Table of contents
1. Parallel imaging
Parallel MR imaging (pMRI) : Phased array receiving coils not only have their own dedicated receiving channels, butalso collect data simultaneously in a sub-sampling manner,Utilizing Spatial Information of Receiving Coil Sensitivity as a Complement to Insufficient Gradient Phase Encoding Lines, using a specific algorithm to reconstruct the final image without aliasing .
Parallel imaging always has a lower SNR due to reduced scan time, and there is an additional SNR penalty . Image aliasing dewarping in SENSE further reduces image noise . In SMASH, the data from the individual coils are combined in complex k-space, resulting in partial signal cancellation resulting in reduced SNR in some parts of the image . In parallel imaging, usuallyThe noise variance is also spatially varying.
Another way to use parallel imaging is to reduce artifacts of echo train pulse sequences (eg, EPI, fSE) . EPI suffers from geometric distortion artifacts due to off-resonance effects, while fSE suffers from blurring due to T2 attenuation . For a fixed echo spacing, the longer the echo train, the heavier the artifact. Parallel imaging reduces the number of k-space lines, which in turn reduces the echo train length, thereby reducing artifacts .
Parallel imaging is image reconstruction from multi-coil undersampled k-space data, can be compared and checked with the image reconstructed with the full-sampled K-space data of the array coil to check its consistency.
(a) The figure shows the full sampling space .
(b) Parallel imaging of K-space with an acceleration factor of R=3, the dotted line represents the unacquired K-space line, and the two missing lines between the two measured lines Km and Km+1 are Km+ΔK, Km+2ΔK(ΔK is the phase encoding step required by the Nyquist sampling theorem)。
In Cartesian acquisition the sweep time is proportional to the number of phase encoding steps .Undersampling Simultaneous Data Acquisition Using Multiple Coils, thus saving time and increasing scanning speed. Increasing the spacing of K-space rows by R times while keeping the maximum covered K-space range (spatial resolution) unchanged , the scanning time will be reduced by R times, soIn parallel imaging R is called the "speed-up factor" or (of scan time) "reduction factor" 。
Increasing the pitch of the phase-encoding lines results in a reduction in the FOV , and if objects extend outside the reduced FOV , aliasing or wrapping artifacts can occur .
Parallel imaging can be divided into K-space methods and image space methods .SENSE reconstructionis to process data in the image space to solve problems , known as "image space method";andSMASH rebuildIt is to process data and solve problems in K space , so it is called "K-space method”。
2. SENSE reconstruction
The SENSE method isDiscrete Fourier transform is performed on the K-space data of each receiving coil, to obtain an image with aliasing, and then use the sensitivity map information of each coil in the image space to form a frame-by-frame intermediate image without aliasing through the de-aliasing algorithm for the aliased image of the corresponding coil, and finallyCombine these intermediate images into a complete image with a full field of view using the square sum method.
Sensitivity encoding (SENSE) is a reconstruction method in technical image domain . SENSE method Use of multiple receive coils for signal acquisition, allowing undersampling of K-space. Some spatial encoding information will be lost in the process of undersampling , so when SENSE is reconstructed, the spatial sensitivity of each coil is used to restore the lost information, and the aliased image is expanded to obtain an image without artifacts .
SENSE rebuild canReduce the number of K-space acquisition lines and shorten the scanning time. However, due to the undersampling, the signal-to-noise ratio of the image is reduced ; and the SENSE reconstruction needs to estimate the coil sensitivity information, and the reconstruction process is cumbersome ; and the coil sensitivity information is difficult to estimate accurately, and its error will lead to reconstructed image artifacts .
(a) The picture shows the normal FOV .
(b) The graph shows the reduced FOV for R=3 for parallel imaging. The number of copies NA aliased at each position in the y direction depends on the edge position and the acceleration factor R.
The SENSE method requires an estimate of coil sensitivity , which can be estimated with additional calibration scans covering the entire volume scanned with parallel imaging . Another approach is by measuring the central part of K-space with full Nyquist sampling ,Get Low Resolution Sensitivity Estimates; while measuring the peripheral region of K-space with undersampling to speed up the acquisition , these steps can be built and integrated in a parallel imaging sequence.
