RECONSTRUCTION OF SPIRAL K-SPACE SAMPLED MAGNETIC RESONANCE IMAGES

Abstract

Disclosed herein is a medical system (100, 300, 500) comprising: a memory (110) storing machine executable instructions (120) and a processor (104). Execution of the machine executable instructions causes the processor to: receive (200) magnetic resonance imaging data (122), wherein the magnetic resonance imaging data has a spiral k-space sampling pattern; reconstruct (202) at least one preliminary magnetic resonance image (124) from the magnetic resonance imaging data; construct (204) a first set of equations comprising (130) each of the at least one preliminary magnetic resonance image being equal to an image transformation of at least one clinical image, wherein the image transformation makes use of a first spatially dependent kernel for each of the at least one clinical image (126, 126′, 126″); construct (206) a second set of equations (134) comprising at least one regularization matrix (132, 132′, 132″) times the at least one clinical image; and numerically (208) solve the first set of equations and the second set of equations simultaneously for the at least one clinical image.

Claims

1. A medical system a memory configured to store machine executable instructions; a processor, wherein execution of the machine executable instructions causes the processor to: receive magnetic resonance imaging data (K.sub.T(u,v)) by sampling k-space, in particular along a spiral trajectory, Fourier transform the acquired MR-data (K.sub.T(u,v)) to form a preliminary artefact (blurring, ringing) image (I.sub.b(m,n)), access a pre-determined artefacting matrix (C) which transforms the diagnostic image into the blurred image and recover a diagnostic image (I(m,n)) through solving an optimisation problem in image space that connects the diagnostic image to the preliminary artefact image by the transformation by the matrix C to the preliminary artefact image, which optimisation involves a regularisation with which an off-diagonal regularisation matrix is associated in image space and involved convolution with a kernel wherein kernel having a support that is at least an order of magnitude more narrow than the matrix size of the image.

2. The medical system of claim 1, wherein the off-diagonal regulation's convolution kernel's pattern varies from location to location.

3. The medical system of claim 1, wherein the off-diagonal regularisation matrix has spatially varying weights that depend on the magnetic resonance imaging method's main magnetic field's spatial variation.

4. The medical system of claim 1, wherein the off-diagonal regularisation matrix has limited support of 3 to 7 voxels in image space.

5. The medical system 1, wherein the at least one clinical image is a Dixon water image and a Dixon fat image, wherein the image transformation comprises a convolution of the Dixon water image and the Dixon fat image using the first spatially dependent kernel for each of the at least one clinical image.

6. The medical system of claim 5, wherein the at least one regularization matrix comprises a water regularization matrix and a fat regularization matrix, and wherein the construction of the second set of equations is performed by multiplying the water regularization matrix times a matrix representation of the Dixon water image; and wherein construction of the second set of equations is further performed by multiplying the fat regularization matrix time a matrix representation of the Dixon fat image.

7. The medical system of claim 6, wherein execution of the machine executable instructions further causes the processor to perform at least one of the following: multiply the water regularization matrix times a water mask before constructing the second set of equations; multiply the fat regularization matrix times a fat mask before constructing the second set of equations.

8. The medical system of claim 7, wherein the water mask comprises a per voxel water value inversely proportional to a water content of each voxel in the Dixon water image, and wherein the fat mask comprises a fat value inversely proportional to a fat content of each voxel in the Dixon fat image.

9. The medical system of claim 1, wherein execution of the machine executable instructions further causes the processor to construct the least one regularization matrix using a second spatially dependent kernel.

10. The medical system of claim 8, wherein the second spatially dependent kernel is any one of the following: a Gaussian curvature kernel times a spatially dependent factor; a mean curvature kernel times the spatially dependent factor; a Laplacian kernel times the spatially dependent factor; and a low pass spatial filter kernel times the spatially dependent factor; a second derivate kernel times the spatially dependent factor.

11. The medical system of claim 8, wherein the spatially dependent factor comprises any one of the following: a spatially dependent signal to noise estimate; a spatially dependent estimate of a B.sub.0 magnetic field gradient; a spatially dependent estimate of a B0 magnetic field error.

12. The medical system of claim 9, wherein the second spatially dependent kernel further comprises an identity term times an additional spatially dependent factor.

