# Three-Dimensional Krawtchouk Descriptors for Protein Local Surface Shape Comparison

## TL;DR

This paper introduces 3D Krawtchouk descriptors that enable efficient, invariant comparison of local surface regions of 3D objects, specifically applied to protein surface shape analysis for ligand binding prediction.

## Contribution

It extends 2D Krawtchouk moment invariants to 3D, creating descriptors that are invariant under translation, rotation, and scaling for local surface comparison.

## Key findings

- Descriptors successfully compare local surface regions.
- Application to protein surfaces predicts ligand binding.
- Invariant properties improve comparison robustness.

## Abstract

Direct comparison of three-dimensional (3D) objects is computationally expensive due to the need for translation, rotation, and scaling of the objects to evaluate their similarity. In applications of 3D object comparison, often identifying specific local regions of objects is of particular interest. We have recently developed a set of 2D moment invariants based on discrete orthogonal Krawtchouk polynomials for comparison of local image patches. In this work, we extend them to 3D and construct 3D Krawtchouk descriptors (3DKD) that are invariant under translation, rotation, and scaling. The new descriptors have the ability to extract local features of a 3D surface from any region-of-interest. This property enables comparison of two arbitrary local surface regions from different 3D objects. We present the new formulation of 3DKD and apply it to the local shape comparison of protein surfaces in order to predict ligand molecules that bind to query proteins.