Math Notation

The documentation of deepinv uses a unified mathematical notation that is summarized in the following table:

Table 2 List of mathematical symbols

Symbol	Meaning
\(x\in\xset\)	Underlying image or signal to reconstruct of \(n\) elements.
\(y\in\yset\)	Observed measurement vector of size \(m\).
\(p(x)\)	Distribution of images \(x\) (often referred to as prior distribution).
\(p(y)\)	Distribution of measurements \(y\).
\(\forw{x}\)	Deterministic mapping that captures the physics of the imaging system.
\(A^\top\colon\yset\to\xset\)	Adjoint of the measurement operator.
\(\noise{y}\)	Stochastic mapping adding noise to measurements.
\(\inverse{y}\)	Reconstruction network that maps measurements to images \(y\mapsto x\).
\(\denoiser{x}{\sigma}\)	Gaussian denoiser for noise of standard deviation \(\sigma\).
\(\datafid{x}{y} = \distance{A(x)}{y}\)	Data fidelity term, enforcing measurement consistency \(y\approx A(x)\), depending on the choice of the distance function (see below).
\(\distance{u}{y}\)	Distance function measuring the discrepancy between \(u\) and \(y\). It is linked to the noise model (likelihood).
\(\reg{x}\)	Regularization term that promotes plausible reconstructions. It is linked to \(p(x)\).
\(\lambda\)	Hyperparameter controlling the amount of regularization.