SYSTEMS AND METHODS FOR IMAGE RECONSTRUCTION IN MAGNETIC RESONANCE IMAGING.

Abstract

Some modalities of the present description describe robust magnetic resonance image reconstruction systems and methods that can model for all or almost all components in the magnetic resonance imaging system, that have compressibility aspects to accelerate reconstructions, and that can be optimized so that reconstruction can be performed within a short period of time.

Description

The present description generally refers to systems and methods for the efficient reconstruction of magnetic resonance images from magnetic resonance imaging signals, and more specifically, to systems and methods that allow generalization to incorporate all or almost all aspects of the magnetic resonance imaging system and compressibility and optimization to expedite the image reconstruction procedure. BACKGROUND OF THE INVENTION In magnetic resonance imaging (MRI), a signal from objects of interest or samples is measured by applying a sequence of pulses. In a traditional MRI system, the measured signals are acquired on a rectilinear grid of frequencies (also referred to as a Cartesian scan), and the image can be reconstructed by a 2D Inverse Fast Fourier Transform (FFT). Another common method is to acquire samples on a polar grid and use the 2D Inverse FFT after interpolating this data to a rectangular grid. BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is an example schematic illustration of a linear system model for modeling image reconstruction in magnetic resonance imaging (MR) systems, according to some modalities of the present description. FIG. 2 is an example illustration of a cross-section profile of a main magnetic field in an MRI system, according to some modalities of the present description. Figures 3A to 3D are example illustrations of an extracted surface corresponding to a shear excitation, according to some modalities of the present description. FIGS. 4A to 4B show example illustrations of nonlinearly varying spatial magnetic field gradients in the xj and y directions, respectively, according to some modalities of the present description. FIG. 5 shows an example flowchart illustrating a method for magnetic resonance imaging reconstruction based on magnetic resonance imaging signals, according to some modalities of the present description. FIG. 6 is a block diagram illustrating a computer system on which the modalities of these teachings can be implemented. FIG. 7 shows an example block diagram illustrating a system for magnetic resonance imaging reconstruction based on magnetic resonance imaging signals, according to some modalities of the present description. It should be understood that the figures are not necessarily drawn to scale, nor are the objects in the figures necessarily drawn to scale relative to one another. The figures are representations intended to provide clarity and understanding of the various types of apparatus, systems, and methods described herein. Whenever possible, the same reference numbers will be used throughout all the drawings to refer to the same or similar parts. It should also be noted that the drawings are not intended to limit the scope of the teachings herein in any way. BRIEF DESCRIPTION OF THE INVENTION The following summarizes some aspects of this description to provide a basic understanding of the technology discussed. This summary is not a comprehensive overview of all aspects of the description, nor does it attempt to identify key or critical elements of every aspect or to define the scope of any or all aspects. Its sole purpose is to present some concepts from one or more aspects of the description in summary form as a prelude to the more detailed description presented later. In some modalities described herein, methods and systems for the effective reconstruction of magnetic resonance images from magnetic resonance imaging signals are outlined. In some modalities, a total magnetic field, including magnetic field gradients and one or more RF pulse sequences, can be applied using a transmitting coil and a radio frequency (RF) source, respectively, to measure one or more states of a sample. Furthermore, the MRI measurement data acquired by the receiving coils during an acquisition window can be received by a processor from one or more receiving coils; this MRI measurement data includes magnetic resonance signal data emitted by the sample.Furthermore, a subset of the MRI measurement data can be linearly combined, via a processor, to generate a first coding matrix. This first coding matrix is ​​a sub-matrix of a second coding matrix configured to (i) be generated by linearly combining all the MRI measurement data, and (ii) represent one or more states of the sample, transmit coil, and receive coils. Additionally, a feature of the MRI measurement data can be determined by the processor based on a calculation of the first coding matrix. DETAILED DESCRIPTION OF THE INVENTION The following description of various embodiments is merely illustrative and explanatory and should not be interpreted as limiting or restrictive in any way. Other embodiments, aspects, objectives, and advantages of the present teachings will be evident from the accompanying description and drawings, and from the claims. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by a person skilled in the art to which this invention pertains. All publications mentioned herein are incorporated for reference purposes to describe and disseminate the devices, compositions, formulations, and methodologies described in this publication and that may be used in connection with this description. As used in this document, the terms comprise, comprise, which comprises, contain, contain, which contains, have, has, include, includes, and which includes, and their variants, are not intended to be limiting, are inclusive or open, and do not exclude additional components, whole numbers, elements, or method steps not explicitly stated. For example, a procedure, method, system, composition, kit, or apparatus comprising a list of aspects is not necessarily limited to only those aspects but may include other aspects not expressly listed or inherent in such procedure, method, system, composition, kit, or apparatus. In some modalities, the term signal components may refer to parts of a magnetic resonance imaging (MRI) system responsible for generating a signal. This may include, but is not limited to, the pulse sequence construct, transmitting coils, and receiving coils. For each pulse sequence state, all or nearly all of the receiving coils can acquire data over an acquisition window. In some modalities, the term acquisition window may refer to the time within the MRI pulse sequence during which the MRI signal is recorded. In some modalities, the term pulse sequence can refer to a set of radio frequency (RF) and gradient pulses, repeated multiple times during an acquisition to encode spatial and amplitude information describing the object of interest (e.g., a sample). In some cases, these repetitions may vary in the degree of gradients, or the shapes of the individual pulses applied, and this may ultimately be determined by the desired image quality. In MRI systems, a gradient is applied to the z-axis to select a slice, and gradients are applied to the xy-axis and y-axis to spatially locate the image, while RF pulses are used for excitation. In some modalities, the term transmitting coils may refer to coils of conductive wire in an MRI system that can generate an oscillating or rotating magnetic field perpendicular to the main static magnetic field (Bo) of the MRI system. In some modalities, the term receiving coils may refer to coils of conductive wire in an MRI system that detect the magnetic resonance (MR) signal. In some respects, receiving coils may differ in their spatial sensitivity (i.e., amplitudes) as well as in phase with respect to each other. Furthermore, in some respects, transmitting coils may also differ in their spatial sensitivity (i.e., amplitudes) as well as in phase with respect to each other. In some modalities, the term image acquisition time may refer to the time required to perform an MRI imaging procedure, comprising only the acquisition time data. In some cases, the total image acquisition time may equal the product of the repetition time, the number of averaged signals, and the number of distinct (position-coded) signals to be acquired for image reconstruction. The additional image reconstruction time is also important in determining how quickly the image can be viewed. When comparing sequential-slice and volume imaging techniques, the equivalent image acquisition time per slice, as well as the actual image acquisition time, can be considered. In some applications, the term frequency shift can refer to the difference between the