Sparse Calibration of Magnetic Fields Generated by Coils in Metal-Rich Environments

Abstract

The calibration method includes receiving measured magnetic field values ​​generated by a plurality of real magnetic transmitters (26) at a plurality of locations on a grid within a region including a magnetic field perturbation element. Approximate locations of the real magnetic transmitters are received. Using the approximate locations, a respective plurality of virtual magnetic sources (55) are characterized within the magnetic field perturbation element (40). Using the measured magnetic field values, the approximate locations, and the characterized virtual sources, (i) actual locations of the real and virtual magnetic sources within the region, and (ii) modeled magnetic field values ​​resulting from the real and virtual magnetic sources at the actual locations. The calculated locations and the modeled magnetic field values ​​at the plurality of locations on the grid are used to derive a magnetic field calibration function for the region.

Description

[Technical field]

[0001] The present invention relates generally to sensing the position of an object placed within a living body, and in particular to compensating for magnetic disturbances that affect the position sensor. [Background technology]

[0002] Techniques to aid in tracking of invasive probes in organ cavities using magnetic fields have been previously proposed in the patent literature. For example, US Patent Application Publication No. 2012 / 0165656 describes a method that uses multiple magnetic transmitters to generate a magnetic field in a region and introduces a magnetic field perturbation element into the region. The method includes characterizing multiple images of each magnetic transmitter in the magnetic field perturbation element and calculating a reaction magnetic field in the region based on the characterized images. The method further includes positioning a probe in the region and measuring the perturbed magnetic field at the probe and determining a position of the probe corresponding to the measured perturbed magnetic field and the calculated reaction magnetic field.

[0003] US Patent Application Publication No. 2016 / 0011288 describes a medical device comprising a magnetic resonance imaging system, a magnetic compensation coil for compensating for magnetic inhomogeneities in an imaging field, a gantry operable to rotate about the imaging field, a position sensor for measuring the angular position and angular velocity of the gantry, at least one magnetic field distortion component in the gantry, and a memory storing machine executable instructions and field correction data. The instructions cause a processor to receive position and angular velocity data from the position sensor, determine coil control commands for controlling the magnetic compensation coil using the field correction data, the position data, and the angular velocity data, control the magnetic compensation coil to compensate for magnetic inhomogeneities in the imaging field using the coil control commands, and acquire magnetic resonance data. Summary of the Invention [Means for solving the problem]

[0004] One embodiment of the present invention provides a calibration method that includes receiving measured magnetic field values ​​at a plurality of locations on a grid in a region that is generated by a plurality of real magnetic transmitters and includes a magnetic field perturbation element. Approximate locations of the real magnetic transmitters are received. Using the approximate locations, a respective plurality of virtual magnetic sources are characterized within the magnetic field perturbation element. Using the measured magnetic field values, the approximate locations, and the characterized virtual sources, (i) actual locations of the real and virtual magnetic sources in the region, and (ii) modeled magnetic field values ​​resulting from the real and virtual magnetic sources at the actual locations are iteratively calculated. The calculated locations and the modeled magnetic field values ​​at the plurality of locations on the grid are used to derive a magnetic field calibration function for the region.

[0005] In some embodiments, receiving the magnetic field values ​​includes positioning a probe in the region and measuring the magnetic field generated at the probe across a plurality of positions.

[0006] In some embodiments, characterizing the virtual magnetic sources includes estimating a position of each of the virtual magnetic sources within the perturbation element.

[0007] In one embodiment, the magnetic field calibration function is provided as a three-dimensional array of values ​​on a grid of calibrated positions that is denser than the grid used to measure the magnetic field values.

[0008] In another embodiment, deriving the magnetic field calibration function includes modeling the magnetic fields generated by the real and virtual magnetic sources as a linear combination of spherical harmonics and evaluating the modeled magnetic fields at the actual locations.

[0009] According to another embodiment of the present invention, there is further provided an apparatus including a memory and a processor, the memory being configured to store magnetic field values ​​generated by a plurality of real magnetic transmitters and measured at a plurality of locations on a grid in a region including a magnetic field perturbation element, and to store approximate locations of the real magnetic transmitters, the processor being configured to (a) characterize a respective plurality of virtual magnetic sources in the magnetic field perturbation element using the approximate locations of the real magnetic transmitters, (b) iteratively calculate (i) actual locations of the real and virtual magnetic sources in the region, and (ii) modeled magnetic field values ​​resulting from the real and virtual magnetic sources at the actual locations using the measured locations, and (c) derive a magnetic field calibration function for the region using the calculated locations and the modeled magnetic field values ​​for the plurality of locations on the grid.

