Abstract
We consider a setting in which it is desired to find an optimal complex vector {mathbf {x}}in {mathbb {C}}^N that satisfies {mathcal {A}}({mathbf {x}}) approx {mathbf {b}} in a least-squares sense, where {mathbf {b}} in {mathbb {C}}^M is a data vector (possibly noise-corrupted), and {mathcal {A}}(cdot ): {mathbb {C}}^N rightarrow {mathbb {C}}^M is a measurement operator. If {mathcal {A}}(cdot ) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where {mathcal {A}}(cdot ) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering {mathbf {x}} as a vector in {mathbb {R}}^{2N} instead of {mathbb {C}}^N. While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms.
Highlights
Consider a generic complex-valued finite-dimensional inverse problem scenario in which the forward model is represented as b = A(x) + n, (1)where b ∈ CM represents the measured data, A(·) : CN → CM is the measurement operator, n ∈ CM represents noise, and x ∈ CN represents the unknown signal that we wish to estimate based on knowledge of b and A(·)
Even though the descriptions we present in this paper are generally applicable to arbitrary real-linear operators, we were initially motivated to consider such operators because of specific applications in magnetic resonance imaging (MRI) reconstruction
Inputs: A(·) : CN → CN, b ∈ CM, and x0 ∈ CN Initialization: r0 = A∗(b − A(x0)); p0 = r0; k = 0; Iteration: While stopping conditions are not met: zk = A∗(A(pk )); αk =/real(pkH zk ); xk+1 = xk + αk pk ; rk+1 = rk − αk zk ; βk =/(rkH rk ); pk+1 = rk+1 + βk pk ; k = k + 1; While we have only shown complex-valued adaptations of the Landweber and conjugate gradient (CG) algorithms, this same approach is applied to other related algorithms like LSQR (Paige and Saunders 1982)
Summary
These are nonlinear least-squares problems because the operators involved are nonlinear, previous work has benefitted from the fact that this kind of inverse problem can be transformed into an equivalent higher-dimensional real-valued linear least-squares problem (Bydder and Robson 2005; Willig-Onwuachi et al 2005; Lew et al 2007; Hoge et al 2007; Haldar et al 2013; Blaimer et al 2016; Haldar 2014; Haldar and Setsompop 2020; Haldar 2015; Kim and Haldar 2018) This can be done by replacing all complex-valued quantities with real-valued quantities, e.g., separating x ∈ CN into its real and imaginary components, and treating this as an inverse problem in R2N rather than the original space CN. This can enable both improved computation speed and simplified algorithm implementations
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