System and method for image reconstruction in magnetic resonance imaging

Abstract

To provide a system and a method for effectively reconstructing a magnetic resonance image from a magnetic resonance imaging signal.SOLUTION: A system and a method for reconstructing a robust magnetic resonance image enable modeling in terms of all or almost all components within a magnetic resonance imaging system, and enable optimization so that reconstruction can be executed within a short period of time, by possessing a compressibility function for speeding up the reconstruction.SELECTED DRAWING: Figure 5

Description

[Technical field]

[0001] The present disclosure relates generally to systems and methods for efficiently reconstructing magnetic resonance images from magnetic resonance imaging signals, and more particularly to systems and methods that allow generalization to incorporate all or nearly all aspects of a magnetic resonance imaging system, as well as compressibility and optimization to speed up the image reconstruction process. [Background technology]

[0002] In magnetic resonance imaging (MRI), signals from an object or sample are measured by applying a train of pulses. In a conventional MRI system, the measured signals are acquired over a linear grid of frequencies (also called a Cartesian scan), and an image can be reconstructed by a 2D inverse fast Fourier transform (FFT). Another common method acquires samples on a polar grid and uses a 2D inverse FFT after interpolating this data to a rectangular grid. [Brief description of the drawings]

[0003] [Figure 1] 1 is an exemplary schematic diagram of a linear system model that models image reconstruction in a magnetic resonance imaging (MRI) system, according to some embodiments of the present disclosure.

[0004] [Diagram 2] 1 is an exemplary diagram of a slice profile of a main magnetic field in an MRI system, according to some embodiments of the present disclosure.

[0005] [Figure 3A-3D] 1A-1C are exemplary diagrams of extracted planes corresponding to slice excitation, according to some embodiments of the present disclosure.

[0006] [Figure 4A-4B] 1A-1C are exemplary diagrams of nonlinearly varying spatial magnetic field gradients in each of the x and y directions, according to some embodiments of the present disclosure.

[0007] [Diagram 5] 1 is an exemplary flowchart illustrating a method for reconstructing a magnetic resonance image based on a magnetic resonance imaging signal according to some embodiments of the present disclosure.

[0008] [Figure 6] FIG. 1 is a block diagram illustrating a computer system on which an embodiment of the present teachings may be implemented.

[0009] [Figure 7] 1 is an exemplary block diagram illustrating a system for reconstructing a magnetic resonance image based on a magnetic resonance imaging signal according to some embodiments of the present disclosure.

[0010] It should be understood that the figures are not necessarily drawn to scale, and that objects in the figures are not necessarily drawn to scale in relation to each other. The figures are illustrations intended to bring clarity and understanding to various embodiments of the devices, systems, and methods disclosed herein. Wherever possible, the same reference numbers will be used throughout the figures to refer to the same or similar parts. Additionally, it should be appreciated that the drawings are not intended to limit the scope of the present teachings in any way. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0011] The following summarizes some aspects of the present disclosure to provide a basic understanding of the described technology. This summary is not an extensive overview of all envisioned features of the present disclosure, and is not intended to identify key or critical elements of all aspects of the present disclosure or to delineate the scope of any or all aspects of the present disclosure. Its sole purpose is to present some concepts of one or more aspects of the present disclosure in summary form as a prelude to the more detailed description that is presented later.

[0012] In some embodiments of the present disclosure, a method and system for effectively reconstructing a magnetic resonance image from a magnetic resonance imaging signal is disclosed. In some embodiments, a transmit coil and a radio frequency (RF) source can be used to apply a total magnetic field including a magnetic field gradient and one or more RF pulse trains, respectively, to measure one or more states of a sample. Also, MRI measurement data acquired by the receive coil during an acquisition window can be received by a processor from one or more receive coils, the MRI measurement data including magnetic resonance signal data emitted by the sample. Also, a first encoding matrix can be generated by linearly combining a subset of the MRI measurement data, the first encoding matrix being a submatrix of a second encoding matrix configured to (i) be generated by linearly combining all the MRI measurement data and (ii) represent one or more states, transmit coils, and receive coils of the sample. Furthermore, a characteristic of the MRI measurement data can be determined by the processor based on the calculation of the first encoding matrix. Detailed Description

[0013] The following description of various embodiments is illustrative and explanatory only and should not be construed as limiting or restrictive in any sense. Other embodiments, features, objects, and advantages of the present teachings will be apparent from this description and the accompanying drawings, as well as from the claims. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which the present invention belongs.

[0014] All publications mentioned herein are incorporated by reference for the purpose of describing and disclosing the devices, compositions, formulations and methods that are described in the publications and that can be used in connection with the present disclosure.

[0015] As used herein, "comprising," "containing," "having," "including," and variations thereof are not intended to be limiting and are inclusive or open-ended and do not exclude additional, unrecited additives, components, integers, elements, or method steps. For example, a process, method, system, composition, kit, or apparatus that includes a list of features is not necessarily limited to only those features and may include other features not expressly recited or inherent to such process, method, system, composition, kit, or apparatus.

[0016] In some embodiments, the term "signal component" may refer to the portion of the magnetic resonance imaging (MRI) system responsible for generating the signal. This may include, but is not limited to, at least the pulse train structure, the transmit coils and the receive coils. For each state of the pulse train, all or almost all the receive coils may acquire data over an acquisition window. In some embodiments, the term "acquisition window" may refer to the time within the magnetic resonance (MR) pulse train during which MR signals are recorded.

[0017] In some embodiments, the term "pulse train" may refer to a set of radio frequency (RF) and gradient pulses that are repeated multiple times during an acquisition period to encode spatial and amplitude information that describes an object (e.g., a sample). In some cases, these repetitions may vary in the magnitude of the gradients or the shape of the individual pulses applied, and may ultimately be determined by the quality of the image desired. In an MRI system, a z-axis gradient is applied to select the slice to excite, x-axis and y-axis gradients are applied to spatially localize, and an RF pulse is applied to perform the excitation.

[0018] In some embodiments, the term "transmit coil" may refer to a coil of wire in an MRI system that can generate an oscillating or rotating magnetic field perpendicular to the main static magnetic field ("B0") of the MRI system. In some embodiments, the term "receive coil" may refer to a coil of wire in an MRI system that detects magnetic resonance (MR) signals. In some aspects, the receive coils may be different such that their spatial sensitivities (i.e., amplitude) and phases cross. Additionally, in some aspects, the transmit coils may also be different such that their spatial sensitivities (i.e., amplitude) and phases cross.

