BO-corrected sensitivity encoding magnetic resonance imaging

Abstract

A magnetic resonance imaging system (200, 300, 400) includes a radio-frequency system (216, 214) with multiple coil elements (214) for acquiring magnetic resonance data (264) and a memory (250) for storing machine executable instructions (260) and pulse sequence commands (262). The pulse sequence commands are configured for controlling the magnetic resonance imaging system to acquire the magnetic resonance data according to a SENSE imaging protocol. Execution of the machine executable instructions causes a processor (244) to: control (500) the magnetic resonance imaging system to acquire the magnetic resonance data using the pulse sequence commands; reconstruct (502) a set of folded magnetic resonance images (266) from the magnetic resonance data; calculate (504) a voxel deformation map (270) from a magnetic field inhomogeneity map; and calculate (506) a set of unfolding matrices (274) using a least partially a coil sensitivity matrix (272) for the multiple coil elements, wherein the set of unfolding matrices includes at least one modified unfolding matrix which is calculated at least partially using the a coil sensitivity matrix and the voxel deformation map. Undistorted magnetic resonance image data (276) is calculated (508) using the set of folded magnetic resonance images and the set of unfolding matrices.

Claims

1. A magnetic resonance imaging system comprising: a radio-frequency system comprising multiple coil elements for acquiring magnetic resonance data; a non-transitory computer readable memory configured to store machine executable instructions and pulse sequence commands, wherein the pulse sequence commands are configured for controlling the magnetic resonance imaging system to acquire the magnetic resonance data according to a SENSE imaging protocol that is associated with folding effects due to under sampling in k-space of the magnetic resonance data, a processor for controlling the magnetic resonance imaging system, wherein execution of the machine executable instructions causes the processor to: control the magnetic resonance imaging system to acquire the magnetic resonance data using the pulse sequence commands; reconstruct a set of folded magnetic resonance images from the magnetic resonance data; calculate a voxel deformation map from a magnetic field inhomogeneity map; calculate a set of unfolding matrices using at least partially a coil sensitivity matrix for the multiple coil elements, wherein the set of unfolding matrices comprises at least one modified unfolding matrix, wherein the at least one modified unfolding matrix is calculated using the coil sensitivity matrix and the voxel deformation map; wherein the at least one modified unfolding matrix corrects folding-like artifacts due to inhomogeneities in the magnetic field; calculate undistorted magnetic resonance image data using the set of folded magnetic resonance images and the set of unfolding matrices including the at least one modified unfolding matrix; and calculate modified magnetic resonance image data by transforming the undistorted magnetic resonance image data with the voxel deformation map to incorporate the magnetic field inhomogeneity back into the undistorted magnetic resonance image data.

2. The magnetic resonance imaging system of claim 1, further including a display device configured to selectively display an undistorted image and an image with the magnetic field homogeneity incorporated back in.

3. The magnetic resonance imaging system of claim 1, wherein calculating the modified unfolding matrix includes transforming the coil sensitivity matrix with the voxel deformation map.

4. The magnetic resonance imaging system of claim 1, wherein the SENSE imaging protocol is an echo planar imaging protocol, wherein the echo planar imaging protocol is configured for acquiring magnetic resonance data with phase encoding in at least one phase encoding direction, wherein the voxel deformation map is descriptive of a local magnetic field distortion in the at least one phase encoding direction.

5. The magnetic resonance imaging system of claim 1, wherein the at least one phase encoding direction is two phase encoding directions, wherein the SENSE imaging protocol is a three-dimensional SENSE imaging protocol, wherein one of the two orthogonal phase encoding directions is configured for slice selection.

6. The magnetic resonance imaging system of claim 1, wherein the SENSE imaging protocol is a simultaneous multiple slice acquisition imaging protocol.

7. The magnetic resonance imaging system of claim 1, wherein the modified unfolding matrix is formulated in a feed forward format.

