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SNR+CNR.lyx
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SNR+CNR.lyx
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#LyX 1.6.7 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
\textclass article
\begin_preamble
\usepackage{amsmath}
\newcommand{\argmax}{\operatornamewithlimits{argmax}}
\newcommand{\gini}{\mathtt{gini}}
\newcommand{\rce}{\mathtt{rce}}
\end_preamble
\use_default_options true
\language english
\inputencoding auto
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\graphics default
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\use_hyperref false
\papersize a4paper
\use_geometry true
\use_amsmath 1
\use_esint 1
\cite_engine basic
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\paperorientation portrait
\leftmargin 3cm
\topmargin 3cm
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\headheight 3cm
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\paragraph_separation skip
\defskip bigskip
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\author ""
\author ""
\end_header
\begin_body
\begin_layout Section*
Signal to noise ratio (SNR) and contrast to noise ratio (CNR)
\end_layout
\begin_layout Standard
This is a note to try to understand what is meant by the terms
\emph on
signal to noise ratio
\emph default
(SNR) and
\emph on
contrast to noise ratio
\emph default
(CNR) in the context of diffusion magnetic resonance imaging (dMRI).
We assume for the moment that we are measuring dMRI signal in a homogeneous
voxel where an axial mono-exponential model holds [ref?].
Let
\begin_inset Formula $S_{0}$
\end_inset
represent the measured signal with no diffusion weighting, and let
\begin_inset Formula $S_{\Vert}=S_{0}\exp(-bdr^{2})$
\end_inset
and
\begin_inset Formula $S_{\theta}=S_{0}\exp(-bdr^{2}\cos^{2}\theta)$
\end_inset
represent the measured signals with gradients along, and at an angle
\begin_inset Formula $\theta$
\end_inset
to, the axis.
[Do we need a diagram?]
\end_layout
\begin_layout Standard
Then
\begin_inset Formula $\mathrm{SNR}=(S_{0}-S_{\Vert})/S_{0}=1-\exp(-bdr^{2})$
\end_inset
and
\begin_inset Formula $\mathrm{CNR}=(S_{\theta}-S_{\parallel})/S_{0}=\exp(bdr^{2}(1-\cos^{2}\theta))-1.$
\end_inset
These two functions are graphed below as a function of
\begin_inset Formula $r$
\end_inset
:
\end_layout
\begin_layout Standard
This illustrates the comment by Aganj that
\begin_inset Quotes eld
\end_inset
high signal-to-noise ratio and high angular contrast-to-noise ratio ...
at respectively low and high
\begin_inset Formula $b$
\end_inset
-values
\begin_inset Quotes erd
\end_inset
.
\end_layout
\end_body
\end_document