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README.md

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-# ImageMetrics [![Build Status](https://github.com/emmt/ImageMetrics.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/emmt/ImageMetrics.jl/actions/workflows/CI.yml?query=branch%3Amain) [![Coverage](https://codecov.io/gh/emmt/ImageMetrics.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/emmt/ImageMetrics.jl)
+# Image metrics
+[![Build Status](https://github.com/emmt/ImageMetrics.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/emmt/ImageMetrics.jl/actions/workflows/CI.yml?query=branch%3Amain)
+[![Coverage](https://codecov.io/gh/emmt/ImageMetrics.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/emmt/ImageMetrics.jl)
+
+This repository collect notes and tools for image metrics that is means to
+quantitatively compare images.
+
+## References
+
+
+## Image Metrics Principles
+
+Assessing image quality for optical interferometry has been studied[^Gomnes+2016]
+specific:
+
+* bright object on a dark background;
+* extrapolation of the synthesized filed of view can be naturally done by zero-padding (i.e. assuming that missing pixels have a zero value).
+
+
+The quality of an image has to be assessed by an objective quantitative
+criterion. The literature shows that establishing a universal image quality
+criterion is a very controversial subject. What is the best criterion also
+largely depends on the context. Here we will assume that the *metric*
+$\def\V#1{\boldsymbol{#1}}\def\M#1{\mathbf{#1}}\def\Dist{\mathcal{D}}\Dist(\V{x}, \V{y})$ is used to estimate the discrepancy between a reconstructed
+image $\V{x}$ and a reference image $\V{y}$. To simplify the discussion, we also assume
+that the lower $\Dist(\V{x}, \V{y})$ the better the agreement between $\V{x}$ and $\V{y}$. In
+other words $\Dist(\V{x}, \V{y})$ can be thought as a measure of the distance between
+$\V{x}$ and $\V{y}$.
+
+
+When assessing image quality it is important that the result does not depend on
+irrelevant changes. This however depends on the type of images and on the
+context. For instance, for object detection or recognition, the image metric
+should be insensitive to the background level, to a geometrical transform
+(translation, rotation, magnification, etc.) or to a multiplication of the
+brightness by some positive factor which does not affect the shape of the
+object. In cases where image reconstruction has some degeneracies, these should
+not have any incidence on the metric. The easier is then to minimize the metric
+with respect to the degenerated parameters. These parameters include, but are
+not limited to, brightness scaling, geometric transformation of coordinates,
+etc. For optical interferometry and when only power-spectrum and closure phase
+data are available, the images to be compared may have to be shifted for best
+matching.
+
+When comparing a true image (with potentially an infinitely high resolution) to
+a restored image, the effective resolution achievable by the instrument and the
+image restoration process must be taken into account. Otherwise and because
+image metrics are in general based on pixel-wise comparisons, the slightest
+displacement of sharp features would lead to large loss of quality (according
+to the metric) whereas the images may look very similar at a lower and more
+realistic resolution. The easiest solution is then to define the reference
+image to be the true image blurred by an effective point spread function (PSF)
+whose shape corresponds to the effective resolution. The choice of the
+effective resolution is then a parameter of the metric.
+
+To summarize and to be more specific, using the distance $\Dist(\V{x}, \V{y})$
+between the restored image $\V{x}$ and the reference image $\V{y}$, the discrepancy
+between the restored image $\V{x}$ and the true image $\V{z}$ would be given by:
+
+``` math
+\label{eq:general-discrepancy}
+d(\V{x}, \V{z}) = \min_{\alpha,\beta,t}
+\Dist\bigl(\alpha\,\V{x} + \beta, h_{\sigma,t} \ast \V{z} \bigr),
+```
+
+with $\alpha$ a brightness scale, $\beta$ a bias and $h_{\sigma,t}$ an
+effective PSF of *width* $\sigma$ and centered at position $t$. The symbol
+$\ast$ denotes the convolution. Assuming an effective PSF of Gaussian shape,
+the parameter $\sigma > 0$ can be chosen to be the standard deviation of the
+PSF; the full width at half maximum ($\def\FWHM{\operatorname{FWHM}}\FWHM$) is
+then given by:
+
+``` math
+\label{eq:gaussian-fwhm}
+\FWHM = \sqrt{8\,\log2}\,\sigma \approx 2.355\,\sigma.
+```
+
+Note that the merit function could be minimized with respect to the width of
+the PSF in order to estimate the effective resolution achieved by a given
+restored image. Our choice to assigning the translation to the PSF is to avoid
+relying on some particular method to perform sub-pixel interpolation (of $\V{x}$,
+$\V{y}$ or $\V{z}$) for fine tuning the position. Not doing so would add another
+ingredient to the metric.
+
+In the following, we first review the most common metrics found in the
+literature and argue whether they are appropriate or not in the context of
+optical interferometry. We then propose a family of suitable metrics.
+
+**Notations:** In this document, we denote scalars by Greek letters (e.g.
+$\alpha$, $\beta$), collection of values, a.k.a. *vectors*, by italic bold
+lower case Latin letters (e.g. $\V{x}$, $\V{y}$, $\V{z}$), and linear
+operators, a.k.a. *matrices*, by upright bold upper case Latin letters (e.g.,
+$\M{W}$).
+
+[^Gomnes+2016]: N. Gomes, P. J. V. Garcia & É. Thiébaut, *Assessing the quality
+of restored images in optical long-baseline interferometry* in Monthly Notices
+of the Royal Astronomical Society, vol. **465**, pp. 3823-3839 (2016).
+
+[^Sanchez+2016]: J. Sanchez-Bermudez, É. Thiébaut, K.-H. Hofmann, M. Heininger,
+D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary,
+J. Young & F. Malbet, F., *The 2016 interferometric imaging beauty contest* in
+Optical and Infrared Interferometry and Imaging V, SPIE International
+Conference, **9907**, 99071D (2016) [doi](https://doi.org/10.1117/12.2231982).