficient extent whenever the geometrical extension of the object, the frequency and the modulus of the complex conductivity K are low enough. This is the case for human body parts with diameters up to 50 cm, typical biological conductivities up to several S/m and frequencies up to several MHz. The method does not apply for metallic or strongly magnetic (e. g. ferromagnetic) objects with magnetic permeability μ » 1.

The spatial variation of the signal can be omitted for the discussion here; it is of course recovered from the spatial reconstruction after determination of the phase angles.

Furthermore, according to the invention, we introduce a periodic modulation of the main field. This periodic modulation is, for instance, generated by periodically moving the trans- mitter coils or the receiver coils by a small distance. This will cause a modulation signal m which is purely real in the frame (Reix , Imτχ) .

The modulation by the periodic movement may also affect the signal S. However, for the geometries of the measurement apparatus (Figs. 1 to 3) the modulation of the signal S will be much smaller than m, so S can be considered as unmodulated in the further analysis ("selective modulation condition"). This condition can always be met to very good extent. Even in the (rare) cases where it is not fulfilled sufficiently with the setup used, an additional stationary object made of a well-conducting material could be positioned near the coils so that its virtually purely real modulation signal dominates any imaginary contribution from the object.

Now both s and m further propagate through the receiver coils and suffer a phase shift φ. Finally the signals encounter a further shift G₂ caused by the following amplifiers and the digital-analog converter (DAC) before they can be processed. Thus, we have rotated signals s' and m'. When demodulating the rotated signals s' and m' with respect to the reference frame ReLo and ImLo we will obtain a strong spurious modulation signal iris in the imaginary axis, although in reality there is (nearly) no modulation of the original imaginary part according to the above modulation selectivity condition. As can be seen from Fig.4, the imaginary signal s_s is the projection of s' to the axis ImLo and does not any more reflect correctly the magnitude of the original imaginary signal S.

In order to cancel out this phase error, the signals should be back-rotated in the complex plane by the inverse angle, i.e. -(Θj+θ₂+φ) = -ψ.

With one approach, as described in WO 2008/011649 Al, a reference signal is used to determine the phase θi (e.g. from a separate coil). This leaves the determination of the remaining phase shift, and the combined angle (φ+θ₂) has to be determined separately, for instance empirically in a calibration step.

According to the invention the correction is done by determination of the phase angle of the modulation signal m, and applying the inverse angle to the measured signals. The actual signal is then the imaginary part of the measured signals. This is equivalent to rotating the signals back until the imaginary projection iris of the modulation signal vanishes in the LO frame. This rotation takes account of the complete phase angle -ψ, including the error phases φ, O₂ and the unknown reference phase Oj at once. The invention is based on the realisation that the modulation signal m can be separated from the actual signal S, using its different time-dependent behaviour according to its very nature: The actual signal is virtually stationary with respect to the modulation (selective modulation condition), which in turn will have typical frequencies much lower than the frequency of the AC magnetic field. For instance, if the modulation is introduced by means of vibrating a receiver (or transmitter) coil, the modulation signal has a specific frequency according to the vibration frequency and will produce specific sidebands. Another possibility of great interest is to use a random (stochastic) modulation, which is treated by statistical methods, taking a sample of measurements while the object is not changing. This enables to uniquely recognize the modulation signal and use its phase (the modulation signal phase) for determination of the phase error in the presence of the actual signal. The invention is operable at a single frequency of the AC magnetic field, and no additional reference channel is needed for measuring G₁.

2. TX - gradiometer combination:

With a gradiometer receiver coil the signal at the terminals of the gradiometer is the sum of two voltages V₁ and V₂ generated as electromotoric forces (EMFs) in the respective coil halves. These voltages are essentially equal in magnitude but opposed in phase so that they virtually cancel out mutually. However, as depicted in Fig. 5, there are unavoidable parasitic capacitances, resistances and inductances L, R, C as well as the impedances of the termination network R₃ and C₃ . Therefore, V| and V₂ are only filtered copies of the induced EMFs which consist of the directly induced Vi_ι0 and V_2ιo and the wanted signal voltages ηi , η₂, respectively. Even in the case that the two EMFs are equal, V| and V₂ will differ in magnitude and phase unless all parasitic elements are exactly equal.

Fig. 6 illustrates the signals obtained with a gradiometer coil and the relation between the actual signals and the different coordinate frames in the gradiometer. We denote the phase angles of the two filtered voltages as φi and <p2, and the corresponding magnitudes as a and b. For the further calculation the additional phase shift Q₂ caused by the following amplifiers and the DAC can be added to φi and φ₂ so as to give the resulting angles ψi and ψ₂ . Upon modulation of the system as explained above, additional signals mi and ma arise which are the modulations of the two rotated gradiometer voltages a and b. The sum modulation m corresponds to the signal m in Fig. 4. The original EMFs, which are parallel to the axes Imτx and Re_TX, are rotated by their respective angles so as to give the components with magnitudes a and b.

