General image metrics

A general discussion about measuring discrepancy between images with a specific focus on optical interferometry is developed in^[Gomes2016]. A proper image metric shall provide a quantitative score suitable to order restored images. This score must reflect the perception by an expert of the fidelity of the images with a given ground truth or reference image. The score shall be insensitive to changes that may be considered as irrelevant for the considered context. For example, image quality indices widely used in the signal processing community are insensitive to an affine transform of the pixel values. For optical interferometry, the pixel size and the field of view of a restored image are, to some extend, free parameters and may thus be different from the pixel size of the ground truth image. Moreover, when only power spectrum and phase closure data (vis2data and t3phi in OI_VIS2 and OI_T3 data-blocks of OI-FITS files) are used to reconstruct an image, the position of the object in the field of view is not constrained by the data and the image metric should not depend on this position.

Accounting for all these remarks, a possible definition of the distance between $\Vy$, a reference image, and $\Vx$, a reconstructed image, is given by:

\[\begin{align*} \Dist(\Vx,\Vy) = \min_{\Params} \Bigl\{ &\sum_{j \in |\MR_{\Vtheta}\cdot\Vx| \cap |\Vy|} d\!\left(α\,(\MR_{\Vtheta}\cdot\Vx)_j + β, y_j\right) \notag\\ &+ \sum_{j \in |\MR_{\Vtheta}\cdot\Vx| \backslash (|\MR_{\Vtheta}\cdot\Vx| \cap |\Vy|)} d\!\left(α\,(\MR_{\Vtheta}\cdot\Vx)_j + β, \eta\right) \notag\\ &+ \sum_{j \in |\Vy| \backslash (|\MR_{\Vtheta}\cdot\Vx| \cap |\Vy|)} d\!\left(\eta,y_j\right) \Bigr\} \end{align*}\]

where $d(x,y)$ is some pixel-wise distance, $\MR_{\Vtheta}$ is a linear operator which implements resampling with a given magnification, translation, and blurring, and $\eta$ is the assumed out-of-field pixel value. $|\Vy|$ denotes the list of pixels of the image $\Vy$ and $\Vtheta = \{\rho,\Vt,\omega\}$ accounts for all parameters defining the operator $\MR_{\Vtheta}$: the magnification $\rho$, the translation $\Vt$, and the blur width $\omega$.

The distance is minimized in the set $\Params$ of parameters which are irrelevant for judging of the image quality. These parameters depend on the context. For image reconstruction from interferometric data, $\beta = 0$ and $\eta = 0$ are natural settings while $\alpha \in \mathbb{R}$ and the translation $\Vt \in \mathbb{R}^2$ must be adjusted to reduce the mismatch between the images. Hence, $\Params = \{\alpha,\Vt\}$ in this context.

A score may be defined by normalizing the distance and such that the higher the score, the better the restored image $\Vx$:

\[\Score(\Vx) = 1 - \frac{\Dist(\Vx,\Vy)}{\Dist(\eta\,\One,\Vy)}\]

where $\One$ is an image of the same size as $\Vy$ but filled with ones, hence $\eta\,\One$ is an image of the same size as $\Vy$ but filled with $\eta$, the assumed out-of-field pixel value. The score may be negative but the maximal score is 1.

Denoting by $\mathbb{K}$ the set of possible pixel values, the following properties must hold for the pixel-wise distance:

1. $d(x,x) = 0$ for any $x \in \mathbb{K}$;
2. $d(x,y) > 0$ for any $(x,y) \in \mathbb{K}^2$ such that $x \not= y$;
3. $d(y,x) = d(y,x)$ for any $(x,y) \in \mathbb{K}^2$.

To remain general, a possible pixel-wise distance for which the above properties hold is given by:

\[d(x, y) = \left|\Gamma(x) - \Gamma(y)\right|^p\]

where the exponent $p$ and the function $\Gamma: \mathbb{K}\to\mathbb{R}$ are introduced to make the distance more flexible. $\Gamma$ is a brightness correction monotonic function to emphasize the interesting parts of the images amd $p > 0$ to have a non-decreasing distance with respect to the absolute value of the difference $\Gamma(x) - \Gamma(y)$. For example:

\[\Gamma(x) = \Sign(x)\,|x|^\gamma,\]

where $\Sign(x)$ is the sign of $x$:

\[\Sign(x) = \begin{cases} -1 & \text{if $x < 0$}\\ +1 & \text{if $x > 0$}\\ \phantom{+}0 & \text{if $x = 0$}\\ \end{cases}\]

Metric parameters $p$ and $\gamma$ can be chosen depending on the context. According to a human panel^[Gomes2016], $p = 1$ with $\gamma = 1$ best reflect the human perception of image quality.