Patent ID: 12253578

MODES FOR CARRYING OUT THE INVENTION

A description will now be given of an embodiment of the present invention referring to drawings.

FIG.1is a perspective view showing voxels V and magnetic sensors MS according to an embodiment of the present invention.FIG.2is a functional block diagram showing the configuration of a signal vector derivation apparatus1according to the embodiment of the present invention.

Referring toFIG.1, signal sources S1and S2output signals. The signals are each represented by a vector “m” having a predetermined direction. The vector “m” is, for example, a magnetic dipole moment. It is noted that the number of signal sources is, for example, two, but may be three or more as long as it is less than the number of magnetic sensors MS. Note here that signals output from the signal sources have their respective frequencies or phases different from each other.

The positions within a space at which the signal sources S1and S2exist are also represented by voxels V (e.g. 10×10×10=1000 voxels). The signal sources S1and S2are positioned within their respective different voxels V. It is noted that the 1000 voxels V are denoted as V1to V1000.

Multiple (e.g., 64 in 8 rows and 8 columns) magnetic sensors MS are arranged to receive signals (e.g., magnetic dipole moments) and measure X, Y, and Z triaxial components Bx, By, and Bzorthogonal to each other. It is noted that the 64 magnetic sensors MS are denoted as MS1to MS64.

Here given a directional vector “r” from a signal source (magnetic dipole) to a magnetic sensor MS, the magnetic flux density B (function of the vector “r”) measured by the magnetic sensor MS is expressed by Biot-Savart's law as in formula (1), where μ0is the magnetic constant. The vector “r” can also represent the positional relationship between each of the voxels V (V1to V1000) and each of the magnetic sensors MS (MS1to MS64).

B⁡(r→)=μ04⁢π⁢{3⁢(m→·r→)❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢r→-m→❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3}(1)

From formula (1), Bxis expressed as in formula (2) below, where rx, ry, and rzare x-, y-, and z-components of the vector “r”, respectively and mx, my, and mzare x-, y-, and z-components of the vector “m”, respectively.

Bx=μ04⁢π⁢{(3⁢rx2❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5-1❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3)⁢mx+3⁢rx⁢ry❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢my+3⁢rz⁢rx❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢mz}(2)Bx=μ04⁢π⁢(vx⁢1⁢mx+vx⁢2⁢my+vx⁢3⁢mz)(2′)

Here, when the coefficients of mx, my, and mzin formula (2) are replaced, respectively, with vx1, vx2, and vx3, the formula (2) is expressed as in formula (2′). The measurement result Bxfrom each magnetic sensor MS is then proportional to the sum (vx1mx+vx2my+vx3mz) of the X, Y, and Z triaxial components mx, my, and mzof the vector “m” multiplied, respectively, by vx1, vx2, and vx3(first coefficients).

From formula (1), Byis expressed as in formula (3) below.

By=μ04⁢π⁢{3⁢rx⁢ry❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢mx+(3⁢ry2❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5-1❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3)⁢my+3⁢ry⁢rz❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢mz}(3)By=μ04⁢π⁢(vy⁢1⁢mx+vy⁢2⁢my+vy⁢3⁢mz)(3′)

Here, when the coefficients of mx, my, and mzin formula (3) are replaced, respectively, with vy1, vy2, and vy3, the formula (3) is expressed as in formula (3′). The measurement result Byfrom each magnetic sensor MS is then proportional to the sum (vy1mx+vy2my+vy3mz) of the X, Y, and Z triaxial components mx, my, and mzof the vector “m” multiplied, respectively, by vy1, vy2, and vy3(first coefficients).

From formula (1), Bzis expressed as in formula (4) below.

