fracdim function

Calculation of Fractal Dimension of Lef Veins Based on the Box-Counting Method

fracdim is used to calculate the fractal dimension of leaf veins based on the box-counting method. UTF-8

fracdim(x, y, frac.fig = TRUE, denomi.range = seq(8, 30, by=1), 
        ratiox = 0.02, ratioy = 0.08, main = NULL)

Arguments

• x: the $x$ coordinates of leaf-vein pixels.
• y: the $y$ coordinates of leaf-vein pixels.
• frac.fig: the option of drawing the results of the linear fitting.
• denomi.range: the number of equidistant segments of the maximum range between the range of the $x$ coordinates and that of the $y$ coordinates.
• ratiox: the the $x$ coordinate of the location parameter for positioning the legend.
• ratioy: the the $y$ coordinate of the location parameter for positioning the legend.
• main: the main title of the figure.

Details

The box-counting approach uses a group of boxes (squares for simplicity) with different sizes ($\delta$) to divide the leaf vein image into different parts. Let $N$ represent the number of boxes that include at least one pixel of leaf vein. The maximum of the range of the $x$ coordinates and the range of the $y$ coordinates for leaf-vein pixels is defined as $z$. Let $\delta$ represent the vector of $z$/denomi.range. Then, we used the following equation to calculate the fractal dimension of leaf veins:

where $b$ is the theoretical value of the fractal dimension. We can use its estimate as the numerical value of the fractal dimension for a leaf venation network.

Returns

• a: the estimate of the intercept.

• sd.a: the standard deviation of the estimated intercept.

• lci.a: the lower bound of the 95% confidence interval of the estimated intercept.

• uci.a: the upper bound of the 95% confidence interval of the estimated intercept.

• b: the estimate of the slope.

• sd.b: the standard deviation of the estimated slope.

• lci.a: the lower bound of the 95% confidence interval of the estimated slope.

• uci.a: the upper bound of the 95% confidence interval of the estimated slope.

• r.sq: the coefficient of determination.

• delta: the vector of box sizes.

• N: the number of boxes that include at least one pixel of leaf vein.

Note

Here, x and y cannot be adjusted by the adjdata function because the leaf veins are not the leaf's boundary data.

Author(s)

Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .

References

Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516, 123$-$134. tools:::Rd_expr_doi("10.1111/nyas.14862")

Shi, P., Yu, K., Niinemets, Ü., Gielis, J. (2021) Can leaf shape be represented by the ratio of leaf width to length? Evidence from nine species of Magnolia and Michelia (Magnoliaceae). Forests 12, 41. tools:::Rd_expr_doi("10.3390/f12010041")

Vico, P.G., Kyriacos, S., Heymans, O., Louryan, S., Cartilier, L. (1998) Dynamic study of the extraembryonic vascular network of the chick embryo by fractal analysis. Journal of Theoretical Biology 195, 525$-$532. tools:::Rd_expr_doi("10.1006/jtbi.1998.0810")

See Also

veins

Examples

data(veins)

dev.new()
plot(veins$x, veins$y, cex=0.01, asp=1, cex.lab=1.5, cex.axis=1.5, 
     xlab=expression(italic("x")), ylab=expression(italic("y")))

fracdim(veins$x, veins$y)

graphics.off()