[0010] The invention will be more fully understood from consideration of the following detailed description taken in conjunction with the drawings, in which: [Brief description of the drawings]

[0011] [Figure 1] FIG. 1 is a schematic, pictorial diagram of an ear-nose-throat (ENT) system including a magnetic position tracking subsystem, in accordance with an embodiment of the present invention. [Diagram 2] FIG. 2 is a schematic, pictorial front view of a magnetic radiator assembly mounted on the metal base of FIG. 1 in accordance with one embodiment of the present invention. [Diagram 3] 3 is a schematic depiction of real and respective virtual magnetic sources within the metal base of FIG. 2, according to an embodiment of the present invention; [Figure 4] 11 is a graph plotting convergence of iteratively calculated positions both with and without the use of a virtual magnetic source, in accordance with an embodiment of the present invention; [Diagram 5] 11 is a plot of convergence of positions of real and virtual magnetic sources during iterative calculations, according to one embodiment of the present invention. [Figure 6] 1 is a flow chart that generally illustrates a method for sparse calibration of a magnetic field in the presence of metal-rich objects, in accordance with an embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0012] Overview Tracking a probe within a patient's organ using a magnetic position sensor at the distal end of the probe requires calibration of an external magnetic tracking system. In some cases, the magnetic field generated by the system is perturbed by the presence of field-perturbing metal-rich elements, such as equipment near the system's field generators. The perturbations complicate the calibration, as they require the use of a dense grid of calibrated locations in space. Typically, such a complex calibration may require measuring tens of thousands of magnetic field values ​​in 3D space over the course of a day using a calibration probe.

[0013] An embodiment of the present invention provides a method for rapid calibration of perturbing magnetic fields in the vicinity of a static metal-rich element, such as a metal base that provides mechanical support for a set of coil field generators (hereinafter also referred to as the "real magnetic source" or "coil transmitter") of a position tracking system.

[0014] To take into account the presence of perturbing elements, e.g., the metallic base of a coil, an embodiment of the present invention provides a calibration model that assumes that each coil transmitter generates eddy currents in the perturbing element. These eddy currents can be modeled as one or more imaginary (or virtual) magnetic sources. The calibration model further assumes that each virtual magnetic source generates a respective reaction field that causes a perturbation of the magnetic field based on linear superposition.

[0015] The disclosed rapid calibration method includes: (a) receiving measured magnetic field values ​​generated by a plurality of real magnetic transmitters at a plurality of locations on a grid in a region including a magnetic field perturbation element; (b) receiving approximate locations of the real magnetic transmitters; (c) characterizing a plurality of virtual magnetic sources, each within the magnetic field perturbation element, using the approximate locations; (d) iteratively calculating (i) actual locations of the real and virtual magnetic sources in the region and (ii) modeled magnetic field values ​​resulting from the real and virtual magnetic sources at the actual locations using the measured magnetic field values, the approximate locations, and the characterized virtual sources; and (e) deriving a magnetic field calibration function for the region using the calculated locations and the modeled magnetic field values ​​for the plurality of locations on the grid.

[0016] In some embodiments, each virtual magnetic source is characterized as a magnetic multipole, i.e., a combination of dipoles, quadrupoles, and / or higher-order multipoles. The characteristics of each virtual magnetic source also depend, among other things, on the actual transmitter fields that generate the virtual magnetic source. The calibration model calculates the multi-reaction magnetic field from each of the multipole virtual magnetic sources by assuming that the magnetic field can be represented by a spherical harmonics expansion according to the characteristics of the virtual magnetic source. Spherical harmonics are particularly convenient for this application, although other extensions such as wavelets are also valid.

[0017] The actual position of the real coil transmitter and the modeled position of each virtual magnetic source are left as parameters to be solved by using iterative calculations. Thus, the disclosed method involves adjusting a pre-specified position of the real coil transmitter, which may be inaccurate due to, for example, the lack of mechanical repeatability in manufacturing the transmitter base assembly and therefore may vary from system to system.