Regularization means to add some restrictions to the loss function, and through this rule to regulate them in the next loop iterations, do not self-inflate (Regularization is to prevent overfitting)。
Singular value decomposition is to find the low-dimensional matrix M closest to the original matrix A in the low-dimensional space. To put it bluntly, it isData dimensionality reduction。
With the traditional sum of squares reconstruction , the degree or number of image aliasing increases with the increase of R due to the undersampling of each coil . And useSENSE reconstructed images without any aliasing artifacts, but the noise distribution is not uniform in the case of high R. The noise difference of the SENSE images in the two phase encoding directions is obvious, which is due to the difference in their geometric factors. For the coil arrangement shown in the figure below, it is obviously more advantageous to choose vertical phase encoding. The SENSE theory does not impose any constraints on coil configuration, relative arrangement position, or k-space sampling .
SENSEIncluding strong reconstruction techniques and weak reconstruction techniques , the former strictly optimizes the shape of each voxel, while the latter pays more attention to the signal-to-noise ratio . Because both algorithms arehybrid code reconstruction, so the fast Fourier transform (FFT) cannot be used directly .
The first step of SENSE technology is to make each coil data of the coil array undergo discrete Fourier transform to form images with aliasing artifacts ; the second step is toThe aliased image obtained in the first step is de-aliased and stitched together to form a complete frame of image.
The time reduction factor is from 1 to 4, and the phase is encoded in the horizontal direction. The left column is the traditional sum of squares image ; the middle column is the image reconstructed by the same data SENSE ; the right column is the noise level map expected by SENSE theory .
SENSE can also be applied to non-Cartesian K-space trajectories , such as radial trajectories , spiral trajectories , etc. Parallel imaging reconstruction algorithms with optimal image quality for spiral scans and other non-rectilinear k-space data require matrix inversions that are two orders of magnitude larger than those produced by rectilinear trajectories. Such matrices generally require iterative inversion methods , such as the conjugate gradient method.
conjugate gradient methodis one of the most popular and well-known iterative techniques for solving sparse symmetric positive definite systems of linear equations . A method for solving the linear equation A x = b, especially the state-of-the-art method for iterative solution of sparse linear systems, is called the linear conjugate gradient method. Later, people gradually extended this method to the solution of nonlinear problems, called the nonlinear conjugate gradient method .
3. SMASH reconstruction
The SMASH method puts the sensitivity information of the coil and the undersampled data of each coil in the K space for processing, constructs the low-order spatial harmonic components by fitting the coil sensitivity , and uses the K space data of each coil undersampled to construct the corresponding full field of view The composite K-space data , and then undergo a Fourier transform to obtain the final image without artifacts.
The SMASH reconstruction algorithm uses a linear combination of raw data and sensitivities acquired by,Recover K-space phase-encoded line data lost due to undersampling. Using coil sensitivities to provide spatial weights for received signals, a linear combination of coil sensitivities can approximate the complex exponential function corresponding to the missing spatially encoded row .
Harmonics of the same order as the number of coils in the coil array are sometimes difficult to construct, and the promotion of SMASH can replace the second positive harmonic near the Km row with the first negative harmonic near the Km+1 row. More accurate than synthesizing higher harmonics due to synthesizing lower harmonics .
Another promotion isblock reconstruction, synthesize the missing harmonic row Km+pΔK with multiple measurement rows . This generalization improves the fitting accuracy, especially for coil arrangements of complex exponential functions that are difficult to approximate with linear combinations of coil sensitivities.The more measurement lines used to construct missing k-space lines, the better the image quality at the expense of slower reconstruction.
(a) The picture shows the four-coil array that can be used in SMASH
(b) The picture shows the coil sensitivity
© The picture shows the sum of the coil sensitivities (1)
(d), (e), (f) are the sensitivity combinations
The optimal coil geometry for SMASH is probably an array of coils placed along a line, although SMASH can produce high-quality images with any coil geometry. Parallel imaging can also be applied to the second phase encoding of the 3D scan . Parallel imaging is most useful at high-field SNR, and one of the most successful applications is contrast-enhanced MRA. Reducing the scan time is beneficial for capturing the peak of the bolus passage and shortening the breath-hold time . Cardiac scans with SSFP gradient echo sequences (such as True FISP) give sufficient SNR to allow an acceleration factor R>=2. This results in shorter breath-hold times, better spatial or temporal resolution.