13. The medical system of claim 1, wherein the medical system further comprises a magnetic resonance imaging system configured for acquiring the magnetic resonance imaging data from an imaging zone, wherein the memory further contains pulse sequence commands configured for acquiring the magnetic resonance imaging data using the spiral k-space sampling pattern, wherein execution of the machine executable instructions further cause the processor to control the magnetic resonance imaging system with the pulse sequence commands to acquire the magnetic resonance imaging data.

14. The medical system of claim 13, wherein the pulse sequence commands are configured to control the magnetic resonance imaging system according to a magnetic resonance imaging protocol, wherein the magnetic resonance imaging protocol is any one of the following: a Dixon magnetic resonance imaging protocol, a two point Dixon magnetic resonance imaging protocol, a three point Dixon magnetic resonance imaging protocol, a four point Dixon magnetic resonance imaging protocol, a greater than four point Dixon magnetic resonance imaging protocol, a sensitivity-encoded parallel magnetic resonance imaging protocol, and an echo-planar imaging magnetic resonance imaging protocol.

15. A computer program product comprising machine executable instructions stored on a non-transitory computer readable medium for execution by a processor controlling a medical system, wherein execution of the machine executable instructions causes the processor to: receive magnetic resonance imaging data (K.sub.T(u,v)) by sampling k-space, in particular along a spiral trajectory, Fourier transform the acquired MR-data (K.sub.T(u,v)) to form a preliminary artefact (blurring, ringing) image (I.sub.b(m,n)), access a pre-determined artefacting matrix (C) which transforms the diagnostic image into the blurred image and recover a diagnostic image (I(m,n)) through solving an optimisation problem in image space that connects the diagnostic image to the preliminary artefact image by the transformation by the matrix C to the preliminary artefact image, which optimisation involves a regularisation with which an off-diagonal regularisation matrix is associated in image space, and involves convolution with a kernel wherein kernel having a support that is at least an order of magnitude more narrow than the matrix size of the image.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0060] In the following preferred embodiments of the invention will be described, by way of example only, and with reference to the drawings in which:

[0061] FIG. 1 illustrates an example of a medical system;

[0062] FIG. 2 shows a flow chart which illustrates a method of operating the medical system of FIG. 1;

[0063] FIG. 3 illustrates an example of a medical system;

[0064] FIG. 4 shows a flow chart which illustrates a method of operating the medical system of FIG. 3;

[0065] FIG. 5 illustrates an example of a medical system;

[0066] FIG. 6 shows a flow chart which illustrates a method of operating the medical system of FIG. 5; and

[0067] FIG. 7 illustrates examples of a spiral k-space sampling pattern.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0068] 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.

[0069] FIG. 1 illustrates an example of a medical system 100. In this example the medical system comprises a computer 102 with a processor 104. The processor is intended to represent one or more processors. The processors for example may be multiple cores. In other examples the processors 104 may be distributed between multiple computer systems and/or server units. The processor 104 is shown as being connected to an optional hardware interface 106. The hardware interface may for example be a network connection that enables the processor 104 to communicate with other computer systems and/or to communicate or control other components of the medical system 100. For example, the processor in some examples may be useful for controlling a magnetic resonance imaging system. The processor 104 is also shown as being connected to an optional user interface 108. The processor 104 is shown as being further connected to a memory 110.

[0070] The memory 110 may be any combination of memory which is accessible to the processor 104. 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 110 may be considered to be a non-transitory computer-readable medium.

[0071] The memory stores machine-executable instructions 120. The machine-executable instructions 120 enable the processor 104 to perform various data and image manipulation operations. In some examples the machine-executable instructions 120 also enable the processor 104 to control additional components via the hardware interface 106. The memory 110 is further shown as containing magnetic resonance imaging data 122 that has a spiral k-space sampling pattern. The magnetic resonance imaging data 122 may be received in various ways. It may for example be transferred to the memory 110 from another memory device or it may for example be received from a magnetic resonance imaging system or transferred via network connection.

[0072] The memory 110 is further shown as containing at least one preliminary magnetic resonance image 124 that was reconstructed from the magnetic resonance imaging data 122. The memory 110 is further shown as containing at least one clinical image 126 that was calculated numerically from the at least one preliminary magnetic resonance image 124. The memory 110 is further shown as containing a first spatially dependent kernel 128 that was used to generate a first set of equations 130. There is at least one regularization matrix 132 also stored in the memory 110 which is used to generate a second set of equations 134 which is also shown as being stored in the memory 110. The first set of equations 130 and the second set of equations 134 are solved numerically to obtain the at least one clinical image 126.