8. The magnetic resonance imaging system of claim 7, wherein the modified unfolding matrix is at least partially calculated using a regularization term.

9. The magnetic resonance imaging system of claim 1, wherein the SENSE imaging protocol is a multi-shot SENSE imaging protocol.

10. The magnetic resonance imaging system of claim 1, wherein the voxel deformation map is calculated using the pulse sequence commands and the magnetic field inhomogeneity map.

11. The magnetic resonance imaging system of claim 1, wherein execution of the machine executable instructions further causes the processor to: acquiring preliminary magnetic resonance data using a coil sensitivity measuring magnetic resonance imaging protocol; and calculate the coil sensitivity matrix using the preliminary magnetic resonance data.

12. The magnetic resonance imaging system of claim 1, wherein execution of the machine executable instructions further cause the processor to: acquiring magnetic field magnetic resonance data using a magnetic field measuring magnetic resonance imaging protocol; and calculate the magnetic field inhomogeneity map using the magnetic field magnetic resonance data.

13. A computer program product comprising: machine executable instructions stored on a non-transitory computer readable medium for execution by a processor for controlling a magnetic resonance imaging system, wherein the magnetic resonance imaging system comprises a radio-frequency system comprising multiple coil elements acquiring magnetic resonance data, to control the magnetic resonance imaging system to perform the method comprising: controlling the magnetic resonance imaging system to acquire the magnetic resonance data using pulse sequence commands, wherein the pulse sequence commands are configured for controlling the magnetic resonance imaging system to acquire the magnetic resonance data according to a SENSE imaging protocol that is associated with folding artifacts due to under sampling in k-space of the magnetic resonance data due to magnetic field inhomogeneities; reconstructing a set of folded magnetic resonance images from the magnetic resonance data; calculating a voxel deformation map from a magnetic field inhomogeneity map; calculating at least one unfolding matrix using a coil sensitivity matrix for the multiple coil elements; calculating at least one modified unfolding matrix using the coil sensitivity matrix and the voxel deformation map to correct for the folding artifacts due to inhomogeneities in the magnetic field; calculating undistorted magnetic resonance image data using the folded magnetic resonance images, the at least one unfolding matrix, and the at least one modified unfolding matrix; controlling a display to selectively display an image corrected for folding artifacts attributable to the magnetic field inhomogeneities as well as the under sampling in k-space and an image incorporating the magnetic field inhomogeneity; and calculating modified magnetic resonance image data by transforming the undistorted magnetic resonance image data with the voxel deformation map to incorporate the magnetic field inhomogeneity back into the undistorted magnetic resonance image data.

14. A method of magnetic resonance imaging using a magnetic resonance imaging system including a radio-frequency system with multiple coil elements for acquiring magnetic resonance data, the method comprising: controlling the magnetic resonance imaging system to acquire the magnetic resonance data using pulse sequence commands, wherein the pulse sequence commands are configured for controlling the magnetic resonance imaging system to acquire the magnetic resonance data according to a SENSE imaging protocol that is associated with folding artifacts due to under sampling in k-space of the magnetic resonance data due to magnetic field inhomogeneities; reconstructing a set of folded magnetic resonance images from the magnetic resonance data; calculating a voxel deformation map from a magnetic field inhomogeneity map; calculating at least one unfolding matrix using a coil sensitivity matrix for the multiple coil elements; calculating at least one modified unfolding matrix using the coil sensitivity matrix and the voxel deformation map to correct for the folding artifacts due to inhomogeneities in the magnetic field; calculating undistorted magnetic resonance image data using the folded magnetic resonance images, the at least one unfolding matrix, and the at least one modified unfolding matrix; controlling a display to selectively display an image corrected for folding artifacts attributable to the magnetic field inhomogeneities as well as the under sampling in k-space and an image incorporating the magnetic field inhomogeneity; and calculating modified magnetic resonance image data by transforming the undistorted magnetic resonance image data with the voxel deformation map to incorporate the magnetic field inhomogeneity back into the undistorted magnetic resonance image data.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) In the following preferred embodiments of the invention will be described, by way of example only, and with reference to the drawings in which:

