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标题中提到的“Accelerated MRI by SPEED with generalized sampling schemes”指出了一项关于磁共振成像（MRI）的研究，主题是通过一种名为SPEED（跳过相位编码和边缘去鬼影）的技术来加速MRI扫描，同时还引入了广义采样方案的概念。加速MRI成像技术的核心目标是减少数据采集时间，改善患者的舒适度，以及提供更灵活的数据采集选择。描述部分提供了文章的背景目的，它强调了在临床上，MRI常常受限于相对较长的数据采集时间，因此研究者们提出了各种方法来减少成像时间和提高患者舒适度。最近开发的快速成像方法SPEED，不仅可以有效地通过采集跳过的k空间数据线来减少扫描时间，而且还能够基于解析解快速重建图像。本研究的目的是增强这一快速成像技术，使之具备更多通用的采样选择，从而增加实现和应用上的灵活性。从方法部分来看，SPEED通过跳过相位编码（PE）步骤来加速MRI扫描，以前PE跳过的大小仅限于素数选择。本研究放宽了这个限制，允许选择任何整数，而不仅仅是素数。文章中还研究了在所有可能的PE跳过大小和PE移位组合下，不同的采样模式，并引入了一个基于鬼影相位矩阵的秩值的标准来评估SPEED重建质量。从结果来看，重建质量与鬼影相位矩阵的秩值以及跳过的PE大小N相关。一个低秩值意味着一个奇异矩阵，这会导致SPEED重建失败。通过选择合适的PE移位，发现任何整数，包括素数和复合数，都可以作为PE跳过的大小用于SPEED。这一发现允许更灵活的数据采集选择，可能在实际应用中带来更多的自由度。在结论部分，作者总结了通过适当选择PE移位，任何整数，包括素数和复合数，都可以用作SPEED中的PE跳过大小，从而使得数据采集更加灵活，并可能导致在实际应用中更多的自由度。关键词中提到了压缩感知、优化采样快速成像、秩、边缘去鬼影、SPEED以及MRI。压缩感知是一个相对较新的信号处理理论，它挑战了奈奎斯特采样定律，认为可以通过求解优化问题在远低于传统采样率的情况下精确重建信号。优化采样是指在尽可能少的采样点中获取必要的信息，以减少数据采集时间和提高效率。而边缘去鬼影技术用于处理MRI采集数据中的运动伪影问题。通过全文的描述，我们可以得到以下知识点： 1. MRI成像技术中加速采集的重要性及其在临床应用中的限制。 2. SPEED技术的概念，包括跳过相位编码和边缘去鬼影，以及它在减少扫描时间和快速重建图像方面的优势。 3. 通用采样方案（Generalized Sampling Schemes）的引入，以及如何通过放宽对PE跳过大小选择的限制来提高采样模式的灵活性。 4. 鬼影相位矩阵秩值在评估SPEED重建质量中的作用。 5. 素数和复合数在PE跳过大小中的应用，并且它们对重建结果的影响。 6. 压缩感知在MRI成像中的应用及其对传统采样理论的挑战。 7. 优化采样快速成像技术的目标和方法，以及其在提高MRI效率方面的作用。 8. 如何通过选择合适的PE移位来提高SPEED技术中采样方案的灵活性和实用性。

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Accelerated MRI by SPEED with Generalized

Sampling Schemes

Zhaoyang Jin

* and Qing-San Xiang

Purpose: To enhance the fast imaging technique of skipped

phase encoding (PE) and edge deghosting (SPEED) for more

general sampling options, and thus more flexibility in imple-

mentations and applications.

Methods: SPEED uses skipped PE steps to accelerate MRI

scan. Previously, the PE skip size was chosen from prime

numbers only. This restriction has been relaxed in this study to

allow choice of any integers rather than merely prime num-

bers. Various sampling patterns were studied under all possi-

ble combinations of PE skip size and PE shifts. A criterion

based on the rank values of ghost phasor matrices was intro-

duced to evaluate SPEED reconstruction.