The resulting received carrier signal at frequency ωc can be written in complex form as:

where t denotes the time, and the quantities a, b, ψi and ψ2 where defined above.

According to the invention, the position X of the coils is modulated with a certain time pattern along the spatial direction X. Fig. 7 illustrates one possible configuration of the modulation. The gradiometer coil is well adjusted so the coil halves are exactly symmetric with respect to the axis of the TX coil when X=O. The curve | Bo(X) | depicts the profile of the magnetic flux density along the displacement axis X. In this position both gradiometer halves receive exactly the same magnetic flux. Shifting the gradiometer by Δx as indicated will increase the flux in half 1 and decrease it in half 2. Since the voltage change in one gradiometer half will be (approximately) the same as the one in the other half but with opposite sign. Assuming that the modulation of the magnetic flux is sinusoidal in time with a frequency ωv the modulated signal can be modeled as

Here k-i and k₂ are the modulation depths which may be slightly different due to a slight misalignment of the gradiometer. χ is the starting phase angle of the modulation signal. The negative sign in front of k₂ reflects the above mentioned fact that the sign of the voltage change in the gradiometer halves are opposite to each other.

This can also be seen Fig. 6, where the corresponding modulation signals are depicted as phasors rn-i and m₂. Note that phasors mi and m₂ are in phase because due to the counter- phase action of both gradiometer halves the negative sign in front of k₂ is compensated by the negative sign of b in eq. (M 2). Therefore both signals mi and m₂ sum up constructively to the overall modulation m.

Fig. 8 illustrates the phase angles ψi , ψ2 of the modulation signal components as well as the modulation signal phase angle Ψ of the sum modulation m. The projection of m into the imaginary axis IΠILO vanishes if (Tl is exactly orthogonal to IniLO. This happens if all signals are rotated by the angle Ψ , which angle can be determined using the geometric condition for cancellation of the projection:

which is equivalent to the condition:

The modulation signal phase angle can be found by taking the inverse tangent of the right- hand side of this equation. Thus, the invention provides analytical solutions which allow the direct calculation of the correction angle from the knowledge of the measured real and imaginary spectra.

Figs. 9 to 12 illustrate various possible implementations of how to introduce the modulation. Each of Figs. 9 to 12 shows one of the transmitter coils, SSj, and the corresponding receiver coil ESi which is realized movable as explained hereinafter; it will be clear to the person skilled in the art, that it could be the transmitter coil as well that is moved.

In Fig. 9 the receiver coil ESi is rotatable around an axis, according to a rotation vibration motion with a small amplitude angle. This can be done with a motor or actuator ACT generating a periodic movement. Preferably, the vibration frequency is known and can be tuned, in order to allow a phase-locked signal processing.

Fig. 10 shows another possibility for the modulation, namely by a translatory movement (translatory vibration); in other respects the same considerations as with Fig. 9 apply. This mode corresponds to Fig. 7, although the discussion given above for Fig. 5 is obviously easily adapted to the rotatory movement mode of Fig.9.

Not only the active introduction of a modulation may be suitable, but also a stochastic motion may be used for the generation of the modulation. This is illustrated in Fig. 11, where the receiver coils ESi suffer spatial variation of position with regard to the transmitter coil SSj, for instance, caused by vibrations from outside which cause slight distortions of the measurement setup of translatory and/ or rotatory kind. In order to enhance the mechanic response of the coil ESi, the support ELA may comprise elastic components.

Instead of a change of the geometry of coils as shown in Figs. 9 to 11, a modulation of the mutual inductance is also possible through a change in the field configuration, for instance by the provision of a conducting auxiliary body STK which is actuated by actuator ACT (see fig. 12). The modulation imparted by the auxiliary body STK may be deterministic or stochastic as already explained earlier. The auxiliary body STK may be positioned between the transmitter and receiver coils as shown, or may by positioned outside if its size and electromagnetic properties allow to do so. 3. Reconstruction of the correction angle from periodic movements:

In the following the method of reconstruction is derived for the case of a gradiometer coil. (The result can be easily taken to the case of a simple receiver coil simply by setting b=0 and omitting the angle Ψ2; Ψi is then equivalent to Ψ .) Using complex notation eq. (M 2) can be re-written as

Omitting the non-modulated carrier contributions we get upper side-band (USB) and lower side band (LSB) signals according to

The Fourier coefficients CUSB, CLSB of the two sidebands are therefore: H)

Considering that (the asterisk * denotes the complex conjugate )

we can write

Taking the ratio of real and imaginary part yields:

The right hand side is exactly the same as in eq. (M 3) so that we can conclude:

Thus, the angle is easily derived by taking the inverse tangent of the right-hand side of this equation. The Fourier coefficients CUSB and CLSB can be easily obtained from a FFT of the received signal as the complex peak values of the sidebands in case that the sampling frequency is an exact multiple of the modulation frequency.

