The 2016' Interferometric Beauty Contest

For the 2016' Interferometric Imaging Beauty Contest^{[Sanchez2016]} held in Edinburgh, the objective of the contest was to reconstruct multi-wavelength images. A metric was thus specifically designed to compare multi-channel images so as to be insensitive to irrelevant differences due to:

• orientation: image axes may be inverted, notably the E-W axis;
• translation: there may be an arbitrary shift between images (possibly a different shift in every spectral channel);
• pixel size: the size of the pixels used for the restored image is arbitrary even though it should be sufficiently small to account for the highest measured frequencies;
• brightness: not all entries are normalized according to the channel-wise flux given in OI_FLUX data-block;
• out-of-field values: after all geometric transformations, the fields of view of the reference and reconstructed images may be different, it is assumed that missing pixel values are equal to zero. This rationale leads to impose that $β = 0$ and $η = 0$.

In addition, the comparison must take into account that the measurements have limited angular resolution. Thus the reference image $\Vy_λ$ in spectral channel $λ$ is the ground truth image $\Vz_λ$ convolved with an effective Point Spread Function (PSF) whose Full Width at Half Maximum (FWHM) is $\omega_{\mathrm{ref}}$ chosen to match the interferometric resolution:

\[\Vy_λ = \MR_{\Vtheta^{\mathrm{ref}}_λ}\cdot\Vz_λ\]

with $\MR_{\Vtheta^{\mathrm{ref}}_λ}$ the linear operator used to resample images for measuring image distances but with parameters:

\[\Vtheta^{\mathrm{ref}}_λ = \{ \rho_{\mathrm{ref}} = 1, \Vt_{\mathrm{ref}} = \Zero, \omega_λ = λ/(2\,B_{\mathrm{max}}) \}\]

With $\rho_{\mathrm{ref}} = 1$, it is assumed that there is no translation between the ground truth images $\Vz_λ$ and the reference images $\Vy_λ$, with $\rho_{\mathrm{ref}} = 1$, the pixel size (3 mas/pixel) is kept the same, and $\omega_λ = λ/(2\,B_{\mathrm{max}})$ is the FWHM of the interferometric resolution at wavelength $λ$ and maximal (projected) baseline $B_{\mathrm{max}}$.

Since $\Gamma(η) = \Gamma(0) = 0$ and $\Gamma(\alpha\,x) = \Gamma(\alpha)\,\Gamma(x)$ (whatever $x$ and $\alpha$), the distance between the restored and the reference images in a given spectral channel can be written as:

\[\Dist_{\Gamma,p}(\Vx_λ,\Vy_λ) = \min_{\tilde{α}_λ,\Vt_λ} \sum\limits_{j \in Ω_λ} \left| \tilde{\alpha}_λ\,[\tilde{\Vx}_λ]_{j} - [\tilde{\Vy}_λ]_j \right|^p\]

where $Ω_λ = |\MR_{\Vtheta_λ}\cdot\Vx_λ| \cup |\Vy_λ|$ is the union of the fields of view of $\MR_{\Vtheta_λ}\cdot\Vx_λ$ and $\Vy_λ$, and with $\tilde{α}_λ = \Gamma(\alpha_λ)$,

\[\left[\tilde{\Vy}_λ\right]_j = \left\{\begin{array}{ll} \Gamma\bigl(\bigl[\Vy_λ\bigr]_j\bigr) & \text{if }j \in |\Vy_λ|\\ \Gamma(η) = 0 & \text{else} \end{array}\right,\]

the brightness corrected extrapolated reference image, and:

\[\left[\tilde{\Vx}_λ\right]_j = \left\{\begin{array}{ll} \Gamma\bigl(\bigl[\MR_{\Vtheta_λ}\cdot\Vx_λ\bigr]_j\bigr) & \text{if }j \in |\MR_{\Vtheta_λ}\cdot\Vx_λ|\\ \Gamma(η) = 0 & \text{else} \end{array}\right.\]

where

\[\Vtheta_λ = \{\rho_λ,\Vt_λ,\omega_λ\}\]

are the settings for resampling the restored image in spectral channel $λ$. The magnification $\rho_λ$ is computed as the ratio of the known pixel sizes of $\Vx_λ$ and $\Vy_λ$ and, usually, does not depend on $λ$.

For the 2016' Interferometric Imaging Beauty Contest^{[Sanchez2016]}, $γ = 0.7$ and $p = 2$ were the metric parameters chosen for the brightness correction function. When $p = 2$, minimizing the distance with respect to $\tilde{α}_λ$ has the following closed-form solution:

\[\tilde{α}_λ = \Gamma(α_λ) = \frac{ \sum_{j \in Ω_λ} [\tilde{\Vx}_λ]_j\,[\tilde{\Vy}_λ]_j }{ \sum_{j \in Ω_λ} \left([\tilde{\Vx}_λ]_j\right)^2 }\]

and the distance (for $p = 2$) simplifies to:

\[\Dist_{\Gamma,2}(\Vx_λ,\Vy_λ) = \sum_{j \in |\tilde{\Vy}_λ|} \left([\tilde{\Vy}_λ]_j\right)^2 - \max_{\Vt_λ} \frac{ \left( \sum_{j \in Ω_λ} [\tilde{\Vx}_λ]_{j}\,[\tilde{\Vy}_λ]_j \right)^2 }{ \sum_{j \in Ω_λ} \left([\tilde{\Vx}_λ]_{j}\right)^2 }\]

and the score is:

\[\begin{align} \Score_{\Gamma,2}(\Vx_λ) = 1 - \frac{ \Dist_{\Gamma,2}(\Vx_λ,\Vy_λ) }{ \sum_{j \in |\tilde{\Vy}_λ|} \left([\tilde{\Vy}_λ]_j\right)^2 } &= \max_{\Vt_λ} \frac{ \left( \sum_{j \in Ω_λ} [\tilde{\Vx}_λ]_{j}\,[\tilde{\Vy}_λ]_j \right)^2 }{ \left(\sum_{j \in Ω_λ} \left([\tilde{\Vx}_λ]_{j}\right)^2\right)\, \left(\sum_{j \in Ω_λ} \left([\tilde{\Vy}_λ]_j\right)^2\right) }\notag\\ &= \frac{ \max\limits_{\Vt_λ} \left( \sum_{j \in Ω_λ} [\tilde{\Vx}_λ]_{j}\,[\tilde{\Vy}_λ]_j \right)^2 }{ \left(\sum_{j \in |\MR_{\Vtheta_λ}\cdot\Vx_λ|} \left([\tilde{\Vx}_λ]_{j}\right)^2\right)\, \left(\sum_{j \in |\Vy_λ|} \left([\tilde{\Vy}_λ]_j\right)^2\right) }\notag \end{align}\]

The score for a multi-spectral image $\Vx$ is the sum of the scores in all spectral channels:

\[\Score_{\Gamma,p}(\Vx) = \sum_λ \Score_{\Gamma,p}(\Vx_λ).\]