[0019] In some embodiments, the term "image acquisition timing" may refer to the time to perform an MR imaging procedure, including only data acquisition time. In some cases, the total image acquisition time may be equal to the product of the repetition time, the average number of signals, and the number of distinct signals (encoded for position) that must be acquired for use in image reconstruction. Additional image reconstruction time is also important to determine how quickly an image can be viewed. When comparing sequential planar and volumetric imaging techniques, the equivalent image acquisition time and the actual image acquisition time per slice may be considered.

[0020] In some embodiments, the term "frequency offset" can refer to the difference between a given signal frequency and a reference frequency. In some aspects, the frequency offset can be the center frequency for the slice excitation.

[0021] In some embodiments, an object imaged by an MRI system can emit a signal that can be measured by the MRI system, including the amplitude of the measured frequency. In some cases, the measured signal can be acquired over a linear grid or Cartesian scan of frequencies, and the image can be reconstructed by a 2D inverse Fourier transform (FFT). In some cases, samples may be acquired on a polar grid, and a 2D non-uniform FFT is used after interpolating this data to a rectangular grid. In such an approach, the problem can be defined as follows: s(t)=∫m(r)e -j2πk(t)r dr

[0022] where s(t) is the measured signal, k represents the k-space trajectory, r corresponds to the sampled spatial location, and the problem can be formulated as the reconstruction of image m by observing a number of samples at s.

[0023] However, such techniques may have various limitations, including requiring the signal to be adequately sampled or regridded in the case of non-Cartesian grids or undersampling before performing a non-uniform FFT, i.e., to have sufficient bandwidth and adequate sample spacing to adequately capture the object without limiting the field of view or aliasing. Non-Cartesian acquisitions may also require frequency sampling density correction to minimize over-representation of the k-space region, which may result in artifacts. Furthermore, these methods may make several assumptions that may simplify or under-capture the MRI system, including that the gradients are linear, that the magnet's main magnetic field B0 is uniform, that the receive coils have similar spatial sensitivity profiles, etc. Also, these methods may mathematically model the signals in the pulse train without practical considerations such as rise and fall times, the true shape of the gradient profile, etc. As a result, image reconstruction for multi-parametric MRI may not adequately model these factors. Furthermore, image reconstruction for multi-parametric MRI may take several hours for high quality images.

[0024] Although several methods have been proposed to overcome these limitations (e.g., non-uniform fast Fourier transform, sensitivity coding, generalized auto-calibrated partially parallel acquisition, compressed sensing techniques, etc.), the above limitations may still exist and there is a need for robust image reconstruction that can be modeled for all or almost all components in the system, has compressibility features to speed up reconstruction, and can be optimized so that reconstruction is achievable in minutes rather than hours. The present disclosure discloses embodiments configured for magnetic resonance image reconstruction that address the above challenges and limitations to provide a more robust system and method for image reconstruction.

[0025] Some embodiments of the present disclosure disclose an image reconstruction technique or method that can be generalized, compressed, and / or optimized. The method is generalizable in that the image reconstruction technique can model all or almost all components in the system, including at least the magnetic field gradient, the main magnetic field B0, the phase and amplitude sensitivity of the transmit and receive coils, the shape of the gradient profile, the phase during measurement, which may include the phase induced by the readout gradient and the x and y gradients in the presence of a permanent main magnetic field. The technique or method can also have compressibility features to speed up the reconstruction, for example, image reconstruction without dense sampling in a manner that can exploit the characteristics of the imaged object. In some embodiments, compressibility can be achieved by various mechanisms such as random undersampling, parallel reconstruction using multiple spatially sensitive coils, sparsity constraints using wavelet compression, L1 and kernel type norm objective functions, multiple slice excitation within a single iteration time, and broadband chirped pulse excitation to excite large regions. Furthermore, the image reconstruction technique or method can be optimized such that reconstruction is feasible in a few minutes or within an hour.

[0026] FIG. 1 illustrates an example schematic diagram of a linear system model for modeling image reconstruction in a magnetic resonance imaging (MRI) system, according to some embodiments of the present disclosure. In general, a model for reconstructing an image from given parameter measurements can be generally described as linear. As shown in FIG. 1, the linear system model 100 includes a signal vector signal(s) 102, which includes all or almost all measurements obtained on the system (i.e., on the object or sample) from multiple receive coils. The linear system model 100 also includes a matrix E 104, which acts as an encoding matrix that includes all or almost all generalizations of a given system. In some cases, the matrix E is a block matrix E0, E1, ..., E N-1 Each block can be constructed from N=n s corresponds to a single line of pulse trains separated by repetition times (RT), s is the number of pulse train lines. Furthermore, each block matrix may be constructed from a separate row corresponding to a single time point for that line and one receive coil in the pulse train from the signal window being recorded. The total number of rows in matrix E is the number of states (i.e., pulse train lines), n states , the number of receiver coils used n coils and the number of points in the acquisition window, n t (i.e., n states ×n coils ×n t ) The linear system model 100 further includes an image vector m106 to be solved by making observations on the signal s102, where the matrix E104 describes the state of the system. That is, the state of the MRI system 100, including system features such as magnetic field gradients, receive and transmit coils, main magnetic field B0 configuration for the pulse sequence, etc., is encoded in the matrix E104.

[0027] In some embodiments, the element E in the E matrix 104 ij can be modeled as follows:

number

[0028] The phase term can be calculated as follows:

number

[0029] In these definitions, the main magnetic field B0 and the gradient magnetic field may be expressed in units of radians (premultiplied by gamma). These formulas allow for the calculation of phases from the gradient profile matrix (A) and the phase matrix (B) while incorporating the non-uniform magnetic field B0 and the center frequency offset matrix F. In some embodiments, the above encoding matrix E may have a particular structure. In some cases, the encoding matrix E may be completely determined by the transmit and receive coil maps, the A and B matrices, and may not require complete storage in memory to generate or reconstruct a magnetic resonance image. In some embodiments, the encoding matrix E may be exceptionally large, for example, on the order of hundreds of thousands of rows and a similar number of columns. However, in some cases, determining individual elements in the matrix as needed without requiring storage of the entire matrix allows for advantageous exploitation of the above structure in the encoding matrix E.

[0030] In some embodiments, the measured signal may contain a small global shift, and the reconstructed image may contain a central bright dot artifact. To remove this, a column of 1s may be added to the encoding matrix and the image reconstructed with the estimated shift coefficients.