(2) FIG. 1 illustrates a shift in voxel location due to magnetic field inhomogeneities;

(3) FIG. 2 illustrates an example of a magnetic resonance imaging system;

(4) FIG. 3 illustrates a further example of a magnetic resonance imaging system;

(5) FIG. 4 illustrates a further example of a magnetic resonance imaging system;

(6) FIG. 5 illustrates a method of using the magnetic resonance imaging system of FIG. 2, 3 or 4;

(7) FIG. 6 shows a plot of a map of the B0 magnetic field in a magnetic resonance imaging system; and

(8) FIG. 7 shows an example of voxel deformation map calculated using the Map of the B0 magnetic field of FIG. 6.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(9) Like numbered elements in these figures are either equivalent elements or perform the same function. Elements which have been discussed previously will not necessarily be discussed in later figures if the function is equivalent.

(10) In MRI, particularly when using an Echo Planar Imaging (EPI) sequence, significant geometric distortions may occur due to local deviations of the magnetic field. This may be particularly seen close to metal implants or close to cavities, e.g. the ear or nose cavitiesor, in general, with all tissue-to-air interfaces. In many cases, the distortion itself is clinically not very relevant nor disturbing.

(11) However, when using SENSE, there may be indirect consequences of the distortion. At the folding position relative to the distorted region (e.g. at a distance of half the field-of-view with SENSE_factor=2), artifacts may appear of the distorted region, even if that folding-position is rather free from distortion. This may for example result in artifacts in otherwise undistorted, and clinically relevant, regions. Examples may have the effect of reducing or eliminating artifacts in SENSE images that originate from regions of geometric distortions caused by magnetic field inhomogeneities.

(12) Examples may have one or more of the following features:

(13) 1. When applying SENSE, the coil sensitivities are taken from the true geometric positions that contribute to a folded point, as opposed to taking the coil-sensitivities from their nominal (distorted) positions.

(14) 2. Due to the distortion, we may get a larger number of locations that contribute to a folded point than the nominal SENSE factor: in that case, a (locally) extended set of SENSE-equations is resolved.

(15) 3. After performing the SENSE reconstruction, the image may be re-distorted using the voxel deformation map to aide a physician in interpreting the image.

(16) FIG. 1 shows a plot of the true geometric location of a voxel versus the distorted or nominal location 102. FIG. 1 is used to illustrate the effects of folding caused by magnetic field inhomogeneities when performing a SENSE-type magnetic resonance imaging protocol. In FIG. 1 a SENSE factor of 2 is assumed. The diagonal line 104 shows the relationship between the true geometric location 102 and the distorted nominal location 102 if there were no effects due to magnetic field distortions. The curve 106 shows the actual geometric location 100 as a function of the distorted or nominal location 102. This can be seen as a curve which overlaps itself in the y or true geometric location 100. This results in a pseudo folding effect. With the SENSE factor of 2 some points are imaged more than once within the field of view.

(17) The point a 110 and the point A 112 correspond to folded points. The distance 114 is the folding distance. The points A 110 and A 112 are projected onto the curve 106. This gives the true geometric location 100. Point a 110 is projected onto point y 116 and point A 112 is projected onto point X 118. For A and A 110 and 112 there is only one corresponding point 116 and 118. The field of view is represented by the line 108. Within the field of view 108 there is a second point B 120 that is mapped onto points 130, 128, and 126. The corresponding folded point B 122 is mapped onto a single point one 124. The point B as it is mapped onto three different points cannot be corrected using a normal SENSE reconstruction. This however can be corrected using a voxel deformation map. If the curve 106 is discretized it may be interpreted as a voxel deformation map.

