Devices and methods for determining a magnetic field

Abstract

A method and device for determining values of a magnetic field component of a magnetic vector field. A method for determining values of a magnetic field component of a magnetic vector field, comprising: determining first distribution data comprising values of the magnetic field component, for a first predetermined area defined along a predetermined surface; determining second distribution data comprising second values of the component of the magnetic field for a second predetermined area defined along a second predetermined surface, wherein the first and the second predetermined surfaces are parallel;
wherein determining second distribution data comprises manipulation of the first distribution data based on making use of intrinsic physical properties of the magnetic field; and associated device.

Claims

1. A method of inspecting the magnetic properties of a magnetic object to determine its quality, comprising: placing a first predetermined surface of the magnetic object in a sensing area of a magnetic camera device comprising a plurality of magnetic field sensors, arranged as a two-dimensional (2D) grid on an xy plane at a position Z.sub.0 along a z-axis perpendicular to the xy plane, and adapted to receive a magnetic field component from the magnetic object, wherein the magnetic object is placed in the sensing area at a position z.sub.1 Z.sub.0; measuring the magnetic object using the magnetic camera device, to obtain a 2D distribution of a magnetic field on the 2D grid, the magnetic object under inspection acting as a source of the magnetic field; obtaining a first magnetic field component from the magnetic object at Z.sub.0 by determining a first distribution data comprising values of said first magnetic field component, for the sensing area, wherein determining first distribution data comprises measuring values of said component on the 2D grid by means of the magnetic camera device to obtain the 2D distribution of the magnetic field; obtaining a second magnetic field component from the magnetic object by determining second distribution data comprising values of said second magnetic field component, for a second area of a second plane parallel to the sensing area and at a position z.sub.2 defined as a distance z from the sensing area, said determining second distribution data comprising manipulation of said first distribution data based on intrinsic physical properties of at least one of: the magnetic field vector, and said magnetic field component, from the magnetic object under inspection; said determining second distribution data further comprising computer-implemented steps of: performing a Fourier transformation of said first distribution data resulting in Fourier transformed data; followed by performing data manipulation comprising multiplying said Fourier transformed data obtained at z.sub.0 by e.sup.|k|z, resulting in manipulated Fourier transformed data; followed by performing an inverse Fourier transformation of said manipulated Fourier transformed data; and determining a magnetic field distribution in a three-dimensional (3D) volume, wherein: |k| in said e.sup.|k|z is a magnitude of a spatial frequency vector determined by first and second directions defining the second plane of the sensing area in a Fourier domain; and determining the magnetic quality of the magnetic object under inspection based on the determined second distribution data.

2. The method according to claim 1, wherein said first magnetic field component comprises a magnitude of projection of a magnetic field vector on an axis or on a surface, or a magnitude of the magnetic field vector.

3. The method according to claim 1, wherein determining first distribution data further comprises modeling said first magnetic field component based on a predetermined model and/or predetermined input parameters.

4. The method according to claim 1, wherein said first distribution data of said first magnetic field component comprises non-zero values corresponding to a location at an outer border of said first predetermined surface.

5. The method according to claim 1, further comprising: generating additional distribution data of said magnetic field component, said additional distribution data comprising expected values for said magnetic field component in an extension area, said extension area adjacent to said sensing area and along said first predetermined surface; and determining second distribution data for an extended set of distribution data, said extended set of distribution data comprising said first distribution data and said additional distribution data.

6. The method according to claim 5, wherein said expected values of said magnetic field component for the additional distribution data are set to be monotonously decreasing to zero in said extension area when moving from an outer boundary of said first predetermined surface, away from said first predetermined area, towards an outer border of said extension area.