frequency of a given signal and a reference frequency. In some cases, the frequency shift can be the center frequency for the cutoff excitation. In some modalities, the object of interest being imaged by an MRI system can emit a signal that can be measured by the MRI system and contains amplitudes of the measured frequencies. In some cases, the measured signals can be acquired on a rectilinear frequency grid or a Cartesian scan, and the image can be reconstructed by a 2D inverse Fourier transform (FFT). In some cases, the samples can be acquired on a polar grid, and the 2D non-uniform FFT used after interpolating this data to a rectangular grid. In such approaches, the problem can be defined as s(t) = J m(r)e~i2nk^rdr where s(t) is the measured signal, k represents the path in k-space, and r corresponds to the ML / a / ZUZl / Ul 0Ó0Z spatial positions mastered, and the problem can be formulated as the recovery of the image m, when observing a number of samples in s. Such approaches, however, can have several limitations, including requiring that the signal be fully sampled, or re-gridded in cases of non-Cartesian grids or undersampling, before performing a non-uniform FFT. This means having sufficient bandwidth and appropriate sample spacing to fully capture the object of interest without limiting the field of view or overlapping the image. Non-Cartesian acquisitions may also require frequency sampling density correction to minimize overrepresentation of k-space regions that can produce artifacts. Furthermore, these methods can make many assumptions that may simplify or fail to fully capture the MRI system, including assumptions that gradients are linear, that the main field of the Bo magnet is uniform, that the receiving coils have similar spatial sensitivity profiles, and so on.Furthermore, these methods can model signals in pulse sequences mathematically without practical considerations of rise and fall times, the true shape of the gradient profile, etc. As a result, multi-parametric MRI image reconstructions may not adequately model these factors. Additionally, multi-parametric MRI image reconstructions can take hours to produce high-quality images. Although several methods have been proposed to overcome these limitations (e.g., fast non-uniform Fourier transforms, sensitivity coding, partial parallel acquisition with generalized self-calibration, compressed detection approaches, etc.), the limitations described above may still exist. There is a need for robust image reconstruction that can model all or nearly all components in the system, possesses compressibility features to accelerate reconstructions, and can be optimized so that reconstruction is feasible in minutes, not hours. This description outlines configured modalities for magnetic resonance imaging reconstruction that address the challenges and limitations discussed above, providing more robust systems and methods for image reconstruction. Some modalities of the present description describe image reconstruction approaches or methods that can be generalized, compressed, and / or optimized. The methods can be generalized in that the image reconstruction approach can model all or nearly all components in the system, including at least the magnetic field gradients, the main magnetic field Bo, the phase and amplitude sensitivities of the transmitting and receiving coils, the shape of the gradient profile, the readout gradient, and the phase during measurement, which may include phases induced by the xyy gradient in the presence of a permanent main field. Furthermore, the approaches or methods may have compressibility aspects for MA / a / 4U41 0304 Accelerating reconstructions, for example, can enable image recovery without dense sampling in modes that can exploit aspects of the objects being imaged. In some modalities, compressibility can be achieved via various mechanisms: random undersampling, parallel reconstruction using multiple spatially sensitive coils, scarcity constraints using wave train compression, objective nuclear and L1 norm functions, multiple-slice excitations within a single time repetition, and bandwidth noise pulse excitations to excite large regions. Furthermore, image reconstruction approaches or methods can be optimized so that reconstruction is feasible in minutes or within an hour. Figure 1 shows a schematic example of a linear system model for modeling image reconstruction in magnetic resonance imaging (MR) systems, according to some modalities of the present description. In general, a model for reconstructing an image from provided parameter measurements can be described as linear. As illustrated in Figure 1, the linear system model 100 includes a signal vector 102, which contains all or nearly all measurements made on the system (i.e., on the object of interest or sample) from multiple receiving coils. The linear system model 100 also includes an array E 104 that serves as the encoding matrix, containing all or nearly all generalizations of the given system. In some cases, the array E may be comprised of block arrays Eo, Ei, ..., En-i, where each block corresponds to a single line of the pulse sequence separated by repetition time (RT) for a total of N=ns, where ns is the number of pulse sequence lines. Each block matrix may further be comprised of individual rows that correspond to a single time point for that line and a receiving coil in the pulse sequence of the recorded signal window. The total number of rows in the matrix E can be expressed as the product of the number of states (i.e., pulse sequence lines) noted, the number of receiving coils used (n coils), and the number of points in the acquisition window (nt) (i.e., noted x n coils x nt). The linear system model 100 further includes the image vector m 106, which is to be resolved by making observations on the signal s 102, where the matrix E 104 describes the state of the system.That is, the status of the MRI 100 system, including system aspects such as magnetic field gradients, receiving and transmitting coils, main magnetic field Bo configurations for the pulse sequence, and / or similar, are encoded in the E 104 matrix. In some modalities, a component in the matrix E 104 Eij can be modeled as follows: = r + ,c~eíy^PA+PB+Pc+PB0^lJ H(ts)lc where pA is the accumulated phase when a magnetic gradient is present during the reading of wiA / ai¿v¿i iv io the signal, pB is the accumulated phase when the gradient pulse before the reading induces a phase change, pc is the phase of the transmitting and receiving coils, and finally pB0 is the phase of the Bo field that is static (e.g., always present or on during the measurement). In some modes, the parameter H may represent the selection of the transmitting coil for the measurements and is a function of the state, e.g., for a given state (e.g., state = 10), the transmitting coil selected by H(state0) is selected for transmission. The phase terms can be calculated as Pa = < A{is,0,it)Bgx(J) + A(is, l,it)Bgy(j\B^(j) >, Pb = < B(is,0)Bgx(j) + B(is,l)B9gy(J),B^(j) >, pc= ^C+(j) +¿C~, and Peo = (^oO) -F(is))tit. In these definitions, the principal field Bo and the gradient fields can be expressed in radian units (with gamma pre-multiplied). These expressions allow the calculation of the phases of the gradient profile matrix (δ / I) and the phase matrix (δ), while incorporating the non-uniform field Bo and the center frequency shift matrix F. In some modalities, the coding matrix E may have a specific structure. In some cases, the coding matrix E may be completely determined by the transmit and receive coil maps, matrices A and B, and may not need to be fully stored in memory to generate or reconstruct magnetic resonance images. In some modalities, the coding matrix E may also be exceptionally large, for example, on the order of hundreds of thousands of rows and a similar number of columns.However, in some cases, constructing individual elements in the matrix as needed without having to store the entire matrix allows one to advantageously exploit the above structure in the encoding matrix E. In some modalities, the measured signal may contain a small overall shift. The reconstructed image may contain a bright spot artifact in the center. To remove this, a column of ones can be appended to the encoding matrix, with the image being reconstructed along with the estimated coefficient of change. The index / cor corresponds to a single acquisition for a single pulse sequence line, coil (e.g., receive or transmit), and time point. The indices íse and icson are extracted from i and are defined as follows: for / en [0, nestados x nabinas x nt], isse refers to a pulse sequence line, Ase refers to the recorded signal for a specific coil, and ícor corresponds to the time when the signal was recorded within the signal window. Furthermore, y corresponds to