Results: The reconstruction quality was found to correlate

with the rank value of the ghost phasor matrix and the skipped

PE size N. A low-rank value indicates a singular matrix that

causes failure of the SPEED reconstruction. Composite num-

bers combined with appropriately chosen PE shifts yielded

satisfactory reconstruction results.

Conclusion: With properly chosen PE shifts, it was found that

any integers, including both prime numbers and composite

numbers, could be used as PE skip size for SPEED. This find-

ing allows much more flexible data acquisition options that

may lead to more freedom in practical implementations and

applications. Magn Reson Med 000:000–000,

Wiley Periodicals, Inc.

Key words: fast imaging; rank; edge deghosting; SPEED;

compressed sensing; optimized sampling

MRI is often limited by its relatively long data acquisi-

tion time in clinical applications. Various approaches

have been proposed for both reduced imaging time and

improved patient comfort level (1–21). The recently

developed fast imaging method of skipped phase encod-

ing (PE) and edge deghosting (SPEED) is able to not only

effectively reduce scan time by acquiring skipped

k-space data lines (22) but also reconstruct images very

fast based on analytical solutions. Typically, three inter-

leaved datasets with different shifts in PE allow recon-

struction by using a two-layer signal model for ghosted

edge maps. A few variations and applications have been

developed for SPEED, given its simplicity and flexibility.

SPEED has been further developed with array coil

enhancement (ACE) by combing itself with parallel imag-

ing (23). SPEED-array coil enhancement partially sam-

ples k-space with a PE skip size of N by using multiple

receiver coils in parallel, achieving scan reduction fac-

tors even greater than the number of receive coils. Sim-

plified SPEED (S-SPEED) was proposed to accelerate

magnetic resonance angiography by taking advantage of

the sparsity of vasculature (24,25). For sparse objects

with dark or modest tissue background, simplified

SPEED partially samples k-space with only two inter-

leaved datasets and models the sparse ghosted images

with a single-layer structure. The efficient multiple

acquisitions-SPEED method was proposed for more effi-

cient scan time reduction by sharing similar spatial in-

formation among multiple acquisitions, leading to accel-

eration factors greater than that achievable with single

acquisition (26).

The PE skip size N in previous versions of SPEED

except SPEED-array coil enhancement, was restricted to

be prime numbers only, such as N ¼ 5, 7, or 11, avoid

reconstruction difficulties due to possible ghost phase

degeneracy (22–26). In this study, this restriction was

very much relaxed, and more generalized sampling

schemes of SPEED (G-SPEED) were proposed. This gen-

eral approach will be called G-SPEED. It was demon-

strated that the PE skip size N does not have to be lim-

ited to prime numbers, and composite numbers with

appropriately selected PE shifts can also result in satis-

factory reconstruction. In fact, combinations between the

PE skip size N and PE shifts determine the quality of

reconstruction, which is reflected in the rank value of a

ghost phasor matrix. High quality reconstruction was

achieved with composite skip sizes at N ¼ 4, 6, 8, 9 and

10. These new possibilities in data acquisition provide

SPEED with more flexibility in terms of practical imple-

mentations and applications.

METHODS

SPEED with Generalized Sampling Schemes

Similar to the original SPEED (22), the full k-space data

were selectively sampled into three interleaved datasets

(k), S

(k), and S

(k), denoted as N (d

, d

), where

N is the PE skip size and d

, d

, and d

are different PE

shifts. We have used k ¼ (k

, k

) to indicate k-space data

location. As the data-sampling pattern is periodic, only

the relative PE shifts are important. With d

¼ 0, d

Institute of Information and Control, Hangzhou Dianzi University,

Hangzhou, Zhejiang, People’s Republic of China.

Department of Radiology, University of British Columbia, Vancouver,

British Columbia, Canada.

Grant sponsor: National Natural Science Foundation of China; Grand num-

ber: 60901032.

Grant sponsor: Children’s and Women’s Health Centre of British Columbia.

*Correspondence to: Zhaoyang Jin, Ph.D., Institute of Information and Con-

trol, Hangzhou Dianzi University, Hangzhou, Zhejiang, People’s Republic of

China. E-mail: jinzhaoyang@hdu.edu.cn

Received 24 August 2012; revised 28 November 2012; accepted 28

November 2012

DOI 10.1002/mrm.24605

Published online

in Wiley Online Library

(wileyonlinelibrary.com).