[0073] FIG. 2 shows a flowchart which illustrates a method of operating the medical system 100 of FIG. 1. First in step 200 the magnetic resonance imaging data 122 is received. The magnetic resonance imaging data has a spiral k-space sampling pattern. Next in step 202 the at least one preliminary magnetic resonance image 124 is reconstructed from the magnetic resonance imaging data 122. Then in step 204 the first set of equations 130 is constructed. The first set of equations comprise each of the at least one preliminary magnetic resonance image 124 being equal to an image transformation of at least one clinical image 126. The image transformation makes use of a first spatially dependent kernel 128 for each of the at least one clinical image 126. Next in step 206 a second set of equations 134 is constructed. The second set of equations comprise at least one regularization matrix 132 times the at least one clinical image 126.

[0074] In some examples the regularization matrix 132 may be the same for each clinical image 126. In other examples there may be an individual regularization matrix for each clinical image 126. Finally, in step 208 the first set of equations 130 and the second set of equations 134 are solved simultaneously to numerically obtain the at least one clinical image 126.

[0075] A possible modification of the technique illustrated in FIG. 2 is that the regularization matrix 132 is constructed using a non-diagonal matrix. The construction of the second set of equations 134 would then be performed using the non-diagonal regularization matrix. This modification could for example be achieved by using a second spatially dependent kernel to construct the regularization matrix. The kernel may be used for measuring for example the local curvature and/or derivative within a particular location. This may be very useful for removing spatially dependent or repeating artefacts. For example, the second spatially dependent kernel could be a Gaussian curvature kernel times a spatially dependent factor, a mean curvature kernel times spatially dependent factor, a Laplacian kernel times a spatially dependent factor, and a second derivative kernel time the spatially dependent factor.

[0076] The spatially dependent factor may be useful in calculating how well the image is known at a particular location. For example, the spatially dependent factor could be a spatially dependent signal-to-noise estimate, a spatially dependent estimate of a B.sub.0 magnetic field gradient, and a spatially dependent estimate of the B.sub.0 magnetic field error.

[0077] FIG. 3 illustrates a further example of a medical system 300. The medical system 300 illustrated in FIG. 3 is similar to the medical system 100 illustrated in FIG. 1. In FIG. 3 the medical system 300 has been specifically constructed or programmed to reconstruct images according to a Dixon magnetic resonance imaging protocol. The at least one clinical image 126 in this example is a Dixon water image 126′ and a Dixon fat image 126″. The at least one regularization matrix 132 is a separate water regularization matrix 132′ for the Dixon water image 126′ and a fat regularization matrix 132″ for the Dixon fat image 126″. The memory 110 is further shown as containing a water mask 320 and a fat mask 322.

[0078] The water mask 320 is used to identify the location or partial volume of water in individual voxels of the Dixon water image 126′. The fat mask 122 is used to identify the location of fat or partial volume of fat within voxels of the Dixon fat image 126″. The water mask 320 and the fat mask 322 could be obtained in different ways. For example, they could be obtained using a lower resolution scan. In some instances, previous magnetic resonance imaging images may be segmented and this information may be used for constructing a realistic water mask 320 and/or fat mask 322.

[0079] In this example the first set of equations 130 is a convolution of the Dixon water image 126′ and a Dixon fat image 126″ being equal to each of the at least one preliminary magnetic resonance image 124.

[0080] In this example the construction of the second set of equations is performed by multiplying the water regularization matrix times a matrix regularization of the Dixon water image 126′. This for example may be set equal zero or a constant. The second set of equations is further constructed by multiplying the fat regularization matrix 132″ times the Dixon fat image 126″. This again may for example be set equal to zero or a constant.

[0081] To use the water mask 320 and the fat mask 322 the water mask 322 is multiplied by the water regularization matrix 132′ and the fat mask 322 is multiplied by the fat regularization matrix 132″. For example, the water mask 320 may have an element for each of the water regularization matrix 132′. Likewise, the fat mask 322 may have an element for each of the fat regularization matrix 132″. In this example the multiplication would then be the multiplication of corresponding elements.