the voxel j° within the image vector m. The parameters Bgx, Bgy, and Bgz can represent the measured field of the gradients x, y, and z, respectively. When subscripted by x, y, and z, the parameter B represents the field generated by the gradient in that direction. For example, Bgx corresponds to the field produced by the gradient z, which is a gradient applied predominantly along the x-direction, and is a 3D vector field, where each location of Bgx in the image contains x, y, and z components. The matrix Bgx can thus be expressed as an n image x 3 matrix, where n image = Xn x Yn x Zn, where Xn, Yn, and Zn are the number of voxels reconstructed along each dimension. As another example, Bgy corresponds to the field produced by the gradient y, which is a gradient applied predominantly along the y-direction, and is a 3D vector field, where each location of Bgy in the image contains x, y, and z components. The Bgy matrix can thus be expressed as an nimage x 3 matrix.As yet another example, the matrix Bgx can thus be expressed as an nimage x 3 matrix, where the number of voxels in the reconstructed image is Xn x Yn y, Zn y, where Xn, Yn, and Zn are the number of voxels reconstructed along each dimension. As another example, Bgz corresponds to the field produced by the gradient z, which is a gradient applied predominantly along the z-direction, and is a 3D vector field, where each location of Bgz in the image contains x, y, and z components. The matrix Bgz can thus be expressed as an nimage x 3 matrix. In some modalities, the dimensionality of the linear system model 100 in FIG. 1 can be defined based on the measured parameters, which can be identified or defined as follows. In some aspects, the repetition of the pulse sequence can be referred to as a single state in the system, where the total number of such states is defined as ns (or notated). For each state, a measured signal array of length nt is measured, representing the length of a single acquisition. In some modalities, the dimensions of the image being reconstructed can be expressed as (Xn, Yn, Zn), where Xn, Yn, and Zn are the number of voxels reconstructed along each dimension. The image vector m 106 illustrated in FIG. 1 can be expressed as the linear arrangement (vectorization) of all voxels such that the total length is the product of the dimensions, i.e., nimage = Xn x Yn x Zn. In some configurations, the MRI system may be equipped with multiple coils (receiving coils), each of which takes measurements simultaneously (e.g., independently). The number of such receiving coils can be represented by the parameter ncoils. In some modalities, the matrix A (e.g., in the expression for PA) is provided with dimensions represented by ns x 2 x nm, where nm is the number of points in a readout window (during acquisition). For each pulse sequence state in the first dimension, and for each gradient direction in the second dimension, the matrix A can contain the gradient profile stored along the third dimension for the duration of the acquisition (e.g., along the readout), thus representing the normalized shape of the applied readout gradient. In a simplified configuration, the matrix A can have identical profiles for all pulse sequence states; that is, the same gradient can be applied during the readout. For example, a rectangular pulse can be applied during the readout so that the entire signal recorded during the readout is affected by this gradient (along the gradient direction). In some modalities, the matrix B (for example, in the expression for pe) provides dimensions represented by ns x 2. For each pulse sequence state in the first dimension, the matrix B can contain the stored phase for each gradient. These are the gradient pulses applied before acquiring the signal so that the turns are endowed with the phase corresponding to this gradient pulse during signal acquisition. As a non-limiting example illustration, in a simple phase imaging experiment acquiring 30 x 30 phase encodings, there are 900 states in the pulse sequence, or 30 variable gradient forces along one direction and 30 along the other. Each row of the matrix B can contain accumulated phase due to each gradient for the pulse sequence line to which it corresponds. In some configurations, the C matrix (e.g., in the PC expression), which can be referred to as a receive coil sensitivity matrix, can be provided with the dimensions represented by nimage x 2 x ncoils. For each voxel in the image referenced by the first dimension, the second dimension corresponds to the amplitude and phase for that voxel, and the index of each receive coil (for multiple coils) is specified in the third dimension. As such, the C matrix stores the two-component (amplitude and phase) sensitivity map for each receive coil over the entire image. For each voxel location, the amplitude and phase sensitivities are stored for each receive coil in the matrix. The C+ matrix (also referred to as a transmit coil matrix) can be constructed in the same way as C but for the transmit coils.In some modes, the parameter tw represents the time window over which the echo is measured, centered near the echo time. In defining the dimensionality of the linear system in Figure 1, the calculation for the measured signals can be provided, which contains the recorded signal from all states and all coils: s = (s0·1)7-, (s0·2)7, (s0-3)r.....(s™-1·1)7; (s^-1·2)7; (s^-1'3)7·) where siJ corresponds to the measured acquisition window for state i and coil j, respectively. That is, since each state is a single pulse sequence line, coil J refers to coil j° that is recording a signal when pulse sequence line i° is represented. The signal thus contains all possible measurements of the system, including all coils, all states, and the entire acquisition window. That is, each siJ is a vector (points recorded along the reading), and they are all stacked together to form much larger vectors, which contain all states, all coils, and all points—that is, the entire signal recorded by the system. With reference to Figure 1, the length of the vectors and image vector m are, respectively, lengths) = n 5* nabinas*· nty length(m) = n¡magen.In some modes, the vector m is equal to the number of voxels. That is, it can be a linear arrangement of all the voxel intensities that can be determined by the reconstruction or the image size selected by the user to reconstruct the image on it. Therefore, taking into consideration these two lengths, the dimensions of the encoding matrix E, i.e., the number of rows and the number of columns of the encoding matrix are represented by nsx nbobinas x nt, n¡magen. In some modalities, the data from the receiving coils may have already collapsed before reconstruction. In such pre-collapse situations, the above formulation can be generalized to the case of a single coil containing the adjusted phase and averaged measurements from all coils. In other cases, individual images can be reconstructed using one coil at a time and subsequently combined in the image domain. Some methods of image-domain combination are discussed in the non-patent literature entitled "Adaptive reconstruction of phased array MR imagery," by D.O. Walsh, A.F. Gmitro, and M.W. Marcellin, Magnetic Resonance in Medicine (May 1, 2000), which is incorporated herein by reference. In some modalities, methods for providing enhanced magnetic resonance image reconstruction include encoding the matrix construction with constraints. These methods provide a generalized framework for image reconstruction that advantageously implements a universal encoding matrix configuration that successfully accounts for nonlinear gradients, a non-uniform Bo field (i.e., a non-uniform main magnetic field), global signal changes during image acquisition, variable sensitivity amplitude and phase for the transmitting and receiving coils, and excitation frequency and bandwidth of the received signal that supports all or nearly all pulse sequences. Additional constraints for wavetrain-domain compressibility and reconstructed image uniformity can also be optimized simultaneously. Furthermore, the method advantageously solves curvilinear 3D image sections excited by the main magnetic field Bo using simultaneous iterative reconstruction, conjugate gradients, and compressed detection solvers configured to use the universal encoding matrix. Moreover, the provided solvers can be advantageously optimized on a graphics processing unit (GPU) without the need to construct the universal encoding matrix all at once, enabling the solution of very large linear systems, such as those with more than, for example, 500,000 rows and columns each. This can be achieved by exploiting the structure of the encoding matrix by storing its underlying matrices (e.g., A, B, C+, C-, etc.) and constructing them dynamically.Likewise, the methods also