Bz=μ04⁢π⁢{3⁢rz⁢rx❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢mx+3⁢ry⁢rz❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5⁢my+(3⁢rz2❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"5-1❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3)⁢mz}(4)Bz=μ04⁢π⁢(vz⁢1⁢mx+vz⁢2⁢my+vz⁢3⁢mz)(4′)

Here, when the coefficients of mx, my, and mzin formula (4) are replaced, respectively, with vz1, vz2, and vz3, the formula (4) is expressed as in formula (4′). The measurement result Bzfrom each magnetic sensor MS is then proportional to the sum (vz1mx+vz2my+vz3mz) of the X, Y, and Z triaxial components mx, my, and mzof the vector “m” multiplied, respectively, by vz1, vz2, and vz3(first coefficients).

It is noted that referring to formulae (2) to (4) and (2′) to (4′), vx1, vx2, vx3, vy1, vy2, vy3, vz1, vz2, and vz3(first coefficients) are defined based on the vector “r”.

Referring toFIG.2, the signal vector derivation apparatus1according to the embodiment of the present invention includes a relative position recording section11, a first coefficient deriving section12, a transfer function deriving section13, a noise eigenvector deriving section14, a spectrum deriving section16, a direction deriving section18, and a position deriving section19.

The signal vector derivation apparatus1is arranged to receive measurement results from the multiple sensors MS1to MS64and derive the direction of the vector “m”.

The relative position recording section11is arranged to record a vector “r” as a relative position between each of the 1000 voxels V and each of the magnetic sensors MS (MS1to MS64).

The first coefficient deriving section12is arranged to read the vector “r” out of the relative position recording section11and derive first coefficients vx1, vx2, vx3, vy1, vy2, vy3, vz1, vz2, and vz3(see formulae (2) to (4) and (2′) to (4′)).

For example, the first coefficient vx1is expressed as in formula (5) below.

The vector “r” is determined by the position of each voxel V and the position of each magnetic sensor MS and thereby has 1000×64 different candidate values. Accordingly, the first coefficient vx1also has 1000×64 different candidate values. In formula (5), the 1st row denotes vx1for the magnetic sensor MS1, the 2nd row denotes vx1for the magnetic sensor MS2, . . . , and the 64th row denotes vx1for the magnetic sensor MS64. Further, in formula (5), the 1st column denotes vx1for the voxel V1, the 2nd column denotes vx1for the voxel V2, . . . , and the 1000th column denotes vx1for the voxel V1000. For example, the element vx1(1, 1000) of the 1 st row and the 1000th column in formula (5) denotes vx1for the magnetic sensor MS1and the voxel V1000. That is, vx1(1, 1000) can be obtained by substituting the vector “r” as a directional vector from the voxel V1000to the magnetic sensor MS1into the coefficient of mxin formula (2).

The other first coefficients vx2, vx3, vy1, vy2, vy3, vz1, vz2, and vz3also each have 1000×64 different candidate values.

The first coefficient vx1is normalized as in formula (6) below for subsequent processing, though may be used without normalization.
vx1(h,n)/(vx1(1,n)2+vx1(2,n)2+ . . . +vx1(64,n)2)1/2(6)

Here, “h” and “n” represent row and column, respectively. That is, the first coefficient vx1of the h-th row and the n-th column is divided by the square root of the sum of the squares of the first coefficients vx1of the 1st, 2nd, . . . , 64th rows and the n-th column to be a new first coefficient vx1of the h-th row and the n-th column.

The other first coefficients vx2, vx3, vy1, vy2, vy3, vz1, vz2, and vz3are also normalized.

The first coefficient deriving section12is arranged to output the thus normalized first coefficients.

The noise eigenvector deriving section14is arranged to obtain eigenvectors of a noise subspace from the measurement results Bx, By, and Bzfrom each magnetic sensor MS according to the MUSIC method.

X(t)xis first obtained from the measurement result Bxfrom each magnetic sensor MS as in formula (7) below, where, “t” is the time of measurement and T represents transposition.