[0031] Index i corresponds to a single acquisition for a unique pulse train line, coil (e.g., receive or transmit) and time point. s and i c is extracted from i and defined as: [0,n states ×n coils ×n t For i in ], i s refers to the pulse train line, and i c refers to the signal recorded for a particular coil, and i t corresponds to the time when the signal was recorded within the signal window, and j corresponds to the jth voxel in the image vector m.

[0032] Parameter B gx , B gy and B. gz can represent the magnetic field measured from the x, y, and z gradients, respectively. When subscripted with x, y, and z, the parameter B represents the magnetic field generated by the gradient in that direction. For example, B gx corresponds to the magnetic field caused by the x-gradient, which is a gradient applied primarily along the x-direction, and is a 3D vector field, and each B gx A position contains x, y and z components. Hence the matrix B gx is image can be expressed as a × 3 matrix, where n image =x n xy n xz n and x n , y n and z n is the number of voxels to be reconstructed along each dimension. gycorresponds to the magnetic field caused by the y-gradient, which is a gradient applied primarily along the y-direction, and is a 3D vector field, and each B gy A position contains x, y and z components. Hence the matrix B gy is image As another example, the matrix B gx is image can be expressed as a × 3 matrix, where the number of voxels in the reconstructed image is n image =x n xy n xz n and x n , y n and z n is the number of voxels to be reconstructed along each dimension. gz corresponds to the magnetic field caused by the z-gradient, which is a gradient applied primarily along the z-direction, and is a 3D vector field, and each B gz A position contains x, y and z components. Hence the matrix B gz is image It can be expressed as a ×3 matrix.

[0033] In some embodiments, the dimensions of the linear system model 100 of FIG. 1 can be defined based on the parameters that are measured, and the parameters that are measured can be identified or defined as follows: In some aspects, a repetition of the pulse train can be referred to as a single state in the system, and the total number of such states is n s (or n states ) for each state. t In some embodiments, a single array of n ,y n ,z n ), and x n , y n and z n is the number of voxels to be reconstructed along each dimension. The image vector m106 shown in Figure 1 has a total length equal to the product of the dimensions, i.e., n image =xn xy n xz n The vector can be represented as a linear array (vectorized) of all voxels such that

[0034] In some embodiments, the MRI system may be provided with multiple coils (receiving coils), each coil performing measurements simultaneously (e.g., and independently). The number of such receiving coils is determined by the parameter n coils It can be represented by:

[0035] In some embodiments, the matrix A (e.g., p A In the formula for s x2xn m Given a dimension denoted by n m is the number of points in the readout window (during the acquisition). For each pulse train state in the first dimension and for each gradient direction in the second dimension, matrix A can represent the normalized shape of the applied readout gradient by containing a gradient profile stored along the third dimension (e.g. along the readout) for the duration of the acquisition. In a simplified setup, matrix A may have an identical profile for all states of the pulse train, i.e. the same gradient can be applied during the readout. For example, a rectangular pulse during the readout such that all signals recorded during the readout are affected by this gradient (along this gradient direction).

[0036] In some embodiments, the matrix B (e.g., p B In the formula for sThe dimensions of the matrix B are given by x2. For each pulse train state in the first dimension, matrix B may contain the phases stored for each gradient. These are gradient pulses applied before acquiring a signal such that the spins are given a phase corresponding to this gradient pulse during signal acquisition. As a non-limiting example, in a simple single-phase imaging experiment acquiring 30x30 phase encoding, there are 900 states in the pulse train, or 30 along one direction and 30 along the other direction with varying gradient strengths. Each row of the B matrix may contain the accumulated phases due to each gradient for the corresponding pulse train line.

[0037] In some embodiments, the matrix C - (For example, p C (in the equation for ) may be referred to as the receive coil sensitivity matrix, and n image x2xn coils For each voxel in the image referenced by the first dimension, the second dimension corresponds to the amplitude and phase for that voxel and for each receive coil index specified in the third dimension (in the case of multiple coils). Thus, the matrix C - stores a two-component (amplitude and phase) sensitivity map for each reception across the entire image. For each voxel location, the amplitude and phase sensitivities are C - The matrix C is stored for each receiving coil. + (also called the transmit coil matrix) is C - However, the parameter t may be configured for the transmit coil. w represents the time window in which the echo is measured, centered around the echo time.

[0038] In defining the dimensions of the linear system of Figure 1, we can provide a calculation for the measured signal s, which includes the signals recorded from all states and all coils: s=((s 0,0 ) T ,(s 0,1 ) T ,(s 0,2 ) T,(s 0,3 ) T ,...,(s ns-1,0 ) T ,(s ns-1,1 ) T ,(s ns-1,2 ) T ,(s ns-1,3 ) T ) Here, s i,j Each s corresponds to the acquisition window measured for state i and coil j. That is, since each state is a single pulse train line, coil j refers to the j-th coil that is recording a signal when the i-th pulse train line ends. Thus, the signal contains all possible measurements of the system including all coils, all states, and the entire acquisition window. That is, each s i,j are vectors (points recorded along the readout) that are all stacked together to form a much larger vector s, which contains all the states, all the coils and all the points, i.e. all the signals recorded by the system. Referring to Figure 1, the length of the vector s and the image vector m, respectively, are length(s)=n s ×n coils ×n t and length(m)=n image In some embodiments, the vector m is equal to the number of voxels, i.e., the vector m may be a linear array of all voxel intensities that may be determined by reconstruction or an image size selected by a user to reconstruct the image.

[0039] Therefore, taking these two lengths into account, the dimension of the encoding matrix E, i.e. the number of rows and columns of the encoding matrix, is n s ×n coils ×n t , n image It is expressed as:

[0040] In some embodiments, data from the receive coils may already be collapsed prior to reconstruction. In such a pre-collapsed situation, the above formulation can be generalized to the case of only one coil with adjusted phase and averaged measurements from all coils. In other instances, separate images may be reconstructed using one coil at a time and then combined in the image domain. Several methods of combining in the image domain are discussed in "Adaptive reconstruction of phased array MR imagery," DOWalsh, AFGmitro, and MWMarcellin, Magnetic Resonance in Medicine (May 1, 2000), which is incorporated by reference in its entirety.

[0041] In some embodiments, a method for providing enhanced magnetic resonance image reconstruction includes encoding matrix configuration with constraints. The method provides a generalized framework for image reconstruction that advantageously provides a universal encoding matrix setting that takes into account nonlinear gradients, nonuniform B0 magnetic field (i.e., nonuniform main magnetic field), global signal shift during image acquisition, sensitivity amplitude and phase that vary in transmit and receive coils, and excitation frequency and bandwidth of receive signals that support all or almost all pulse trains. Additional constraints for compressibility in the wavelet domain and smoothness of the reconstructed image can also be optimized simultaneously.