[0018] In one embodiment, the position tracking system includes five three-axis magnetic transmitters, with each three-axis transmitter including three mutually orthogonal coils, for a total of 15 real coil transmitters. Each of the 15 real coil transmitters generates a virtual coil transmitter in the metal base of the position tracking system's location pad. Applying calibration measurements and assuming that the real and virtual transmitter fields are given by a spherical harmonics model, the calibration provides effective positions of the real and virtual transmitters, and these positions are used to determine the magnetic field at any point in the range of motion. The use of a sparse grid in the calibration process requires at least ten times (x10) fewer measurement data points over the entire area than previous calibration processes. In this system, the disclosed technique reduces the typical duration of the calibration procedure from about one day to about one hour.

[0019] In one embodiment, the processor is configured to provide the magnetic field calibration function as a three-dimensional array of values ​​on a grid of calibrated locations that is more densely populated than the sparse grid used in the measurements.

[0020] Typically, the processor is programmed with software that contains specific algorithms that enable the processor to perform each of the processor-related steps and functions outlined above.

[0021] Once the system is calibrated, the processor is able to track with high accuracy the position of the medical probe within the magnetically perturbed region during a medical session.

[0022] System Description 1 is a schematic, pictorial diagram of an ear, nose and throat (ENT) system 20 including a magnetic position tracking subsystem 23, according to one embodiment of the present invention. In the following description, during a medical procedure with an ENT tool 21 in the head of a patient 28, the tool 21 includes one or more magnetic sensors 34, typically single-axis or three-axis coils, that are tracked by the magnetic position tracking subsystem 23 during the procedure.

[0023] To enable tracking, the system 20 registers the coordinate systems of a CT (Computed Tomography) image of the patient 28 and the magnetic position tracking subsystem 23. Both before and during the medical procedure, a magnetic tracking system consisting of a magnetic emitter assembly 24 is positioned under the patient's head.

[0024] To use a magnetic field transmission system as described herein, the magnetic emitter assembly 24 must be mounted on a metal base 40. (The metal base 40 shields the transmitter from interfering metal that may be present in the bed or chair on which the patient 28 lies or sits.) However, the transmitter and base (which themselves introduce perturbations to the magnetic field) must be calibrated.

[0025] The assembly 24 includes five magnetic field triple coil transmitters 26, which are fixed in position and transmit alternating magnetic fields into a region 30 in which the head of a patient 28 is located. Electrical potentials generated in response to the magnetic fields by sensors 34 in the region 30 allow measurement of its position and its orientation within the reference frame of a magnetic tracking system. The position can be measured in three linear dimensions (3D), while the orientation of the distal end of the tool 21 can be determined using one of the coils of the sensors 34, whose axis is aligned with the longitudinal axis of the distal end.

[0026] As an example, the five coil transmitters 26 of assembly 24 are arranged in a generally horseshoe shape around the head of patient 28. However, alternative configurations for the coil transmitters of assembly 24 may be used, and all such configurations are assumed to fall within the scope of the present invention. Each of the coil transmitters 26 includes three mutually orthogonal coils that generate a magnetic field. Thus, system 20 has a total of 15 transmitter coils.

[0027] Prior to the procedure, the coordinate system of the magnetic tracking system may be aligned with the CT image by positioning a magnetic sensor at a known location in the image (eg, the top of the patient's head).

[0028] Elements of the system 20, such as the coil transmitter 26 and the sensors 34, are under the overall control of a system processor 41. The processor 41 may be mounted on a console 50, which includes a motion control 58, which typically includes a keypad and / or a pointing device, such as a mouse or trackball. The console 50 connects to the coil transmitter 26 and the sensors 34 wirelessly and / or via one or more cables. A physician 54 uses the motion control 58 to interact with the processor while performing an ENT procedure using the system 20. While the procedure is being performed, the processor may display the results of the procedure on a screen 56.

[0029] System 20 further comprises a memory 42. Processor 41 operates system 20 using software stored in memory 42. The software may be downloaded in electronic form to processor 41, for example over a network, or alternatively or additionally, the software may be provided and / or stored on a non-transitory tangible medium, such as magnetic, optical, or electronic memory. In particular, processor 41 executes dedicated algorithms as disclosed herein, including in FIG. 6, that enable processor 41 to perform the disclosed steps, as further described below.

[0030] In some embodiments, memory 42 is also used to store measured magnetic field values ​​and the approximate position of coil transmitter 36 as part of the disclosed calibration scheme, as described below.

[0031] Sparse Calibration of Magnetic Fields Generated by Coils in Metal-Rich Environments Figure 2 is a front view of magnetic emitter assembly 24 mounted on metal base 40 of Figure 1, according to one embodiment of the present invention. Also visible are eddy currents generated in metal base 40 by coil transmitter 26, which cause perturbations in the magnetic field generated by coil transmitter 26.

