Proteins are involved in almost all functions in a living cell, and functions of proteins are realized by their tertiary structures. Obtaining a global perspective of the variety and distribution of protein structures lays a foundation for our understanding of the building principle of protein structures. In light of the rapid accumulation of low-resolution structure data from electron tomography and cryo-electron microscopy, here we map and classify three-dimensional (3D) surface shapes of proteins into a similarity space. Surface shapes of proteins were represented with 3D Zernike descriptors, mathematical moment-based invariants, which have previously been demonstrated effective for biomolecular structure similarity search. In addition to single chains of proteins, we have also analyzed the shape space occupied by protein complexes. From the mapping, we have obtained various new insights into the relationship between shapes, main-chain folds, and complex formation. The unique view obtained from shape mapping opens up new ways to understand design principles, functions, and evolution of proteins.

We propose a new set of moment invariants based on Krawtchouk polynomials for comparison of local patches in 2D images. Being computed from discrete functions, these moments do not carry the error due to discretization. Unlike many orthogonal moments, which usually capture global features, Krawtchouk moments can be used to compute local descriptors from a region-of-interest in an image. This can be achieved by changing two parameters, and hence shifting the center of interest region horizontally or vertically or both. This property enables comparison of two arbitrary local regions. We show that Krawtchouk moments can be written as a linear combination of geometric moments, so easily converted to rotation, size, and position independent invariants. We also construct local Hu-based invariants using Hu invariants and utilizing them on images localized by the weight function given in the definition of Krawtchouk polynomials. We give the formulation of local Krawtchouk-based and Hu-based invariants, and evaluate their discriminative performance on local comparison of artificially generated test images.