| Acronyms |
| MRI | Magnetic resonance imaging |
| (p)MR(I) | (Parallel) magnetic resonance (imaging) |
| FOV | Field of view |
| ROI | Region of interest |
| S(E)(N)R | Signal-to-(error) (noise) ratio |
| (N)(R)MSE | (Normalized) (root-)mean-squared error |
| SL | Shepp-Logan |
| TV | Total variation |
| EPI | Echo planar imaging |
| ISTA | Iterative shrinkage/thresholding algorithm |
| (F)(W)(S)ISTA | (fast) (weighted) (subband adaptive) ISTA |
| CS | Compressed sensing |
| D(W)(C)(F)T | Discrete (wavelet) (cosine) (Fourier) transform |
| CG | Conjugate gradient |
| IRLS | Iteratively reweighted least squares |
| MSE | Mean-squared error |
| RS | Random shifting |
| Continuous Domain and Functions |
|
r | ∈ℝ2 | spatial coordinates (XY plane) |
| k | ∈ℝ2 | k-space coordinates (XY plane) |
| ρ(r) | ∈ℝ+ | object (proton density) in space |
| m(k) | ∈ℂ | observation of the object in k-space |
| ϕ(r) | ∈ℝ | generating function |
| f^(k) | ∈ℂ | function f in the k-space domain |
| C | ∈ℂMℝ | cost function of a vector representing an image |
| Tτ | ℂM↦ ℂM | shrinkage operator with thresholds τ |
| ω | ∈ℝ2 | Fourier angular frequency |
| mS(k) | ∈ℂ | k-space observation from receiving coil S |
| S(r) | ∈ℂ | spatial sensitivity of the receiving coil |
| f^(ω) | = | ∫ℝd f(x)e−jω·xd x∈ℂ (Fourier transform) |
| χ R(r) | ∈{0,1} | characteristic function of a region R |
| ∂ R | ⊂ℝd | closed contour of a region R |
| Jn | ∈ℝℝ | n-th order Bessel function of the first kind |
| erf | ∈ℂℂ | error function of a complex argument |
| γ(s,z) | ∈ℂℝ×ℂ | lower incomplete gamma function |
| Discrete Data and Linear Algebra |
|
j | ∈ℂ | imaginary unit such that j2=−1 |
| p | ∈ℤ2 | discrete spatial coordinates |
| M | ∈ ℕ | number of pixels in the ROI |
| N | ∈ ℕ | number of k-space samples |
| R | ∈ ℕ | number of receiving channels |
| kn | ∈ ℝ2 | nth k-space sampling position |
| mn | ∈ ℂ | nth k-space observation |
| m | ∈ ℂN | measurement vector |
| b | ∈ ℂN | noise vector |
| E0 | ∈ Mℂ(N,M) | Fourier encoding matrix (single homogeneous coil) |
| E | ∈ Mℂ(RN,M) | SENSE encoding matrix |
| M | ∈ Mℂ(RN,M) | system matrix (wavelet domain to k-space) |
| W | ∈ Mℂ(M,M) | DWT matrix |
| c[p] | ∈ ℂ | reconstructed spatial coefficient |
| c | ∈ ℂM | vector of spatial coefficients |
| w | ∈ ℂM | vector of wavelet coefficients |
| XH | ∈ Mℂ(N,M) | Hermitian transpose of the matrix X∈ Mℂ(M,N) |
| λmax(X) | ∈ ℝ+ | largest eigenvalue of a symmetric matrix X |
| κ(X) | ∈[1,+∞[ | ℓ2-condition number of a matrix X |
| ⟨x , y ⟩ | ∈ ℂ | regular inner product |
| x·y | ∈ ℝ | regular inner product |
| ⟨x , y ⟩Λ | ∈ ℂ | weighted inner product (xHΛy) |
| ||x ||ℓ2 | ∈ ℝ+ | regular quadratic norm |
| ||x ||Λ | ∈ ℝ+ | weighted quadratic norm (√xHΛx) |
| ||x ||ℓ1 | ∈ ℝ+ | ℓ1 norm (∑|xi|) |
| ei | ∈ℝd | the canonical vectors such that x·ei = xi |
| δk,l | ℕ2↦{0,1} | Kronecker’s delta (1 if k=l and 0 otherwise) |
| Operators on Vectors |
|
a× b | ∈ ℝ3 | vector product for a,b∈ℝ3 |
| ∇ f | ℝd↦ℝd | gradient operator for f:ℝd↦ℝ |
| ∇·f | ℝd↦ℝ | divergence operator for f:ℝd↦ℝd |
| ∇× f | ℝ3↦ℝ3 | curl operator for f:ℝf↦ℝ3 |
| Multi-Index Notations |
|
zα | = | ∏ziαi ∈ ℝ for z∈ℝd and α∈ℕd |
| |α| | = | ∑αi ∈ ℕ for α∈ℕd |
| p! | = | ∏pi! ∈ ℕ for p∈ℕd |
| Cpq | = | ∏Cpiqi = p!/((p−q)!q!) ∈ ℕ for p,q∈ℕd |
| ∑p = ab | = | ∑p1 = a1b1∑p2 = a2b2⋯∑pd = adbd for p,a,b∈ℕd |
