3D Zernike Moments Research Articles

ZMPY3D: accelerating protein structure volume analysis through vectorized 3D Zernike moments and Python-based GPU integration.

Volumetric 3D object analyses are being applied in research fields such as structural bioinformatics, biophysics, and structural biology, with potential integration of artificial intelligence/machine learning (AI/ML) techniques. One such method, 3D Zernike moments, has proven valuable in analyzing protein structures (e.g., protein fold classification, protein-protein interaction analysis, and molecular dynamics simulations). Their compactness and efficiency make them amenable to large-scale analyses. Established methods for deriving 3D Zernike moments, however, can be inefficient, particularly when higher order terms are required, hindering broader applications. As the volume of experimental and computationally-predicted protein structure information continues to increase, structural biology has become a "big data" science requiring more efficient analysis tools. This application note presents a Python-based software package, ZMPY3D, to accelerate computation of 3D Zernike moments by vectorizing the mathematical formulae and using graphical processing units (GPUs). The package offers popular GPU-supported libraries such as CuPy and TensorFlow together with NumPy implementations, aiming to improve computational efficiency, adaptability, and flexibility in future algorithm development. The ZMPY3D package can be installed via PyPI, and the source code is available from GitHub. Volumetric-based protein 3D structural similarity scores and transform matrix of superposition functionalities have both been implemented, creating a powerful computational tool that will allow the research community to amalgamate 3D Zernike moments with existing AI/ML tools, to advance research and education in protein structure bioinformatics. ZMPY3D, implemented in Python, is available on GitHub (https://github.com/tawssie/ZMPY3D) and PyPI, released under the GPL License.

MCI Conversion Prediction Using 3D Zernike Moments and the Improved Dynamic Particle Swarm Optimization Algorithm

Mild cognitive impairment (MCI) conversion prediction is a vital challenge in the area of Alzheimer’s disease (AD) as it could determine possible treatment pathways for AD patients. In this work, we presented a robust MCI conversion prediction framework based on the 3D-Zernike Moment (3D-ZM) method that generates statistical features (e.g., shape, texture, and symmetry information) from 3D-MRI scans and improved dynamic particle swarm optimization (IDPSO) that finds an informative sub-set of Zernike features for MCI conversion prediction. We quantified the efficiency of the proposed prediction framework on a large sample of MCI patients including 105 progressive-MCI (pMCI) and 121 stable-MCI (sMCI) at the baseline from the ADNI dataset. Using the proposed MCI conversion prediction framework, pMCI patients were distinguished from sMCI patients with an accuracy exceeding 75% (sensitivity, 83%, and specificity, 68%), which is well comparable with the state-of-the-art MCI conversion prediction approaches. Experimental results indicate that the 3D-ZM method can represent informative statistical patterns from 3D-MRI scans and IDPSO has a great capability to find meaningful statistical features for identifying MCI patients who are at risk of conversion to the AD stage.

Stable Evaluation of 3D Zernike Moments for Surface Meshes

The 3D Zernike polynomials form an orthonormal basis of the unit ball. The associated 3D Zernike moments have been successfully applied for 3D shape recognition; they are popular in structural biology for comparing protein structures and properties. Many algorithms have been proposed for computing those moments, starting from a voxel-based representation or from a surface based geometric mesh of the shape. As the order of the 3D Zernike moments increases, however, those algorithms suffer from decrease in computational efficiency and more importantly from numerical accuracy. In this paper, new algorithms are proposed to compute the 3D Zernike moments of a homogeneous shape defined by an unstructured triangulation of its surface that remove those numerical inaccuracies. These algorithms rely on the analytical integration of the moments on tetrahedra defined by the surface triangles and a central point and on a set of novel recurrent relationships between the corresponding integrals. The mathematical basis and implementation details of the algorithms are presented and their numerical stability is evaluated.