X⁡(t)x=(Bx⁢1(t⁢1)Bx⁢2(t⁢1)⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢Bx⁢64(t⁢1)Bx⁢1(t⁢2)⁢Bx⁢2(t⁢2)⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢Bx⁢64(t⁢2)‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐Bx⁢1(t⁢N)Bx⁢2(t⁢N)⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢‐⁢Bx⁢64(t⁢N))T(7)

X(t)xis a transposed matrix describing Bxmeasured at time t1 in the 1st row, Bxmeasured at time t2 in the 2nd row, . . . , Bxmeasured at time tN in the Nth row and Bxmeasured by the magnetic sensor M1in the 1st column, Bxmeasured by the magnetic sensor M2in the 2nd column, . . . , Bxmeasured by the magnetic sensor M64in the 64th column.

X(t)xis used to obtain a correlation matrix as in formula (8) below.
E{X(t)xX(t)xT}  (8)

Here, E represents the ensemble average. A matrix of 64 rows and 64 columns is obtained from formula (8). Eigenvalues and eigenvectors are then obtained of the correlation matrix obtained from formula (8). Among the thus obtained eigenvalues, the ones for the number of the signal sources (2) are large, while the remaining (64−2=62) eigenvalues are small. Eigenvectors ex of the noise subspace are then obtained correspondingly to the smaller eigenvalues. Each of the eigenvectors ex of the noise subspace is a vector of 64 rows and 1 column. 62 eigenvectors ex of the noise subspace exist correspondingly to the smaller eigenvalues.

It is noted that eigenvectors ey of the noise subspace can also be obtained similarly. First, Bxin formula (7) is replaced with Byand X(t)xin formulae (7) and (8) is replaced with X(t)y, and formula (8) is used to obtain a correlation matrix. Then, in a similar manner as above, eigenvectors ey of the noise subspace are obtained correspondingly to the smaller eigenvalues. Each of the eigenvectors ey of the noise subspace is a vector of 64 rows and 1 column. 62 eigenvectors ey of the noise subspace exist correspondingly to the smaller eigenvalues.

Eigenvectors ez of the noise subspace can also be obtained similarly. First, Bxin formula (7) is replaced with Bzand X(t)xin formulae (7) and (8) is replaced with X(t)z, and formula (8) is used to obtain a correlation matrix. Then, in a similar manner as above, eigenvectors ez of the noise subspace are obtained correspondingly to the smaller eigenvalues. Each of the eigenvectors ez of the noise subspace is a vector of 64 rows and 1 column. 62 eigenvectors ez of the noise subspace exist correspondingly to the smaller eigenvalues.

The transfer function deriving section13is arranged to derive transfer functions vx, vy, and vzas in formulae (9), (10), and (11) below. The sum of the first coefficients vx1, vx2, and vx3multiplied, respectively, by the second coefficients ax, bx, and cxis derived (see formula (9)). The derivation result is the transfer function vx. The sum of the first coefficients vy1, vy2, and vy3multiplied, respectively, by the second coefficients ay, by, and cyis derived (see formula (10)). The derivation result is the transfer function vy. The sum of the first coefficients vz1, vz2, and vz3multiplied, respectively, bythe second coefficients az, bz, and czis derived (see formula (11)). The derivation result is the transfer function vz.
vx=axvx1+bxvx2+cxvx3(9)
vy=ayvy1+byvy2+cyvy3(10)
vz=azvz1+bzvz2+czvz3(11)

It is noted that the transfer functions vx, vy, and vzare the transfer functions in the MUSIC method.

Also, the second coefficients may each be a value other than zero. For example, ax=bx=cx=1 (i.e., vx=vx1+vx2+vx3) may be set, ay=by=1 and cy=−1 (i.e., vy=vy1+vy2−vy3) may be set, or az=1, bz=−1, and cz=1 (i.e., vz=vz1−vz2+vz3) may be set.

Note here that any one or two of the second coefficients may be zero. For example, ax=1 and bx=cx=0 (i.e., vx=vx1) may be set or ax=bx=1 and cx=0 (i.e., vx=vx1+vx2) may be set.