[0042] The method also advantageously solves for curved 3D image slices excited by the main magnetic field B0 using a simultaneous iterative reconstruction, conjugate gradient and compressive sensing solver configured to use a universal encoding matrix. Moreover, the provided solver can be advantageously optimized on a graphics processing unit (GPU) without the need to create a universal encoding matrix all at once, making it possible to solve very large linear systems, e.g., with rows and columns exceeding 500,000 each. This is because the underlying matrices (e.g., A, B, C)+ , C - This can be done by leveraging the structure of the encoding matrix by storing multiple image slices (e.g., 3D slices) and dynamically creating them. Additionally, the method further advantageously provides an interpolation method (e.g., a volumetric interpolation method) for combining overlapping curvilinear 3D image slices to construct a complete rectilinear 3D volume.

[0043] Referring now to FIG. 2, the main magnetic field B0 can generally be designed to optimize linearity and homogeneity within a slice. However, the contours of points that receive roughly the same degree of excitation across the entire main magnetic field for a slice excitation frequency can be in the form of a curved surface as shown in FIGS. 3A-3D. In some embodiments, the voxel intensities at all points within the curved cross section can be solved. The curved surface can be extracted by a linear regression fit to the data, assuming an elliptical geometry. In some cases, the surface can be curved as a result of the shape of the main B0 magnetic field; that is, its contours can generate this shape. Thus, when a frequency is excited, curved regions whose resonant frequencies match this excitation frequency are excited in the same way, allowing an image to be reconstructed in this region. In the case of multiple slices, the reconstructions have similar curved geometry and can appear as surfaces of some thickness, called slabs. Many such slabs can be reconstructed across the field of view and interpolated to create a 3D rectilinear volume.

[0044] In some embodiments, the following exemplary workflow can be applied to identify a particular slab to be reconstructed: First, a field of view and resolution of the slab, defined by x, y and z parameters, is selected from which a mesh grid is created in the x and y parameters. The field of view in the z parameter may depend on the bandwidth of the transmit and receive coils.

[0045] Secondly, a plane corresponding to the center of the slab is determined from the slice excitation frequency, for example using a regression model. For example, when creating a regression model, the features given to fit the regression model are the x, y parameters and the frequency (γB0) for known values ​​of z. In this case, the measurement data of the main magnetic field is represented by B0(x,y,z) (where the magnetic field is represented at all measurement positions of x, y and z) and the frequency is represented by γB0. From this, the coefficients of the model can be estimated. Using the coefficients of the model and the mesh grid across x and y, the z positions corresponding to these x and y grid points can be determined. This constitutes the desired slab center plane.

[0046] Then, using the determined slab center plane, additional planes parallel to the center plane can be created to span the field of view in z. After this is complete, a grid G fl (x,y,z) is given, where f l is the lth excited frequency.

[0047] From there, G fl At each grid point in the grid, a slab can be reconstructed. As a result, the main magnetic field B can be extracted at every point along the grid, which we call B G In addition, the gradients at these locations may also be extracted. According to various embodiments, these gradients are spatially nonlinear, as shown below in FIG. 4A for the x-axis and FIG. 4B for the y-axis, so that the magnetic fields from these gradients in x and y are expressed as B x G and B. y G Finally, the spatial sensitivity of the transmit coil and receive coil maps may also be determined in a similar manner.

[0048] Thus, a method is provided for identifying a first plane to be reconstructed. The first plane may be a slab. The first plane may have a curved geometry. The method may include selecting a field of view for the first plane and selecting a resolution in x, y and z. The method may further include creating a mesh grid in x and y. The method may further include determining a second plane corresponding to the center of the first plane for a particular slice excitation frequency. The determining may include creating a regression model. The method may further include creating an additional second plane parallel to the central plane. The creation may include creating a grid G ​​for the entire first plane by spanning the field of view in z. fl (x,y,z), where f l is the lth excited frequency.

[0049] Reconstruction

[0050] In some embodiments, after the linear system is set up as described above, the image m (represented as a curved slab as described above) may be solved for a series of slice excitation positions. Each such reconstructed image is represented as m sl where sl corresponds to the index of the excited slice. We obtain a set of reconstructions {m s1 ,m s2 ,...,m sn After generating}, the final volume V may be constructed as described in more detail below.

[0051] 1, with the solved matrix E, reconstruction begins after the entire signal vector s is acquired, or signal data is collected depending on the method used. Reconstruction can be achieved using multiple reconstruction methods (individually or in combination), including, for example, simultaneous iterative reconstruction technique (SIRT) and conjugate gradient methods, such as conjugate gradients least squares (CGLS) and conjugate gradient solver with compressed sensing constraints (CG-CS).

[0052] Thus, there is provided a method for performing image reconstruction at a series of slice excitation positions, the method comprising constructing an encoding matrix E, obtaining a signal vector s, and applying a reconstruction method selected from the group consisting of SIRT, CGLS, CG-CS, and combinations thereof.

[0053] In some embodiments, SIRT can also be used for computed tomography reconstruction, where each observation in the measurements and the corresponding row in the encoding matrix causes an update to the image. Essentially, projections from all rows are calculated and the image is updated continuously in parallel. Additionally, several other measurements may be updated simultaneously.

[0054] Conjugate gradient methods can be described in comparison with steepest descent methods. Steepest descent methods generally begin with an initialization of the solution and converge iteratively by taking steps in directions that reduce the cost. This iterative convergence can continue until a sufficiently close solution is reached, e.g., until the image does not change enough to require further iterations. However, steepest descent methods can take steps in directions previously passed in the encoding matrix, resulting in inefficiencies. This can be especially true for poorly tuned and slow-converging systems.

[0055] In contrast, conjugate gradient methods generally rely on orthogonal search directions, taking one step in each direction until convergence. Conjugate gradient methods can be used with an additional regularization term that must be minimized simultaneously. The gradient can be estimated, for example, using the derivative of the combined cost function. Other options include the use of wavelet functions and a total variation term that measures the gradient variation in the image.

[0056] optimization

[0057] The above-mentioned solvers may be particularly slow due to the large size and low rank of the encoding matrix, making general-purpose tools impractical. For example, simply storing a matrix of size 300,000×200,000 in memory is an unfeasible approach. Therefore, according to various embodiments, the provided solver can be advantageously optimized on a graphics processing unit (GPU) without the need to create a universal encoding matrix all at once, making it possible to solve very large linear systems, for example, with more than 500,000 rows and columns.