7. A device for inspecting a magnetic object, comprising: a magnetic camera device comprising a plurality of magnetic field sensors arranged as a two-dimensional (2D) grid on an xy plane at a position Z.sub.0 along a z-axis perpendicular to the xy plane, and adapted to measure a first distribution data comprising values of a magnetic field component of the magnetic object, placed in a sensing area of the magnetic camera device at a position z.sub.1Z.sub.0, wherein said magnetic camera device is configured to measure the magnetic object to obtain a 2D distribution of a magnetic field on the 2D grid, the magnetic object under inspection acting as a source of the magnetic field; means for obtaining a second magnetic field component from the magnetic object by determining a second distribution data comprising values of said second magnetic field component, for a second area of a second plane parallel to the sensing area and at a position z.sub.2 defined as a distance z from the sensing area, wherein determining second distribution data comprises manipulation of said first distribution data based on intrinsic physical properties of at least one of: the magnetic vector field, and said magnetic field component, from the magnetic object under inspection, and wherein, for determining the second distribution data, said means for obtaining the second magnetic field component is configured to: perform a Fourier transformation of said first distribution data resulting in Fourier transformed data; followed by performing data manipulation comprising multiplying said Fourier transformed data obtained at z.sub.0 by e.sup.|k|z, resulting in manipulated Fourier transformed data; followed by performing an inverse Fourier transformation of said manipulated Fourier transformed data; and determining a magnetic field distribution in a three-dimensional (3D) volume, wherein: |k| in said e.sup.|k|z is a magnitude of a spatial frequency vector determined by first and second directions defining the second plane of the sensing area in a Fourier domain; and the determined second distribution data is applied to a determination of the magnetic quality of the magnetic object under inspection.

8. The method of claim 1, wherein the method is a method of inspecting a permanent magnet or magnet assembly, and the method further comprises outputting a result of the permanent magnet or magnet assembly inspection.

9. The method of claim 8, wherein the outputting step relies on the first and second distribution data.

10. The method of claim 1, wherein the spacing of the 2D grid is less than 1 mm in orthogonal directions parallel the xy plane.

11. The method of claim 1, wherein the spacing of the 2D grid is less than 0.2 mm in orthogonal directions parallel the xy plane.

12. The method of claim 1, wherein z is greater than zero.

13. The method of claim 1, wherein z is less than zero.

14. The method of claim 1, wherein the computer-implemented step of determining second distribution data further comprise defining the second area of the second plane.

15. The method of claim 14, wherein the computer-implemented step of determining second distribution data further comprise defining the second area of the second plane at a location free from magnetic field sources.

16. The method of claim 1, wherein the computer-implemented step of determining second distribution data further comprise defining a magnitude of the distance z.

17. The method of claim 1, wherein said measuring the magnetic object comprises scanning the magnetic object in one direction using the magnetic camera device to obtain the 2D distribution of the magnetic field on the 2D grid.

18. The method of claim 17, wherein the one direction for said scanning is one of: in a Cartesian direction, in an axis in a cylindrical coordinate system, and in a spherical coordinate system.

19. The method of claim 17, wherein the magnetic object includes a magnet, and wherein said scanning the magnetic object comprises scanning the magnetic object including the magnet in the one direction using the magnetic camera device.

20. The method of claim 19, wherein said determining the magnetic quality of the magnetic object comprises determining the magnetic quality of the magnetic object including the magnet.

21. The method of claim 17, wherein the magnetic object is a magnet, and wherein said scanning the magnetic object comprises scanning the magnet in the one direction using the magnetic camera device.

22. The method of claim 21, wherein said determining the magnetic quality of the magnetic object comprises determining the magnetic quality of the magnet.

23. The method of claim 1, wherein z.sub.1 is less than Z.sub.0.

24. The method of claim 1, wherein the magnetic field distribution includes a z-component, the z-component being out of plane with respect to a plane defined by an x-axis and a y-axis in the 3D volume.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The disclosure will be further elucidated by means of the following description and the appended figures.

(2) FIGS. 1 to 8 illustrate embodiments of the present invention. FIGS. 3 to 8 illustrate examples of how additional distribution data for the component of the magnetic field vector can be generated, according to embodiments of the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

(3) The present disclosure will be described with respect to particular embodiments and with reference to certain drawings but the disclosure is not limited thereto but only by the claims. The drawings described are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. The dimensions and the relative dimensions do not necessarily correspond to actual reductions to practice of the disclosure.