Of course this formula is also applicable with any periodic (non-sinusoidal) modulation patterns, in this case CUSB and CL_SB are the Fourier coefficients at the fundamental modulation frequency.

4. Reconstruction of the correction angle from random movements:

This relationship is now derived for a simple coil system but it also applies to the gradiom- eter by replacing Ψ with Ψ

The imaginary part in the reference system is

lm_L0 = -Re'sinΨ + lm'cosΨ

(V 1)

We calculate now the variance of this imaginary part

The optimal correction angle Ψ is found if muo and RΘLO are maximally decorrelated, i. e. if all systematic variations which are not true channel noise disappear in the axis Imto. Denoting

the eq. (V 2) becomes:

We now use the following shorthand:

Maximum decorrelation means the minimization of the variance of the imaginary part with respect to the angle Ψ, which is equivalent to the minimum of y = Var(ImLo) with respect to a:

Excluding the trivial case that ψ = ±π/2 this condition is equivalent to:

Writing we obtain

The minimum is found by testing that the second derivative is positive (d^/da² > 0). This condition is fulfilled for the value of C in eq. (V 6) which obeys

The correction angle Ψ follows with:

Ψ = arc cot(C)

The statistical variables U, V, X are estimated from a series of measurements of sufficient statistical size, e. g. 20, so as to calculate a reasonable statistics. One very important condition is that the modulation signal, the carrier signal and the data acquisition window must be exactly phase-locked throughout the measurement of a statistic set. Therefore, all data acquisition circuits in the SYD and all signal generation circuits in SIG (fig. 1) must either be strictly synchronous by running from the same master clock or they must be triggered with a low trigger jitter from one master trigger source.

Claims

1. A method for correcting measured signals in magnetic induction tomography, wherein

- an object (OBJ) having inhomogeneous passive electrical properties is exposed to alternating magnetic fields by means of transmitter coils (SPl, SP2, SP3) located at specific transmitter positions,

- alternating voltage signals are picked up through receiver coils (ESl, ES2, ES3) located at specific receiver positions, wherein said voltage signals contain information relating to the electric conductivity and its spatial distribution within the object, and

- an image of the spatial distribution of the electrical properties within the object is reconstructed from the alternating voltage signals based on their amplitudes and phases, characterized in that at least one of the receiver positions is subject to a modulation with regard to the transmitter positions, causing an alternating modulation signal superimposing the corresponding picked-uρ alternating voltage signal, a phase of said modulation signal is determined, and the alternating voltage signal is corrected by applying a phase shift compensating said modulation signal phase.

2. Method according to claim 1, wherein the imaginary part of the corrected voltage signal is taken and the imaginary part thus obtained is used for the reconstruction of the image of spatial distribution.

3. Method according to claim 1 or 2, wherein the modulation includes a generally linear movement of the receiver position of a receiver coil, preferably along a direction within a plane of the receiver coil.

4. Method according to any one of the previous claims, wherein the modulation includes a periodic movement of the receiver position with respect to a corresponding transmitter position at a modulation frequency by an actuating means (ACT) of at least one of the associated transmitter and receiver coils, and the modulation signal phase is determined by means of the Fourier coefficients of the frequency sidebands relating to the modulation frequency.

5. Method according to claim 4, wherein the modulation signal phase is determined as a value equivalent to the inverse tangent of the expression

wherein C-US_ES and C_LSB denote the Fourier coefficients of the upper and lower sidebands, respectively.

6. Method according to any one of the previous claims, wherein the modulation includes a random movement of the receiver position with respect to a corresponding transmitter position caused by external vibrations, and the modulation signal phase is determined by taking a statistical sample of measurements of the corresponding voltage signals and evaluating measures of the statistic dispersion in the statistical sample of the imaginary and real parts of the voltage signals.

7. Method according to claim 6, wherein the statistical measures include the variances of the imaginary and real parts of the voltage signals as well as the covariance of said imaginary and real parts.

8. Method according to claim 7, wherein the modulation signal phase is determined as a value equivalent to the inverse cotangent of C, where C is a solution of the equation

V - U - X ( C - VC) = O, wherein V and U represent the variances of the real and imaginary parts of the voltage signals, respectively, and X the covariance of said imaginary and real parts.

9. Method according to any one of the previous claims, wherein the receiver and transmitter coils are provided outside the object.

10. Magnetic induction tomography system comprising

- transmitter coils (SPl, SP2, SP3) located at specific transmitter positions, said transmitter coils being configured to expose an object (OBJ) having inhomogeneous passive electrical properties to alternating magnetic fields,

- receiver coils (ESl, ES2, ES3) located at specific receiver positions, said receiver coils being configured to pick up alternating voltage signals, which contain information relating to the electric conductivity and its spatial distribution within the object, and - an image reconstruction means (BIR) for reconstructing an image of the spatial distribution of the electrical properties within the object from the alternating voltage signals based on their amplitudes and phases, characterized by a preprocessing means (PRP) which is configured to determine a phase shift for correcting measured signals according to any one of claims 1 to 9.