It is here assumed that (ak, bk, ck) (where k=x, y, z) can have the following 13 different candidate combinations: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, −1, 0), (0, 1, 1), (0, 1, −1), (1, 0, 1), (−1, 0, 1), (1, 1, 1), (−1, 1, 1), (1, −1, 1), and (1, 1, −1). vkconsists of vk1, vk2, . . . , and vk13, accordingly.

For example, if (ax, bx, cx)=(1, 0, 0), then vx=vx1. If (ax, bx, cx)=(1, 1, 0), then vx=vx4=vx1+vx2. If (ax, bx, cx)=(1, 1, −1), then vx=vx13=vx1+vx2−vx3.

For example, if (ay, by, cy)=(1, 0, 0), then vy=vy1. If (ay, by, cy)=(1, 1, 0), then vy=vy4=vy1+vy2. If (ay, by, cy)=(1, 1, −1), then vy=vy13=vy1+vy2−vy3.

For example, if (az, bz, cz)=(1, 0, 0), then vz=vz1. If (az, bz, cz)=(1, 1, 0), then vz=vz4=vz1+vz2. If (az, bz, cz)=(1, 1, −1), then vz=vz13=vz1+vz2−vz3.

The spectrum deriving section16is arranged to derive a spectrum having local maximum values within the voxels V in which the signal sources S1and S2exist. Such a spectrum is obtained according to the MUSIC method. The spectrum has two local maximum values correspondingly to the number of signal sources. It is noted that if the number of signal sources is three or more, the spectrum also has three or more local maximum values accordingly.

Spectrums are derived by the spectrum deriving section16based on (the eigenvectors ex, ey, and ez of the noise subspace obtained from) the measurement results Bx, By, and Bzfrom each magnetic sensor MS and the sum of the first coefficients multiplied, respectively, bythe second coefficients (i.e. transfer functions vx, vy, and vz) (formulae (9), (10), (11)). The spectrum deriving section16is arranged to derive spectrums based on the transfer functions vx, vy, and vzoutput from the transfer function deriving section13and the eigenvectors ex, ey, and ez of the noise subspace output from the noise eigenvector deriving section14.

The spectrum deriving section16is arranged to derive the spectrum Px1as follows.(1) There are 62 eigenvectors ex (of 64 rows and 1 column) in the noise subspace, and these vectors ex are arranged in 62 columns to be a matrix of 64 rows and 62 columns.(2) the matrix ex is transposed and bywhich the transfer function vx1(a matrix of 64 rows and 1000 columns) is multiplied. That is, exTvx1is obtained. This is a matrix of 62 rows and 1000 columns.(3) Each element of the matrix obtained in (2) is squared.(4) The elements of the matrix obtained in (3) are summed for each column and arranged in a row to obtain a matrix of 1 row and 1000 columns. For example, (1, Q) element+(2, Q) element+ . . . +(62, Q) element of the matrix obtained in (3) results in the (1, Q) element of the matrix of 1 row and 1000 columns obtained in (4) (where Q is an integer of 1 to 1000).(5) Each element of the matrix obtained in (4) is inverted to obtain a spectrum Px1(a matrix of 1 row and 1000 columns).

It is noted that the columns of the spectrum Px1corresponds, respectively, to the voxels V1to V1000. The same applies to the other spectrums.

The spectrum deriving section16is also arranged to derive the spectrums Px2, Px3, . . . , and Px13. The spectrums Px2, Px3, . . . , and Px13can be derived by replacing the transfer function vx1in (2) above, respectively, with vx2, vx3, . . . , and vx13.

The spectrum deriving section16is arranged to derive the spectrums Py1, Py2, Py3, . . . , and Py13. The spectrums Py1, Py2, Py3, . . . , and Py13can be derived by replacing the eigenvector ex of the noise subspace in (1) above with ey and replacing the transfer function vx1in (2) above, respectively, with vy1, vy2, vy3, . . . , and vy13.