(4) Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequential or chronological order. The terms are interchangeable under appropriate circumstances and the embodiments of the disclosure can operate in other sequences than described or illustrated herein.

(5) Moreover, the terms top, bottom, over, under and the like in the description and the claims are used for descriptive purposes and not necessarily for describing relative positions. The terms so used are interchangeable under appropriate circumstances and the embodiments of the disclosure described herein can operate in other orientations than described or illustrated herein.

(6) Furthermore, the various embodiments, although referred to as preferred are to be construed as exemplary manners in which the disclosure may be implemented rather than as limiting the scope of the disclosure.

(7) A method is described for determining values of a magnetic field component of a magnetic vector field, comprising: determining first distribution data comprising values of the magnetic field component, for a first predetermined area defined along a first predetermined surface; determining second distribution data comprising second values of the component of the magnetic field for a second predetermined area defined along a second predetermined surface, wherein the first and the second predetermined surfaces are parallel;
wherein determining second distribution data comprises manipulation of the distribution data based on making use of intrinsic physical properties of the magnetic field.

(8) According to preferred embodiments of the present invention, determining second distribution data of the component comprises performing a Fourier transformation of the first distribution data resulting in Fourier transformed data, followed by performing data manipulation on the Fourier transformed data resulting in manipulated Fourier transformed data, followed by an inverse Fourier transformation of the manipulated Fourier transformed data.

(9) Below, a derivation is presented demonstrating this aspect.

(10) It is known to persons skilled in the art of physics that a magnetic vector field {right arrow over (B)}=(B.sub.x,B.sub.y,B.sub.z) can be expressed as
{right arrow over (B)}=(B.sub.x,B.sub.y,B.sub.z)=(.sub.x.sup.,.sub.y.sup.,.sub.z.sup.)=,Equation 1
where is the magnetic potential, and is the del operator, also called nabla operator known to the skilled person.

(11) One set of intrinsic physical properties of a harmonic potential field are the Green's Identities, which are known by persons skilled in the art of mathematics. Consider U to be a closed region in three-dimensional space, and U the boundary surface of this region. Consider and to be harmonic continuous functions with continuous partial derivatives of first and second orders in the region U. From Green's Second Identity it then follows that

(12) $\begin{array}{cc}\frac{1}{4}{}_U\left(\frac{}{n}-\frac{}{n}\right)\mathrm{dS}=0,& \mathrm{Equation}2\end{array}$
where

(13) $\frac{}{n}$
is the directional derivative of in the direction of the outward pointing normal {right arrow over (n)} to the surface element dS.

(14) If P is a point inside the closed region U, then it follows from Green's Third Identity that

(15) $\begin{array}{cc}\left(P\right)=\frac{1}{4}{}_U\left(\frac{1}{r}\frac{}{n}-\frac{}{n}\frac{1}{r}\right)\mathrm{dS},& \mathrm{Equation}3\end{array}$
where r is the distance between the surface portion dS and the point P.

(16) It is known to persons skilled in the art of physics, that the magnetic potential of Equation 1 is a harmonious function with continuous partial derivatives of first and second orders in a region U in three-dimensional space where no sources of magnetic field are present.

(17) Consider a Cartesian coordinate system XYZ, in which one measures the magnetic field on a XY plane at a certain position z.sub.0 along the Z axis. It is supposed that all sources of magnetic field are located at z<z.sub.0, that means, below the measurement surface. Suppose that one wants to know the magnetic field at a point P with coordinates (x,y,z.sub.0+z), that means, at a distance z above the measurement surface. One defines a region R in space enclosed by a disk with radius p lying in the measurement plane and a half sphere with the same centre and radius as the disk, extending from the disk circumference in the positive Z direction. It will be apparent to those skilled in the art that in this closed region Equation 3 is valid, whereby the function is understood to be the magnetic potential. The integral in Equation 3 can thereby be expressed as the sum of a first integral over the disk lying in the XY plane, and a second integral over the hemisphere. Next it is considered that the value of evolves to infinity, in which case it is easily shown that the second integral, namely the one over the hemisphere, evolves to the value zero, and that the integration surface U reduces to the complete XY plane (x,y,z.sub.0). In the limit .fwdarw., Equation thus becomes:

(18) $\begin{array}{cc}\left(x,y,z_0+z\right)=\frac{1}{4}{}_{-}^{+}{}_{-}^{+}\left(\frac{1}{r}\frac{\left(x^{},y^{},z_0\right)}{z^{}}-\left(x^{},y^{},z_0\right)\frac{}{z^{}}\frac{1}{r}\right){\mathrm{dx}}^{}{\mathrm{dy}}^{},& \mathrm{Equation}4\end{array}$
where r={square root over ((xx).sup.2+(yy).sup.2+(z.sub.0+zz).sup.2)}, and where z>0.

(19) Equation 4 contains a term with

(20) $\frac{0\left(x^{},y^{},z_0\right)}{z^{}},$
which can be eliminated in the following way. By adding together Equation 2 and Equation 3, one obtains:

(21) $\begin{array}{cc}\left(P\right)=\frac{1}{4}{}_U\left[\left(+\frac{1}{r}\right)\frac{}{n}-\frac{}{n}\left(+\frac{1}{r}\right)\right]\mathrm{dS}.& \mathrm{Equation}5\end{array}$

(22) One now defines the point P as the mirror image of the point P with respect to the plane z=z.sub.0, that means with P having coordinates (x,y,z.sub.0z), and one defines =1/, where ={square root over ((xx).sup.2+(yy).sup.2+(z.sub.0zz).sup.2)}. With this definition following conditions are satisfied: firstly,

(23) $+\frac{1}{r}=0$
on the XY surface where z=z.sub.0; secondly,

(24) $+\frac{1}{r}$
vanishes on the hemisphere in the limit .fwdarw.; and thirdly, is harmonic. In the limit .fwdarw. Equation 5 then becomes:

(25) $\begin{array}{cc}\left(x,y,z_0+z\right)=-\frac{1}{4}{}_{-}^{+}{}_{-}^{+}\left(x^{},y^{},z_0\right)\frac{}{z^{}}\left[\frac{1}{r}-\frac{1}{}\right]{\mathrm{dx}}^{}{\mathrm{dy}}^{}.& \mathrm{Equation}6\end{array}$

(26) When the derivative in this equation is calculated and z is evolving to the XY plane, one obtains:

(27) 0$\begin{array}{cc}\left(x,y,z_0+z\right)=-\frac{z}{2}{}_{-}^{+}{}_{-}^{+}\frac{\left(x^{},y^{},z_0\right)}{{\left[{\left(x-x^{}\right]}^2+{\left(y-y^{}\right)}^2+z^2\right]}^{3/2}}{\mathrm{dx}}^{}{\mathrm{dy}}^{},& \mathrm{Equation}7\end{array}$
where z>0. Equation 7 can be used to calculate the potential at a point (x,y,z.sub.0+z) based on measurements in a plane (x,y,z.sub.0).

(28) In order to be useable in practice, Equation 7 can be considered in the Fourier domain. It is noted that Equation 7 has the form of a two-dimensional convolution:
(x,y,z.sub.0+z).sub..sup.+.sub..sup.+(x,y,z.sub.0)u(xx,yy,z)dxdy,Equation 8
where

(29) $\begin{array}{cc}u\left(x,y,z\right)=\frac{z}{2}\frac{1}{{\left(x^2+y^2+z^2\right)}^{3/2}}.& \mathrm{Equation}9\end{array}$

(30) A convolution in the spatial domain corresponds to a multiplication in the Fourier domain. Therefore Equation 8 can be written as
{circumflex over ()}(k.sub.z,k.sub.y,z+z)={circumflex over ()}(k.sub.x,k.sub.y,z.sub.0){circumflex over (u)}(k.sub.x,k.sub.y,z),Equation 10
Where {umlaut over ()}(k.sub.x,k.sub.y,z+z) is the two-dimensional Fourier transform of the magnetic potential in the XY plane at z=z.sub.0+z. The notation {circumflex over (F)} denotes the Fourier transform of the function F.