Most coil arrays are designed so that the sum of sensitivities is approximately a constant in order toGives an approximately uniform image signal (for uniform objects)。
4. Sensitivity Calibration
The image quality of all parallel acquisition imaging depends on an estimate of the coil sensitivity , a poor estimate will result in some aliasing that cannot be corrected and low SNR .
(1) Direct sensitivity measurement : measure data with lower spatial resolution to save time ,The volume of image data acquired must encompass the entire area reconstructed with parallel imaging, the calibration scan is usually a two-dimensional (2D) or three-dimensional (3D) gradient echo. It is also possible to sample with variable density in parallel imaging scans , i.e.Rows close to the center of k-space are sampled at the Nyquist frequency, while high-k-space regions are undersampled, and low-k-space data are used to reconstruct low-resolution sensitivity maps.
(2) Indirect sensitivity measurement : In the indirect sensitivity measurement method, one or more Nyquist rows are (extra) sampled in the center of K-space , similar to the self-calibration method. This set of Nyquist sampling lines is called the self-calibration signal and avoids inconsistencies in patient motion between the calibration scan and the parallel imaging scan.
Indirect sensitivity measurements, becauseThe calibration line is fully sampled, which is less than the total time of the separate calibration and parallel imaging scans combined . Self-calibrating scans can also be used in parallel imaging reconstruction, reducing the amount of inherent aliasing that must be corrected and giving higher SNR.
3) Objective function normalization : Regardless of SENSE or SMASH, the intensity weights of the final image can be manipulated by properly normalizing the surface coil sensitivity data . Each final image is weighted by the target sensitivity, allowing its surface coil calibration image to be divided by the body coil calibration image, and multiplied by the SENSE image with body coil sensitivity weighting will produce a relatively uniform image. Objective function normalization does not affect the SNR in SENSE images (Equivalent to multiplying both signal and noise by it)。
Sensitivity Normalization Problem: The image reconstructed by the parallel imaging technique always has some coil sensitivity weight. Since all MR data cannot avoid being weighted by coil sensitivity , pure coil sensitivity measurements are not possible.
If the spatial sensitivity map of the coil can be obtained reasonably accurately , SENSE should be preferred . If the acquired coil sensitivity map is poor , a self-calibration technique such as GRAPPA should be used (for example, patient motion includes respiratory motion , resulting in poorly acquired coil sensitivity map , especially at the edge of the coil, where the sensitivity changes sharply ). For example, in single-shot EPI, some areas are severely distorted, and the coil sensitivity map is also poor.
Optimal sensitivity calibration depends on the application . If patient breathing motion is a concern, forMinimize CalibrationandInconsistencies between parallel imaging data, self-calibration may be superior. However, if motion inconsistency is not a major concern and scan time is important , such as in contrast-enhanced volumetric MRA or live imaging,Separate calibration may be superior。
Separate calibration scans should also be selected if higher acceleration factors (e.g. R>=4) are used. Because the self-calibration scan occupies a higher percentage of the total scan time as the acceleration factor increases. GRAPPA and AUTO-SMASH are themselves limited to automatic calibration. In this case, there must be such a difference between the number of ACS lines and the scan time. More ACS lines generally result in better image quality, but at the expense of longer scan times .
5. AUTO-SMASH and VD-AUTO-SMASH
AUTO-SMASH is an improved version of SMASH with self-calibration function . During SMASH scanning, the R-1 row is used for automatic calibration.
If the K-space measurement line and the nearby missing line form a block (block),In AUTO-SMASH, only the sampling in the center of K-space satisfies the Nyquist criterion, while (variable density) VD-AUTO-SMASH satisfies the Nyquist criterion when there are more than one samples in the central region of K-space.