The spectrum deriving section16is arranged to derive the spectrums Pz1, Pz2, Pz3, . . . , and Pz13. The spectrums Pz1, Pz2, Pz3, . . . , and Pz13can be derived by replacing the eigenvector ex of the noise subspace in (1) above with ez and replacing the transfer function vx1in (2) above, respectively, with vz1, vz2, vz3, . . . , and vz13.

The position deriving section19is arranged to derive the positions of the voxels V in which the signal sources S1and S2exist based on the spectrums Px1, Px2, Px3, . . . , Px13, Py1, Py2, Py3, . . . , Py13, Pz1, Pz2, Pz3, . . . , Pz13.

The spectrums output from the spectrum deriving section16are expressed as in formula (12) below.

P⁢Matrix=(Px⁢1⋮Px⁢1⁢3Py⁢1⋮Py⁢1⁢3Pz⁢1⋮Pz⁢1⁢3)(12)

The maximum values P of each spectrum within the respective voxels (i.e. the maximum values in each column of formula (12)) are obtained (see formula (13)).
P=max(PMatrix)  (13)

Since the number of columns (corresponding to voxels) having local maximum values in P corresponds to the number of signal sources (2), the signal sources S1and S2exist within the voxels corresponding to the columns. A method of detecting columns with local maximum values will hereinafter be described.

FIG.3is an example graph of the maximum values P. InFIG.3, the vertical axis represents the spectrum value, while the horizontal axis represents the voxels (V1to V1000).

Referring toFIG.3, it is provided that (the spectrum with) the maximum value P has a value of SP1 (local maximum value) within the voxel V750and (the spectrum with) the maximum value P has a value of SP2 (local maximum value) within the voxel V250. Note here that SP1 is larger than SP2.

The position deriving section19is arranged to obtain the weighted center of the voxels having the maximum values P within a predetermined range from the maximum SP1 of the maximum values P (e.g. the maximum values P of 0.95SP1 or more), while increasing the predetermined range (e.g. the predetermined range is extended by 0.05SP1, such as the maximum values P of 0.95SP1 or more→0.90SP1 or more→0.85SP1 or more→ . . . ), until the number of times of the weighted center changing over a predetermined amount added by 1 reaches the number of the signal sources (2).

The weighted center of the voxels in the vicinity of the maximum SP1 of the maximum values P is around the voxel V750. However, when the predetermined range extends from the maximum SP1 to include SP2, the weighted center of the voxels shifts lower from voxel V750. The weighted center then changes over a predetermined amount and the number of times (1) added by 1 reaches the number of the signal sources (2), where obtaining the weighted center of the voxels is completed.

Next, the voxels for which the weighted center is thus obtained is clustered into the number of the signal sources (2). For example, Kmeans clustering, which is unsupervised machine learning, is performed for labeling by the number of the signal sources.

Finally, the positions of the ones of the clustered voxels with the maximum spectrum are determined as the positions of the voxels in which the respective signal sources exist.

It is noted that the position deriving section19may further reduce the size of each voxel, based on the positions of the voxels in which the thus derived signal sources exist, to derive the positions of the voxels in which the respective signal sources exist. It is thus possible to calculate the positions of the voxels in which the signal sources exist with high accuracy and speed.

The direction deriving section18is arranged to receive Pkj(where k=x, y, z and j=1, 2, 3, . . . ) corresponding to the signal sources S1and S2(i.e. having local maximum values in P) from the position deriving section19. The direction deriving section18is further arranged to derive the direction of the vector “m” based on the second coefficients used to obtain Pkjcorresponding to the signal sources S1and S2.

For example, it is assumed that Px13(Py13or Pz13) of the 750th column (voxel V750) and Px4(Py4or Pz4) of the 250th column (voxel V250) are provided as spectrums corresponding to the respective signal sources S1and S2from the position deriving section19to the direction deriving section18.

