[Paper Review] Bispectral technique for reconstruction the astronomical images with an intensity interferometer

B. E. Zhilyaev|ArXiv.org|Jun 2, 2008

Optical Polarization and Ellipsometry5 references3 citations

TL;DR

This paper proposes a bispectral technique to reconstruct astronomical images using intensity interferometry by recovering Fourier phase information from third-order intensity correlations, overcoming the phase ambiguity inherent in second-order correlations. By combining magnitude from second-order correlations and phase from bispectral analysis, the method enables complete Fourier reconstruction and image recovery without prior assumptions about the source.

ABSTRACT

An extension may be proposed to the intensity interferometer of Hanbury Brown and Twiss to provide the Fourier phase measurement by the use of third-order intensity correlations. It is well known that interferometric reconstruction of astronomical images can be obtained from second-order correlations only when a priory information is used about the object. The third-order intensity correlations contain information about the Fourier phase and need no such assumptions. In the ordinary way we can make measurements of the second-order intensity correlations with an intensity interferometer. We can also calculate the third-order intensity correlations from the data set of measured intensities for each distance triplet. After that we can reconstruct the Fourier phase from third-order correlations by using bipectral technique. When this is combining with the Fourier magnitude obtained from the second-order intensity correlations, we have the ultimate Fourier transform of the source brightness distribution. An inverse Fourier transform recovers the source image.

Motivation & Objective

• To address the phase ambiguity in astronomical image reconstruction using intensity interferometry, which traditionally relies on second-order correlations and lacks phase information.
• To develop a method that extracts Fourier phase from third-order intensity correlations using bispectral analysis.
• To enable complete image reconstruction by combining Fourier magnitude from second-order correlations with phase from bispectral techniques.
• To demonstrate that third-order correlations contain sufficient phase information to recover source brightness distributions without a priori assumptions.
• To extend the Hanbury Brown and Twiss intensity interferometer framework to include phase-sensitive reconstruction via bispectrum analysis.

Proposed method

• Measure second-order intensity correlations using an intensity interferometer to obtain the Fourier magnitude of the source brightness distribution.
• Calculate third-order spatial correlations from measured intensities for each distance triplet using the formula $ I^{(3)}(x_1,x_2) = ∫ I(x)I(x+x_1)I(x+x_2)dx $.
• Compute the bispectrum as the 2D Fourier transform of the third-order correlation: $ B(f_1,f_2) = ∫∫ I^{(3)}(x_1,x_2) e^{-2\pi i(f_1x_1 + f_2x_2)} dx_1 dx_2 $.
• Use the bispectral phase relationship $ \psi(f_1,f_2) = \phi(f_1) + \phi(f_2) - \phi(f_1 + f_2) $ to reconstruct the Fourier phase from the bispectral phase.
• Apply a recursive reconstruction formula $ \phi_k = \frac{1}{k-1} \sum_{i=1}^{k-1} (\phi_i + \phi_{k-i} - \psi_{i,k-i}) $ to recover the Fourier phase sequence.
• Combine the reconstructed Fourier phase with the magnitude from second-order correlations to form the complete Fourier transform, then apply inverse Fourier transform to recover the source image.

Experimental results

Research questions

• RQ1 Can third-order intensity correlations provide phase information sufficient for astronomical image reconstruction when second-order correlations alone cannot?
• RQ2 How can the bispectral technique be applied to reconstruct the Fourier phase of a source brightness distribution from intensity interferometry data?
• RQ3 What is the mathematical relationship between the bispectral phase and the Fourier phase in the context of intensity interferometry?
• RQ4 How can the sign ambiguity and phase offset in bispectral reconstruction be resolved using data-driven or a priori constraints?
• RQ5 To what extent can the bispectral method recover source images without assuming prior knowledge of the source structure?

Key findings

• The bispectral technique enables recovery of the Fourier phase from third-order intensity correlations, which contain phase information absent in second-order correlations.
• The phase reconstruction formula $ \phi_k = \frac{1}{k-1} \sum_{i=1}^{k-1} (\phi_i + \phi_{k-i} - \psi_{i,k-i}) $ allows recursive recovery of the Fourier phase sequence, with $ \phi_1 $ as an arbitrary initial value.
• The bispectral phase is independent of linear phase components, explaining the phase offset ambiguity and justifying the arbitrary choice of $ \phi_1 $.
• The sign ambiguity in the bispectral phase, arising from the $ \pm \arccos $ expression in Eq. (8), can be resolved using data-driven selection.
• Combining the reconstructed phase with the magnitude from second-order correlations yields the complete Fourier transform of the source brightness distribution.
• The method enables full image reconstruction via inverse Fourier transform, demonstrating feasibility for astronomical imaging without prior assumptions.