[0082] FIG. 4 shows a flowchart which illustrates a method of operating the medical system 300 of FIG. 3. In this method steps 200, 202, and 204 are performed in an analogous way using the medical system 300 of FIG. 3. Next in step 400 the water regularization matrix 132′ is multiplied by the water mask 320 before constructing the second set of equations. Next in step 402 the fat regularization matrix 132″ is multiplied times the fat mask 322 before constructing the second set of equations. After this the method then performs steps 206 and 208 which are analogous to steps 206 and 208 in the method of FIG. 2.

[0083] FIG. 5 illustrates a further example of a medical system 500. The medical system 500 in FIG. 5 is similar to the medical system 100 in FIG. 1 except that the medical system additionally comprises a magnetic resonance imaging system 502.

[0084] The magnetic resonance imaging system 502 comprises a magnet 504. The magnet 504 is a superconducting cylindrical type magnet with a bore 506 through it. The use of different types of magnets is also possible; for instance it is also possible to use both a split cylindrical magnet and a so called open magnet. A split cylindrical magnet is similar to a standard cylindrical magnet, except that the cryostat has been split into two sections to allow access to the iso-plane of the magnet, such magnets may for instance be used in conjunction with charged particle beam therapy. An open magnet has two magnet sections, one above the other with a space in-between that is large enough to receive a subject: the arrangement of the two sections area similar to that of a Helmholtz coil. Open magnets are popular, because the subject is less confined. Inside the cryostat of the cylindrical magnet there is a collection of superconducting coils.

[0085] Within the bore 506 of the cylindrical magnet 504 there is an imaging zone 508 where the magnetic field is strong and uniform enough to perform magnetic resonance imaging. A field of view 509 is shown within the imaging zone 508. The magnetic resonance data that is acquired typically acquired for the field of view 509. A subject 518 is shown as being supported by a subject support 520 such that at least a portion of the subject 518 is within the imaging zone 508 and the field of view 509.

[0086] Within the bore 506 of the magnet there is also a set of magnetic field gradient coils 510 which is used for acquisition of preliminary magnetic resonance data to spatially encode magnetic spins within the imaging zone 508 of the magnet 504. The magnetic field gradient coils 510 connected to a magnetic field gradient coil power supply 512. The magnetic field gradient coils 510 are intended to be representative. Typically magnetic field gradient coils 510 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 510 is controlled as a function of time and may be ramped or pulsed.

[0087] Adjacent to the imaging zone 508 is a radio-frequency coil 514 for manipulating the orientations of magnetic spins within the imaging zone 508 and for receiving radio transmissions from spins also within the imaging zone 508. The radio frequency antenna may contain multiple coil elements. The radio frequency antenna may also be referred to as a channel or antenna. The radio-frequency coil 514 is connected to a radio frequency transceiver 516. The radio-frequency coil 514 and radio frequency transceiver 516 may be replaced by separate transmit and receive coils and a separate transmitter and receiver. It is understood that the radio-frequency coil 514 and the radio frequency transceiver 516 are representative. The radio-frequency coil 514 is intended to also represent a dedicated transmit antenna and a dedicated receive antenna. Likewise the transceiver 516 may also represent a separate transmitter and receivers. The radio-frequency coil 514 may also have multiple receive/transmit elements and the radio frequency transceiver 516 may have multiple receive/transmit channels. For example if a parallel imaging technique such as SENSE is performed, the radio-frequency could 514 will have multiple coil elements.

[0088] The transceiver 516 and the gradient controller 512 are shown as being connected to the hardware interface 106 of a computer system 102. The memory 110 is further shown as containing pulse sequence commands. The pulse sequence commands 530 are commands or data which may be translated into such commands which control the magnetic resonance imaging system 502 to acquire magnetic resonance imaging data according to a Dixon magnetic resonance imaging protocol.

[0089] The memory 110 is shown as additionally comprising a set of pulse sequence commands 530. The pulse sequence commands 530 enable the processor 104 to control the magnetic resonance imaging system 502 to acquire the magnetic resonance imaging data 122 with a spiral k-space sampling pattern.

[0090] The features of the medical system 500 in FIG. 5 may also be freely combined with the features of the medical system 300 of FIG. 3. That is to say the pulse sequence commands 530 may be used to acquire the magnetic resonance imaging data 122 according to a Dixon magnetic resonance imaging protocol.

