PLAN DE SEGURIDAD, EVACUACIÓN Y PROTECCIÓN

As discussed in Sections 3.1 and 3.2, many NMR and MRI reconstruction problems are regularization problems. These are convex optimization problems, for the solution of which there exist a plethora of different algorithms [45]. One family of algorithms that tackles these problems is proximal algorithms [46,47]. In these algorithms, a large convex optimization problem is split into small convex optimization problems and a proximal operator function is applied to all these small problems. A proximal operator function, which is described in more detail in Section 3.3.2, effectively moves a search point towards a domain and minimum of a function in a similar way that Newton’s gradient descent method does.

In recent years, a class of the family of proximal algorithms, primal-dual algorithms, has gained much interest [48-51]. The main advantage of the primal-dual algorithms is the efficient splitting of the large problem, and consequently the lower computational cot involved. In this work, the Primal-Dual Hybrid Gradient Method (PDHGM) developed by Chambolle and Pock [50], which has been very successful in imaging applications, is used. The PDHGM method is an efficient algorithm for solving problems of the form:

(𝒖, 𝒗) = min

𝒖 max𝒗 𝑓(𝒖) − 𝑔(𝒗) + 𝒗

T_{𝑹 𝒖 , (3.17)}

where u and v are vectors referred to as the primal and the dual variable, respectively, R is a matrix, and f and g are convex functions. The PDHGM is also applicable to the case when the primal and dual variables are matrices, rather than vectors; this case is not

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discussed in the following sections. Practically, the only difference is that the third term in Eq. (3.17) would be substituted by the inner product of v and R u, 〈𝒗, 𝑹 𝒖〉.

A rigorous mathematical treatment, including a convergence proof is found in the work of Chambolle and Pock [50]. The derivation of the algorithm given in Section 3.3.1 is much simplified and is based on the treatment of Benning [52]. Section 3.3.2 introduces the proximal operator which is the main building block of algorithm. Three examples of the application of the proximal operator are given in Section 3.3.3. Some practicalities regarding the implementation of the PDHGM are given in Section 3.3.4.

3.3.1. Iteration scheme

Let the derivative of f(u) with respect to u be p(u) and the derivative of g(v) with respect to v be q(v). Then, the optimum of the problem in Eq. (3.17) is achieved when:

𝑝(𝒖) + 𝑹T𝒗 = 𝟎 , (3.18a)

𝑞(𝒗) − 𝑹 𝒖 = 𝟎 . (3.18b)

A potential iteration scheme for solving Eqs. 3.18(a) and 3.18(b) is as follows:

[𝑝(𝒖 (𝑘+1)_{) + 𝑹}T_𝒗(𝑘+1) 𝑞(𝒗(𝑘+1)) − 𝑹 𝒖(𝑘+1)] + 𝑻 [𝒖 (𝑘+1)_{− 𝒖}(𝑘) 𝒗(𝑘+1)− 𝒗(𝑘)] = [ 𝟎 𝟎] , (3.19)

where k refers to the iteration step and T is a matrix to be chosen. It is seen that as the iterations converge, i.e. u(k+1)_→u(k) _{and v}(k+1)_→v(k)_{, Eqs. (3.18a) and (3.18b) are obeyed}

and, therefore, an optimum to Eq. (3.17) is obtained. Chambolle and Pock [50] made the following choice for T:

𝑻 = [1 𝜏⁄ 𝑰 −𝑹T

𝑹 1 𝜎⁄ 𝑰] , (3.20)

where τ and σ are algorithmic parameters and I is the identity matrix. This choice of T makes the system of equations in Eq. (3.19) explicit in terms of u(k+1) and v(k+1):

𝒖(𝑘+1)+ 𝜏 𝑝(𝒖(𝑘+1)) = 𝒖(𝑘)− 𝜏 𝑹T_𝒗(𝑘)_{, (3.21a)}

𝒗(𝑘+1)+ 𝜎 𝑞(𝒗(𝑘+1)) = 𝒗(𝑘)+ 𝜎 𝑹(2𝒖(𝑘+1)− 𝒖(𝑘)) . (3.21b)

3.3.2. Proximal operator

It is common to write Eq. (3.21) in operator format:

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(𝑰 + 𝜎 𝜕𝑔T_)(𝒗(𝑘+1)_{) = 𝒗}(𝑘)_{+ 𝜎 𝑹(2𝒖}(𝑘+1) _{− 𝒖}(𝑘)_{) . (3.22b)}

Solving for u(k+1) and v(k+1):

𝒖(𝑘+1) = (𝑰 + 𝜏 𝜕𝑓T)−𝟏_(𝒖(𝑘)_{− 𝜏 𝑹}T_𝒗(𝑘)_{) , (3.23a)}

𝒗(𝑘+1) = (𝑰 + 𝜎 𝜕𝑔T)−𝟏_(𝒗(𝑘)_{+ 𝜎 𝑹(2𝒖}(𝑘+1)_{− 𝒖}(𝑘)_{)) . (3.23b)}

The operator (I + ∂fT₎-1 _{is known as a proximal operator. Therefore, each iteration step in}

the PDHGM algorithm consists in the application of two proximal operators for the updating of the primal and dual variables. The proximal operator is computed with the help of the following observation:

(𝑰 + 𝜏 𝜕𝑓T₎−𝟏_{(𝒂) = arg min} 𝒖 (

1

2‖𝒖 − 𝒂‖2

2_{+ 𝜏 𝑓(𝒖)) . (3.24)}

To prove Eq. (3.24), it is noted that the minimum of the r.h.s. expression is achieved when its derivative with respect to u is equal to zero, which gives:

𝒂 = 𝒖 +𝜕𝑓T(𝒖)

𝜕𝒖 . (3.25)

When Eq. (3.25) is written in an operator format, it gives the l.h.s. in Eq. (3.24).