advantageously provide an interpolation method (e.g., volume interpolation method) to combine overlapping curvilinear 3D image sections to construct a complete curvilinear 3D volume. Referring now to Figure 2, the main magnetic field Bo can generally be designed to optimize linearity and homogeneity within a cross-section. However, for a cut-off excitation frequency, the iso-contour on the main field where all points experience approximately the same degree of excitation can be in the form of a curvilinear surface, as illustrated in Figures 3A to 3D. In some modalities, voxel intensities at all points in the curvilinear section can be resolved. The curvilinear surface can be extracted by a linear regression fit on the data, assuming an elliptical geometry. In some cases, the surface may be curvilinear as a result of the shape of the main magnetic field Bo; that is, its contour can create this shape.As such, when a frequency is excited, the curved region whose resonant frequency matches this excitation frequency can also be excited, and the image can be reconstructed in this region. For multiple slices, the reconstruction has a similar curvilinear geometry and may appear as a surface of some thickness, referred to as a block. Many such blocks can be reconstructed across the entire field of view and interpolated to construct a 3D rectilinear volume. In some modes, the following example workflow can be applied to identify specific blocks for reconstruction. First, a field of view of a block and a resolution defined in the x, y, and z parameters are selected from which to construct a mesh grid in the % and y parameters. The field of view in the z parameter may depend on the bandwidths of the transmit and receive coils. Second, the surface corresponding to the center of the block is determined from the cutoff excitation frequency using, for example, a regression model. For instance, to construct a regression model, the parameters provided for fitting are x, y, and the frequency (yB0) for a known value of z. In this case, the measured principal field data is represented by B0(x, y, z) (where the field is expressed at all measured x, y, and z locations), and the frequency is represented by yB0. From this, the model coefficients can be estimated. Using the model coefficients and the grid over the x, y, and z locations, the z locations corresponding to those x, y, and z grid points can be determined. This constitutes the desired central surface of the block. Next, using the central surface of the given block, additional surfaces parallel to the central surface can be constructed to cover the field of view in z. With the completion of the construction, a grid &'(χ, y, z) is provided for the entire block, where A is the excited frequency A. From here, the block can be reconstructed at each of the grid points. As a result, the main field Bo can be extracted at each point along the grid, referenced by Bfi. Likewise, the gradients at these locations can also be extracted. According to various modalities, these gradients are spatially nonlinear, as shown below in Figure 4A for the xy axis and Figure 4B for the y axis, whereby the field of these gradients in xy and can be referred to as BxGy and BYG respectively at the grid locations. Finally, the spatial sensitivities of the transmit and receive coil maps can also be determined in a similar manner. Accordingly, a method is provided for identifying a first surface to be reconstructed. The first surface may be a block. The first surface may have a curvilinear geometry. The method may comprise selecting a field of view of the first surface and selecting a resolution in x, y, and z. The method may further comprise constructing a mesh grid in x, y, and z. The method may further comprise, for a specific cutoff excitation frequency, determining a second surface that corresponds to the center of the first surface. The determination may comprise constructing a regression model. The method may further comprise constructing additional second surfaces parallel to the central surface. This construction may encompass the field of view in z to provide a grid @(x, y, z) for the entire first surface, where x is the excited frequency Γ. Reconstruction In some modalities, after the linear system is set up as discussed earlier, the image m (represented as a curvilinear block described earlier) can then be resolved into a series of shear excitation locations. Each of such reconstructed images is denoted as msi, where si corresponds to the index of the excited shear. After solving these multiple linear systems for these block excitations to generate a set of reconstructions {msi, mS2,... , m5n}, a final volume V can be constructed, as detailed below. With reference to Figure 1, with the E matrix solved, reconstruction begins after the entire signal vector s is acquired, or as signal data is gathered, depending on the method used. Multiple reconstruction methods (individually or in combination) can be used to achieve reconstruction, including, for example, simultaneous iterative reconstruction (SIRT) and conjugate gradient methods such as the conjugate gradient least squares (CGLS) method and the conjugate gradient solver with compressed detection constraints (CG-CS). Accordingly, a method is provided for performing an image reconstruction at a series of cut excitation locations; the method comprises constructing an E-coding matrix, acquiring signal vectors, and applying a reconstruction method selected from the group consisting of SIRT, CGLS, CG-CS, and a combination thereof. In some modalities, SIRT can also be used for computed tomography reconstruction, where each observation in the measurement(s), and the corresponding row in the encoding matrix, produces an update to the image. Essentially, the projections of all rows are calculated, and the image is updated successively and concurrently. Similarly, a number of other measurements can be updated simultaneously. Conjugate gradient methods can be explained in relation to steepest descent methods. Steepest descent methods typically start with a solution starting point and converge iteratively by taking steps in directions that reduce cost. This iterative convergence can proceed until a sufficiently close solution is reached; for example, the image no longer changes enough to require further iterations. However, steepest descent methods can take steps in directions already traversed in the encoding matrix, creating inefficiencies. This can be especially true for a poorly conditioned system that has slow convergence. In contrast, conjugate gradient methods typically rely on orthogonal search directions, proceeding one step in each direction until convergence. Conjugate gradient methods can be employed with additional regularization terms to be minimized simultaneously. Gradients can be estimated using, for example, derivatives of the combined cost function. Other options include using a wave train function and a total variations term that measures gradient variations in the image. Optimization The solvers discussed earlier can be particularly slow due to the large size of the encoding matrix and low range, making them impractical for immediate use. For example, simply storing a 300,000 x 200,000 matrix in memory is not a feasible approach. Therefore, in accordance with several methods, the provided solvers can be advantageously optimized on a graphics processing unit (GPU) without the need to construct the universal encoding matrix all at once, allowing the possibility of solving very large linear systems, such as those with more than, for example, 500,000 rows and columns each. An example of such an optimization regime is the Unified Compute Device Architecture (CUDA®), Nvidia's approach to parallel data processing. This approach provides the ability to create hundreds of thousands of threads concurrently. These threads can cooperate by sharing a section of shared memory within a thread block. CUDA 'cores' are blocks of code that each thread executes. Depending on the threads' own IDs, these cores can process different parts of the data. An example of this is vector addition for vectors of length 10,000. In this example, each of the 10,000 components can be added independently, and each component can constitute its own thread. The core in this case can simply add corresponding scalar values ​​at an index location between 1 and 10,000, depending on the thread's index. The kernels can be specified by choosing the number of threads per block and the number of blocks. Modifying these parameters can produce some differences in the level of optimization. The two common tasks for parallelization when solving linear systems are to calculate E times E times v, where E is the encoding matrix and v is any vector. Since the matrix E can be very large, this tool allows for