(18) The folded pixel A 110, which (in a folded image) is identical to A 112, has been contributed to by true geometric locations X 118 and Y 116. For this set of locations, a regular SENSE set of two equations has to be solved. However, when considering folded pixel B 120 (identical to B 122 in the folded image), there are actually four contributing locations in the true world: one 124, two 126, three 128 and four 130. This is resolved by a set of extended SENSE equations, i.e. having to resolve four unknowns rather than two, from as many equations as we have coil elements.

(19) Yet, since some of the points (e.g. two 126 and three 128) are very close to each other, the SENSE-unfolding between these two will exhibit a very high level of noise amplification, since their coil sensitivities are rather similar. Yet, the sum of these two results will be of a normal noise level.

(20) So, to prevent excessive noisiness of some image areas, the result may be distorted according to the distortion that was present in the original acquisition (by using the voxel deformation map). E.g., the results of locations two 126, three 128 and four 130 are summed and mapped to location B, while the result of 1 is mapped to location B. In this way, the end-result image is still geometrically distorted, but reconstructed pixel B may now contain fewer SENSE-artifacts originating from regions two 126, three 128 and four 130.

(21) FIG. 2 illustrates an example of a magnetic resonance imaging system 200. The magnetic resonance imaging system 200 comprises a main magnet 204, which may be referred to as the magnet. The magnet 204 is a superconducting cylindrical type magnet 104 with a bore 206 through it. The use of different types of magnets is also possible. Inside the cryostat of the cylindrical magnet, there is a collection of superconducting coils. Within the bore 206 of the cylindrical magnet 204 there is an imaging zone 208 where the magnetic field is strong and uniform enough to perform magnetic resonance imaging.

(22) Within the bore 206 of the magnet there is also a set of magnetic field gradient coils 210 which is used for acquisition of magnetic resonance data to spatially encode magnetic spins within the imaging zone 208 of the magnet 204. The magnetic field gradient coils 210 are connected to a magnetic field gradient coil power supply 212. The magnetic field gradient coils 210 are intended to be representative. Typically magnetic field gradient coils 210 contain three separate sets of coils for spatially encoding in three orthogonal spatial directions. A magnetic field gradient power supply supplies current to the magnetic field gradient coils. The current supplied to the magnetic field gradient coils 210 is controlled as a function of time and may be ramped or pulsed.

(23) Adjacent to the imaging zone 208 are multiple coil elements 214 that each function as radio-frequency antennas for manipulating the orientation of magnetic spins within the imaging zone 208 and for receiving radio transmissions from spins also within the imaging zone 208. The radio frequency coils may also be referred to as a radio frequency antennas or as antennas. The multiple coil elements may also be referred to as a antenna elements. The radio frequency antennas may also be referred to as channels. The multiple coil elements 114 are connected to a radio frequency transceiver 116. The multiple coil elements 114 and radio frequency transceiver 116 may have separate transmitters and receivers for each the multiple coil elements 214.

(24) The coil elements 214 may be used to acquire magnetic resonance data separately. The coil elements 214 may therefore be used for a parallel imaging magnetic resonance technique. Although it is not shown in this Fig, the magnetic resonance imaging system 200 may also comprise a body coil. The body coil would be useful in the parallel imaging technique as it could take acquired data at the same time as the individual coil elements 214 and be used for calculating a set of coil sensitivities.

(25) The magnetic resonance data may be acquired from within the imaging zone 208. The location of a slice 209 is visible within the imaging zone 208. The pulse sequence commands for acquiring the magnetic resonance data may in some examples be done with phase encoding in a phase encoding direction 222. The arrow 222 indicates one possible direction that is parallel or coplanar with the slice 209. If phase encoding in the direction 222 is used the resulting image may be susceptible to distortions in the main magnetic field produced by the magnet 204.

(26) It can be seen that different coil elements 214 are different distances from different regions of the slice 209. Different coil elements 214 will therefore be more or less sensitive to various portions of the slice 209.

(27) Within the bore 206 of the magnet 204 there is a subject support 220 which supports the subject in the the imaging zone 208.