[0032] 3 is a schematic depiction of a real magnetic source 26 (e.g., coil transmitter 26) and respective virtual magnetic sources 55 in the metal base 40 of FIG. 2 according to another embodiment of the present invention. The oscillating magnetic field generated by the coil 26 induces eddy currents in the metal base 40. The eddy currents generate a reaction magnetic field that is modeled by assuming a single virtual source 55. For calibration, the method uses the sparse measured total magnetic field to iteratively find the exact real position of the coil 26 and the modeled positions of the virtual sources 55 to have a self-consistent calibration function.

[0033] The disclosed calibration model assumes that both the real source 26 and the virtual source 55 can be described as a linear combination of spherical harmonics that produces a magnetic field given by:

[0034]

number

[0035] For an array of N real sources 26, the model computes 2N different B(r,θ,φ) functions (including one for each virtual source), where in FIG. 2 N=5. In practice, each real source 26 is a three-axis coil, and each coil transmits a magnetic field that oscillates at a unique frequency. Typically, for each frequency there are 15 unknown spherical harmonic coefficients, which is the 15 Y in Eq. l,m It corresponds to taking (θ,φ) terms, where l runs from 1 to 3 and m runs from 0 to l, representing the magnetic field at a given frequency. For each three-axis coil, three unknown position coordinates are added to it. A virtual source has the same number of unknowns as a real source. Thus, there are 48 unknowns (i.e., 45 spherical harmonic coefficients + 3 position coefficients) for each three-axis coil. Thus, for the configuration of Figure 2, i.e., for N=5, the total number of unknowns that can be detected is 480.

[0036] The optimization procedure uses inputs from P calibration data points. At each point in space, three field components are measured. Thus, there are a total of 3P equations per frequency (transmit coil), where P is in the hundreds, meaning that there are typically over 1000 data points. This is overdetermined, for example, by a factor of 5 or more, so a set of equations (with, for example, 480 unknowns) is solved by the optimization.

[0037] The initial positions of the virtual magnetic sources 55 are estimated based on the mechanical drawings and iteratively calculated using the disclosed techniques. Furthermore, the position of each real magnetic source 26 is only approximately known using the mechanical drawings within a given tolerance that is too wide for sufficiently accurate tracking, for example, of the distal end of the tool 21 within the patient 28, but is sufficiently accurate as an initial condition.

[0038] The disclosed iterative calculations can be performed using library functions of commercial software such as MATLAB. For example, the exact location of the magnetic source can be found, for example, using the f_min_search function of MATLAB, where the solution of the aforementioned set of equations can be performed, for example, using the pinv (Moore-Penrose pseudoinverse) function. A typical cost function of the error after the Mth iteration is:

[0039]

number

[0040]

number

[0041] In one embodiment, using Equation 1, the processor is configured to provide the magnetic field calibration function as a three-dimensional array of values ​​on a grid of calibrated positions that is denser than the sparse grid used for the measurements.

[0042] 4 is a graph plotting convergence of iteratively calculated position without virtual magnetic source 55 (70) and with virtual magnetic source 55 (72) in accordance with another embodiment of the present invention. Thus, by using virtual magnetic sources and doubling the number of equations solved in each iteration, the number of iterations until the calibration error falls below a pre-specified acceptance value is reduced by more than a factor of 10 (e.g., M=8K iterations vs. approximately M less than 0.8K iterations, where K is an integer).

[0043] Figure 5 is a plot of the convergence of the positions of real and virtual magnetic sources during iterations, according to one embodiment of the present invention. The results shown in Figure 5 are for the disclosed calibration process applied to the system 20 of Figure 1. The open circles are intermediate positions. The solid circles 126 are the final positions of the real transmitter 26 of the system 20, and the solid circles 150 are the final positions of the respective virtual sources 55 used by the calibration model.

[0044] FIG. 6 is a flow chart that generally illustrates a method for sparse calibration of a magnetic field in the presence of metal-rich objects, in accordance with another embodiment of the present invention.

[0045] The algorithm according to the present embodiment performs a process that begins with uploading the disclosed model incorporating a virtual magnetic transmitter in a magnetic field perturbation element into the processor in a model upload step 70. Next, in a model initialization step 72, an initial position of the real magnetic source is input into the model in the processor, for example by a person running a calibration algorithm according to the disclosed model. The initial position may be obtained, for example, from an engineering drawing showing the approximate position of the real transmitter and the dimensions of the perturbation element. The initial position is typically stored in memory 42.

