The distribution of image intensity values \(S_i\) divided by the noise standard deviation in \(K\)-space \(\sigma\) in dMRI experiments is assumed to follow a non-central chi-distribution with \(2L\) degrees of freedom and noncentrality parameter \(\eta\), where \(L\) refers to the number of receiver coils in the system and \(\sigma \eta\) is the signal of interest. This is an idealization in the sense that each coil is assumed to have the same contribution at each location. For realistic modeling \(L\) should be a locally smooth function in voxel space that reflects the varying local influence of the receiver coils in the the reconstruction algorithm used.
The functions assume \(L\) to be known and estimate either a local
(function awslsigmc) or global ( function awssigmc)
\(\sigma\) employing an assumption of local homogeneity for
the noncentrality parameter \(\eta\).
Function afsigmc implements estimates from Aja-Fernandez (2009).
Function aflsigmc implements the estimate from Aja-Fernandez (2013).
awssigmc(y, steps, mask = NULL, ncoils = 1, vext = c(1, 1), lambda = 20,
h0 = 2, verbose = FALSE, sequence = FALSE, hadj = 1, q = 0.25,
qni = .8, method=c("VAR","MAD"))
awslsigmc(y, steps, mask = NULL, ncoils = 1, vext = c(1, 1), lambda = 5, minni = 2,
hsig = 5, sigma = NULL, family = c("NCchi"), verbose = FALSE,
trace=FALSE, u=NULL)
afsigmc(y, level = NULL, mask = NULL, ncoils = 1, vext = c( 1, 1),
h = 2, verbose = FALSE, hadj = 1,
method = c("modevn","modem1chi","bkm2chi","bkm1chi"))
aflsigmc(y, ncoils, level = NULL, mask = NULL, h=2, hadj=1, vext = c( 1, 1))a list with components
either a scalar or a vector of estimated noise standard deviations.
the estimated mean structure
3D array, usually obtained from an object of class dwi as
obj@si[,,,i] for some i, i.e. one 3D image from an dMRI experiment.
number of steps in adapive weights smoothing, used to reveal the unerlying mean structure.
restrict computations to voxel in mask, if is.null(mask) all voxel are used.
In function afsigmc mask should refer to background for method %in% c("modem1chi","bkm2chi","bkm1chi") and to voxel within the head for
method=="modevn".
number of coils, or equivalently number of effective degrees of freedom of non-central chi distribution divided by 2.
voxel extentions
scale parameter in adaptive weights smoothing
initial bandwidth
if verbose==TRUE density plots
and quantiles of local estimates of sigma are provided.
if trace==TRUE intermediate results for each step are
returned in component tergs for all voxel in mask.
if sequence=TRUE a vector of estimates for the noise
standard deviation sigma for the individual steps is returned
instead of the final value only.
adjustment factor for bandwidth (chosen by bw.nrd) in mode estimation
quantile to be used for interquantile-differences.
quantile of distribution of actual sum of weights \(N_i=\sum_j w_{ij}\) in adaptive smoothing. Only voxel i with \(N_i > q_{qni}(N_.)\) are used for variance estimation. Should be larger than 0.5.
in case of function awssigmc the
method for variance estimation, either "VAR" (variance) or "MAD" (mean absolute deviation). In function afsigmc see last column in Table 2 in Aja-Fernandez (2009).
threshold for background separation. Used if !is.null(level)
to redefine mask
bandwidth for local avaeraging
Minimum sum of weights for updating values of sigma.
Bandwidth of the median filter.
Initial estimate for sigma
One of "Gauss" or "NCchi" (default) defining the
probability distribution to use.
if verbose==TRUE an array of noncentrality paramters for
comparisons. Internal use for tests only
J\"org Polzehl polzehl@wias-berlin.de
K. Tabelow, H.U. Voss, J. Polzehl, Local estimation of the noise level in MRI using structural adaptation, Medical Image Analysis, 20 (2015), pp. 76--86.