The direction deriving section18then derives that the second coefficient (ak, bk, ck) (where k=x, y, z) to be (1, 1, −1), which is used to obtain Px13(Py13or Pz13) as the direction of the vector “m” at the signal source S1within the voxel V750. Accordingly, the direction deriving section18derives the direction of the vector “m” at the signal source S1within the voxel V750to be parallel to the vector (1, 1, −1). Note here that the vector (1, 1, −1) is a vector having an X component of 1, a Y component of 1, and a Z component of −1.

The direction deriving section18further derives that the second coefficient (ak, bk, ck) (where k=x, y, z) to be (1, 1, 0), which is used to obtain Px4(Py4or Pz4) as the direction of the vector “m” at the signal source S2within the voxel V250. Accordingly, the direction deriving section18derives the direction of the vector “m” at the signal source S2within the voxel V250to be parallel to the vector (1, 1, 0). Note here that the vector (1, 1, 0) is a vector having an X component of 1, a Y component of 1, and a Z component of 0.

Next will be described an operation according to the embodiment of the present invention.

The first coefficient deriving section12reads the vector “r” out of the relative position recording section11and derives first coefficients vx1, vx2, vx3, vy1, vy2, vy3, vz1, vz2, and vz3(see formulae (2) to (4) and (2′) to (4′)).

It is noted that the first coefficients, which have 1000×64 different candidate values (see formula (5)), are normalized (see formula (6)) and provided to the transfer function deriving section13.

The transfer function deriving section13derives transfer functions vx, vy, and vzbased on the first coefficients and the second coefficients ax, bx, cx, ay, by, cy, az, bz, cz(see formulae (9), (10), and (11)).

The noise eigenvector deriving section14derives eigenvectors ex, ey, and ez of a noise subspace from the measurement results Bx, By, and Bzfrom each magnetic sensor MS according to the MUSIC method.

The spectrum deriving section16derives spectrums Px1, Px2, Px3, . . . , Px13, Py1, Py2, Py3, . . . , Py13, Pz1, Pz2, Pz3, . . . , Pz13based on the transfer functions vx, vy, and vzand the eigenvectors ex, ey, and ez of the noise subspace (see formula (12)).

The position deriving section19obtains the maximum values P of each spectrum within the respective voxels (i.e. the maximum values in each column of formula (12)) (see formula (13) andFIG.3). The voxels250,750in which the signal sources S1and S2exist are derived based on the maximum values P.

The direction deriving section18derives the direction of the vector “m” based on the second coefficients used to obtain Pkjcorresponding to the signal sources S1and S2.

The embodiment of the present invention improves the accuracy of measurement of a signal such as a magnetic field.

For example, if the transfer function vkonly consists of vk1, vk2, and vk3, the direction of the vector “m” can be measured only if in parallel with the X, Y, or Z direction. The direction of the vector “m”, if in parallel with directions other than above (e.g. vector (1, 1, 0) (i.e. vector having an X component of 1, a Y component of 1, and a Z component of 0)), cannot be measured.

However, in accordance with the embodiment of the present invention, since the transfer function vkconsists of many types including vk1, vk2, vk3, . . . , and vk13, the direction of the vector “m” can be measured even if not in parallel with the X, Y, and Z directions.

It is noted that the signal vector is not limited to a magnetic dipole moment, though have been descried as a magnetic dipole moment in the embodiment of the present invention. The signal vector may be, for example, an electric dipole moment (vector “p”).

The magnetic flux density B (function of the vector “r”) measured by the magnetic sensor MS is expressed as in formula (14).

B⁡(r→)=μ04⁢π⁢(p→×r→)❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3(14)

From formula (14), Bxis expressed as in formula (15) below, where px, py, and pzare x-, y-, and z-components of the vector “p”, respectively.

