Inverse Problem in Quantitative Susceptibility Mapping From Theory to Application. Jae Kyu Choi.

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1 Inverse Problem in Quantitative Susceptibility Mapping From Theory to Application Jae Kyu Choi Department of CSE, Yonsei University Hangzhou Zhe Jiang University Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

2 1 Introduction 2 Theories in Inverse Problem 3 Applications 4 Conclusions Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

3 Introduction Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

4 Quantitative Susceptibility Mapping (QSM) Magnetic susceptibility χ: an intrinsic property of the material relating the magnetization M and the magnetic field H via M = χh. QSM: aims to visualize magnetic susceptibility χ from MR data. MR Magnitude Image Susceptibility Image Applications: Disease diagnosis (Parkinson s, Alzheimer, stroke, blood calcification, etc.) Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

5 Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

6 Inverse Problem of QSM Solve the deconvolution problem for χ: ˆR3 ψ(x) = pv d(x y)χ(y)dy d(x) = 2x2 3 x 2 1 x 2 2 4π x 5 ( ) 1 Ψ(ξ) = }{{} 3 ξ2 3 X (ξ) = D(ξ)X (ξ). ξ 2 }{{} F(ψ)(ξ) F(χ)(ξ) (pv: the principal value of the integral, F: Fourier transform) Data: relative difference field (RDF) ψ (noisy) Integral kernel d: singular (d(rx) = r 3 d(x) for r > 0). = Data ψ (Noisy) Unit dipole field d Susceptibility χ Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

7 Challenging Issue ˆ d(x y)χ(y)dy! ξ32 1 Ψ(ξ) = X (ξ) 3 ξ 2 {z } {z } {z } F(χ)(ξ) F(ψ)(ξ) ψ(x) = pv (IP-I) R3 (IP-F) D(ξ) ill-posed since D(ξ) = 0 in Γ0 = ξ R3 : ξ12 + ξ22 2ξ32 = 0, and this leads to the streaking artifacts. Data ψ Jae Kyu Choi (Yonsei University) Γ0 in Fourier domain Inverse Problem in QSM Reconstructed χ using (IP-F) / 23

8 Theoretical Results Existence and Uniqueness Result Characterization of Streaking Artifacts Tool Microlocal Analysis & Hörmander s Theory Both of the results are established for the first time. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

9 Source of Error Propagation For a given measurement ψ E, we aim to obtain χ E using (IP-I) or (IP-F): ψ(x) = lim d(x y)χ(y)dy ε 0 ˆ x y >ε ( ) 1 Ψ(ξ) = D(ξ)X (ξ) = 3 ξ2 3 X (ξ) ξ 2 (IP-I) (IP-F). (D : space of distributions, E : space of compactly supported distributions, S : space of tempered distributions) If ψ E satisfies (IP-F) for some χ E, then ψ must lie in E := { u E : û(ξ)/p(ξ) is bounded near Γ 0 }. Here, P(ξ) is the polynomial defined as P(ξ) = 4π2 3 (ξ2 1 + ξ2 2 2ξ2 3 ). Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

10 Theorem (Existence and Uniqueness [J.K.Choi et al ]) If ψ E, we have the unique χ E satisfying (IP-F), and X = F(χ) can be represented as X (ξ) = 4π 2 ξ 2 Ψ(ξ) P(ξ) 9ξ3 4 (Proof follows from Paley-Wiener-Schwartz theorem.) if ξ Γ 0 Ψ ξ 3 (ξ) if ξ Γ 0 \ {0}. ( ) Reference χ ψ E Reconstructed χ using ( ) Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

11 Cause of Streaking Artifacts Streaking artifacts: closely related with the PDE ) 13 2 P(D)χ = ( + χ = ψ x 2 3 (IP-PDE) The solution χ D to (IP-PDE) is expressed as ˆ χ (x) = E ( ψ)(x) = E(x y) yψ(y)dy ψ E (S-PDE) R 3 where E(x) is the fundamental solution of P(D): 3 4π x E(x) = 2 3 2(x2 1 + x2 2 ) if 2(x x 2 2) < x otherwise. E(x) has the singular support along { x R 3 : 2(x x2 2 ) = x2 3}. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

12 Theorem (Characterization of Artifacts [J.K.Choi et al ]) For ψ E, the wave front set of χ = χ satisfies WF(χ) \ WF(ψ) { (t P(ξ) + x, ξ) : ξ Γ 0 \ {0}, t 0, (x, ξ) WF(ψ) } Moreover, if (x, ξ) WF(χ) \ WF(ψ), then 1 ξ Γ 0 2 for any open interval (a, b) containing 0 such that { (x + t P(ξ), ξ) : t (a, b) } WF(ψ) =, we have { (x + t P(ξ), ξ) : t (a, b) } WF(χ). Simulated ψ E \ E Fundamental solution E χ = E ( ψ) Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

13 Our Proposed Approach Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

14 Description Want to develop a reconstruction method w/o regularization. = Artifacted Reconstructed Image χ + True Susceptibility χ Streaking Artifacts χv Based on the decomposition of a measured data = Measured data ψ Jae Kyu Choi (Yonsei University) + True data ψ = d χ Inverse Problem in QSM Violator ψv / 23