[0058] One example of such an optimization scheme is compute unified device architecture (CUDA®), Nvidia's approach to data-parallel computing. This approach provides the ability to create hundreds of thousands of threads in parallel. These threads may also cooperate by sharing a shared memory section within a thread block. A CUDA "kernel" is a block of execution code per thread. Depending on the unique ID of the thread, these kernels may execute different portions of the data. An example of this may be vector addition on a vector of length 10,000. In this example, each of the 10,000 components may be added independently, with each component configuring its own thread. The kernel in this example may simply add the corresponding scalar values ​​at index positions 1 to 10,000 depending on the thread's index.

[0059] A kernel can be specified by choosing the number of threads per block and the number of blocks. Varying these parameters can result in some differences in the level of optimization. Two common tasks for parallelization in solving linear systems are E times v, and E H The goal of the algorithm is to compute v times E, where E is the encoding matrix and v is an arbitrary vector. Since the E matrix can be very large, the tool allows for dynamic computing by choosing appropriate values ​​for all matrices (e.g., the A-G matrices above) and incorporating them into the encoding matrix formulation.

[0060] Also, to speed up the computation time, a pre-computed exponent map is used instead of computing the exponential function on the processing unit.

[0061] Volume Interpolation

[0062] As mentioned above, each curved slab is assigned a separate slice excitation {m s1 ,m s2 ,...,msn} from which the final volume V can be constructed. These are separate curvilinear slabs that have been independently reconstructed and then interpolated into a single 3D rectilinear volume. In the volume interpolation stage, these slabs can overlap so that they can be reconstructed into a 3D volume. The volume interpolation step takes into account overlapping intensities from multiple slabs that may jointly contribute to the intensity at a voxel. The steps for performing volume interpolation are described in detail below.

[0063] According to various embodiments, a method for performing volumetric interpolation is provided. The method may include defining a volume V from which an image is to be reconstructed. The defining may be specified by a field of view and a resolution in x, y and z directions (bounding a slab). The method may further include: s1 ,m s2 ,...,m sn For each slab in {{, v ...

[0064] 5 shows an exemplary flow chart illustrating a method for reconstructing a magnetic resonance image based on a magnetic resonance imaging signal according to some embodiments of the present disclosure. At block 510, in some embodiments, a transmit coil and an RF source of the MRI system can be used to apply a total magnetic field, including a magnetic field gradient, and one or more RF pulse trains to the sample or object to measure one or more conditions of the sample. That is, the sample or object can be placed in the MRI system such that a magnetic resonance image of the sample or object can be reconstructed based on the measurements of the MRI system. In some cases, the total magnetic field can include a main magnetic field B0, which may not be uniform.

[0065] At block 520, a receive coil of the MRI system may detect or acquire MRI measurements including magnetic resonance signals emitted by the sample or object during an acquisition window or period. In some embodiments, there may be multiple receive coils, each acquiring MRI measurements (e.g., independently). In some cases, the sensitivity of the receive coils may differ from the sensitivity of the transmit coils.

[0066] At block 530, in some embodiments, a linear system model may be used to generate an encoding matrix by linearly combining at least some of the measurements from the receive coils, where the encoding matrix may model or represent all or almost all components in the MRI system, including at least the magnetic field gradients during the measurement, the main magnetic field B0, the phase and amplitude sensitivities of the transmit and receive coils, the shape of the gradient profile, the readout gradients, and the phases, which may include phases induced by the x and y gradients in the presence of the permanent main magnetic field, etc. That is, a processor of a computing device (e.g., a GPU) may generate a first encoding matrix by linearly combining a subset of the MRI measurement data, where the first encoding matrix is ​​a submatrix of a second encoding matrix configured to (i) be generated by linearly combining all the MRI measurement data and (ii) represent one or more states of the sample, the transmit coils, and the receive coils. In some embodiments, the encoding matrix may include one or more encoding submatrices that encode or represent some of all or almost all components represented by the encoding matrix. For example, sub-matrices can be generated by linearly combining some (e.g., but not all) of the measurement data and can encode some but not all components of the MRI system (e.g., readout gradients and phases).

[0067] At block 540, in some embodiments, characteristics of the MRI measurement data may be determined based on the calculation of the encoding sub-matrix. The characteristics may include one or more of the components, such as the phase acquired during the MRI measurement. In some embodiments, the subsets of the MRI measurement data that are linearly combined to generate the encoding sub-matrix include MRI measurement data for a single one of the magnetic field gradients and a single one of the one or more states of the sample, and the determined characteristics of the MRI measurement include a shape of a readout gradient of the magnetic field gradient. In some embodiments, the subsets of the MRI measurement data that are linearly combined to generate the encoding sub-matrix include MRI measurement data for a single one of the receive coils or a transmit coil, and the determined characteristics of the MRI measurement include phase information of the sample during the measurement of the one or more states of the sample. In some embodiments, the subsets of the MRI measurement data that are linearly combined to generate the first encoding sub-matrix include MRI measurement data related to the amplitude and phase of the sample for a single one of the receive coils or a transmit coil, and the determined characteristics of the MRI measurement include the respective amplitude and phase sensitivity of the single receive coil or a transmit coil. In some embodiments, the characteristics of the MRI measurement data are determined without generating a second encoding matrix.

[0068] In some embodiments, the method further includes reconstructing, by the processor, a magnetic resonance image of the sample based on the determined characteristics of the MRI measurements. In some embodiments, reconstructing the magnetic resonance image includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) technique, or a conjugate gradient solver with compressed sensing constraints (CG-CS) to at least the determined characteristics of the MRI measurements. In some embodiments, the determined characteristics of the MRI measurements are selected from the group consisting of a shape of a readout gradient of a magnetic field gradient, phase information of the sample during measurement of one or more states of the sample, and amplitude and phase sensitivity of each of a single receive coil or transmit coil.

[0069] In some embodiments, the amplitude and phase sensitivity of the single receive coil is different from the amplitude and phase sensitivity of the transmit coil. In some embodiments, the magnetic field gradient is a non-linear magnetic field gradient. In some embodiments, the total magnetic field comprises a non-uniform static magnetic field.

[0070] Computer Implemented Systems

[0071] In various embodiments, the method of reconstructing a magnetic resonance image from a magnetic resonance imaging signal may be implemented by computer software or hardware. That is, the method disclosed herein (e.g., 500 in FIG. 5) may be implemented on a computing device that includes a processor and an engine that receives inputs and generates outputs. In various embodiments, the computing device may be communicatively connected to a data storage device or memory and a display device via a direct connection or through an Internet connection.