[0091] FIG. 6 shows a method of operating the medical system 500 of FIG. 5. The method in FIG. 6 is similar to the method illustrated in FIG. 2. The method in FIG. 6 starts with step 600. In step 600 the processor 104 controls the magnetic resonance imaging system with the pulse sequence commands 530 to acquire the magnetic resonance imaging data 122. After step 600 the method proceeds to steps 200, 202, 204, 206, and 208 as is illustrated in FIG. 2.

[0092] In spiral scan reconstruction (of magnetic resonance imaging data with a spiral k-space sampling patter), a regularization step within the ‘deblurring’ step can be beneficial for image quality. Examples may implement this regularization as a high-pass filtering; that may be spatially dependent. The regularization level (or its exact shape) may depend on the gradient of the estimated field-deviation in that position.

[0093] Examples may relate to the reconstruction of spiral scans (using a spiral k-space sampling pattern). In particular, it may provide an improvement upon what is known as “deblurring” correcting for blurring that occurred due to an offset of the magnetic field (using the B.sub.0 magnetic field gradient and/or error), whereby the field at that location may be assumed to be known.

[0094] Examples may have the benefit that the prevention of over-enhancement of high-spatial frequencies. Nowadays, this is prevented by “stopping to iterate the numerical solution before it gets too bad”, which is a pretty inexact solution to the problem.

[0095] In some examples, the deblurring may be performed by applying a regularization (constructing 206 a second set of equations) that consists of a high-pass filtering; more specifically, the regularization may comprise of a position-dependent high-pass filtering; preferably, the regularization level (or its exact shape) depend on the gradient of the estimated field-deviation in that position.

[0096] This technique is discussed in the context of a Dixon magnetic resonance imaging technique described below:

[0097] In Dixon magnetic resonance imaging a multiplicity of spiral sampling patterns are acquired in k-space. Examples are equally viable for a single echo, but mostly two or three different echo times are acquired, so the invention is written against that situation. So one acquires spiral k-space samples with slightly different echo times TE.sub.1, . . . , TE.sub.L, . . . TE.sub.E. Mostly, these will be distributed over “one water-fat cycle” (e.g. 2.3 ms for 3 T), so for three echoes, the echo times may be chosen to differ by 0.77 ms—but this is no strict prerequisite. All of these images (the preliminary magnetic resonance images 124) will, at least locally, be unshar.

[0098] A process of image reconstruction is as follows: as input the (complex) data from the three aforementioned echoes, which will be called Ĩ.sub.b1, . . . , Ĩ.sub.bi, . . . , Ĩ.sub.bE. The tilde symbol (˜) indicates the presence of blurring. The Fourier transform of these images (the preliminary magnetic resonance images 124) is denoted as K.sub.Ti(u,v). One may denote the “true” water image (Dixon water image 126′), which one does not know but would like to derive, as I.sub.W(m,n). Equivalently, the unknown fat image (Dixon fat image 126″) is called I.sub.F(m,n).

[0099] The unregularized “forward model” is expressed as:

[00002]$K_{Ti}\left(u,v\right)=\underset{m,n}{.Math.}{I}_W\left(m,n\right).Math.\exp \left(\frac{j2\pi \left(mu+nv\right)}{N}\right).Math.\exp \left(j\mathrm{\gamma \Delta }{B}_0\left(m,n\right).Math.\left(T{E}_i+t\left(u,v\right)\right)\right)+\underset{m,n}{.Math.}{I}_F\left(m,n\right)\exp \left(\frac{j2\pi \left(mu+nv\right)}{N}\right).Math.\exp \left(j\gamma \Delta B_0\left(m,n\right).Math.\left(T{E}_i+t\left(u,v\right)\right)\right).Math.\exp \left(j\Delta f_F.Math.\left(T{E}_i+t\left(u,v\right)\right)\right)$

[0100] Read K.sub.Ti(u,v) as “the k-space signal observed at echo i at k-space positions (u,v)”; read I.sub.W(m,n) as “the true water-proton concentration at spatial positions (m,n)” and similarly I.sub.F(m,n) for “fat”; read ΔB.sub.0(m,n) as “the deviation of the field at spatial position (m,n) relative to the nominal magnetic field” (which information is provided from ‘outside’, e.g. a field-map generated from a pre-scan); read t(u,v) as “the time at which the k-space sample (u,v) is acquired, relative to the first sample of the spiral-readout” (t(u,v) is assumed to have the same shape for all echoes); read TE.sub.i as “the difference of the time-point of the first sample of echo i, relative to the spin-echo time point of the sequence”; and read Δf.sub.F as “the difference in frequency between fat and water-protons” (a somewhat simplified formula, since fat actually has more than one spectral component, but one dominates).

