Radio interferometric gain calibration as a complex optimization problem

TL;DR

This paper introduces a complex optimization framework using Wirtinger calculus for radio interferometric gain calibration, enabling new algorithms for polarized and direction-dependent calibration regimes with improved efficiency.

Contribution

It extends traditional calibration algorithms by applying complex Jacobian formalism, unifying existing methods, and developing new algorithms for advanced calibration scenarios.

Findings

Unified calibration algorithms under complex Jacobian formalism

Efficient implementation of direction-dependent calibration algorithms

Demonstrated applicability with real data results

Abstract

Recent developments in optimization theory have extended some traditional algorithms for least-squares optimization of real-valued functions (Gauss-Newton, Levenberg-Marquardt, etc.) into the domain of complex functions of a complex variable. This employs a formalism called the Wirtinger derivative, and derives a full-complex Jacobian counterpart to the conventional real Jacobian. We apply these developments to the problem of radio interferometric gain calibration, and show how the general complex Jacobian formalism, when combined with conventional optimization approaches, yields a whole new family of calibration algorithms, including those for the polarized and direction-dependent gain regime. We further extend the Wirtinger calculus to an operator-based matrix calculus for describing the polarized calibration regime. Using approximate matrix inversion results in computationally efficient implementations; we show that some recently proposed calibration algorithms such as StefCal and peeling can be understood as special cases of this, and place them in the context of the general formalism. Finally, we present an implementation and some applied results of CohJones, another specialized direction-dependent calibration algorithm derived from the formalism.