In VD-AUTO-SMASH, one or more sets of additional ACS rows are taken to improve the accuracy of the fit, resulting in better linear weighting factors aj,p. The same aj,p can be used to calculate the filling of missing k-space rows. Instead of synthesizing the missing rows with positive harmonics of Km, as for traditional SMASH, these missing rows can be synthesized with negative harmonics of Km+1 to maximize the accuracy of the fit .
The solid line represents the measured k-space line, the small dotted line represents the missing k-space line, and the additionally measured Nyquist sampling line (big dotted line) in the middle area, namely the ACS line, is used to determine the synthetic missing line (small dotted line) The weighting factor of , phase encoding in the vertical direction
((a)R=2, (b)R=3, ©R=4)
These additional acquired ACS lines are encoded with conventional phase encoding gradients, these reference lines SjACS and Usually the relationship between SMASH signal lines can be used to determine the linear weighting factor aj,p
6. GRAPPA reconstruction
GRAPPA is inAUTO-SMASHandVD-AUTO-SMASHBased on a parallel imaging method improved . Instead of synthesizing the missing k-space lines from a single measurement line, the missing k-space lines are synthesized from a set of measurement lines; and in GRAPPA, the k-space lines of each single coil are synthesized, rather than the composite corresponding to the full field of view The k-space line .
This process is repeated for each coil in the array to generate Nc single-coil images, and thenUsing the traditional sum of squares (SOS) to reconstruct and stitch together a complete final image, which has better fitting accuracy and higher SNR .
The sum of squares (SOS) algorithm is considered to be the optimal image synthesis method without knowing the exact sensitivity of the individual phased array coils . However, the sum of squares algorithm uses equal weights to synthesize each coil image , and at the same timeDoes not suppress external noise well, leading to problems such as signal deviation and low signal-to-noise ratio in the final image obtained .
The K-space ACS line data reconstruction mechanism of AUTO-SMASH and VD-AUTO-SMASH synthesizes the full field of view coincident correction line data from(in this example, the collected 4 lines of data are used to fit the full field of view. composite data). GRAPPA assumes that:
Each data point in the k-space of a single coil can be represented by a linear combination of adjacent data points in the k-space of all coils, and the weight set of the linear combination is unchanged in the k-space displacement.
GRAPPA’s K-space ACS line data reconstruction mechanism, the multi-block measurement lines collected in each coil in the array are fitted to the ACS line collected in a single coil in the array (the 4 lines of data collected in this example are used for fitting into 4 A single ACS line in the number coil, each circle represents a data line acquired in a single single coil).
When using multiple measured k-space lines to synthesize missing lines, considering that the signals contributed by nearby lines play a dominant role , oftenMove the measurement row block according to the position of the row to be composited, known as " sliding block reconstruction ".
Because each block of measurement lines generally gives different estimates for missing lines , and this estimate can be combined with a weighted average to give higher SNR and better fit as the criteria for selecting a combination.
GRAPPA is a reconstruction algorithm based on K-space , which collects the center data of K-space at the frequency required by the Nyquist sampling law as the automatic correction data (Auto-Calibration Signal, ACS),K-space undersampling of each channel is filled using the linear correlation of adjacent points in multi-channel K-space, get the full K-space of each channel, and finally get the final wrinkle-free image through channel fusion (SOS or ACC).
In order to reconstruct the missing row data in K-space, we needin two steps. The first step is calibration , that is, the reconstruction weight A is calibrated using the fully sampled data rows near the center of K-space ; the second step is to use the calibrated A (with K-space displacement invariance) and the measured row data to synthesize high K-space missing data .
(a)GRAPPA data collection method. Each K-space data in the four coils is collected every other phase encoding line to speed up the acquisition (R=2). Additional ACS lines were sampled at the center of K-space to form a fully sampled calibration region (II). The high-k space (I) is undersampled by a reduction factor R=2.
(b) Assuming the sampling method shown in (a), the spatial representation of the reconstruction weight A generated by GRAPPA.
Displacement invariance means thatNo matter where the object is in the image, the recognition result of the convolutional neural network should be the same. If the position of the object in the image does not have any effect on the prediction results of the model , then it can be said that it hasdisplacement invariance. Because CNN uses a kernel to continuously step on the entire image to complete the convolution operation, and the parameters of the kernel are shared during this process . In this way, it has displacement invariance in theory (of course, limited bystep span、 Convolution kernel sizeInfluenced by other factors, CNN may also have "leakage" under certain conditions).
