Image processing often includes looking into shapes and quantifying them in a meaningful way accessible to both humans and machines. This is the case in many fields, from astronomy to life science.
Many tools have been developed for this task, among them Minkowski functionals and their tensorial counterparts, Minkowski tensors. They are powerful and comprehensive shape descriptors, containing all shape information for convex bodies and enough information for non-convex bodies to be useful. They are additive and consequently very useful for pixel images, where one can calculate them from small 2×2 pixel windows.
One representation, the irreducible Minkowski tensors, are particularly handy for characterizing symmetries of a body in a way that is accessible to humans:Â The usual Cartesian form forces one to deal with many components per tensor that somehow need to be brought into a scalar form. This has only been satisfyingly solved for rank 2 tensors that encode elongation. On the other hand, the irreducible Minkowski tensors boil their respective symmetry information down to one complex number.
The amplitude of a rank s tensor describes the extent of s-fold symmetry and its phase describes the corresponding orientation. An elongated shape will have a large rank 2 amplitude, whereas rank 4 reacts to “square-ness”. The effects of various such amplitudes and phases can be tested in the morphometer, a small browser tool for gaining some intuition with Minkowski tensors.
Previously, the irreducible Minkowski tensors have been used for analyzing regular, flat images. However, some datasets exist on the surface of the sphere, where projection into flat space would distort the shapes. This is famously the case for the Cosmic Microwave Background (CMB). It can be seen in any direction, rendering flat images useless, and it contains small temperature fluctuations that encode information on the early universe. The details of these fluctuations impact which fundamental cosmological models are viable. This means that powerful morphometric tools are needed.
Basic information on, e.g., the size distribution of these fluctuations is given in the power spectrum, but for more extensive analyses more unwieldy and abstract tools such as the three point correlation were used. On the contrary, Minkowski tensors provide local and real-space shape information, but their irreducible form was unavailable on the sphere. We implemented them for the HEALPix scheme commonly used for CMB data and made our code publicly available.
We looked at the data by creating Minkowski maps; images that show the desired Minkowski tensor of the local surroundings of each output pixel. With this technique masking data is not an issue – Minkowski map pixels with too many masked input pixels are simply marked as masked as well. Smaller masked input areas can be ignored when adding up the shapes for one output pixel. Additionally, local structures only affect local output pixels, creating a straightforward way for the search for local anisotropies.
The creation of a Minkowski map: For the output map, a circular region is chosen and all sections of the underlying shape within are added up (left). As this window moves across the image, a Minkowski map is created. The right side shows Minkowski maps for several rectangles (white) and their corresponding rank 2 anisotropy. Longer rectangles are shown to have higher anisotropy. As the window (red) moves across the shape, it sometimes catches only part of the contour, leading to characteristic patterns.
With maps like this, we tested our methods on Planck CMB temperature data, looking for unusual 2-fold symmetries at length scales of several degrees. We compared Minkowski maps of the data to those of 999 simulations provided by the Planck Collaboration and found two regions where the data appears more anisotropic than the simulations. The origin of these anisotropies is unclear and needs further investigation.
Deviation between Minkowski maps generated for CMB data and simulations in units of standard deviation. Two regions stand out: one near (80, 27), above the Galactic equator to the left, and near (225,-69), to the bottom right.
Furthermore we checked whether the different tensor ranks provide independent information for such image data, and rank 4 and 2 anisotropies appear uncorrelated for pixels in the simulations. This means that there is more to be learned from such an analysis with higher ranks.
Minkowski tensors thus now enable us to quantify spherical data in a comprehensive and versatile way, and the necessary tools are freely available for the research community.