[0046] Typically, an algorithm automatically generates a set of respective positions of the virtual magnetic source, for example as a mirror reflection of the real source relative to an axis of symmetry, as described above.

[0047] Steps 70 and 72 are typically common to all systems of a given model and can be performed in advance without requiring hardware dependent data.

[0048] At begin system calibration step 74, calibration data from the particular system being calibrated is uploaded to a processor, for example by the person performing the calibration. This data typically involves uploading a file that stores magnetic field readings in the form of voltages over a sparse grid of locations in space, the data in the file being measured independently of the steps associated with running the calibration model, for example by performing measurements on the manufacturing floor of the system, as described below in steps 90-94.

[0049] Using the calibration data, the processor iteratively calculates the required calibration functions for the particular system, in steps 76-80.

[0050] In a first iteration step 76, the processor calculates the positions of the real magnetic sources (26) and the virtual magnetic sources (55) within the region and the respective magnetic fields they generate. Then, in an error calculation step 78, the processor calculates the error between the calculated and measured magnetic fields using a cost function as described above.

[0051] In an error check step 80, the processor compares the error to a prespecified value. If the error is greater than the prespecified value, the processor loops to step 76 to refine the calculation in the next iteration. If the error is less than the prespecified value, the processor uses the calculated positions of the real and virtual magnetic sources to derive a magnetic field calibration function for the system being calibrated in a calibration step 82. Finally, the processor stores the derived magnetic field calibration function in memory 42 in a calibration storage step 84.

[0052] As described above, the disclosed model is applied to calibrate a particular system. In a preliminary step 90, the system is operated, for example by real transmitter 26, to generate a magnetic field in a region that includes a perturbation element (e.g., plate 40), for example region 30 of system 20.

[0053] Next, in a measurement step 92, the magnetic field is measured across multiple locations on the grid within the region using a calibration probe with a three-axis magnetic sensor, such as magnetic sensor 34 of ENT tool 21. Finally, the measured calibration data is stored in memory 42 in a data storage step 94 for subsequent use in calibration steps 74-84.

[0054] Once the above-described calibration process is complete, the processor 41 of the system 20 applies the calibration data in accurately tracking the distal end of the tool 21 during a medical procedure.

[0055] Although the embodiments described herein primarily address ENT medical navigation systems, the methods and systems described herein can be used for other applications such as cardiac, pulmonary, and digestive system procedures, as well as for any magnetic tracker with static conductors in the environment.

[0056] Therefore, it will be understood that the above-described embodiments are cited by way of example, and that the present invention is not limited to what has been particularly shown and described above. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described in the above specification, as well as variations and modifications thereof not disclosed in the prior art that would occur to one skilled in the art upon reading the foregoing description. Documents incorporated by reference into this patent application shall be considered an integral part of this application, except that to the extent that any term is defined in such incorporated documents in a manner that is inconsistent with the definition expressly or impliedly made herein, only the definition in this specification shall be considered.

[0057] [Embodiment] (1) A calibration method comprising the steps of: receiving magnetic field values ​​generated by a plurality of real magnetic transmitters and measured at a plurality of locations on a grid within a region including the magnetic field perturbation element; receiving an approximate location of the real magnetic transmitter; characterizing a respective plurality of virtual magnetic sources within the magnetic field perturbation element using the approximate locations; using the measured magnetic field values, the approximate locations, and the characterized virtual sources to iteratively calculate (i) actual locations of the real and virtual magnetic sources within the region, and (ii) modeled magnetic field values ​​resulting from the real and virtual magnetic sources at their actual locations; and using the calculated positions and the modeled magnetic field values ​​for the plurality of positions on the grid to derive a magnetic field calibration function for the region. (2) A calibration method as described in embodiment 1, wherein receiving the magnetic field values ​​includes positioning a probe in the area and measuring the magnetic field generated at the probe across the multiple locations. (3) A calibration method as described in embodiment 1, wherein characterizing the virtual magnetic sources includes estimating a position of each of the virtual magnetic sources within the perturbation element. (4) A calibration method as described in embodiment 1, wherein the magnetic field calibration function is provided as a three-dimensional array of values ​​on a grid of calibrated positions that is denser than the grid used to measure the magnetic field values. (5) The calibration method of claim 1, wherein deriving the magnetic field calibration function includes modeling the magnetic fields generated by the real magnetic source and the virtual magnetic source as a linear combination of spherical harmonic functions, and evaluating the modeled magnetic field at the actual location.