Magnetic Resonance in Medicine 00:000–000 (2013)

2013 Wiley Periodicals, Inc.

2013.

2013

, and d

< d

, there were a total number of (N  2)(N

 1)/2 possible combinations of N (d

, d

) for each N.

The sampled datasets were first Fourier transformed into

three ghosted images, I

(r), I

(r), and I

(r), each of which

is associated with N-fold aliasing ghosts and thus has

ghost overlapping up to N layers. Now, r ¼ (x, y) repre-

sents position in image space. Three sparse ghosted-edge

maps, E

(r), E

(r), and E

(r), were generated after a differ-

ential operation applied along the PE direction for each

dataset. By modeling them with a double-layer structure,

each pixel in ghosted edge maps can be described by

three ghost equations, written as

¼ P

þ P

[1]

¼ P

þ P

[2]

¼ P

þ P

[3]

where G

and G

are two complex-valued ghosts of dif-

ferent orders, P

are ghost phasors, representing possi-

ble vector rotations, or relative phase shifts of the com-

plex-valued ghosts in the three edge maps E

(r), E

(r),

and E

(r). The ghost phasor P

is known to have a form

¼ e

ið2pdn=NÞ

; d ¼ 0; 1; 2; ; N  1; n ¼ 0; 1; 2; ; N  1;

[4]

where d, n, and N are all integers representing the rela-

tive sampling shift in PE, the order of ghost, and the PE

skip size, respectively.

Equations [1]–[3] can also be written in matrix form as

E ¼ PG [5]

where E ¼ (E

, E

)

is a known vector, G ¼ (G

)

is an unknown vector, and P is a 3  2 ghost pha-

sor matrix defined as

P ¼

[6]

To simplify the equations, let

P ¼

Equations [1]–[3] or 5 describe an overdetermined lin-

ear system with three equations and two unknowns. An

analytical approach to solving such an overdetermined

system can begin by multiplying both sides by P





[7]

where the superscript “þ” represents conjugate-transpose

of a matrix, and the superscript “*” means conjugate.

This reduces the system to a square, 2  2 system known

as the normal equations:

E ¼ P

PG [8]

If the column vectors of P are linearly independent,

then P

P is nonsingular and

G ¼ LE ¼ P

½

1

E [9]

where the superscript “1” represents inverse of a ma-

trix, and L ¼ P

P½

1

As P

P ¼





3 R



þ R



þ R



þ R



þ R





;

if we define M ¼ R



þ R



þ R



, then,

P ¼

3 M





; [10]

P½

1

9  M



3 M

M





; [11]

L ¼

9  M



 MR



 MR



 MR



M



þ 3R



M



þ 3R



M



þ 3R





: [12]

The solution candidate of G with constant pair (n

, n

)

can be calculated directly based on Eqs. [9] and [12].

There can be a least-square error (LSE) solution for such

overdetermined system, for example, LSE solution finds

G which minimizes r  |PG  E| [13]

where r is the residual of the solution, which is defined

as the two-norm or square sum of all the vector

elements.

The correct solution of (G

, G

) along with parame-

ters (n

, n

), can be simultaneously determined by test-

ing all possible (n

, n

) pairs while comparing the result-

ing LSE until a minimum r is found. By minimizing

LSE, the ghost-order index pair (n

, n

) is determined

through a multiple, but limited number of trials. Consid-

ering the symmetric positions of n

and n

in Eqs. [1]–

[3], we have a total of N(N  1)/2 trials to run through

with LSE solution, for example, there are 15 trials for N

¼ 6.

With the determined (n

, n

) pair, ghost solutions (G

) are then simultaneously obtained. Subsequently, the

edge ghosts (G

, G

) can be sorted out according to

their associated (n

, n

) values and can thereby yield N

separate edge ghosts. For example, there are six G

maps

when N ¼ 6, with n ¼ 0, 1, 2, 3, 4, and 5. They can be

individually shifted along PE direction by n/N  field of

view to its source location. The six separate edge ghosts

2 Jin and Xiang

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