## High Resolution Projection Reconstruction MR Imaging using FOCUSS

### BibTeX

@MISC{A_highresolution,

author = {Sungho Tak A and Jaeheung Yoo A and Jong Chul Ye A},

title = {High Resolution Projection Reconstruction MR Imaging using FOCUSS},

year = {}

}

### OpenURL

### Abstract

This paper is concerned about high resolution reconstruction of projection reconstruction MR imaging from angular under-sampled k-space data. A similar problem has been recently addressed in the framework of compressed sensing theory. Unlike the existing algorithms used in compressed sensing theory, this paper employs the FOCal Underdetermined System Solver(FOCUSS), which was originally designed for EEG and MEG source localization to obtain sparse solution by successively solving quadratic optimization. We show that FOCUSS is very effective for the projection reconstruction MRI, because the medical images are usually sparse in image domain, and the center region of the under-sampled radial k-space data still provides a meaningful low resolution image, which is essential for the convergence of FOCUSS. We applied FOCUSS for projection reconstruction MR imaging using single coil. Extensive experiments confirms that high resolution reconstruction with virtually free of angular aliasing artifacts can be obtained from severely under-sampled k-space data.

### Citations

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Citation Context ...x||22 = xT (W −1 l ) H W −1 l x = x T ⎛ ⎜ ⎝ |xl−1;1| −1 0 0 |xl−1;2| ··· 0 −1 . . . . ··· . .. 0 . . 0 0 ··· |xl−1;N | −1 ⎞ ⎟ ⎠ x N� ∼ |xl−1,i| i=1 ∼ ||x||1 as l →∞ (12) SPIE-IS&T/ Vol. 6498 64981A-3 =-=(6)-=-swhere the superscript H denotes the Hermitian transpose, and ∼ implies the asymptotic equality as l →∞.This implies that the FOCUSS is asymptotically equivalent to the L1 minimization problem. Since ... |

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Citation Context ... cost function Eq. (4) can be modified into the following form: C(q)=||υ − A¯ρ − AWq|| 2 2 + λ||q||2 2 where ρ = ¯ρ + Wq, and the optimal solution is given by ρ = ¯ρ + ΘA H � AΘA H + λI �−1 (υ − A¯ρ) =-=(7)-=- The novelty of FOCUSS algorithm comes from that the weighting matrix W can be continuously updated using the previous solution (hence, Θ = WW H is updated accordingly). More specifically, if the (n −... |

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Citation Context ...f FOCUSS can be calculated by the following procedure: 10 1. Compute the weighting matrix Wn: ⎛ |ρn−1(1)| ⎜ Wn = ⎜ ⎝ p 0 0 |ρn−1(2)| ··· 0 p . . ··· . .. 0 . 0 0 ··· |ρn−1(N)| p ⎞ ⎟ , ⎠ 1/2 ≤ p ≤ 1 . =-=(9)-=- 2. Compute Θn = WnW H n . 3. Compute the n-thFOCUSSestimate: 4. If converges, stop. Otherwise, increase n andgotoStep1. ρ n = ¯ρ + ΘnA H � AΘnA H + λI �−1 (υ − A¯ρ) . (10) In order to understand why ... |

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Citation Context ...ll that the well-known Fourier slice theory13 tells us that Eq. (13) is indeed the Fourier transform of the following Radon transform with respect to s: � y(s, θ)= m(x, y)δ(xcos θ + y sin θ − s)dxdy, =-=(14)-=- where y(s, θ) is obtained using the inverse Fourier transform of the k-space samples y(k, θ) along the radial direction. Under this formulation, the measurement vector y is constructed using discreti... |

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Citation Context ...(N)| p ⎞ ⎟ , ⎠ 1/2 ≤ p ≤ 1 . (9) 2. Compute Θn = WnW H n . 3. Compute the n-thFOCUSSestimate: 4. If converges, stop. Otherwise, increase n andgotoStep1. ρ n = ¯ρ + ΘnA H � AΘnA H + λI �−1 (υ − A¯ρ) . =-=(10)-=- In order to understand why the FOCUSS provides a sparse solution, consider the l-thFOCUSSupdate. Using ql = W −1 l xl with Eq. (2) and Eq. (3), the l-th FOCUSS update can be written by min ||W −1 l x... |

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Citation Context ...wing asymptotic relation: ||W −1 l x||22 = xT (W −1 l ) H W −1 l x = x T ⎛ ⎜ ⎝ |xl−1;1| −1 0 0 |xl−1;2| ··· 0 −1 . . . . ··· . .. 0 . . 0 0 ··· |xl−1;N | −1 ⎞ ⎟ ⎠ x N� ∼ |xl−1,i| i=1 ∼ ||x||1 as l →∞ =-=(12)-=- SPIE-IS&T/ Vol. 6498 64981A-3 (6)swhere the superscript H denotes the Hermitian transpose, and ∼ implies the asymptotic equality as l →∞.This implies that the FOCUSS is asymptotically equivalent to t... |