[0058] (6) An apparatus comprising: a memory for storing magnetic field values ​​generated by a plurality of real magnetic transmitters and measured at a plurality of locations on a grid within an area including magnetic field perturbation elements, and for storing approximate locations of said real magnetic transmitters; 1. A processor comprising: characterizing a respective plurality of virtual magnetic sources within the magnetic field perturbation element using the approximated locations of the real magnetic transmitters; using the measured magnetic field values, the approximate locations, and the characterized virtual sources to iteratively calculate (i) actual locations of the real and virtual magnetic sources within the region, and (ii) modeled magnetic field values ​​resulting from the real and virtual magnetic sources at their actual locations; a processor configured to use the calculated positions and the modeled magnetic field values ​​for the plurality of positions on the grid to derive a magnetic field calibration function for the region. (7) The apparatus of embodiment 6, wherein the processor is configured to characterize the virtual sources by estimating a position of each of the virtual magnetic sources within the perturbation element. (8) The apparatus of embodiment 6, wherein the processor is configured to provide the magnetic field calibration function as a three-dimensional array of values ​​on a grid of calibrated positions that is denser than the grid used to measure the magnetic field values. (9) The apparatus of embodiment 6, wherein the processor is configured to derive the magnetic field calibration function by modeling the magnetic fields generated by the real magnetic sources and the virtual magnetic sources as a linear combination of spherical harmonics and evaluating the modeled magnetic fields at the actual locations.

Claims

1. A method for operating a device having a processor, comprising: receiving, by the processor, magnetic field values ​​generated by a plurality of real magnetic transmitters and measured at a plurality of locations on a grid within a region including magnetic field perturbation elements; receiving an approximate location of the real magnetic transmitter; using the approximate positions, the processor estimating an initial position of each of a plurality of virtual magnetic sources within the magnetic field perturbation element; the processor using the measured magnetic field values, the approximate positions, and the estimated initial positions of the plurality of virtual magnetic sources; (i) the locations of the real magnetic transmitters and the virtual magnetic sources within the region; and (ii) modeled magnetic field values ​​resulting from the real magnetic transmitter and the virtual magnetic source at the locations of the real magnetic transmitter and the virtual magnetic source; said processor iteratively calculating and using the calculated positions and the modeled magnetic field values ​​of the plurality of positions on the grid, the processor deriving a magnetic field calibration function for the region.

2. The method of claim 1 , wherein the magnetic field calibration function is provided as a three-dimensional array of values ​​on a grid of calibrated positions that is denser than the grid used to measure the magnetic field values.

3. The method of claim 1, wherein the processor deriving the magnetic field calibration function comprises the processor modeling the magnetic fields generated by the real magnetic transmitter and the virtual magnetic source as a linear combination of spherical harmonic functions, and the processor evaluating the modeled magnetic field at the calculated position.

4. An apparatus comprising: a memory for storing magnetic field values ​​generated by a plurality of real magnetic transmitters and measured at a plurality of locations on a grid within an area including magnetic field perturbation elements, and for storing approximate locations of said real magnetic transmitters; 1. A processor comprising: using the approximated positions of the real magnetic transmitters to estimate initial positions of each of a plurality of virtual magnetic sources within the magnetic field perturbation element; using the measured magnetic field values, the approximate positions, and the estimated initial positions of the plurality of virtual magnetic sources; (i) the locations of the real magnetic transmitters and the virtual magnetic sources within the region; and (ii) modeled magnetic field values ​​resulting from the real magnetic transmitter and the virtual magnetic source at the locations of the real magnetic transmitter and the virtual magnetic source; Iteratively calculate a processor configured to use the calculated positions and the modeled magnetic field values ​​for the plurality of positions on the grid to derive a magnetic field calibration function for the region.

5. 5. The apparatus of claim 4, wherein the processor is configured to provide the magnetic field calibration function as a three-dimensional array of values ​​on a grid of calibrated positions that is denser than the grid used to measure the magnetic field values.

6. 5. The apparatus of claim 4, wherein the processor is configured to derive the magnetic field calibration function by modeling the magnetic fields generated by the real magnetic transmitter and the virtual magnetic source as a linear combination of spherical harmonics and evaluating the modeled magnetic fields at the calculated positions.

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