(31) Equation 9 can also be written as

(32) $\begin{array}{cc}u\left(x,y,z\right)=-\frac{1}{2}\frac{}{z}\frac{1}{r},& \mathrm{Equation}11\end{array}$
where r={square root over (x.sup.2+y.sup.2+z.sup.2)}. It can be shown that the Fourier transform of the function 1/r is given by:

(33) $\begin{array}{cc}=2\frac{e^{-.Math.k.Math.z}}{.Math.k.Math.},\left|k\right|0.& \mathrm{Equation}12\end{array}$

(34) The Fourier transform (k.sub.x,k.sub.y,z) of the function u(x,y,z) is then calculated as follows:

(35) $\begin{array}{cc}\begin{array}{c}\hat{u}\left(k_x,k_y,z\right)=\frac{1}{2}\frac{}{z}\\ =\frac{}{z}\frac{e^{-.Math.k.Math.z}}{.Math.k.Math.}\\ =e^{-.Math.k.Math.z},z>0,\end{array}\mathrm{where}.Math.k.Math.=\sqrt{k_x^2+k_y^2}.& \mathrm{Equation}13\end{array}$

(36) Equation 10 was derived for the magnetic potential . It is now shown that it is also valid for each magnetic field component B.sub.x, B.sub.y and B.sub.z separately. From Equation 1, Equation 10 and Equation 13 it follows that

(37) $\begin{array}{cc}& \mathrm{Equation}14\\ \mathrm{and}& \\ \begin{array}{c}\left(k_x,k_y,z_0+z\right)=-\frac{\stackrel{\_}{}\stackrel{\_}{}}{z}\left(k_x,k_y,z_0+z\right)\\ =-.Math.k.Math.\hat{}\left(k_x,k_y,z_0+z\right)\\ =-.Math.k.Math.\hat{}\left(k_x,k_y,z_0\right)\hat{u}\left(k_x,k_y,z\right)\\ =-\left(k_x,k_y,z_0\right)\hat{u}\left(k_x,k_y,z\right)\\ =\left(k_x,k_y,z_0\right)\hat{u}\left(k_x,k_y,z\right)\\ =\left(k_x,k_y,z_0\right)e^{-.Math.k.Math.z}\end{array}& \mathrm{Equation}1\end{array}$

(38) In Equation 14 and Equation 15 the following properties were used, which are known to persons skilled in the art:

(39) $\begin{array}{cc}={\left({\mathrm{ik}}_x\right)}^n\hat{}={\left({\mathrm{ik}}_y\right)}^n\hat{}={.Math.k.Math.}^n\hat{}& \mathrm{Equation}16\end{array}$

(40) Equation 14 and Equation 15 show that the result of Equation 10 can be applied to each component of the vector field derived from , as summarized here:
(k.sub.x,k.sub.yz.sub.0+z)=(k.sub.x,k.sub.y,z.sub.0)e.sup.|k|z.Equation 17

(41) It follows from Equation 17 that the magnetic field in the XY plane at z=z.sub.0+z can be obtained from the magnetic potential in the XY plane at z=z.sub.0 by first performing a Fourier transform, then multiplying by e.sup.|k|z, and then performing an inverse Fourier transform.

(42) Although in the derivation of Equation 17 it is assumed that z>0, which means that the field is calculated at distances farther away from the magnetic field source, it can also be applied with z<0, that means, to calculate the magnetic field at distances closer to the magnetic field source. A condition is that the position where the field is calculated is still in a region free from magnetic field sources. In practice this means that it is possible to measure the magnetic field at a certain distance from a magnet, and that the field can be calculated at positions closer to the magnet, even up to the magnet surface. Since existing magnetic field sensors and magnetic field camera devices often have a minimum measurement distance from the magnet, it is an advantage of the present invention to be able to calculate the field at closer distances to the magnet.