The total number of target points Nt represents the number of training examples for reconstruction. In general, the larger the number of training examples, the more accurate the fitting process . More ACS rows are used to increase the number of available training examples, but at the cost of sacrificing efficiency. Since the reconstruction is displacement invariant, calibration can be performed at all Nx positions along the Kx direction, which further increases Nt. We call such a calibration strategy " full-width readout calibration ".
Raw GRAPPA reconstruction, calibration and synthesis are all performed in K-space, so it is called " K-space reconstruction ", and the original GRAPPA reconstruction uses K-space 1D neighbor reconstruction. To improve accuracy, it can also be generalized to 2D neighbor reconstruction in K-space . Moreover, due to the linearity of the system and the separability of FT, GRAPPA can also be performed in the mixed space (x, Ky) equivalently. As long as the data is 1DFT along Kx, it can be transformed into the mixed space (x, Kx). , as long as the data is subjected to 1DFT along Kx, it can be transformed into the mixed space. All steps of calibration and synthesis remain the same during hybrid space reconstruction, as do the weighting coefficients. For K-space 1D neighbor coil-by-coil reconstruction, there is no accuracy or efficiency advantage in hybrid spaces. However, when extended to the coil-by-coil reconstruction of K-space 2D neighbors [Dy*dx], there is generally an accuracy advantage.
7. SPACE RIP reconstruction algorithm
The SPACE RIP algorithm expresses pMRI reconstruction as a matrix inversion problem . This algorithm requires a large matrix inversion, so we havelong rebuild time. Its advantage is high flexibility , any K-space trajectory is applicable , and the coil arrangement can also be arbitrary .
Each column in the image is reconstructed separately . If the image has N rows and M columns, x in the matrix equation should take 1-M discrete values, corresponding to M such equations. To reconstruct M columns of images, M sensitivity phase encoding matrices need to be inverted . The matrix here does not have to be a square matrix, and its generalized inverse must be calculated for each column . The choice of phase encoding steps F affects the reconstruction quality. Increasing F leads to an increase in the rank of the matrix, and the condition number of the generalized inverse matrix generated is good, the degree of noise amplification is low, and the signal-to-noise ratio is high, but the cost is The amount of reconstruction calculation increases.
becauseSPACE RIP has no strict requirements on coil array arrangement, so it is very suitable for 3D parallel acquisition.
The left entry is a vector with Nc*F elements, containing F phase encoding values for all Nc coils; the rightmost entry is an N-element vector representing the image of a column of elements; the middle entry is a vector with Nc*F rows and a matrix of N columns based on the sensitivity profile and the phase encoding used.
Reconstruction of the columns of images produced by solving the equations for various positions along the x-axis。
The advantage of phased array coil sensitivity value parallel encoding and reconstruction (SPACE RIP) technology is that it can flexibly select the phase encoding position in K space ,Reconstructed images have fewer aliasing artifacts than other parallel imaging algorithms。
Eight, PILS reconstruction algorithm
The PILS reconstruction algorithm is based on the prior knowledge of the center position y0 of the coil sensitivity area and the local imaging field of view yc along the phase encoding direction. Under the condition that the acceleration factor R is smaller than the number of coils Nc, the number of phase encoding steps Ny is not sufficient for the full FOV . Undersampling that satisfies the Nyquist criterion , howeverFull sampling that satisfies the Nyquist criterion for the local FOV covered by each element coil。
The data collected by each unit coil in the array is mainly contributed by the near area , and the Fourier transform is only performed in the local area .Partial images with virtually no aliasing artifacts, and then compose a full-field image according to the sum of squares method (scissors method) . This algorithm works well for **No overlap of coil sensitivitiesIn the case of **, it is the fastest algorithm for parallel imaging data reconstruction of non-linear K-space trajectories such as spiral scans.