15 Assume that both ψ and ψ are supported in Ω. Since ψ = d χ, it follows that ψ E. However, ψ E \ E due to the non-vanishing compatibility violator ψ v := ψ ψ. This ψ v is the cause of artifacts in the reconstructed image χ: ˆ χ(x) = E(x y) yψ(y)dy. (S-PDE) R 3 Since χ is compactly supported, we have ˆ { χ(x) for x R 3 \ Ω, χ v(x) = E(x y) yψ v(y)dy = R 3 χ(x) χ (x) for x Ω. (1) In theory, it is possible to recover ψ v (with a special form of point sources) from the knowledge of χ = χ v in R 3 \ Ω. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

16 Observation Assume that ψ E with supp(ψ) Ω can be decomposed into ψ = ψ + ψ v with ψ v being of the form ψ v(x) = N c j δ(x x j ) c j R, (2) j=1 where x 1,, x N supp(ψ). Then the source of the streaking artifacts ψ v E is determined uniquely from { P(D)χv = ψ v in R 3, with a given χ D defined as (1) in R 3 \ Ω. χ v = χ in R 3 \ Ω Once ψ v is obtained from ψ, then we are able to recover the streaking artifacts uniquely. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

17 Issues in Zero-Cone Related Data Loss Artifacts All the previous methods of QSM rely on the direct inversion of in the frequency domain. Ψ(ξ) = D(ξ)X (ξ) χ can be obtained formally by the formula χ = F 1 (Ψ/D) with Ψ/D properly defined as a limit near Γ 0. If ψ v 0, then there exist ξ Γ 0 such that Ψ(ξ)/D(ξ) =. Hence, for given data available in discrete sense, we should discard the information of Ψ/D on Γ ε defined as { } Γ ε := ξ R 3 : dist(ξ, Γ 0) < ε/2 where 1 ε the distance between the centers of the neighboring voxels in frequency domain 2 dist(ξ, Γ 0): the distance between ξ and Γ 0. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

18 Most previous methods use regularization with the fidelity of the following approximation: χ F 1( Ψ/D (1 η Γε ) ) (η Γε :the characteristic function of the set Γ ε). The information loss of (Ψ/D)η Γε causes unexpected artifacts in the reconstructed image, different from the streaking artifacts χ v generated by ψ v. In theory, it must be [ χ = lim F 1( Ψ/D(1 η Γε ) ) ] χ v. ε 0 However, in the practical computation with discrete images, it is inevitable to lose the information of Ψ/D near the zero cone Γ 0. Even though we choose ε > 0 so small that Γ ε consists of all voxels containing ξ Γ 0 only, this information loss causes the unexpected artifacts, which makes F 1( Ψ/D(1 η Γε ) ) ψ different from χ. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

19 = χ = χ + χv + χ = F 1 Ψ/D(1 ηγ ) ε Jae Kyu Choi (Yonsei University) χv + F 1 Ψ/D(1 ηγ ) χv ε Inverse Problem in QSM χv / 23

20 The artifacts due to the missing data near Γ 0 are closely related to the singular fiber of the compatible part ψ (or Σ x(ψ ).) We present the key observation about the cause of the artifacts due to the singular fiber of the compatible part ψ. Observation Let ψ = ψ + ψ v be the decomposition of ψ with the non-vanishing ψ v. With the mixed use of discrete and continuous variables, let ε be the distance between the centers of the neighboring voxels in the frequency domain. Define χ ε by χ ε = F 1 (Ψ/D(1 η Γε )). (3) The artifacts of χ ε χ caused by the loss of Ψ/D on Γ ε can be explained by using the wave front set. More precisely, the wave front set of χ ε χ is explained by the set (in the continuous concept) { (t P(ξ) + x, ξ) : ξ Σx(ψ ) (Γ 0 \ {0}), t R }. Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

21 ψ = ψ + ψv ψ ψv χε = χ + χv + χ χ + χ χv Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

22 Conclusions Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23

23 Conclusion We established the theoretical ground for the inverse problem of QSM; the compatibility condition of the data ψ E is founded so that the inverse problem can be solvable uniquely. Any data that violates the condition will cause streaking artifacts due to Γ 0, which can be analyzed using the wave front set. For the first time try to find directly the source of the streaking artifacts. In theory, it is possible to extract the source of the streaking artifacts using their characteristics (e.g., wave front set.) In addition, observed that the non-vanishing ψ v (with degeneracy of D on Γ 0 in the frequency domain) produces two types of streaking artifact in the reconstructed image: one caused by ψ v itself and another by the loss of information of X on Γ 0. Future works may consider Proving whether it is possible to pin-point the source of streaking artifacts ψ v when it is compactly supported in Ω. (Our presented case is based on the analyticity of E. However, for general ψ v E, we may not be able to adopt the analyticity of E any more.) Developing a robust numerical method to extract ψ v out of ψ. (Even in our case, the numerical implementation may be unstable because our theory is currently based on the analyticity of E.) Jae Kyu Choi (Yonsei University) Inverse Problem in QSM / 23