(43) Although Equation 17 was derived for the case of a Cartesian coordinate system, the same principle can be applied in other coordinate systems, such as the cylindrical coordinate system. First, one considers the case in which any component of the magnetic field is recorded on a flat disk or ring surface in a cylindrical coordinate system (R, , Z). Since the predetermined area is in a flat surface, it can be projected onto a plane in the Cartesian coordinate system. Thereby it is noted that the data grid in the cylindrical coordinate system may not transform to a regular grid in the Cartesian coordinate system. However, the transformed grid can be made regular again by interpolation methods which are well known to persons skilled in the art. To the obtained data, Equation 17 can be applied, in order to obtain the magnetic field at another distance in the Z-direction. The coordinates of the obtained data are then transformed back to the cylindrical coordinate system. Thereby, according to preferred embodiments, another interpolation step is performed in order to obtain the data values at the original grid points in the cylindrical coordinate system.

(44) In order to obtain suitable boundary conditions on the outer periphery of the transformed disk surface in Cartesian coordinates, it is preferred to apply extrapolation methods according to methods described in the present disclosure.

(45) The person skilled in the art will recognize that the above method equally applies to a ring surface, where only the area between an inner radius and an outer radius is considered. Thereby, the inner disk surface, on which no data points are present, may be extrapolated according to methods described in the present invention.

(46) The principle of Equation 17 can be used also for data on a curved cylinder surface, in order to calculate the magnetic fields at radial distances different from this surface. This is already apparent from Equation 3, which says that the magnetic field in a closed region can be deduced from the field on the boundary of that region. For the case where the predetermined area is a curved cylinder surface with radius R.sub.0, consider the region bound by two cylinders, one with radius R.sub.0 and one with radius R.sub.1 with R.sub.0<R.sub.1, and both extending from z.sub.0 to +z.sub.0 in the Z-direction. In analogy to the given derivation for the Cartesian coordinate system, it is apparent that the integrand of Equation vanishes on the outer cylinder and on the planar ring surfaces in the limit where R.sub.1,z.sub.0.fwdarw., which means that the field at any point in the region R>R.sub.0 can be derived from the field at R.sub.0 using an expression similar to Equation 17, expressed in cylindrical coordinates.

(47) Furthermore, it will be apparent to those skilled in the art that this method is equally applicable to determine the field at a radius R<R.sub.0, assuming that no magnetic field sources are present in the region between R and R.sub.0.

(48) When the distribution of the measured magnetic field component on the boundaries of the predetermined area is not approaching zero, because for example the magnetic object is larger than the measurement area of the magnetic field camera, it can be needed to extend the predetermined measurement area in order to make sure that the measured distribution is sufficiently approaching zero at its boundaries, so that the methods of the present invention can be applied. For such case, it may be needed to measure a larger area by stepping the measurement area in the first and/or second directions, and at each new location measure the distribution in a new predetermined area which is adjacent to the previous predetermined area, after which all determined distributions are stitched together as to result in one large predetermined area, on the borders of which the magnetic field has sufficiently evolved towards zero as to apply the extrapolation methods described in the present disclosure, with the aim of determining second distribution data comprising second values of the component of the magnetic field for a second predetermined area defined along a second predetermined surface, wherein the first and the second predetermined surfaces are parallel. Methods and devices for accomplishing this are for example described in European patent application EP12188521.4 filed on 15 Oct. 2012 by the applicant of the present application, which are hereby incorporated by reference. Such a device can be described as a device for determining a magnetic field distribution of a magnet along a main surface of the magnet, the device comprising: a. an arrangement of at least two independent magnetic field camera modules being arranged in a fixed relative position with respect to each other, each magnetic field camera module being adapted for measuring a magnetic field distribution to which it is exposed by means of a respective detection surface; b. a means for providing a predetermined relative movement between the main surface and the arrangement to thereby scan the magnetic field distribution of the magnet along the main surface.