(28) The transceiver 216 and the gradient controller 230 are shown as being connected to a hardware interface 242 of a computer system 240. The computer system further comprises a processor 244 that is in communication with the hardware system 242, memory 250, and a user interface 246. The memory 250 may be any combination of memory which is accessible to the processor 244. This may include such things as main memory, cached memory, and also non-volatile memory such as flash RAM, hard drives, or other storage devices. In some examples the memory 250 may be considered to be a non-transitory computer-readable medium.

(29) The computer memory 250 is shown as containing machine-executable instructions 260 which enable the processor 244 to control the operation and function of the magnetic resonance imaging system 200. The computer memory 250 is further shown as containing pulse sequence commands 262 which are configured for acquiring magnetic resonance data according to a SENSE imaging protocol. The computer memory 250 is further shown as containing magnetic resonance data 264 that was acquired by controlling the magnetic resonance imaging system 200 with the pulse sequence commands 262. The magnetic resonance data 264 contains data that was acquired from each of the coil elements 215 separately. The computer memory 250 is further shown as containing a set of folded magnetic resonance images 266. Each of the folded magnetic resonance images corresponds to one of the coil elements 215. Each of the folded magnetic resonance images 266 was reconstructed from the various portions of the magnetic resonance data 264 acquired for each of the coil elements 215.

(30) The computer memory 250 is further shown as containing a magnetic field inhomogeneity map 268. The magnetic field inhomogeneity map 268 could be pre-existing in the memory 250 or it could be acquired and calculated prior to acquiring the magnetic resonance data 264. The computer memory 250 is further shown as containing a voxel deformation map 270 that was calculated from the magnetic field inhomogeneity map 268. In many cases the voxel deformation map 270 will be calculated using the magnetic field inhomogeneity map 268 and data extracted from the pulse sequence commands 262. For example if the pulse sequence commands 262 are for an echo planar imaging protocol where there is phase encoding in the phase encoding direction 222 then the amount of phase error in the acquired magnetic resonance data is directly related to the pulse repetition time and the local magnetic field inhomogeneities.

(31) The computer memory 250 is further shown as containing a set of coil sensitivities 272 for the coil elements 215. The set of coil sensitivities 272 could for example be pre-stored or could have been acquired during a pre-acquisition prior to acquiring the magnetic resonance data 264. The computer memory 250 is further shown as containing a set of unfolding matrices 274 which are calculated from the set of coil sensitivities 272 and the voxel deformation map 270. The set of unfolding matrices 274 is used to combine the set of folded magnetic resonance images 268 into an undistorted magnetic resonance image data 276. In this particular example, the magnetic resonance data is acquired for a single slice 209.

(32) The undistorted magnetic resonance image data 276 may in this case represent a two-dimensional image. The computer memory 250 is shown as containing optionally the modified magnetic resonance image data 278 which is calculated by applying the voxel deformation map 270 to the undistorted magnetic resonance image data 276. This creates a re-distorted but unfolded magnetic resonance image or dataset.

(33) FIG. 3 shows an example of a magnetic resonance imaging system 300. The magnetic resonance imaging system 300 is similar to the magnetic resonance imaging system 200 illustrated in FIG. 2. However in this case, the pulse sequence commands 262 are different. In this example the magnetic resonance data is acquired as a three-dimensional acquisition. Instead of a single slice there are a set of voxels 209 which may be divided into multiple slices 209. In addition to the phase encoding direction 222 there is another direction 322 which is orthogonal to direction 222. This additional phase encoding direction 322 is used to differentiate between the different slices 209. Both the phase encoding direction 222 and 322 would be susceptible to distortions due to magnetic field inhomogeneities. In this example the voxel deformation map 270 is used to correct voxel positions for the acquired three-dimensional dataset.