dynamic calculation by selecting appropriate values ​​for all matrices (for example, matrices A through G described earlier) when incorporating them into the encoding matrix formulation. In addition, pre-calculated exponential maps are also used instead of calculating exponentials in the processing unit to speed up computation times. Volume Interpolation As discussed earlier, individual curvilinear blocks are reconstructed for individual shear excitations {msi, mS2, ... , mSn}, from which the final volume V can be constructed. These are individual curvilinear blocks reconstructed independently and then interpolated into a single 3D rectilinear volume. In the volume interpolation stage, these blocks can overlap and thus be reconstructed into a 3D volume. The volume interpolation steps take into account the overlapping intensities of multiple blocks that together can contribute to the intensity in a voxel. The steps for performing volume interpolation are discussed in detail later. In accordance with several modalities, a method for performing volume interpolation is provided. The method may comprise defining a volume V over which to reconstruct an image. The definition may be specified by a field of view and resolution in the x, y, and z directions (to encompass the block boundaries). The method may further comprise, for each block in {m1, m2, m3n} and for each grid location over a plurality of grid locations (or all grid locations), finding the distance between a specified grid point and the nearest (or adjacent) voxels in a 3D volume, and mapping an intensity (e.g., absolute value of individual cutoff excitations) at the grid point and its distance to neighboring voxels of one or more blocks within a specified proximity of the grid point.The mapping can occur such that the grid point contains a list of intensities and corresponding mapped distances from all blocks. The method can further comprise, for each voxel in volume V, interpolating an intensity value from all intensities and corresponding distances assigned to that intensity. The interpolation can involve performing an inverse distance weighting approach, exponential decay, or a Gaussian weighting approach. Figure 5 shows an example flowchart illustrating a method for magnetic resonance imaging (MRI) reconstruction based on MRI signals, according to some modalities described herein. In block 510, in some modalities, a total magnetic field, including magnetic field gradients and one or more RF pulse sequences, can be applied to a sample or object of interest using a transmitting coil and an RF source of an MRI system, respectively, to measure one or more states of the sample. That is, the sample or object of interest can be placed in an MRI system so that a magnetic resonance image of the sample or object of interest can be reconstructed based on the MRI system measurements. In some cases, the total magnetic field may include a main magnetic field Bo, which may not be uniform. In block 520, the MRI system's receiving coils can detect or acquire MRI measurements, including magnetic resonance signals emitted by the sample or object of interest during an acquisition window or time period. In some modalities, there may be multiple receiving coils, each acquiring MRI measurements (for example, independently). In some cases, the sensitivities of the receiving coils may differ from those of the transmitting coils. In block 530, in some modalities, a linear system model can be used to linearly combine at least some of the measurements from the receiving coils to generate an encoding matrix that can model or represent all or almost all components in the MRI system, including at least the magnetic field gradients, the main magnetic field Bo, phase and amplitude sensitivities of the transmitting and receiving coils, the shape of the gradient profile, the reading gradient and phase during measurement, which may include phases induced by the xy gradient in the presence of a permanent magnet field, etc.That is, a processor in a computing device (e.g., a GPU) can linearly combine a subset of the MRI measurement data to generate a first encoding matrix. This 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 one or more states of the sample, transmit coil, and receive coils. In some modalities, the encoding matrix may include one or more sub-encoding matrices that encode or represent some or nearly all of the components represented by the encoding matrix. For example, a sub-matrix may be generated by linearly combining some (e.g., but not all) of the measurement data and may encode some, but not all, of the components (e.g., reading gradients and phase) of the MRI system. In block 540, in some modes, a characteristic of the MRI measurement data can be determined based on a calculation of the coding submatrix. The characteristic may include one or more components, such as phases, acquired during the MRI measurement. In some modes, the subset of MRI measurement data linearly combined to generate the coding submatrix includes MRI measurement data for a single magnetic field gradient and a single state of one or more sample states; and the determined MRI measurement characteristic includes a shape of a reading gradient of the magnetic field gradients.In some modalities, the subset of MRI measurement data linearly combined to generate the coding submatrix includes MRI measurement data for a single receiving coil or the transmitting coil; and the determined MRI measurement characteristic includes sample phase information during the measurement of one or more sample states. In some modalities, the subset of MRI measurement data linearly combined to generate the first coding submatrix includes MRI measurement data related to the amplitude and phase of the sample for a single receiving coil or the transmitting coil; and the determined MRI measurement characteristic includes amplitude and phase sensitivity of the single receiving coil or the transmitting coil, respectively.In some modalities, the characteristic of the MRI measurement data is determined without generating the second coding matrix. In some modalities, the method further comprises reconstructing, via the processor, a magnetic resonance image of the sample based on a predetermined feature of the MRI measurement. In some modalities, the MRI image reconstruction includes applying one or more simultaneous iterative reconstruction techniques (SIRT), a conjugate gradient least squares (CGLS) method, or a conjugate gradient solver with compressed detection constraints (CG-CS) to at least the predetermined feature of the MRI measurement. In some modalities, the predetermined feature of the MRI measurement is selected from the group consisting of a form of a reading gradient of the magnetic field gradients, phase information of the sample during the measurement of one or more sample states, and the amplitude and phase sensitivity of the single receiving coil or the transmitting coil, respectively. In some modes, the amplitude and phase sensitivity of the single receiving coil differ from the amplitude and phase sensitivity of the transmitting coil. In some modes, the magnetic field gradients are nonlinear. In some modes, the total magnetic field includes a non-uniform static magnetic field. Computer-implemented System In several ways, methods for reconstructing magnetic resonance images from magnetic resonance imaging signals can be implemented via computer software or hardware. That is, the methods (e.g., 500 in FIG. 5) described herein can be implemented on a computing device that includes a processor and an input / output engine. In several ways, the computing device can be communicatively connected to a data store or memory and a display device via a direct connection or through an internet connection. Figure 6 is a block diagram illustrating a computer system 600 on which the modalities of these teachings can be implemented. In several modalities of these teachings, the computer system 600 may include a bus 602 or other communication mechanism for transmitting information and a processor 604 coupled to the bus 602 for processing information. In several modalities, the computer system 600 also includes a memory 506, which may be random access memory (RAM) 606 or another dynamic storage device, coupled to the bus 602 to determine instructions to be executed by the processor 604. The memory may also be used to store temporary variables or other intermediate information during the execution of instructions to be executed by the processor 604.In several configurations, the 600 computer system also includes a 608 read-only memory (ROM) or other static storage device coupled to the 602 bus for storing static information and instructions for the 604 processor. A 610 storage device, such as a magnetic or optical disk, is provided and