tortuosity {SoPhy} | R Documentation

## Variability of the tortuosity

### Description

The function estimates the ratio between the mean tortuosity and the smallest tortuosity from profiles of Brilliant Blue tracer experiments

### Usage

      tortuosity(depth, freq,range.depth, range.xi, len.xi=100,
                 lower.bound.m = 1.005, tolerance.m=0.01, PrintLevel=0)

### Arguments

`depth` | depth in pixels
`freq` | number of stained pixels found in depth `depth`
`range.depth` | range of depths for which 1-H^* is fitted
`range.xi` | range of values of xi for which 1-H^* is fitted
`len.xi` | `range.xi` and `len.xi` define the grid `seq(min(range.xi), max(range.xi), len=len.xi)` for which 1-H^* is fitted.
`lower.bound.m` | lower bound for m used in the numerical optimisation
`tolerance.m` | non-negative value. If `tolerance.m=0` only the optimal xi is used to estimate m. If `tolerance.m>0` then all estimated m(D, xi) are considered for which the span of m is less than or equal to (1 + `tolerance.m`) times the minimal span value.
`PrintLevel` | integer. If the value is greater than 1 then tracing information is given

### Value

`tortuosity` returns a list of the following elements:

`xi` | vector of length `len.xi`. It contains the values of xi used in the algorithm
`raw.m` | matrix with `len.xi` rows. The number of colums equals the number of depth values that are between the given bounds `range.depth`. It contains the fitted values of m
`span.xi` | vector of length `len.xi` contains span of all `raw.m` values for the respective value of xi
`span.m` | vector of length `len.xi`. It contains the difference between the largest and the smallest value of `raw.m` for given xi
`opt` | list of the optimal values: `xi`, `m`, `s`, `mspan` (the `span.xi` found at `xi`) and `D` (threshold depth)
`input` | list of two vectors: `freq` and `depth`
`fitted` | list of two vectors that give the best fitted curve: `depth` and `p`; the latter cannot be given if `tolerance.m`>0

### Author(s)

Martin Schlather, schlath@hsu-hh.de http://www.unibw-hamburg.de/WWEB/math/schlath/schlather.html

### References

Schlather, M. and Huwe, B. (2005) A characteristic for the variability of tortuosity. Submitted.

### Examples

    data(F04)
    path <- paste(system.file(package='SoPhy'),"tracer", sep="/")
    F04$name <- paste(path, "F04.G.tif", sep="/")
    m <- analyse.profile(F04, estimate.all=FALSE, method="fix.m",
                         selected.rate=c(0, 0), selected.dist=NULL,
                         interactive=FALSE, Print=0)
    stained <- m$fct(m$picture, m$param)
    freq <- m$absfr
    truedepth <- m$r.i$data[, 1] + ncol(stained) - m$loc[[1]]$y[2]
    tol = 0

    mt <- tortuosity(truedepth, freq,  range.depth=c(85, 97),
                     range.xi=c(-2.5, -0.6), Print=2, tol=tol)
    str(mt$opt)

    # fig. 3A in Schlather and Huwe (2005)
    matplot(mt$xi, mt$raw.m, xlab=expression(xi), ylab="m",
            type="l", col=1, lty=1)

    # fig. 3B in Schlather and Huwe (2005)
    plot(mt$xi, mt$span.m, xlab=expression(xi), ylab="span m",
         type="l", col=1, lty=1)
    for (cex in c(9, 9.5, 10))
      points(mt$opt$x, mt$opt$mspan, cex=cex, col="darkgrey")

    # fig. 2 in Schlather and Huwe (2005)
    plot(mt$input$depth, mt$input$freq, xaxs="i", yaxs="i",
         pch=16, ylab="stained pixels", xlab="depth [pixels]",
         col="blue", cex=2)
    lines(mt$fitted$depth, mt$fitted$p, lwd=3, col=2)

[Package SoPhy version 1.0.28 Index]