[0101] If one considers the Fourier-transform of K.sub.Ti(u,v), called Ĩ.sub.bi, one could recognize that this can be written as a convolution of I with a position-dependent kernel (corresponding to exp(jγΔB.sub.0(m,n).Math.(TE.sub.i+t(u,v)))) and a convolution of I.sub.F with another position-dependent kernel (corresponding to exp (jγΔB.sub.0(m,n).Math.(TE.sub.i+t(u,v)))exp(jΔf.sub.F(TE.sub.i+t(u,v))), typically known. This can also be written in matrix-notation,

Ĩ.sub.bi=C.sub.WiI.sub.W+C.sub.FiI.sub.F,

which can be read as follows: I.sub.W and I.sub.F are column-vectors, each element corresponding to one pixel location; C.sub.Wi and C.sub.Fi are matrices (number-of-pixels×number-of-pixels) representing the position-dependent ‘blurring’ kernels. In still more of a matrix notation,

[00003]${\tilde{I}}_{bi}=\left[\begin{array}{cc}{C}_{\mathrm{Wi}}& C_{\mathrm{Fi}}\end{array}\right]\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right].$

Or, if all of the equations are stacked:

[00004]${\tilde{I}}_b=\left[\begin{array}{c}{\tilde{I}}_{b1}\\ \mathrm{.Math.}\\ {\tilde{I}}_{\mathrm{bi}}\\ \mathrm{.Math.}\\ {\tilde{I}}_{\mathrm{bE}}\end{array}\right]=\left[\begin{array}{cc}{C}_{W1}& C_{F1}\\ \mathrm{.Math.}& \mathrm{.Math.}\\ C_{Wi}& C_{Fi}\\ \mathrm{.Math.}& \mathrm{.Math.}\\ C_{WE}& C_{FE}\end{array}\right]\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right].$

This can be written, short-hand, as

[00005]${\tilde{I}}_b=C\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right].$

This is an example of the first set of equations 130.

[0102] The solution consists of iteratively resolving I.sub.W and I.sub.F from the knowledge of C and Ĩ.sub.b, e.g. by using, for example, steepest descent of conjugate-gradients.

[0103] The approach above can be improved by using regularization by constructing the second set of equations 134. A regularization of the solution (construction of the second set of equations 134) may be adding by including a constraint to the equations, i.e. by extending:

[00006]$C\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right]={\tilde{I}}_b$

Into the set of equations

[00007]$\hspace{1em}\{\begin{array}{c}C\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right]={\tilde{I}}_b\\ \left[\begin{array}{cc}{Q}_W& Q_F\end{array}\right]\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right]=0\end{array}$

One may choose Q.sub.W and Q.sub.F to be identical as a simple case. Q.sub.W and Q.sub.F are water 132′ and fat 132″ regularization matrices respectively and are used to form the second set of equations 134.

[0104] If the Q matrices were to be diagonal, this would lead to a very plain regularization approach (linking it to more common classical jargon, the regularization matrix R corresponds to (Q.sup.hQ).sup.−1, so one can write “R.sup.−1/2” rather than “Q”). The essence here is to make Q non-diagonal, in the sense that it imposes only the high spatial frequencies of I.sub.W and I.sub.F to be zero.

[0105] Actually, each row of (e.g.) Q.sub.W indicates which combination of pixels from I.sub.W is supposed to be zero. Taking a hypothetical 1-dimensional situation to give an example, if one constructs the 7.sup.th row of Q.sub.W to look as follows,

[0 0 0 0 0 −q/2 q −q/2 0 0 . . . ],

[0106] this actually imposes that the 2.sup.nd order derivative of I.sub.W around the 7.sup.th pixel should be zero. The larger the value of q, the stronger one imposes that constraint.

[0107] There is actually a rationale for such an endeavour. This stems from the following model: one wants to image an object using spiral scanning; that object also contains regions where the main-field deviation exhibits a field-gradient (which is of course unwanted, but it is present nevertheless). Whereas in regions of no field-gradient the spiral is well-behaved (plot 700 in FIG. 7 below), in the local presence of e.g. an x-gradient, the spiral ‘drifts away’ (plot 702 in FIG. 7 below).