coupled to the 602 bus for storing information and instructions. In some configurations, the computer system may be a graphics processing unit (GPU). In several configurations, the computer system 600 can be coupled via bus 602 to a display 612, such as a cathode ray tube (CRT) or liquid crystal display (LCD), to present information to a computer user. An input device 614, including alphanumeric or other keys, can be coupled to bus 602 to communicate information and command selections to the processor 604. Another type of user input device is the cursor control 616, such as a mouse, trackball, or cursor direction keys, to communicate direction information and command selections to the processor 604 and to control the cursor's movement on the display 612. This input device 614 typically has two degrees of freedom on two axes, a first axis (i.e., x) and a second axis (i.e., y), which allows the device to specify positions in a plane.However, it should be understood that the 614 input devices that allow 3-dimensional cursor movement (x, y, yz) are also covered herein. According to certain implementations of the present techniques, the results can be provided by the computer system 600 in response to the processor 604 executing one or more sequences of one or more instructions contained in main memory 606. Such instructions can be read into main memory 606 from another computer-readable medium or computer-readable storage medium, such as storage device 610. The execution of the instruction sequences contained in memory 606 causes the processor 604 to perform the processes described herein. Alternatively, hardwired circuits can be used instead of, or in combination with, software instructions to implement the present teachings. Therefore, the implementations of the present description are not limited to any specific combination of hardware and software. The term computer-readable medium (e.g., data store, data storage, etc.) or computer-readable storage medium as used herein refers to any medium involved in providing instructions to the processor for execution. Such a medium 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. Examples of transmission media include coaxial cables, copper wire, and optical fiber, including the cables comprising the bus. Common forms of computer-readable media include, for example, floppy disk, hard disk, magnetic tape, any other magnetic medium, CD-ROM, any other optical medium, punched cards, paper tape, any other physical medium with hole patterns, RAM, PROM, EEPROM, FLASH-EEPROM, any other memory chip or cartridge, or any other tangible medium from which a computer can read. In addition to computer-readable media, instructions or data can be provided as signals on transmission media included in a communications device or system to deliver sequences of one or more instructions to the computer system's processor for execution. For example, a communications device might include a transceiver that carries signals indicating instructions and data. The instructions and data are configured to cause one or more processors to implement the functions described in this document. Representative examples of data communications transmission connections might include, but are not limited to, telephone modem connections, wide area networks (WANs), local area networks (LANs), infrared data connections, NFC connections, and so on. It should be noted that the methodologies described herein, the flowcharts, diagrams and accompanying description can be implemented using the 600 computer system as a standalone device or in a distributed network of shared computing processing resources, such as a cloud computing network. The methodologies described herein can be implemented through various means depending on the application. For example, these methodologies can be implemented in hardware, firmware, software, or any combination thereof. For a hardware implementation, the processing unit can be implemented in one or more of the following application-specific integrated circuits (ASICs), digital signal processing units (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), general-purpose processors, controllers, microcontrollers, microprocessors, electronic devices, other electronic units designed to perform the functions described herein, or a combination thereof. In various ways, the methods described herein can be implemented as firmware and / or software programs and applications written in conventional programming languages ​​such as C, C++, Python, etc. If implemented as firmware and / or software, the methods described herein can be implemented on a non-transient, computer-readable medium where a program is stored to instruct a computer to perform the methods described above. It should be understood that the various engines described herein can be provided in a computer system, such as the 600 computer system, whereby the 604 processor would execute the analyses and determinations provided by these engines, subject to instructions provided by any of, or a combination of, memory components 606 / 608 / 610 and user input provided through the 614 input device. Figure 7 shows an example block diagram illustrating an MRI 700 system for magnetic resonance image reconstruction based on magnetic resonance imaging signals, according to some of the modalities described herein. In some modalities, the MRI 700 system may include a transmitting coil 710, a radio frequency source 720, a receiving coil 730, and a processor 740. In some modalities, the MRI 700 system may be coupled to or in communication with a computer device 750 configured to receive the output of the MRI 700 system for further analysis, presentation, etc. In some modalities, the processor 740 may be part of the computer device 750, or the computer device 750 may be a component of the MRI 700 system. In some modes, the transmitting coil and the radio frequency (RF) source can be configured to apply a total magnetic field, including magnetic field gradients and one or more RF pulse sequences, respectively, to measure one or more states of a sample. Additionally, in some modes, one or more receiving coils can be configured to acquire MRI measurement data, including magnetic resonance signal data emitted by the sample during an acquisition window. Furthermore, in some modes, the processor is configured to receive the MRI measurement data from one or more receiving coils.Furthermore, the processor can be configured to linearly combine a subset of the MRI measurement data to generate a first coding matrix. This first coding matrix is ​​a sub-matrix of a second coding matrix configured to (i) be generated by linearly combining all the MRI measurement data, and (ii) represent one or more states of the sample, transmit coil, and receive coils. Additionally, the processor can be configured to determine a feature from the MRI measurement data based on a calculation of the first coding matrix. In some modalities, the processor can also be configured to reconstruct a magnetic resonance image of the sample based on the determined feature from the MRI measurement. While these teachings are described in conjunction with several modalities, they are not intended to be limited to such modalities. On the contrary, these teachings encompass various alternatives, modifications, and equivalents, as will be appreciated by those skilled in the technique. In describing the various modalities, the specification may have presented a method and / or procedure as a particular sequence of steps. However, to the extent that the method or procedure is not based on the particular order of steps set forth herein, the method or procedure should not be limited to the particular sequence of steps described, and a person skilled in the art can readily appreciate that the sequences may vary and still remain within the spirit and scope of the various modalities. Although illustrative embodiments have been shown and described, a wide variety of modifications, changes, and substitutions are contemplated in the foregoing description, and in some cases, some aspects of the embodiments can be employed without corresponding use of other aspects. A person of ordinary skill in the art will recognize many variations, alternatives, and modifications. Thus, the scope of the invention should be limited only by the following claims, and it is appropriate that the claims be considered broadly and in a manner consistent with the scope of the embodiments described herein. While this specification contains many specific implementation details, these should not be interpreted as limitations on the scope of any invention or on what may be claimed, but rather as descriptions of specific aspects of particular implementations of particular inventions. Certain aspects described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, various