(34) FIG. 4 illustrates a further example of magnetic resonance imaging system 400. The example shown in FIG. 4 is similar to the examples illustrated in both FIGS. 2 and 3. In the example shown in FIG. 4 the pulse sequence commands 262 are for a SENSE imaging protocol that is a simultaneous multiple slice acquisition imaging protocol. The individual slices 209 are acquired simultaneously but are not acquired with phase encoding as is performed in FIG. 3. The SENSE imaging protocol methodology is used to separate the simultaneously acquired data into distinct slices. The set of coil sensitivities 272 calculated using the voxel deformation map 270 is useful in correctly separating data into the proper slices 209.

(35) FIG. 5 shows a flowchart, which illustrates a method of operating the magnetic resonance imaging systems 200, 300, or 400. First in step 500 the magnetic resonance imaging system 200, 300, 400 is controlled to acquire the magnetic resonance data 264 using the pulse sequence commands 262. Next in step 502 the set of folded magnetic resonance images 266 is calculated for each of the coil elements 215 from the acquired magnetic resonance data 264. Next in step 504 the voxel deformation map 270 is calculated from the magnetic field inhomogeneity map 268. Then in step 506 the set of unfolding matrices 274 is calculated using the set of coil sensitivities 272 and for some of the unfolding matrices the voxel deformation map 270 is also used. Next in step 508 the undistorted magnetic resonance image data 276 is calculated 508 using the set of folded magnetic resonance images 266 and the set of unfolding matrices 274. Step 510 is an optional step where a modified magnetic resonance image data 278 is calculated by transforming the undistorted magnetic resonance image data 276 with the voxel deformation map 270.

(36) Examples may take the deformations due to local B0 variations into account during the unfolding of individual coil elements. In the below example an EPI scan is discussed, however the technique may also be applicable to other types of SENSE image reconstructions.

(37) In one example a B0 map may be acquired as a prescan to the EPI scan. The shift at a pixel p, .sub.p (in reconstructed pixels) is directly related to the variation in B0 at p, B0.sub.p (in Hertz) according to:

(38) $_{p} = \frac{B 0_{p} .Math. N_{pe}}{{BW}_{pe} .Math. R_{pe}}$
Where (in phase encoding direction) BW.sub.pe is the acquisition bandwidth, in Hertz per folded FOV, N.sub.pe the unfolded grid size and Rpe the in-plane sense factor.

(39) The B0 deformation discussed here is a deformation on top of deformations due to gradient non-linearities. It is assumed both the CSM and the clinical scan are affected to the same amount by gradient imperfections.

(40) For a phase encoding line a deformation matrix (also referred to as a voxel deformation map) can be derived which maps undeformed coordinates to deformed or vice versa. A deformation matrix maps every undeformed point to a deformed point. The deformation matrix is a discrete representation (at unfolded grid size) where every undeformed point maps to 1 or more deformed points. The deformation between two successive points is interpolated linear such that a continuous trace of non-zero matrix elements is created.

(41) To ensure that a mapping from deformed to undeformed exists for every deformed point, the deformation at begin and end of each Y-line is set to 0, i.e. no deformation. In practice this is already zero if the anatomy is completely included in the FOV and/or if sufficient intrinsic fold-over suppression is applied.

(42) In case there are regions where the B0 map does not completely cover the unfolded FOV, deformations in some examples are assumed to be zero in those regions.

(43) FIG. 6 illustrates an example of a magnetic field inhomogeneity map 268. In this example the magnetic field inhomogeneity map is a mapping of the spatial dependence of the main magnetic field, which is also referred to as the B0 field. The line 600 indicates where a portion of this data is extracted and used to calculate a voxel deformation map 270. The line 600 from the B0 map (268) from which a deformation matrix (below) can be derived.

(44) FIG. 7 illustrates the voxel deformation map 270 (also referred to as deformation matrix herein) calculated for the data extracted along line 600 in FIG. 6. The plot in FIG. 7 resembles the plot in FIG. 1, however in FIG. 7 the mapping 270 is for discreet voxels.