[0108] FIG. 7 illustrates several spiral k-space sampling patterns 700, 702. In the top Fig. a spiral k-space sampling pattern 700 is represented. In the lower illustration the spiral k-space sampling pattern 702 is shown as being distorted due to a high gradient of the B.sub.0 magnetic field. The area marked 704 is intended to be sampled but is not able to be because of distortions of the sampling pattern 702.

[0109] Without appropriate regularization, the reconstruction algorithm attempts to reconstruct meaningful data corresponding to the spatial frequencies within the circle. It simply does not have enough information to meaningfully reconstruct the information corresponding to the area 704 in plot 702.

[0110] Via regularization, one can ‘tell’ the algorithm that the area 704 is expected to be unreliable. This can be done by an appropriate high-pass filter in that region of the matrix Q. And since one does have an estimate of the local field-gradient, one can estimate the size of the region that the spiral does reach and by that, the extent of the unreliable region. An elaborate implementation would consist of implementing each row of Q by the inverse Fourier transform of the grey-shaded area represented above.

[0111] The regularization matrices 132, 132′, 132″ may be constructed using kernels. This is illustrated in the examples below using a 1-dimensional “image.” This simplifies the explanation and the and extension to two dimensional matrices is straightforward.

[0112] In this simplified example, Multiplying an image with a scalar a can also be seen by multiplying the vector with a diagonal matrix, i.e.:

TABLE-US-00001 a 0 0 0 0 0 0 a 0 0 0 0 0 0 a 0 0 0 0 0 0 a 0 0 0 0 0 0 a 0 0 0 0 0 0 a

[0113] A convolution with a fixed kernel (e.g. a 3-points kernel [a b c]) can also be written as a matrix multiplication, the matrix being:

TABLE-US-00002 b c 0 0 0 0 a b c 0 0 0 0 a b c 0 0 0 0 a b c 0 0 0 0 a b c 0 0 0 0 a b

[0114] Typically, a matrix like C.sub.W1 (from above) could for example look approximately like this:

TABLE-US-00003 1 0 0 0 0 0 0.1 0.8 0.1 0 0 0 0.1 0.2 0.4 0.2 0.1 0 0 −0.2 0.8 −0.2 0.8 −0.2 0 0 −0.2 0.8 −0.2 0.8 0 0 0 0 0.1 0.8

[0115] In this example, in the ‘topmost’ region, there is hardly any field error and so the blurring kernel is represented by a single peak ([0 0 0 1 0 0 0]); further down the ‘image’ (in this example the 4.sup.th pixel) the field substantially deviates, which causes a blurring represented by the kernel [0 0 −0.2 0.8 −0.2 0.8 −0.2 0 0].

[0116] (In practice, the kernel elements would be complex, but this is omitted here for the sake of explanation). In the same way, the regularization matrix Q can also be seen as a matrix as well as a convolution with spatially-dependent kernels.

[0117] Returning to the construction of the second set of equations, a simplified form is to fill each row of Q by a two-dimensional 2nd-derivative kernel. For example:

q/4 q/2 q/4
q/2 q q/2 and derive q by multiplying
q/4 q/2 q/4 [0118] 1/(The estimate of the SNR of the end-result I at that location) [0119] (k.sub.r/k.sub.e).sup.2, where k.sub.r is the nominal k-space extent (the radius of the brown circle) and k.sub.e is the radius of a circle that has been fully “seen” by the spiral scan (the dashed green circle in the right drawing). This can be calculated from the knowledge of the field gradient. E.g. for constant-angular-velocity spirals, k.sub.e=k.sub.r−|∇B|t.sub.a, where t.sub.a is the sampling time of the spiral.

[0120] This can be re-written as (1/(1−|∇B|t.sub.a/k.sub.r)).sup.2, or, in general one has f(|∇B|t.sub.a/k.sub.r) as a factor to q. (“The higher the local gradient of the static field, the higher q, i.e. the less one trusts the high spatial frequencies”.) q may be considered to be an example of the spatially dependent factor in the second spatially dependent kernel.