Almost all methods are equal in terms of speeding up data acquisition by supplementing gradient encoding with coil sensitivity encoding ;are different in how the reconstruction problem is solved to produce the final alias-free image. Based on how the coil sensitivity is encoded from the multi-channel data can be divided into two categories, namely physics-based reconstruction and data-driven reconstruction: the first category of
pMRI methods involves explicit knowledge of the coil sensitivity function to separate aliased signals , such as SENSE , Simultaneous acquisition of space harmonics SMASH, improved SMASH, promoted SMASH, SPACE RIP, PILS, arbitrary K-space trajectory SENSE, etc. This type of pMRI is called "physics-based" reconstruction because its model is closely related to the physical processes that occurred during image acquisition. SENSE is the most typical physics-based reconstruction algorithm ,All coil sensitivity maps must be estimated explicitly, since these maps are used to dealias the image in the image domain. Images reconstructed by physics-based pMRI methods are susceptible to artifacts caused by coil sensitivity calibration. Common sources of calibration errors include insufficient SNR, Gibbs beat, motion artifacts, or limited FOV.
The second type of pMRI method does not require explicit coil sensitivity information, but uses data fitting methods to reconstruct the linear combination weights of the target data by computing the adjacent source data, and then uses the measured data and these corresponding weights to reconstruct the missing K spatial data .Call this type of approach "data-driven" reconstruction, because these methods rely on training data to calibrate the relationship between input (source) data and output (target) based on limited knowledge of the underlying physical process. Such methods include AUTO-SMASH, VD-AUTO-SMASH, GRAPPA, etc. , all of which are data-driven reconstruction methods. Among them, GRAPPA performs coil-by-coil reconstruction, which provides improved image quality. Since the coil sensitivity map is not required, the coil-by-coil data-driven (CCDD) method is superior to the physics-based method, especially for the situation where accurate coil sensitivity estimation is very difficult.
9. PRUNO reconstruction algorithm
GRAPPA reconstruction produces good image quality at low acceleration factors , but its performance degrades significantly at high acceleration factors unless a large number of self-calibration lines (ACS) is imposed. PRUTO is an iterative K-space data-driven pMRI reconstruction algorithm, which is more flexible than GRAPPA .
In PRUNO,Data calibration and image reconstruction as linear algebra problems, the iterative conjugate gradient algorithm is effectively used to solve the reconstruction equation, and the obtained image quality is higher than that of GRAPPA, and the required ACS lines are not many, especially in the high acceleration factor .
The k-space samples from all receptive channels are linearly correlated in nature, a consequence of the inherent nature of sensitivity encoding. Once the non-zero system matrix N can be determined, the equation ND=0 can be used to solve the pMRI reconstruction problem . This is the reason why this step is named PRUNO (parallel reconstruction using null operations).
The acceleration factor R=2-6 is used for comparison. Aberrations are exaggerated by a factor of 10 and ACS data is also included resulting in Reff<R . For GRAPPA, the artifacts are clearly visible when R=3-4; for PRUNO, as long as R<6 (Reff<4), the image quality is good, and the error ( aliasing) is evenly distributed .
Regarding coil sensitivity coding, there are two important assumptions as follows:
Assumption 1: All sensitivity maps are band-limited and have a total width ws in K-space, since coil sensitivity maps are smooth in nature, ws is usually approximated by a reasonably small number。
Assumption 2: The overall sensitivity encoding yields good orthogonality. That is, if Nc>R, pMRI reconstruction can be treated as an over-determined problem, and the K-space sampling is relatively uniform (R is the acceleration factor, Nc is the number of coils)。
Here N represents the non-zero system matrix whose rows are zeroed out by multiplying the k-space samples of all coils.
10. UNFOLD Algorithm
UNFOLD methodApplied to fast scans, similar to parallel imaging. But it can only be used for time series images , sampling Kt space, such as functional imaging series, dynamic imaging series or multiphase cardiac imaging series.UNFOLD uses spatial aliasing to reduce scan time, which is similar to parallel imaging in this respect, but converts spatial aliasing to temporal aliasing and removes it with a temporal filter.
Unlike parallel imaging, UNFOLD does not require multiple coils . But UNFOLD can also be combined with parallel imaging to further increase scanning speed or reduce artifacts .