(45) In an SENSE acquisition, a realistic model of the measured magnetic resonance data can be written as:

(46) $m_{i} (\overset{.fwdarw.}{x}) = \underset{j}{.Math.} \underset{y_{u}}{.Math.} w (y (\overset{.fwdarw.}{x_{j}} (\overset{.fwdarw.}{x})), y_{u}; x (\overset{.fwdarw.}{x_{j}} (\overset{.fwdarw.}{x})), z (\overset{.fwdarw.}{x_{j}} (\overset{.fwdarw.}{x}))) s_{i} (\overset{.fwdarw.}{x_{u, j}}) p (\overset{.fwdarw.}{x_{u, j}})$
Glossary on how to read this:
{right arrow over (x)} The coordinate vector of a folded and deformed voxel.
m.sub.i({right arrow over (x)}) The value seen by coil element i on location {right arrow over (x)}.
j Index to the points that fold onto voxel {right arrow over (x)}; j=1 . . . N, where N is the total SENSE factor.
{right arrow over (x.sub.J)}({right arrow over (x)}) The coordinates of the j-th of the points that fold onto folded location {right arrow over (x)}.
y({right arrow over (x.sub.J)}({right arrow over (x)})) The y-coordinate of the vector {right arrow over (x.sub.J)}. Similarly for x({right arrow over (x.sub.J)}({right arrow over (x)})) and z({right arrow over (x.sub.J)}({right arrow over (x)})).
y.sub.u Undeformed coordinate y.
y.sub.u Summation over all the undeformed points that (may) contribute to a deformed point.
w(y, y.sub.u; x, z) For a given value of x and z, w(y,yu) is the deformation map, explained in the sequel.
{right arrow over (x.sub.u,J)} Undeformed folding point: shorthand for (x({right arrow over (x.sub.J)}({right arrow over (x)})), y.sub.u, z({right arrow over (x.sub.J)}(x))), i.e. a location that contributes (by a weight w) to the deformed point {right arrow over (x.sub.J)}({right arrow over (x)}).
s.sub.i({right arrow over (x.sub.u,J)}) The sensitivity of coil i to an undeformed point that contributes to the deformed point that is the j-th point that folds onto position {right arrow over (x)}.
p({right arrow over (x.sub.u,J)}) The actual value of magnetization density at that undeformed point
In a somewhat shorter form, this is written as:
for each {right arrow over (x)}: m.sub.i=.sub.jy.sub.u w(y.sub.j, y.sub.u)s.sub.i(y.sub.u)p(y.sub.u)
Soper unfolded point, y.sub.u sums over the undeformed points. The contribution of the undeformed points to a deformed point y is the deformation map, denoted by w(y, y.sub.u; x, z), or shorthand w(y,y.sub.u).

(47) Note the measurements m.sub.i({right arrow over (x)}) are deformed, while the sensitivity s.sub.i and the unknown pixel value p are undeformed.

(48) The contributions w(y,y.sub.u) are retrieved from the deformation transformation. A single deformed point will likely map to a limited number of undeformed points, i.e. most of the contributions will be zero. In case the B0 map is 0 everywhere, the deformation matrix becomes the unit transform, and the modified SENSE equation reduces to the normal SENSE equation.

(49) This sparsity allows to define an undeformed sensitivity matrix SU (and similarly RU). For this, we define, for each value of {right arrow over (x)}, the range k=1, . . . , M, where M is the total number of points that substantially contribute to {right arrow over (x)} (via deformation and via folding). M is likely to be (somewhat) larger than the SENSE factor N.

(50) The elements of the matrix SU are written (in shorthand notation) as

(51) $s_{ik}^{U} = \underset{j}{.Math.} w (y_{j}, y_{k}) s_{i} (y_{k})$
And, equivalently,

(52) $r_{kk}^{U} = {.Math. \underset{j}{.Math.} w (y_{j}, y_{k}) r (y_{k}) .Math.}^{2}$

(53) Solving the SENSE equation similar as discussed in the previous paragraph 6.1, in matrix notation, gives:
p=(S.sup.UHS.sup.U+(R.sup.U).sup.1).sup.1S.sup.UHm=Cm
Here, vector p is expressed in undeformed coordinates. It contains M elements. In full notation, we should actually write p({right arrow over (x)}), i.e. the set of (undeformed and unfolded) locations that correspond to the folded location {right arrow over (x)}.