[0072] FIG. 6 is a block diagram illustrating a computer system 600 in which embodiments of the present teachings may be implemented. In various embodiments of the present teachings, the computer system 600 may include a bus 602 or other communication mechanism for communicating information and a processor 604 coupled to the bus 602 for processing information. In various embodiments, the computer system 600 may also include memory, which may be a random access memory (RAM) 606 or other dynamic storage device coupled to the bus 602 for determining instructions to be executed by the processor 604. The memory may also be used to store temporary variables or other intermediate information during execution of instructions to be executed by the processor 604. In various embodiments, the computer system 600 may further include a read only memory (ROM) 608 or other static storage device coupled to the bus 602 for storing static information and instructions for the processor 604. A storage device 610, such as a magnetic or optical disk, may be provided and coupled to the bus 602 for storing information and instructions. In some embodiments, the computer system may be a graphics processing unit (GPU).

[0073] In various embodiments, computer system 600 may be coupled via bus 602 to a display 612, such as a cathode ray tube (CRT) or liquid crystal display (LCD), for displaying information to a computer user. An input device 614, including alphanumeric and other keys, may be coupled to bus 602 for communicating information and command selections to processor 604. Another type of user input device is a cursor control 616, such as a mouse, trackball, or cursor direction keys, for communicating directional information and command selections to processor 604 and controlling cursor movement on display 612. This input device 614 generally has two degrees of freedom in two axes, a first axis (i.e., x) and a second axis (i.e., y), allowing the device to specify a position in a plane. However, it should be understood that input devices 614 that allow three-dimensional (x, y, and z) cursor movement are also contemplated herein.

[0074] Consistent with certain implementations of the present teachings, results may be provided by computer system 600 in response to processor 604 executing one or more sequences of one or more instructions contained in memory 606. Such instructions may be read into memory 606 from another computer-readable medium or computer-readable storage medium, such as storage device 610. Execution of the sequences of instructions contained in memory 606 can cause processor 604 to perform the processes described herein. Alternatively, hard-wired logic can be used in place of or in combination with software instructions to implement the present teachings. Thus, implementation of the present teachings is not limited to any specific combination of hardware circuitry and software.

[0075] The terms "computer-readable medium" (e.g., data store, data storage, etc.) or "computer-readable storage medium" as used herein refer to any medium that participates in providing instructions to processor 604 for execution. Such media may take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Examples of non-volatile media may include, but are not limited to, dynamic memory, such as memory 606. Examples of transmission media may include, but are not limited to, coaxial cables, copper wire, and fiber optics, including the wires that comprise bus 602.

[0076] Common forms of computer readable media include, for example, a floppy disk, a flexible disk, a hard disk, a magnetic tape or any other magnetic medium, a CD-ROM, any other optical medium, punch cards, paper tape, any other physical medium with a pattern of holes, RAM, PROM, and EPROM, flash EPROM, other memory chips or cartridges, or any other tangible medium from which a computer can read.

[0077] In addition to computer-readable media, instructions or data may be provided as signals on a transmission medium included in a communication device or system to provide a sequence of one or more instructions to the processor 604 of the computer system 600 for execution. For example, a communication device may include a transceiver having signals indicative of instructions and data. The instructions and data are configured to cause one or more processors to implement the functions outlined in the disclosure herein. Representative examples of data communication transmission connections may include, but are not limited to, a telephone modem connection, a wide area network (WAN), a local area network (LAN), an infrared data connection, an NFC connection, and the like.

[0078] It should be appreciated that the methods, flowcharts, diagrams and accompanying disclosure described herein can be implemented using computer system 600 as a stand-alone device or on a distributed network of shared computer processing resources, such as a cloud computing network.

[0079] The methods described herein may be implemented by various means depending on the application. For example, the methods may be implemented in hardware, firmware, software, or any combination thereof. For hardware implementations, the processing unit may be implemented in one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processors (DSPDs), programmable logic circuits (PLDs), field programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, electronic devices, other electronic units designed to perform the functions described herein, or combinations thereof.

[0080] In various embodiments, the methods of the present teachings may be implemented as firmware and / or software programs and applications written in conventional programming languages ​​such as C, C++, Python, etc. When implemented as firmware and / or software, the embodiments described herein may be implemented on a non-transitory computer-readable medium having stored thereon a program for causing a computer to execute the above-mentioned methods. It should be understood that the various engines described herein may be provided on a computer system such as computer system 600, such that processor 604 performs the analysis and decisions provided by these engines according to instructions provided by any one or combination of memory components 606 / 608 / 610 and user input provided via input device 614.

[0081] 7 shows an exemplary block diagram illustrating an MRI system 700 for reconstructing a magnetic resonance image based on a magnetic resonance imaging signal, according to some embodiments of the present disclosure. In some embodiments, the MRI system 700 may include a transmit coil 710, a radio frequency source 720, a receive coil 730, and a processor 740. In some embodiments, the MRI system 700 may be coupled to or in communication with a computing device 750 configured to receive output from the MRI system 700 for further analysis, presentation, etc. In some embodiments, the processor 740 may be part of the computing device 750, or the computing device 750 may be a component of the MRI system 700.

[0082] In some embodiments, the transmit coil and the radio frequency (RF) source may be configured to measure one or more states of the sample by applying a total magnetic field including a magnetic field gradient and one or more RF pulse trains, respectively. Also, in some embodiments, the one or more receive coils may be configured to acquire MRI measurement data including magnetic resonance signal data emitted by the sample during an acquisition window. Furthermore, in some embodiments, the processor may be configured to receive the MRI measurement data from the one or more receive coils. Also, the processor may be configured to linearly combine a subset of the MRI measurement data to generate a first encoding matrix, the first encoding matrix being a submatrix of a second encoding matrix configured (i) to be generated by linearly combining all the MRI measurement data and (ii) to represent one or more states, transmit coils, and receive coils of the sample. Furthermore, the processor may be configured to determine a characteristic of the MRI measurement data based on the calculation of the first encoding matrix. In some embodiments, the processor may be further configured to reconstruct a magnetic resonance image of the sample based on the determined characteristic of the MRI measurements.

[0083] While the present teachings have been described in conjunction with various embodiments, it is not intended that the present teachings be limited to such embodiments. On the contrary, the present teachings encompass various alternatives, modifications, and equivalents, as will be recognized by those skilled in the art.

[0084] In describing various embodiments, the specification may present a method and / or process as a particular sequence of steps. However, to the extent that the method or process relies on a particular sequence of steps described herein, the method or process should not be limited to the particular sequence of steps described, and as one of ordinary skill in the art can readily recognize, the sequence may be altered and still remain intact within the spirit and scope of the various embodiments.

