Blind Compressed Sensing Based MRI

Compressed sensing (CS) enables accurate MRI reconstruction from lim- ited k -space or Fourier measurements [44, 71, 72]. However, CS-based MRI suffers from various artifacts at high undersampling factors, when using non-adaptive or analytical sparsifying transforms [73]. Recent works [44] proposed blind compressed sensing (BCS)-based MR image reconstruction methods using learned signal models, and achieved superior reconstruction results. The terminology blind compressed sensing (or image model-blind compressed sensing) is used because the dictionary or sparsifying transform for the underlying image patches is assumed unknown a priori, and is learned simultaneously with the image from the undersampled (compressive) mea- surements themselves. MR image patches also typically contain variously oriented features [26], which have recently been shown to be well sparsifiable by directional wavelets [68]. As an alternative to approaches involving direc- tional analytical transforms, here, we propose an adaptive FRIST-based ap- proach that can adapt a parent transform W while clustering image patches simultaneously based on their geometric orientations. This leads to more accurate modeling of MR image features.

Similar to the previous TL-MRI work [44], we propose a BCS-based MR image reconstruction scheme using the (adaptive) FRIST model, dubbed FRIST-MRI. For computational efficiency, we restrict the parent W to be a unitary transform in the following. The FRIST-blind image recovery problem with a sparsity constraint is formulated as

(P7) min W,y,{xi,Ck} µ kFuy − zk2₂+ K X k=1 X i∈Ck kW ΦkRiy − xik2₂ s.t. WHW = I, kXk₀ ≤ s, kyk₂ ≤ L, {Ck} ∈ Γ ,

where WH_{W = I is the unitary constraint, y ∈ C}P _{is the MR image to}

be reconstructed, and z ∈ CM _{denotes the measurements with the sensing}

matrix Fu ∈ CM ×P, which is the undersampled (single-coil) Fourier encoding

matrix. Here M  P , so Problem (P7) is aimed to reconstruct an MR image y from highly undersampled measurements z. The constraint kyk₂ ≤ L with L > 0, represents prior knowledge of the signal energy/range. The sparsity term kXk₀ counts the number of non-zeros in the entire sparse code matrix

X, whose columns are the sparse codes {xi}. This sparsity constraint enables

variable sparsity levels for individual patches [44].

We use a block coordinate descent approach [44] to solve Problem (P7). The proposed algorithm alternates between (i) sparse coding and clustering, (ii) parent transform update, and (iii) MR image update. We initialize this algorithm, dubbed FRIST-MRI, with an image (estimate for y) such as the zero-filled Fourier reconstruction F_uHz. Step (i) solves Problem (P7) for {xi, Ck} with fixed W and y as

min {xi,Ck} K X k=1 X i∈Ck kW ΦkRiy − xik2₂ s.t. kXk₀ ≤ s, {Ck} ∈ Γ . (3.10)

The exact solution to Problem (3.10) requires calculating the sparsification error (objective) for each possible clustering. The cost of this scales as O(P n2_KP_{) for P patches,}1 _{which is computationally infeasible. Instead,}

we present an approximate solution here, which we observed to work well in our experiments. In this method, we first compute the total sparsification error SEk, associated with each Φk, by solving the following problem:

SEk = P X i=1 SE_ki _{, min} {βk i} P X i=1 W ΦkRiy − βik 2 2 s.t. Bk 0 ≤ s , (3.11)

where the columns of Bk _{are β}k

i . The optimal Bk above is obtained by

thresholding the matrix with columns nW ΦkRiy

oP

i=1

and retaining the s largest magnitude elements. The clusters {Ck} in (3.10) are approximately

computed by assigning i ∈ C_kˆ where ˆk = arg min k

SE_ki. Once the clusters are computed, the corresponding sparse codes ˆX in (3.10) (for fixed clusters) are easily found by thresholding the matrix hW Φˆ_k1R1y | ... | W Φˆ_kPRPy

i and retaining the s largest magnitude elements [44].

Step (ii) updates the parent transform W with the unitary constraint (and with other variables fixed). The optimal solution, which is similar to previous work [23], is computed as follows. First, we calculate the full SVD AXH _{= ˜S ˜Σ ˜V}H_{, where the columns of A are {Φ}

kRiy}P_i=1. The optimal unitary

parent transform is then ˆW = ˜V ˜SH_.

1_{The number of patches is P when we use a patch overlap stride of 1 and include}

patches at image boundaries by allowing them to ‘wrap-around’ on the opposite side of the image [35].

Step (iii) solves for y with fixed W and {xi, Ck} as min y K X k=1 X i∈Ck kW ΦkRiy − xik22+ µ kFuy − zk22 s.t. kyk2 ≤ L. (3.12)

As Problem (3.12) is a least squares problem with an `2 constraint, it can

be solved exactly using the Lagrange multiplier method [74]. Thus (3.12) is equivalent to min y K X k=1 X i∈Ck kW ΦkRiy − xik2₂+ µ kFuy − zk2₂+ ρ(kyk2₂− L) , (3.13)

where ρ ≥ 0 is the optimally chosen Lagrange multiplier. An alternative approach to solving Problem (3.12) is by employing the iterative projected gradient method. However, because of the specific structure of the matrices (e.g., partial Fourier sensing matrix in MRI) involved, (3.12) can be solved much more easily and efficiently with the Lagrange multiplier method as discussed next.

Similar to the previous TL-MRI work [44], the normal equation for Prob- lem (3.13) (for known multiplier ρ) can be simplified as follows, where F denotes the full Fourier encoding matrix assumed normalized (FH_{F = I):}

(F EFH + µF F_uHFuFH + ρI)F y = F K X k=1 X i∈Ck RH_i ΦH_kWHxi+ µF FuHz (3.14) where E , PK k=1 P i∈CkR H

i ΦHkWHW ΦkRi = PP_i=1RHi Ri. When the patch

overlap stride is 1 and all wrap-around patches are included, E = nI, with I the P × P identity. Since F EFH_{, µF F}H

u FuFH [44], and ρI are all diagonal

matrices, the matrix pre-multiplying F y in (3.14) is diagonal and invertible. Hence, (3.14) can be solved cheaply. Importantly, using a unitary constraint for W leads to an efficient update in (3.14). In particular, the matrix E is not easily diagonalizable when W is not unitary. The problem of finding the optimal Lagrange multiplier reduces to solving a simple scalar equation (see, for example, equation (3.17) in [44]) that can be solved using Newton’s method. Thus, the approach based on the Lagrange multiplier method is much simpler compared to an iterative projected gradient scheme to estimate

Cameraman Peppers Man Couple

kodak 5 kodak 9 kodak 18

Figure 3.1: Testing images used in the image denoising and image inpainting experiments.

the (typically large) vector-valued image in Problem (3.12).