As is seen in Eq. (3.24), the proximal operator applied to a vector a, returns a vector that is a compromise between being as close as possible to a while minimizing a cost function,

f, with the relative weight determined by τ.

3.3.3. Examples

Three examples of the proximal operator for common cost functions that are encountered in NMR and MRI applications are now presented.

Example 1

The simplest cost function is a set function:

𝑓(𝒖) = 𝐼_𝐶(𝒖) = {0 𝒖 ∈ 𝐶

+∞ 𝒖 ∉ 𝐶 , (3.26)

where C is a set. Since the cost is infinite if 𝒖 is not in the set, the solution to the proximal operator in Eq. (3.24) is the projection of a into the set C.

Example 2

50 𝑓(𝒖) = 𝜒𝛼(𝒖) = {

0 ‖𝒖‖_∞≤ 𝛼 +∞ ‖𝒖‖∞> 𝛼

, (3.27)

where the L∞-norm is defined in Eq. (3.4); it represents the largest absolute value entry of

u. The function χα(u) is known as the indicator function. The proximal operator of χα(u):

(𝑰 + 𝜏 𝜕𝑓T₎−𝟏_{(𝒂) = arg min} 𝒖 ( 1 2‖𝒖 − 𝒂‖2 2_{+ 𝜏 𝜒} 𝛼(𝒖)) , (3.28)

is best computed by considering each element of a independently, because the minimization problem in Eq. (3.28) is separable. If an element of a is ≤ α, then the corresponding element in u must be equal to that entry of a, because in this way the proximal function equals 0 and this is the smallest value it can reach. If an element of a is > α, the corresponding element in u cannot be greater than α, because the indicator function blows up. The best value the element in u can get is α, because this minimizes the first term while keeping the indicator function at 0. Therefore, the proximal operator is given by:

(𝑰 + 𝜏 𝜕𝑓T₎−𝟏_{(𝒂) =} 𝒂

max(1,𝒂/𝛼) , (3.29)

where all operations are element-wise.

Example 3

A common function encountered in under sampled MRI data acquisitions, as discussed in Section 3.2, is:

𝑓(𝒖) = 1

2‖𝑺 𝓕 𝒖 − 𝒚‖2

2 _{, (3.30)}

where S is a subsampling matrix and ℱ is the Fourier operator. By definition, the proximal operator of such a function is given by:

(𝑰 + 𝜏 𝜕𝑓T₎−𝟏_{(𝒂) = arg min} 𝒖 ( 1 2‖𝒖 − 𝒂‖2 2₊𝜏 2‖𝑺 𝓕 𝒖 − 𝒚‖2 2_{) . (3.31)}

Differentiating the r.h.s. with respect to u and setting it to zero:

𝒖 − 𝒂 + 𝜏 𝓕−1_𝑺T_{𝑺 𝓕 𝒚 − 𝓕}−1_𝑺T_{𝒚 = 𝟎 . (3.32)}

Set u = ℱ u. After some rearrangement, Eq. (3.32) becomes:

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Taking the Fourier transform of both sides and rearranging, the proximal operator becomes:

(𝑰 + 𝜏 𝜕𝑓T₎−𝟏_{(𝒂) = 𝓕}−1_{𝒖 = 𝓕}−1₍ 𝓕 𝐚+𝑺T𝒚

diag(𝑰+𝜏𝑺T_𝑺)) , (3.34)

where diag(I + τ STS) extracts the main diagonal of the matrix in brackets and the division in Eq. (3.34) is element-wise.

3.3.4. Implementation

The PDHGM algorithm has two parameters, τ and σ, which control the convergence, speed and stability of the algorithm. It was proven by Chambolle and Pock [50] that convergence is guaranteed if:

𝜏 𝜎 ‖𝑹‖_2,22 ≤ 1 . (3.35)

The Lp,q norm of the matrix R, ||R||p,q, is defined as:

‖𝑹‖_𝑝,𝑞 = (∑ (∑ |𝑅_{𝑗 𝑖 𝑖𝑗}|𝑝)𝑞/𝑝)

1/𝑞

. (3.36)

There is freedom to choose τ and σ, if Eq. (3.35) is obeyed, and the choice depends on the problem being studied.

Different stopping criteria could be used for the PDHGM algorithm; the criterion used in this work was:

‖𝒖(𝑘+1)_−𝒖(𝑘)_‖ 2

‖𝒖(𝑘)_‖ 2

< TOL , (3.37)

where the tolerance level, TOL, is set heuristically.