[0085] Although exemplary embodiments have been shown and described, a wide range of modifications, variations, and alternatives are contemplated in the above disclosure, and in some instances, some features of the embodiments may be used without the corresponding use of other features. Those skilled in the art will recognize many variations, alternatives, and modifications. Accordingly, the scope of the present invention should be limited only by the scope of the following claims, which should be appropriately construed broadly and consistent with the scope of the embodiments disclosed herein.

[0086] Although this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or possible claimed subject matter, but rather as descriptions of features specific to a particular implementation of a particular invention. Some features described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable subcombination. Furthermore, although features may be described above as acting in some combinations and may initially be claimed as such, one or more features from a claimed combination can be deleted from the combination in some cases, and the claimed combination may be directed to a subcombination or a variation of a subcombination.

[0087] Similarly, although operations are illustrated in the figures in a particular order, this should not be understood as requiring that such operations be performed in the particular or sequential order illustrated, or that all of the illustrated operations be performed, to achieve desirable results. In some situations, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system components in the above implementations should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems may generally be integrated into a single software product or packaged into multiple software products.

[0088] References to "or" may be construed as inclusive such that any term described with "or" may refer to either the single, the plural, and all of the described terms. Labels such as "first," "second," "third," etc. do not necessarily imply an ordering, but are generally used merely to distinguish between like or similar items or elements.

[0089] Various modifications to the implementations described in this disclosure will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other implementations without departing from the spirit or scope of the disclosure. Thus, the claims are not intended to be limited to the implementations shown herein, but are to be accorded the widest scope consistent with the present disclosure, i.e., the principles and novel features disclosed herein. Enumeration of embodiments

[0090] Embodiment 1. A magnetic resonance imaging (MRI) method comprising: measuring one or more states of a sample by applying a total magnetic field including a magnetic field gradient and one or more RF pulse trains using a transmit coil and a radio frequency (RF) source, respectively; and receiving from one or more receive coils MRI measurement data acquired by the receive coils during an acquisition window, the MRI measurement data including magnetic resonance signal data emitted by the sample; the method further comprises generating, by a processor, a first encoding matrix by linearly combining a subset of the MRI measurement data, the first encoding matrix being a submatrix of a second encoding matrix configured (i) to be generated by linearly combining all of the MRI measurement data and (ii) to represent the one or more states of the sample, the transmit coil and the receive coil; the method further comprises determining, by the processor, a characteristic of the MRI measurement data based on calculation of the first encoding matrix.

[0091] Embodiment 2. The method of embodiment 1, wherein the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data for a single one of the magnetic field gradients and a single state of the one or more states of the sample, and the determined characteristics of the MRI measurements include a shape of a readout gradient of the magnetic field gradient.

[0092] Embodiment 3. The method of embodiment 1 or 2, wherein the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data for a single one of the receive coils or the transmit coil, and the determined characteristics of the MRI measurement include phase information of the sample during the measurement of the one or more states of the sample.

[0093] Embodiment 4. The method of any one of embodiments 1 to 3, wherein the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data related to the amplitude and phase of the samples for a single one of the receive coils or the transmit coil, and the determined characteristics of the MRI measurements include the amplitude and phase sensitivity of each of the single receive coil or the transmit coil.

[0094] Embodiment 5. The method of embodiment 4, wherein the amplitude and phase sensitivity of the single receive coil is different from the amplitude and phase sensitivity of the transmit coil.

[0095] Embodiment 6. The method of any one of embodiments 1-5, wherein the magnetic field gradient is a non-linear magnetic field gradient.

[0096] Embodiment 7. The method of any one of embodiments 1-6, wherein the total magnetic field comprises a non-uniform static magnetic field.

[0097] Embodiment 8. The method of any one of embodiments 1 to 7, further comprising reconstructing, by the processor, a magnetic resonance image of the sample based on the determined characteristics of the MRI measurements.

[0098] Embodiment 9. The method of embodiment 8, wherein reconstructing the magnetic resonance image includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) technique, or a conjugate gradient solver with compressed sensing constraints (CG-CS) to at least the determined characteristics of the MRI measurements.

[0099] Embodiment 10. The method of any one of embodiments 1 to 9, wherein the determined characteristics of the MRI measurement are selected from the group consisting of a shape of a readout gradient of the magnetic field gradient, phase information of the sample during the measurement of the one or more states of the sample, and the amplitude and phase sensitivity of each of the single receive coil or the transmit coil.

[0100] Embodiment 11. The method of any one of embodiments 1 to 10, wherein the characteristics of the MRI measurement data are determined without generating the second encoding matrix.

[0101] Embodiment 12. A magnetic resonance imaging (MRI) system comprising: a transmit coil and a radio frequency (RF) source configured to measure one or more states of a sample by applying a total magnetic field including a magnetic field gradient and one or more RF pulse trains, respectively; one or more receive coils configured to acquire MRI measurement data including magnetic resonance signal data emitted by the sample during an acquisition window; and a processor, wherein the processor is configured to receive the MRI measurement data from the one or more receive coils and generate a first encoding matrix by linearly combining a subset of the MRI measurement data, the first encoding matrix being a submatrix of a second encoding matrix configured to (i) be generated by linearly combining all of the MRI measurement data and (ii) represent the one or more states of the sample, the transmit coil and the receive coil; and the processor is further configured to determine a characteristic of the MRI measurement data based on calculation of the first encoding matrix.

[0102] Embodiment 13. The system of embodiment 12, wherein the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data for a single one of the magnetic field gradients and a single state of the one or more states of the sample, and the determined characteristics of the MRI measurement include a shape of a readout gradient of the magnetic field gradient.

[0103] Embodiment 14. The system of embodiment 12 or 13, wherein the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data for a single one of the receive coils or the transmit coil, and the determined characteristics of the MRI measurement include phase information of the sample during the period of the measurement of the one or more states of the sample.

[0104] Embodiment 15. The system of any one of embodiments 12 to 14, wherein the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data related to the amplitude and phase of the samples for a single one of the receive coils or the transmit coil, and the determined characteristics of the MRI measurement include the amplitude and phase sensitivity of each of the single receive coil or the transmit coil.

[0105] Embodiment 16. The system of embodiment 15, wherein the amplitude and phase sensitivity of the single receive coil is different from the amplitude and phase sensitivity of the transmit coil.

[0106] Embodiment 17. The system of any one of embodiments 12 to 16, wherein the magnetic field gradient is a nonlinear magnetic field gradient.