aspects described in the context of a single implementation may also be implemented in multiple separate implementations or in any suitable subcombination.Furthermore, although aspects may above be described as acting in certain combinations and even initially claimed as such, in some cases, one or more aspects of a claimed combination may be removed from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. Similarly, while the operations are depicted in the diagrams in a particular order, this should not be interpreted as requiring that these operations be performed in the specific order shown or sequentially, or that all illustrated operations be performed, to achieve the desired results. In certain circumstances, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system components in the implementations described above should not be interpreted as requiring such separation in all implementations. It should be understood that the program components and systems described can generally be integrated into a single software product or packaged into multiple software products. The ao references can be considered inclusive, so any terms described using them can indicate any one, more than one, or all of the terms described. The labels first, second, third, and so on are not necessarily intended to indicate an order and are generally used simply to distinguish between similar or like items or elements. Several modifications to the implementations described herein may be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other implementations without departing from the spirit or scope of this description. Thus, the claims are not intended to be limited to the implementations shown herein, but shall be in accordance with the broadest scope consistent with this description, the principles, and the novel aspects described herein. Recitation of Modalities Modality 1. A magnetic resonance imaging (MRI) method, comprising: applying, using a transmitting coil and a radio frequency (RF) source, a total magnetic field including magnetic field gradients and one or more RF pulse sequences, respectively, to measure one or more states of a sample; receiving, from one or more receiving coils, MRI measurement data acquired by the receiving coils during an acquisition window, the MRI measurement data including magnetic resonance signal data emitted by the sample;by linearly combining, via a processor, a subset of the MRI measurement data to generate a first coding matrix, the first coding matrix being a sub-matrix of a second coding 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, transmit coil and receive coils; and to determine, by the processor, a feature of the MRI measurement data based on a calculation of the first coding matrix. Mode 2. The method of Mode 1, wherein: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single magnetic field gradient of the magnetic field gradients and a single state of one or more sample states; and the determined feature of the MRI measurement includes a shape of a reading gradient of the magnetic field gradients. Mode 3. The method of Modes 1 or 2, wherein: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single receiving coil of the receiving coils or the transmitting coil; and the determined feature of the MRI measurement includes sample phase information during the measurement of one or more sample states. Modality 4. The method of any of Modalities 1 to 3, wherein: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data relating to the amplitude and phase of the sample for a single receive coil of the receive coils or the transmit coil; and the determined characteristic of the MRI measurement includes an amplitude and phase sensitivity of the single receive coil or the transmit coil, respectively. Mode 5. The Mode 4 method, wherein the amplitude and phase sensitivity of the single receiving coil are different from the amplitude and phase sensitivity of the transmitting coil. Mode 6. The method of any of Modes 1 to 5, wherein the magnetic field gradients are nonlinear magnetic field gradients. Option 7. The method of any of Options 1 to 6, where the field MA / a / 4U41 0304 Total magnetic field includes a non-uniform static magnetic field. Modality 8. The method of any of modalities 1 to 7, which further comprises reconstructing, via the processor, a magnetic resonance image of the sample based on the determined characteristic of the MRI measurement. Modality 9. The method of Modality 8, wherein the magnetic resonance image reconstruction includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) method, or a conjugate gradient solver with compressed detection constraints (CG-CS) to at least the determined feature of the MRI measurement. Mode 10. The method of any of Modes 1 to 9, wherein the determined characteristic of the MRI measurement is selected from the group consisting of a form of a reading gradient of the magnetic field gradients, a phase information of the sample during the measurement of one or more sample states, and an amplitude and phase sensitivity of the single receiving coil or transmitting coil, respectively. Modality 11. The method of any of Modalities 1 to 10, where the characteristic of the MRI measurement data is determined without generating the second coding matrix. Modality 12. A magnetic resonance imaging (MRI) system, comprising: a transmitting coil and a radio frequency (RF) source configured to apply a total magnetic field including magnetic field gradients and one or more RF pulse sequences, respectively, to measure one or more states of a sample; one or more receiving coils configured to acquire MRI measurement data including magnetic resonance signal data emitted by the sample during an acquisition window; and a processor configured to: receive, from the one or more receiving coils, the MRI measurement data;linearly combine a subset of the MRI measurement data to generate a first coding matrix, the first coding matrix being a sub-matrix of a second coding 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, transmit coil, and receive coils; and determine a feature of the MRI measurement data based on a calculation of the first coding matrix. Mode 13. The Mode 12 system, wherein the subset of MRI measurement data linearly combined to generate the first encoding matrix includes MRI measurement data for a single magnetic field gradient of the magnetic field gradients and a single state of one or more sample states; and the determined feature of the MRI measurement includes a form of a reading gradient of the magnetic field gradients. Mode 14. The system of Modes 12 or 13, wherein the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single receiving coil of the receiving coils or the transmitting coil; and the determined characteristic of the MRI measurement includes sample phase information during the measurement of one or more sample states. Mode 15. The system of any of Modes 12 to 14, wherein the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data relating to the amplitude and phase of the sample for a single receiving coil of the receiving coils or the transmitting coil; and the determined characteristic of the MRI measurement includes an amplitude and phase sensitivity of the single receiving coil or the transmitting coil, respectively. Mode 16. The Mode 15 system, wherein the amplitude and phase sensitivity of the single receiving coil are different from the amplitude and phase sensitivity of the transmitting coil. Mode 17. The system of any of Modes 12 to 16, where the magnetic field gradients are non-linear magnetic field gradients. Mode 18. The system of any of Modes 12 to 17, wherein the total magnetic field includes a non-uniform static magnetic field. Mode 19. The system of any of the modes 12 to 18, wherein the processor is further configured to reconstruct a magnetic resonance image of the sample based on the determined characteristic of the MRI measurement. Modality 20. The Modality 19 system, wherein magnetic resonance image reconstruction includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) method, or a conjugate gradient solver with compressed detection constraints (CG-CS) to at least the determined feature of the MRI measurement. Mode 21. The system of any of Modes 12 to 20, wherein the determined characteristic of the MRI measurement is selected from the group consisting of a form of a reading gradient of the magnetic field gradients, a phase information of the sample during the measurement of one or more states of the sample, and an amplitude and phase sensitivity of the single receiving coil or transmitting coil, respectively. Modality 22. The system of any of the Modalities 12 to 21, where the characteristic of the MRI measurement data is determined without generating the second coding matrix.