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Citation Context ...ated using the previous solution (hence, Θ = WW H is updated accordingly). More specifically, if the (n − 1)-th iteration of the image estimate is given by ρ n−1 =[ρn−1(1),ρn−1(2), ··· ,ρn−1(N) ] T , =-=(8)-=- where N denotes the number of discretized voxels in (x, y, f) space. Then, the n-th iteration of FOCUSS can be calculated by the following procedure: 10 1. Compute the weighting matrix Wn: ⎛ |ρn−1(1)... |

36 |
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Citation Context ...ional burden of these steps significantly. 3.2. Extension to Dynamic Imaging Now suppose k-t space measurements for dynamic varying objects are given by � y(k,t)=y(k, θ, t)= m(r)e −j2π(k·r+tf) drdf , =-=(15)-=- where k =(k cos θ, k sin θ) denotes the coordinate vector in k-space where k-space samples are acquired according to the polar coordinate (k, θ), and r =(x, y) is the coordinate vector in spatial dom... |

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Citation Context ...) corresponds to the extension of PR-FOCUSS 8–10 for dynamic imaging applications. Consider the conventional radial k-t BLAST update equation 16 given by: ρ 1 = ¯ρ + Θ0A H � AΘ0A H + λI �−1 (υ − A¯ρ) =-=(16)-=- ∗ 9 Even for p ≥ 0.5, the FOCUSS still guarantees a sparse solution. However, throughout the paper we use p =0.5 for simplicity. SPIE-IS&T/ Vol. 6498 64981A-4swhere Θ0 is the diagonal covariance matr... |

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Citation Context ...optimization problem: find ρ = Wq (2) where ρ is the unknown image, W is a weighting matrix, and q is computed by solving the constrained minimization problem: min ||q||2, subject to ||υ − AWq||2 ≤ ɛ =-=(3)-=- The constrained optimization problem can be converted into an un-constrained optimization problem using Lagrangian multiplier: C(q)=||υ − AWq|| 2 2 + λ||q||22 (4) SPIE-IS&T/ Vol. 6498 64981A-2swhere ... |

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Citation Context ...OCUSS provides a sparse solution, consider the l-thFOCUSSupdate. Using ql = W −1 l xl with Eq. (2) and Eq. (3), the l-th FOCUSS update can be written by min ||W −1 l x||22, subject to ||Ax − y|| ≤ ɛ. =-=(11)-=- Suppose we set p =0.5forWlupdate. Then, we have the following asymptotic relation: ||W −1 l x||22 = xT (W −1 l ) H W −1 l x = x T ⎛ ⎜ ⎝ |xl−1;1| −1 0 0 |xl−1;2| ··· 0 −1 . . . . ··· . .. 0 . . 0 0 ··... |

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Citation Context ...tiplier: C(q)=||υ − AWq|| 2 2 + λ||q||22 (4) SPIE-IS&T/ Vol. 6498 64981A-2swhere λ denotes the appropriate Lagrangian parameter. The optimal solution is then given by ρ = Wq = ΘA H � AΘA H + λI �−1 υ =-=(5)-=- where Θ = WW H . In a slightly different formulation, ρ is initialized with non-zero values ¯ρ. Inthiscase,the cost function Eq. (4) can be modified into the following form: C(q)=||υ − A¯ρ − AWq|| 2 ... |

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Citation Context ... ||q||2, subject to ||υ − AWq||2 ≤ ɛ (3) The constrained optimization problem can be converted into an un-constrained optimization problem using Lagrangian multiplier: C(q)=||υ − AWq|| 2 2 + λ||q||22 =-=(4)-=- SPIE-IS&T/ Vol. 6498 64981A-2swhere λ denotes the appropriate Lagrangian parameter. The optimal solution is then given by ρ = Wq = ΘA H � AΘA H + λI �−1 υ (5) where Θ = WW H . In a slightly different... |

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Citation Context ...pread the energy rather than obtaining the sparse solution. In order to derive FOCUSS, let us consider the following optimization problem: Now consider the following optimization problem: find ρ = Wq =-=(2)-=- where ρ is the unknown image, W is a weighting matrix, and q is computed by solving the constrained minimization problem: min ||q||2, subject to ||υ − AWq||2 ≤ ɛ (3) The constrained optimization prob... |