(49) The associated method is a method for determining a magnetic field distribution of a magnet along a main surface of the magnet, the device comprising: providing the magnet; providing an arrangement of at least two independent magnetic field camera modules being arranged in a fixed relative position with respect to each other, each magnetic field camera module being adapted for measuring a magnetic field distribution to which it is exposed by means of a respective detection surface; providing a predetermined relative movement between the main surface and the arrangement to thereby scan the magnetic field distribution of the magnet along the main surface.

(50) In order to determine the second distribution data comprising second values of the magnetic field component for this stitched distribution, the methods described in the present invention can be applied to the resulting (stitched) predetermined area.

(51) Another method however is to apply the so-called overlap-add and overlap-save methods, which are known to persons skilled in the art of signal processing, whereby the Fourier transform, data manipulation based on intrinsic physical properties of a magnetic vector field, and inverse transform are performed on sub areas of the respective predetermined areas or combination of predetermined area and extension area.

(52) The overlap-add and overlap-save methods can also be applied in the other cases described in the present disclosure in the following way. Instead of first determining the additional distribution data in the extended area, and after that applying data manipulation based on intrinsic physical properties of a magnetic vector field (for instance comprising applying a Fourier transform, manipulating the Fourier transformed data, and performing an inverse Fourier Transform) in order to obtain the second distribution data of the component, the Fourier Transform, data manipulations, and Inverse Fourier Transform are applied separately to the predetermined area and to (sub-regions of) the extended area in a block-wise fashion. Thereby each of the blocks can be zero padded to a certain extent in order to create overlap regions between the back-transformed blocks. The resulting back-transformed blocks are then combined in the final larger matrix, whereby the overlap regions are added together. Additionally, the predetermined area can itself also be treated block-wise, for example in the case of a large magnet where the predetermined area is relatively large and is measured in a block-wise way. This method corresponds to the overlap-add method. A person skilled in the art will readily recognize the analogous possibility of applying the overlap-save method to the same data.

(53) Alternatively, the predetermined and extended areas can be treated block-wise or section-wise, where the manipulations are performed on each block/section separately, and the results of them added together or saved according to overlap-add and overlap-save methods, respectively.

(54) As a further improvement on the previous method, the overlap-add and overlap-save methods can also be used on an infinite extension area, where the extrapolation is expressed as an analytical function which the Fourier transform and other operations can be analytically determined on the interval stretching from infinity to the border of the predetermined area.

(55) Distribution data of the component of the magnetic vector field in the extension area can be determined in many ways, some of which are explained below.

(56) FIGS. 3 to 8 illustrate examples of how additional distribution data for the component of the magnetic vector field can be generated, according to embodiments of the present disclosure. The depicted patterns illustrate the evolution of the values of the component, the z-component of the magnetic field, along to a cross-section S as depicted in FIG. 1. The cross-section can comprise an outer portion of the predetermined area only, but can extend up until the centre of the predetermined area.

(57) According to a first embodiment, illustrated in FIG. 3, the component of the magnetic field vector is set to value zero (0) in the complete extension area. When the measured values of the component of the magnetic vector field on the outer boundary of the predetermined area differ from zero, the method may though introduce a discontinuity in the values on the outer boundary of the predetermined area, which may still introduce errors in a subsequent calculation of the second distribution data of the component of the magnetic field vector.

(58) According to an alternative embodiment, illustrated in FIG. 8, of the present disclosure, the additional distribution data comprising the component of the magnetic field vector is simulated or modeled for the predetermined area and the simulation results or model are/is extrapolated into the extension area.

(59) According to preferred embodiments, illustrated in FIG. 2, the magnetic field component at locations lying in the extension area may be set to the same value as the nearest point in the predetermined area. For a rectangular predetermined area, this would mean that the boundary values of the predetermined area and thus measured area are set as a fixed, constant, value throughout the extension area along a direction which is orthogonal on the outer boundary of the predetermined area (portions 20). In the corner areas which are then remaining (21), the value of the corresponding corner of the predetermined area is set as a constant value. Subsequently, a window function is applied on the predetermined area and the extension area, which has a value 1 in the predetermined or measured area and which evolves continuously to a (near to) zero value on the outer boundary of the predetermined area. For instance, the window function can be a Tukey-window (illustrated in FIG. 5) or a Planck-Taper window. Alternatively a so-called bump-function or a test-function can be used, which have the property that they evolve from a value 1 to a value zero within a limited area, whereby they can be infinitely differentiated, and whereby they thus do not introduce discontinuities in the function or any of its first or higher order derivatives, which makes them suitable for being used as a window function in this context.

