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 | ∈ℝ^{2} | spatial coordinates (XY plane) |

k | ∈ℝ^{2} | 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 | ∈ℂ^{M}^{ℝ} | cost function of a vector representing an image |

T_{τ} | ℂ↦ ^{M}ℂ^{M} | shrinkage operator with thresholds τ |

ω | ∈ℝ^{2} | Fourier angular frequency |

m(_{S}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·x}d x∈ℂ (Fourier transform) |

χ_{ R}(r) | ∈{0,1} | characteristic function of a region R |

∂ R | ⊂ℝ^{d} | closed contour of a region R |

J _{n} | ∈ℝ^{ℝ} | n-th order Bessel function of the first kind |

erf | ∈ℂ^{ℂ} | error function of a complex argument |

γ(s,z) | ∈ℂ^{ℝ×ℂ} | lower incomplete gamma function |

Discrete Data and Linear Algebra | ||

j | ∈ℂ | imaginary unit such that j^{2}=−1 |

p | ∈ℤ^{2} | discrete spatial coordinates |

M | ∈ ℕ | number of pixels in the ROI |

N | ∈ ℕ | number of k-space samples |

R | ∈ ℕ | number of receiving channels |

k_{n} | ∈ ℝ^{2} | nth k-space sampling position |

m _{n} | ∈ ℂ | nth k-space observation |

m | ∈ ℂ^{N} | measurement vector |

b | ∈ ℂ^{N} | noise vector |

E_{0} | ∈ M_{ℂ}(N,M) | Fourier encoding matrix (single homogeneous coil) |

E | ∈ M_{ℂ}(RN,M) | SENSE encoding matrix |

M | ∈ M_{ℂ}(RN,M) | system matrix (wavelet domain to k-space) |

W | ∈ M_{ℂ}(M,M) | DWT matrix |

c[p] | ∈ ℂ | reconstructed spatial coefficient |

c | ∈ ℂ^{M} | vector of spatial coefficients |

w | ∈ ℂ^{M} | vector of wavelet coefficients |

X^{H} | ∈ M_{ℂ}(N,M) | Hermitian transpose of the matrix X∈ M_{ℂ}(M,N) |

λ_{max}(X) | ∈ ℝ^{+} | largest eigenvalue of a symmetric matrix X |

κ(X) | ∈[1,+∞[ | ℓ_{2}-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 (x^{H}Λy) |

||x ||_{ℓ2} | ∈ ℝ^{+} | regular quadratic norm |

||x ||_{Λ} | ∈ ℝ^{+} | weighted quadratic norm (√x^{H}Λx) |

||x ||_{ℓ1} | ∈ ℝ^{+} | ℓ_{1} norm (∑|x|)_{i} |

e_{i} | ∈ℝ^{d} | the canonical vectors such that x·e = _{i}x_{i} |

δ_{k,l} | ℕ^{2}↦{0,1} | Kronecker’s delta (1 if k=l and 0 otherwise) |

Operators on Vectors | ||

a× b | ∈ ℝ^{3} | vector product for a,b∈ℝ^{3} |

∇ f | ℝ↦ℝ^{d}^{d} | gradient operator for f:ℝ↦ℝ^{d} |

∇·f | ℝ↦ℝ ^{d} | divergence operator for f:ℝ↦ℝ^{d}^{d} |

∇× f | ℝ^{3}↦ℝ^{3} | curl operator for f:ℝ↦ℝ^{f}^{3} |

Multi-Index Notations | ||

z ^{α} | = | ∏z_{i}^{αi} ∈ ℝ for z∈ℝ and ^{d}α∈ℕ^{d} |

|α| | = | ∑α ∈ ℕ for _{i}α∈ℕ^{d} |

p! | = | ∏p! ∈ ℕ for _{i}p∈ℕ^{d} |

C_{p}^{q} | = | ∏C = _{pi}^{qi}p!/((p−q)!q!) ∈ ℕ for p,q∈ℕ^{d} |

∑_{p = a}^{b} | = | ∑_{p1 = a1}^{b1}∑_{p2 = a2}^{b2}⋯∑_{pd = ad} for ^{bd}p,a,b∈ℕ^{d} |