[0121] This can be extended by writing f(|∇B|t.sub.a/k.sub.r, σ.sub.Bt.sub.a), where σ.sub.B is the expected error one has to the knowledge of the magnetic field at that location. Preferably, this can be combined as f(|∇B|t.sub.a/k.sub.r,σ.sub.Bt.sub.a)=√{square root over ((|∇B|t.sub.a/k.sub.r).sup.2+(2πσ.sub.Bt.sub.a).sup.2)}, so the factor preferably becomes 1/(1−√{square root over ((|∇B|t.sub.a/k.sub.r).sup.2+(2πσ.sub.Bt.sub.a).sup.2)}).sup.2.

[0122] Examples may also provide for better Dixon imaging by using prior knowledge. A benefit of some examples may be the improved problem conditioning for spiral reconstruction leading to faster convergence and improved image quality (through better water/fat separation). Both elements may be beneficial to the overall success in removing or reducing artefacts in magnetic resonance imaging that uses spiral k-space sampling.

[0123] Some examples may use species dependent regularization. In Dixon magnetic resonance imaging, one can readily obtain prior knowledge on the location of water and fat (or ‘fat fraction’) on lower resolution within the anatomy using a pre-scan. This may for example be performed using water masks 320 and fat masks 322. This information is known in the ‘undeformed/non-blurred’ domain and hence can directly be used in the regularization as described above (written slightly different below, this example combines the first and second set of equations):

[00008]$\left[\begin{array}{c}{\tilde{I}}_b\\ 0\end{array}\right]=[\begin{array}{c}C\\ \left[\begin{array}{cc}{Q}_W& 0\\ 0& Q_F\end{array}\right]\end{array}]\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right]$

[0124] Q.sub.W is an example of a water regularization matrix and Q.sub.F is an example of a fat regularization matrix.

[0125] Although the problem is not solved using a least squares solver, the problem solution can be written as such:

[00009]$\left[\begin{array}{c}{I}_W\\ I_F\end{array}\right]={\left(C^{H}C+\left[\begin{array}{cc}{Q}_W^H{Q}_W& 0\\ 0& Q_F^H{Q}_F\end{array}\right]\right)}^{-1}{C}^H{\tilde{I}}_b$

[0126] Balancing of the regularization with the data ‘consistency’ term can be done similar to how it is done in SENSE.

[0127] 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.

[0128] 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 fulfil 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

[0129] 100 medical system [0130] 102 computer [0131] 104 processor [0132] 106 hardware interface [0133] 108 user interface [0134] 110 memory [0135] 120 machine executable instructions [0136] 122 magnetic resonance imaging data (spiral k-space) [0137] 124 at least one preliminary magnetic resonance image [0138] 126 at least one clinical image [0139] 126′ Dixon water image [0140] 126″ Dixon fat image [0141] 128 first spatially dependent kernel [0142] 130 first set of equations [0143] 132 at least one regularization matrix [0144] 132′ water regularization matrix [0145] 132″ fat regularization matrix [0146] 134 second set of equations [0147] 200 receive magnetic resonance imaging data [0148] 202 reconstruct at least one preliminary magnetic resonance image from the magnetic resonance imaging data [0149] 204 construct a first set of equations comprising each of the at least one preliminary magnetic resonance image being equal to an image transformation of at least one clinical image [0150] 206 construct a second set of equations comprising at least one regularization matrix times the at least one clinical image [0151] 208 numerically solve the first set of equations and the second set of equations simultaneously for the at least one clinical image [0152] 300 medical system [0153] 320 water mask [0154] 322 fat mask [0155] 400 multiply the water regularization matrix times a water mask before constructing the second set of equations [0156] 402 multiply the fat regularization matrix times a fat mask before constructing the second set of equations [0157] 500 medical system [0158] 502 magnetic resonance imaging system [0159] 504 magnet [0160] 506 bore of magnet [0161] 508 imaging zone [0162] 509 field of view [0163] 510 magnetic field gradient coils [0164] 512 magnetic field gradient coil power supply [0165] 514 radio-frequency coil [0166] 516 transceiver [0167] 518 subject [0168] 520 subject support [0169] 530 pulse sequence commands [0170] 600 control the magnetic resonance imaging system with the pulse sequence commands to acquire the magnetic resonance imaging data [0171] 700 spiral k-space sampling pattern [0172] 702 distorted spiral k-space sampling pattern [0173] 704 area inaccessible to sampling