Claims

1. A magnetic resonance imaging (MRI) method comprising: applying, using a transmitting coil and a radio frequency (RF) source, a total magnetic field including magnetic field gradients and one or more RF pulse sequences, respectively, to measure one or more states of a sample; receiving, from one or more receiving coils, MRI measurement data acquired by the receiving coils during an acquisition window, the MRI measurement data including magnetic resonance signal data emitted by the sample; linearly combining, via a processor, a subset of the MRI measurement data to generate a first encoding matrix, the first encoding matrix being a sub-array 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 transmitting coil, and the receiving coils;and to determine, by the processor, a characteristic of the MRI measurement data based on a calculation of the first coding matrix.; 2. The method according to claim 1, further characterized in that: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single magnetic field gradient of the magnetic field gradients and a single state of one or more sample states; and the determined feature of the MRI measurement includes a shape of a reading gradient of the magnetic field gradients.

3. The method according to claim 1, further characterized in that: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single receive coil of the receive coils or the transmit coil; and the determined feature of the MRI measurement includes sample phase information during the measurement of one or more sample states.

4. The method according to claim 1, further characterized in that: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data relating to the amplitude and phase of the sample for a single receiving coil of the receiving coils or the transmitting coil; and the determined characteristic of the MRI measurement includes an amplitude and phase sensitivity of the single receiving coil of the receiving coils or the transmitting coil, respectively.

5. The method according to claim 4, further characterized in that the amplitude and phase sensitivity of the single receiving coil of the receiving coils are different from the amplitude and phase sensitivity of the transmitting coil.

6. The method according to claim 1, further characterized in that the magnetic field gradients are non-linear magnetic field gradients.

7. The method according to claim 1, further characterized in that the total magnetic field includes a non-uniform static magnetic field.

8. The method according to claim 1, further characterized in that it comprises reconstructing, via the processor, a magnetic resonance image of the sample based on the determined characteristic of the MRI measurement.

9. The method according to claim 8, further characterized in that the magnetic resonance image reconstruction includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) method, or a conjugate gradient solver with compressed detection constraints (CG-CS) to at least the determined feature of the MRI measurement.

10. The method according to claim 1, further characterized in that the determined feature of the MRI measurement is selected from the group consisting of a form of a reading gradient of the magnetic field gradients, a phase information of the sample during the measurement of one or more sample states, and an amplitude and phase sensitivity of the single receiving coil of the receiving coils or the transmitting coil, respectively.

11. The method according to claim 1, further characterized in that the characteristic of the MRI measurement data is determined without generating the second coding matrix.

12. A magnetic resonance imaging (MRI) system, comprising: a transmitting coil and a radio frequency (RF) source configured to apply a total magnetic field including magnetic field gradients and one or more RF pulse sequences, respectively, to measure one or more states of a sample; one or more receiving coils configured to acquire MRI measurement data including magnetic resonance signal data emitted by the sample during an acquisition window; and a processor configured to: receive, from the one or more receiving coils, the MRI measurement data;linearly combine a subset of the MRI measurement data to generate a first coding matrix, the first coding matrix being a sub-matrix of a second coding 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, transmit coil, and receive coils; and determine a feature of the MRI measurement data based on a calculation of the first coding matrix.

13. The system according to claim 12, further characterized in that: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single magnetic field gradient of the magnetic field gradients and a single state of one or more sample states; and the determined feature of the MRI measurement includes a form of a reading gradient of the magnetic field gradients.

14. The system according to claim 12, further characterized in that: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data for a single receiving coil of the receiving coils or the transmitting coil; and the determined feature of the MRI measurement includes sample phase information during the measurement of one or more sample states.

15. The system according to claim 12, further characterized in that: the subset of MRI measurement data linearly combined to generate the first encoding matrix includes the MRI measurement data relating to the amplitude and phase of the sample for a single receiving coil of the receiving coils or the transmitting coil; and the determined characteristic of the MRI measurement includes an amplitude and phase sensitivity of the single receiving coil of the receiving coils or the transmitting coil, respectively.

16. The system according to claim 15, further characterized in that the amplitude and phase sensitivity of the single receiving coil of the receiving coils are different from the amplitude and phase sensitivity of the transmitting coil.

17. The system according to claim 12, further characterized in that the magnetic field gradients are non-linear magnetic field gradients.

18. The system according to claim 12, further characterized in that the total magnetic field includes a non-uniform static magnetic field.

19. The system according to claim 12, further characterized in that the processor is additionally configured to reconstruct a magnetic resonance image of the sample based on the determined characteristic of the MRI measurement.

20. The system according to claim 19, further characterized in that the magnetic resonance image reconstruction includes applying one or more of a simultaneous iterative reconstruction technique (SIRT), a conjugate gradient least squares (CGLS) method, or a conjugate gradient solver with compressed detection constraints (CG-CS) to at least the determined feature of the MRI measurement.

21. The system according to claim 12, further characterized in that the determined feature of the MRI measurement is selected from the group consisting of a form of a reading gradient of the magnetic field gradients, a phase information of the sample during the measurement of one or more sample states, and an amplitude and phase sensitivity of the single receiving coil of the receiving coils or the transmitting coil, respectively.

22. The system according to claim 12, further characterized in that the characteristic of the MRI measurement data is determined without generating the second encoding matrix.

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