(54) In theory, we now have the issue that some elements of p({right arrow over (x)}) might be the same location(s) as elements of a vector p pertaining to another folded point. Yet, this is resolved with the operation explained below.

(55) For each folded location {right arrow over (x)}, undeformed points are transformed back to a deformed pixel:

(56) $p_{d, j} = \underset{k}{.Math.} w (y_{j}, y_{k}) p_{k}$

(57) In words: the j-th element of the normal (i.e. deformed) vector pertains to the deformed location yj, is assembled from the elements of the undeformed vector p (whereby the k-th element has value p.sub.k and pertains to the undeformed location y.sub.k), using the weights of the deformation map.

(58) The rationale: in case of signal pile-up a single high-intensity measurement will unfold to several undeformed points. As these points are close to each other, there will be little discrimination between the CSM values, resulting in low SNR in the elements of p. This step allows to recover from that SNR loss.

(59) There can be imperfections in the B0 map, there might be motion between the B0 prescan and the EPI scan, and also additional distortions in the EPI scan itself (e.g. due to Eddy currents). Robustness against artifacts stemming from these imperfections may be increased by shifting the deformation matrix over its diagonal and applying a pixelwise MAX operation on the original deformation matrix and the shifted one. This can be repeated multiple times while shifting in the positive and the negative direction.

(60) While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the invention is not limited to the disclosed embodiments.

(61) Other variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word comprising does not exclude other elements or steps, and the indefinite article a or an does not exclude a plurality. A single processor or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measured cannot be used to advantage. A computer program may be stored/distributed on a suitable medium, such as an optical storage medium or a solid-state medium supplied together with or as part of other hardware, but may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems. Any reference signs in the claims should not be construed as limiting the scope.

LIST OF REFERENCE NUMERALS

(62) 100 true geometric location 102 distorted (nominal) location 104 diagonal line 106 geometric location as a function of distorted locations 108 field of view 110 point A 112 point A 114 folding distance 116 point Y 118 point X 120 point B 122 point B 124 point one 126 point two 128 point three 130 point four 200 magnetic resonance system 204 main magnet 206 bore of magnet 208 imaging zone 209 slice 210 magnetic field gradient coils 212 gradient coil power supply 214 coil element 216 transceiver 218 subject 220 subject support 224 phase encoding direction 240 computer system 242 hardware interface 244 processor 246 user interface 250 computer memory 260 machine executable instructions 262 pulse sequence commands 264 magnetic resonance data 266 set of folded magnetic resonance images 268 magnetic field inhomogeneity map 270 voxel deformation map 272 set of coil sensitivities or coil sensitivity matrix 274 set of unfolding matrices 276 undistorted magnetic resonance image data 278 modified magnetic resonance image data 300 magnetic resonance imaging system 322 phase encoding direction 400 magnetic resonance imaging system 500 control the magnetic resonance imaging system to acquire the magnetic resonance data using the pulse sequence commands; 502 reconstruct a set of folded magnetic resonance images from the magnetic resonance data 504 calculate a voxel deformation map from a magnetic field inhomogeneity map 506 calculate a set of unfolding matrices using a least partially a coil sensitivity matrix for the multiple coil elements and the set of unfolding matrices comprises at least one modified unfolding matrix 508 calculate undistorted magnetic resonance image data using the set of folded magnetic resonance images and the set of unfolding matrices 510 calculate modified magnetic resonance image data by transforming the undistorted magnetic resonance image data with the voxel deformation map 600 line

BO-corrected sensitivity encoding magnetic resonance imaging

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