Spherical harmonics texture extraction for versatile analysis of biological objects
Fig 2
Spherical Texture method design.
A) A C. elegans meiotic nucleus in the pachytene stage, stained with DAPI, shown as maximum intensity projections over Z (left) and X, with the YZ view rescaled isotropically (center) and to square pixels (right) about the XY view. B) Data from A rescaled to 80x80x80 pixels in XY (left) and YZ (right) views. C) A graphic showing the mean intensity projection to spherical space, showing first a subset of the radial rays (left, red lines) used to generate the mean-intensity spherical projection as spherical data and as planar map projection (center). The mean intensities are normalized to mean=0 and variance=1 (right). D) Projections of the spherical harmonics basis functions on the sphere of the first 7 spherical harmonic degrees. E) The spherical harmonics power spectrum of the spherical projection in C shows a distinct peak around approx. the 10th harmonic degree. F) Rescaling the harmonic degrees to approximate wavelength yields a spherical harmonics spectrum, which shows a corresponding peak in the contribution to variance around a wavelength of approx. 0.1 rad/2π. G) The standard output of the Spherical Texture method corresponds to the binned spectrum shown in F. Insets show the spherical projection band-passed to fine, medium and coarse wavelengths, as indicated by dashed lines in the spectrum (7th and 27th harmonic degree). The band-passed regions reflect the part of the signal quantified by each region of the plot, where the region that shows high variance in the quantification corresponds to the scale of the most prominent signal in the data (here: chromosomes, note that this region alone already resembles the spherical projection shown in C). H) The Spherical Texture extraction is implemented as a Python package and it is directly available in ilastik, allowing for its adoption into the Object Classification workflow. In this workflow, users can interactively train a Random Forest machine learning classifier. Shown here is a part of a C. elegans gonad with segmented nuclei, where some nuclei were labeled as Class 1 and others as Class 2 (solid colors). Based on the Spherical Texture of these labels, ilastik predicts the class of all other nuclei (transparent colors).