(60) According to a further embodiment, illustrated in FIG. 4, the values of the component of the magnetic vector field at the outer boundary of the predetermined area are set to be evolving exponentially towards a zero value within the extension area. Preferably the exponential factor is predetermined such that its sufficiently small value or zero value is achieved on the outer boundary of the extension area. In practice, it is enough that the magnetic field distribution on the outer boundary of the extension area has a value which is below the measurement noise. This method guarantees the continuity of the values on the outer border of the predetermined area, but does not guarantee the continuity of the first derivative thereof.

(61) In a further preferred embodiment, the component of the magnetic vector field in the (boundary area of the) predetermined area is represented by a polynomial representation. The order of the extrapolated polynomial function can be reduced, for instance to order two (quadratic polynomial), or to order one (linear polynomial), in order not to obtain instable extrapolation values. This method guarantees that the continuity of the measurement values and of the derivatives is ascertained to the same extent as the order of the used polynomial function. In practice, it is further preferred to further apply a window-function on these extrapolated values, such that it can be guaranteed that the distribution of the component of the magnetic field vector reaches a value which is small enough or zero at the outer boundary of the extension area.

(62) In a further preferred embodiment, the component of the magnetic vector field in the (boundary area of the) predetermined area is represented by a rational function representation. The order of the extrapolated rational function can be chosen, for instance to order 1 (1/x), order 2 (1/x.sup.2) or order 3 (1/x.sup.3). The latter case is particularly useful since it corresponds to the decline rate of the magnetic field of a magnetic dipole, which is an approximation of a magnet at large distances from the magnet field source, e.g. magnet. This method guarantees that the magnetic field approaches zero when moving further into the extension area. In practice, it can further be preferred to apply a window-function on these extrapolated values, such that it can be guaranteed that the distribution of the component of the magnetic field vector reaches a value which is small enough or zero at the outer boundary of the extension area.

(63) In a further preferred embodiment, the component of the magnetic vector field in the (boundary area of the) predetermined area is represented by a spline-representation, illustrated in FIG. 6. The order of the extrapolated spline-function can be reduced, for instance to order two (quadratic spline), or to order one (linear spline), in order not to obtain instable extrapolation values. This method guarantees that the continuity of the measurement values and of the derivatives is ascertained to the same extent as the order of the used spline-function. In practice, it is further preferred to further apply a window-function on these extrapolated values, such that it can be guaranteed that the distribution of the component of the magnetic field vector reaches a value which is small enough or zero at the outer boundary of the extension area. This is illustrated in FIG. 7.

(64) According to a still further embodiment of the present disclosure, the values of the component of the magnetic field at the boundary of the predetermined area can be extrapolated on the basis of a simulated magnetic field distribution for the magnetic object of which the magnetic field distribution was measured. Simulation algorithms can be used to calculate the magnetic field distribution based for instance on one or more properties of the magnetic object such as for instance shape, material, magnetization vector, position and so forth. Also here, use can be made of any of the methods disclosed for instance in EP2508906.

(65) According to another embodiment of the present disclosure, the determination of the second distribution data of the component of the magnetic field distribution of a stitched predetermined area is performed in the following way, by using the overlap-add or overlap-save method. Thereby the Fourier transform, manipulation operations and inverse Fourier transform are performed on each individual predetermined area, which may or may not be zero padded, without applying the extrapolation methods described in the current disclosure. When combining the determined second data distributions of the component into the larger distribution, the overlap-save or overlap-add method can be applied.

(66) According to another embodiment of the present disclosure, the overlap-add or overlap-save method is used on a finite or an infinite extension area.