Three-dimensional Krawtchouk descriptors for protein local surface shape comparison

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. from them do not carry any error due to discretization unlike many other moments related to continuous functions [23]. 2) These polynomials are orthogonal; each moment brings in a new feature of the image, where minimum redundancy is critical in their discriminative performance. Moreover, they are directly defined in the image coordinate space, and hence their orthogonality property is well retained in the computed moments. 3) They are complete with a finite number of functions (equal to the image size) while many other polynomial spaces have infinitely many members. 4) They have the ability to retrieve local image patches by only changing the resolution of reconstruction and using low order moments. 5) The location of the patch can also be controlled by changing three parameters and hence shifting the region-of-interest along each dimension. 6) We also prove that these moments can be transformed into local descriptors, which are invariant under translation, rotation, and scaling. Therefore, using only a small number of invariant descriptors per image will make it possible to develop an efficient method for quick local image retrieval.Moment-based approaches, particularly Krawtchouk moments, are very useful for representing biological and medical images as they are pixelized or voxelized data. In medical imaging, such as computerized tomography (CT) scan and magnetic resonance imaging (MRI), objects are observed at different viewpoints and local images need to be extracted and examined. In digital pathology, for instance, pathologists are interested in information about specific structures rather than the whole image [9]. Thus, it is necessary to construct moment invariants that do not change by translation, rotation, and scaling and can retrieve local image patches or subimages.

Stable Calculation of Krawtchouk Functions from Triplet Relations

“…In the digital age, the discrete polynomials are especially popular and those with a finite support (discrete Chebyshev and Krawtchouk) have been used in various signal processing tasks, but dominantly for image processing. Examples of applications of Krawtchouk polynomials can be found in, e.g., [1][2][3][4][5][6][7].…”

Efficient Translation, Rotation, and Scale Invariants of Discrete Tchebichef Moments

“…An important property of moment functions is that they should be invariant to translation, rotation and scale which can lead to better performances in image analysis [10], [26]- [29]. The translation, rotation and scale invariants of discrete Tchebichef moments are formulated either directly from the moment functions or indirectly from other moments like geometric moments.…”

“…The method is based on invariant proposed by Yap et al [21] for Krawtchouk moments. This indirect approach is simple to formulate and thus has become the most commonly used translation, rotation and scale invariants for discrete orthogonal moments on 2D and 3D patterns to date [21], [24], [25], [29]- [32]. However the computation of transformed central-geometric moments is relatively time consuming because of the appearance of complicated quantities and repetitive calculations in the expressions.…”

2DKD: a toolkit for content-based local image search

“…Moment-based approaches are very useful for representing biological and medical images as they are pixelized [1] or voxelized data [2][3][4]. In medical imaging, such as computerized tomography (CT) scan and magnetic resonance imaging (MRI), objects are observed at different viewpoints and local images need to be extracted and examined.…”