[0107] Embodiment 18. The system of any one of embodiments 12 to 17, wherein the total magnetic field comprises a non-uniform static magnetic field.

[0108] Embodiment 19. The system of any one of embodiments 12 to 18, wherein the processor is further configured to reconstruct a magnetic resonance image of the sample based on the determined characteristics of the MRI measurements.

[0109] Embodiment 20. The system of embodiment 19, wherein reconstructing the magnetic resonance image includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) technique, or a conjugate gradient solver with compressed sensing constraints (CG-CS) to at least the determined characteristics of the MRI measurements.

[0110] Embodiment 21. The system of any one of embodiments 12 to 20, wherein the determined characteristics of the MRI measurement are selected from the group consisting of the shape of a readout gradient of the magnetic field gradient, phase information of the sample during the measurement of the one or more states of the sample, and the amplitude and phase sensitivity of each of the single receive coil or the transmit coil.

[0111] Embodiment 22. The system of any one of embodiments 12 to 21, wherein the characteristics of the MRI measurement data are determined without generating the second encoding matrix.

Claims

1. 1. A method of magnetic resonance imaging (MRI), comprising: measuring one or more conditions of the sample by applying a total magnetic field including a magnetic field gradient and one or more trains of RF pulses using a transmit coil and a radio frequency (RF) source, respectively; receiving, from one or more receive coils, MRI measurement data acquired by the receive coils during an acquisition window, the MRI measurement data including magnetic resonance signal data emitted by the sample; The method further comprises: generating, by a processor, a first encoding matrix by linearly combining a subset of the MRI measurement data, the first encoding matrix being a sub-matrix of a second encoding matrix (i) generated by linearly combining all the MRI measurement data and (ii) configured to represent the one or more states of the sample, the transmit coils and the receive coils; The method further comprises: determining, by the processor, a characteristic of the MRI measurement data based on the calculation of the first encoding matrix. A magnetic resonance imaging method comprising:

2. the subsets of MRI measurement data that are linearly combined to generate the first encoding matrix include the MRI measurement data for a single one of the magnetic field gradients and a single state of the one or more states of the sample; the determined characteristics of the MRI measurement include a shape of a readout gradient of the magnetic field gradient. The method of claim 1.

3. the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data for a single one of the receive coils or the transmit coil; the determined characteristics of the MRI measurements include phase information of the sample during the measurement of the one or more states of the sample. The method of claim 1.

4. the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data associated with the amplitude and phase of the samples for a single one of the receive coils or the transmit coil; the determined characteristics of the MRI measurements include amplitude and phase sensitivity of the single one of the receive coils or each of the transmit coils. The method of claim 1.

5. The method of claim 4 , wherein the amplitude and phase sensitivity of the single one of the receive coils is different from the amplitude and phase sensitivity of the transmit coil.

6. The method of claim 1 , wherein the magnetic field gradient is a nonlinear magnetic field gradient.

7. The method of claim 1 , wherein the total magnetic field comprises a non-uniform static magnetic field.

8. The method of claim 1 , further comprising reconstructing, by the processor, a magnetic resonance image of the sample based on the determined characteristics of the MRI measurements.

9. 9. The method of claim 8, wherein reconstructing the magnetic resonance image comprises applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) technique, or a conjugate gradient solver with compressive sensing constraints (CG-CS) to at least the determined characteristics of the MRI measurements.

10. 2. The method of claim 1, wherein the determined characteristics of the MRI measurement are selected from the group consisting of a shape of a readout gradient of the magnetic field gradient, phase information of the sample during the measurement of the one or more states of the sample, and amplitude and phase sensitivity of the single one of the receive coils or each of the transmit coils.

11. The method of claim 1 , wherein the characteristics of the MRI measurement data are determined without generating the second encoding matrix.

12. 1. A magnetic resonance imaging (MRI) system comprising: a transmit coil and a radio frequency (RF) source configured to measure one or more conditions of the sample by applying a total magnetic field including a magnetic field gradient and one or more RF pulse trains, respectively; one or more receive coils configured to acquire MRI measurement data including magnetic resonance signal data emitted by the sample during an acquisition window; Processor and wherein the processor: receiving the MRI measurement data from the one or more receive coils; generating a first encoding matrix by linearly combining the subset of the MRI measurement data; wherein the first encoding matrix is ​​a sub-matrix of a second encoding matrix configured to (i) be generated by linearly combining all the MRI measurement data, and (ii) represent the one or more states of the sample, the transmit coils, and the receive coils, and wherein the processor further comprises: determining a characteristic of the MRI measurement data based on the calculation of the first encoding matrix; A magnetic resonance imaging system configured as follows.

13. the subsets of MRI measurement data that are linearly combined to generate the first encoding matrix include the MRI measurement data for a single one of the magnetic field gradients and a single state of the one or more states of the sample; the determined characteristics of the MRI measurement include a shape of a readout gradient of the magnetic field gradient. The system of claim 12.

14. the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data for a single one of the receive coils or the transmit coil; the determined characteristics of the MRI measurements include phase information of the sample during the measurement of the one or more states of the sample. The system of claim 12.

15. the subset of MRI measurement data that is linearly combined to generate the first encoding matrix includes the MRI measurement data associated with the amplitude and phase of the samples for a single one of the receive coils or the transmit coil; the determined characteristics of the MRI measurements include amplitude and phase sensitivity of the single one of the receive coils or each of the transmit coils. The system of claim 12.

16. The system of claim 15 , wherein the amplitude and phase sensitivity of the single one of the receive coils is different from the amplitude and phase sensitivity of the transmit coil.

17. The system of claim 12 , wherein the magnetic field gradient is a non-linear magnetic field gradient.

18. The system of claim 12 , wherein the total magnetic field comprises a non-uniform static magnetic field.

19. The system of claim 12 , wherein the processor is further configured to reconstruct a magnetic resonance image of the sample based on the determined characteristics of the MRI measurements.

20. 20. The system of claim 19, wherein reconstructing the magnetic resonance image includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) technique, or a conjugate gradient solver with compressive sensing constraints (CG-CS) to at least the determined characteristics of the MRI measurements.

21. 13. The system of claim 12, wherein the determined characteristics of the MRI measurement are selected from the group consisting of a shape of a readout gradient of the magnetic field gradient, phase information of the sample during the measurement of the one or more states of the sample, and amplitude and phase sensitivity of the single one of the receive coils or each of the transmit coils.

22. The system of claim 12 , wherein the characteristics of the MRI measurement data are determined without generating the second encoding matrix.

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