Bx=μ04⁢π⁢(rz❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3⁢py-ry❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3⁢pz)(15)Bx=μ04⁢π⁢(vx⁢1⁢px+vx⁢2⁢py+vx⁢3⁢pz)(15′)

Here, when the coefficients of px, py, and pzin formula (15) are replaced, respectively, with vx1, vx2, and vx3, the formula (15) is expressed as in formula (15′). The measurement result Bxfrom each magnetic sensor MS is then proportional to the sum (vx1px+vx2py+vx3pz) of the X, Y, and Z triaxial components px, py, and pzof the vector “p” multiplied, respectively, byvx1, vx2, and vx3(first coefficients). Note here that the component (px) of the vector in the same direction (X direction) as that of the component of the measurement result Bxis zero, and the first coefficient vx1multiplying it is 1.

From formula (14), Byis expressed as in formula (16) below.

By=μ04⁢π⁢(rx❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3⁢pz-rz❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3⁢px)(16)By=μ04⁢π⁢(vy⁢1⁢px+vy⁢2⁢py+vy⁢3⁢pz)(16′)

Here, when the coefficients of px, py, and pzin formula (16) are replaced, respectively, with vy1, vy2, and vy3, the formula (16) is expressed as in formula (16′). The measurement result Byfrom each magnetic sensor MS is then proportional to the sum (vy1px+vy2py+vy3pz) of the X, Y, and Z triaxial components px, py, and pzof the vector “p” multiplied, respectively, byvy1, vy2, and vy3(first coefficients). Note here that the component (py) of the vector in the same direction (Y direction) as that of the component of the measurement result Byis zero, and the first coefficient vy2multiplying it is 1.

From formula (14), Bzis expressed as in formula (17) below.

Bz=μ04⁢π⁢(ry❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3⁢px-rx❘"\[LeftBracketingBar]"r→❘"\[RightBracketingBar]"3⁢py)(17)Bz=μ04⁢π⁢(vz⁢1⁢px+vz⁢2⁢py+vz⁢3⁢pz)(17′)

Here, when the coefficients of px, py, and pzin formula (17) are replaced, respectively, with vz1, vz2, and vz3, the formula (17) is expressed as in formula (17′). The measurement result Bzfrom each magnetic sensor MS is then proportional to the sum (vz1px+vz2py+vz3pz) of the X, Y, and Z triaxial components px, py, and pzof the vector “p” multiplied, respectively, byvz1, vz2, and vz3(first coefficients). Note here that the component (pz) of the vector in the same direction (Z direction) as that of the component of the measurement result Bzis zero, and the first coefficient vz3multiplying it is 1.

The configuration and operation of the signal vector derivation apparatus1is the same as those when the signal vector is a magnetic dipole moment (vector “m”) and will not be described.

The above-described embodiment may also be implemented as follows. A computer including a CPU, a hard disk, and a medium (USB memory, CD-ROM, or the like) reading device is caused to read a medium with a program recorded thereon that achieves the above-described components (e.g. the relative position recording section11, the first coefficient deriving section12, the transfer function deriving section13, the noise eigenvector deriving section14, the spectrum deriving section16, the direction deriving section18, and the position deriving section19) and install the program in the hard disk. The above-described features can also be achieved in this manner.

DESCRIPTION OF REFERENCE NUMERAL

1Signal Vector Derivation Apparatus11Relative Position Recording Section12First Coefficient Deriving Section13Transfer Function Deriving Section14Noise Eigenvector Deriving Section16Spectrum Deriving Section18Direction Deriving Section19Position Deriving SectionMS Magnetic SensorV VoxelB Magnetic Flux Densityvx1, vx2, vx3, vy1, vy2, vy3, vz1, vz2, vz3First Coefficientsak, bk, ckSecond Coefficientsvk1, vk2, . . . , vk13Transfer FunctionS1, S2Signal Sourcem Vector (Magnetic Dipole Moment)ex, ey, ez Eigenvectors of the Noise SubspaceP Maximum Value