正交字典与非凸正则化驱动的组稀疏MRI重建算法
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本文主要探讨了在磁共振成像（MRI）重建领域中的一个重要进展，即利用正交字典和非凸正则化技术实现组稀疏性，从而提高压缩感知MRI（Compressed Sensing MRI, CS-MRI）的精度和效率。MRI扫描通常依赖于大量数据来获取清晰图像，而传统的高采样率可能导致扫描时间过长。CS-MRI通过采样策略和信号重构技术，能够在减少采集次数的同时仍保持图像质量，从而显著缩短扫描时间。
在文中，作者Shujun Liu等人提出了一个结合了两种先验信息的算法：一是稀疏性，这是指信号在某种基下的系数表示相对较少；二是非局部相似性，它强调信号的不同部分之间存在局部结构的一致性。他们将这两种先验结合在一起，采用正交字典，这是一种特殊的基，它在保持方向分离的同时具有良好的可解释性，有助于捕获信号的特征。
关键在于，他们引入了非凸优化技术，相较于传统的凸优化方法，非凸优化允许更广泛的可能性空间，能够找到全局最优解，这对于捕捉信号的复杂结构和潜在模式非常关键。通过这种方法，组稀疏性得以实现，即信号被分解为多个相互关联的组，每个组内的元素共享相似的特性，这有助于进一步提升重建的精确度。
在具体实施过程中，该算法首先学习一个适合数据的正交字典，然后利用非凸正则化惩罚函数来推广稀疏性和组结构，最后通过迭代优化求解过程，从不完全采样的k空间数据中准确地重构MRI图像。经过实验验证，这个算法在处理实际MRI数据时显示出优秀的性能，不仅提高了重建速度，还显著提高了图像的质量。
总结来说，这篇研究论文提供了一种创新的思路，将正交字典和非凸优化应用于MRI重建，极大地推动了CS-MRI技术的发展，为医疗成像领域的高效和精确成像提供了新的可能性。其潜在的应用价值和对低秩问题的有效处理，使得该方法具有广泛的科学价值和实际应用前景。
164 S. Liu et al. / Information Sciences 451–452 (2018) 161–179
X
i
= [ D α
i, 0
, D α
i, 1
, ···, D α
i,m −1
]
= D [ α
i, 0
, α
i, 1
, ···, α
i,m −1
]
= D A
i
, (5)
where α
i, j
denotes the sparse coefficient of x
i, j
and A
i
is the set of coefficients of all patches in X
i
. Due to the similarity
in the group, the corresponding sparse coefficients α
i, 0
, α
i, j
, . . . , α
i,m −1
have the same sparse profile. Different from inde-
pendent sparse representation of each patch, the nonlocal similarity between each patch is utilized to promote the sparsity
of the group. To conveniently represent the notation usage, a matrix A ∈ C
n ×m
composed of multiple sparse column vectors
with the same sparse profile is used, and a pseudo norm to enhance such group sparsity is developed that jointly regularizes
the sparsity of all columns, given by Mairal et al. [30]
|| A | |
p,q
=
n
k =1
|| α
k
||
p
q
, (6)
where α
k
denotes the k -th row of A . Especially when p = 0 , q = ∞ , || A ||
0, ∞
is the number of nonzero rows of A , which
differs from the l
0
norm counting the number of nonzero elements. Hence, as a regularization, || A ||
0, ∞
will form nonzero
elements of A at the same rows, which is consistent with the group sparsity. Therefore, || • ||
0, ∞
is a fair indication and
regularization to reflect and promote group sparsity.
Unfortunately, minimizing || A ||
0, ∞
is also an NP-hard problem. Therefore, the corresponding convex relaxation || A ||
1, 2
has
been used as the regularization, and some promising results have been reported in image denoising [13,30] . Seeing the bet-
ter performance achieved by non-convex surrogates, the non-convex || A ||
p , 2
(0 < p < 1) should provide better reconstruction
than the utilization of || A ||
1, 2
. Moreover, by means of the equivalence of norms, i.e., || α
k
||
∞
≤ || α
k
||
2
≤m || α
k
||
∞
, the upper
and lower bounds of || A ||
p , 2
are
n
k =1
|| α
k
||
p
∞
≤
n
k =1
|| α
k
||
p
2
≤ m
p
n
k =1
|| α
k
||
p
∞
, which indicates that || A | |
0 , ∞
≤ lim
p→ 0
|| A | |
p, 2
≤
|| A | |
0 , ∞
. According to the Squeeze theorem, one simply deduces that the limit of || A ||
p , 2
at p = 0 is exactly || A ||
0, ∞
. This
fact explains why || A ||
p , 2
(0 < p < 1) better approximates || A ||
0, ∞
than || A ||
1, 2
. Hence, the proposed CS recovery model with
the non-convex of group || A ||
p , 2
is formulated by
min
x , A
i
λ
2
|| y − F
u
x ||
2
2
+
M
i =1
|| A
i
| |
p, 2
s . t .
˜
R
i
x = D A
i
(for i = 1 , 2 , . . . , M )
, (7)
where λ should be inversely proportional to the variance of noise in the k-space data y and M is the number of the extracted
group. By introducing the regularization parameter β, an unconstrained form of Eq. (7) is
min
x , A
i
λ
2
|| y − F
u
x ||
2
2
+
β
2
M
i =1
||
˜
R
i
x −D A
i
||
2
F
+
M
i =1
|| A
i
| |
p, 2
. (8)
Due to the equality constraint of
˜
R
i
x = D A
i
, β should approach infinity. In the implementations, β is usually initialized to
be a small constant, and minimization of Eq. (8) with respect to (w.r.t) x and A
i
is conducted with the increased value of β
at each iteration, which is known as the alternating direction method (ADM) [33,45] . At each iteration, Eq. (8) is decomposed
into two subproblems, and they are
min
A
i
β
2
||
˜
R
i
x −D A
i
||
2
F
+ || A
i
| |
p, 2
(9)
and
min
x
λ
2
|| y − F
u
x ||
2
2
+
β
2
M
i =1
||
˜
R
i
x −D A
i
||
2
F
. (10)
Note that x in Eq. (9) is the image obtained from the previous iteration and that solving Eq. (9) requires a non-convex
optimization known as SSC [30,38,39] . Recent developments show that the orthogonal dictionary with competitive perfor-
mance is better than the redundant dictionary in terms of computation [25,33] . Inspired by this fact, the D in Eq. (9) is set
as an orthogonal dictionary, and then the first Frobenius norm term in Eq. (9) is rewritten as
||
˜
R
i
x −D A
i
||
2
F
= || D
H
˜
R
i
x −A
i
||
2
F
= || W
i
− A
i
||
2
F
, (11)
where the auxiliary variable W
i
= D
H
˜
R
i
x is introduced for simplicity. To discover the form of the optimal solution and sim-
plify the optimization, a decomposition of A
i
and W
i
is conducted as
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