Extraction of Human Motion Information from Digital Video Based on 3D Poisson Equation

Based on the 3D Poisson equation, this paper extracts the features of the digital video human body action sequence. By solving the Poisson equation on the silhouette sequence, the time and space features, time and space structure features, shape features, and orientation features can be obtained. First, we use the silhouette structure features in three-dimensional space-time and the orientation features of the silhouette in three-dimensional space-time to represent the local features of the silhouette sequence and use the 3D Zernike moment feature to represent the overall features of the silhouette sequence. Secondly, we combine the Bayesian classifier and AdaBoost classifier to learn and classify the features of human action sequences, conduct experiments on the Weizmann video database, and conduct multiple experiments using the method of classifying samples and selecting partial combinations for training. Then, using the recognition algorithm of motion capture, after the above process, the three-dimensional model is obtained and matched with the model in the three-dimensional model database, the sequence with the smallest distance is calculated, and the corresponding skeleton is outputted as the results of action capture. During the experiment, the human motion tracking method based on the university matching kernel (EMK) image kernel descriptor was used; that is, the scale invariant operator was used to count the characteristics of multiple training images, and finally, the high-dimensional feature space was mapped into the low-dimensional to obtain the feature space approximating the Gaussian kernel. Based on the above analysis, the main user has prior knowledge of the network environment. The experimental results show that the method in this paper can effectively extract the characteristics of human body movements and has a good classification effect for bending, one-foot jumping, vertical jumping, waving, and other movements. Due to the linear separability of the data in the kernel space, fast linear interpolation regression is performed on the features in the feature space, which significantly improves the robustness and accuracy of the estimation of the human motion pose in the image sequence.

3D Human Activity Classification with 3D Zernike Moment Based Convolutional, LSTM-Deep Neural Networks

In this paper, we propose a method for classification 3D human activities using the complementarity of CNNs, LSTMs, and DNNs by combining them into one unified architecture called CLDNN. Our approach is based on the prediction of 3D Zernike Moments of some relevant joints of the human body through Kinect using the Kinect Activity Recognition Dataset. KARD includes 18 activities and each activity consists of real-world point clouds that have been carried out 3 times by 10 different subjects. We introduce the potential for the 3D Zernike Moment feature extraction approach via a 3D point cloud for human activity classification, and the ability to be trained and generalized independently from datasets using the Deep Learning methods. The experimental results obtained on datasets with the proposed system has correctly classified 96.1% of the activities. CLDNN has been shown to provide a 5% relative improvement over LSTM, the strongest of the three individual models.

Real time structural search of the Protein Data Bank.

Detection of protein structure similarity is a central challenge in structural bioinformatics. Comparisons are usually performed at the polypeptide chain level, however the functional form of a protein within the cell is often an oligomer. This fact, together with recent growth of oligomeric structures in the Protein Data Bank (PDB), demands more efficient approaches to oligomeric assembly alignment/retrieval. Traditional methods use atom level information, which can be complicated by the presence of topological permutations within a polypeptide chain and/or subunit rearrangements. These challenges can be overcome by comparing electron density volumes directly. But, brute force alignment of 3D data is a compute intensive search problem. We developed a 3D Zernike moment normalization procedure to orient electron density volumes and assess similarity with unprecedented speed. Similarity searching with this approach enables real-time retrieval of proteins/protein assemblies resembling a target, from PDB or user input, together with resulting alignments (http://shape.rcsb.org).

Real-time Terrain Matching Based on 3D Zernike Moments

Since the descriptors based on Three-Dimensional (3D) Zernike moments are robust to geometric transformations and noise, they have been proposed for terrain matching. However, terrain matching algorithms based on 3D Zernike Moments (3DZMs) are often difficult to implement in practice since they are computationally intensive. This paper presents a more efficient real-time terrain matching algorithm based on 3DZMs for land applications. Two efficient methods based on coordinate transformation and symmetry are proposed to compute the geometric moments. The impact of the sample difference on the matching result due to heading angle is investigated to prove the feasibility of using a circular template. Consequently, the terrain feature vectors of the reference map can be computed off-line with the circular template to significantly reduce on-line computation. Numerical experiments on a real digital elevation model demonstrate that the proposed algorithm is robust to the heading angle and can be implemented for real-time terrain matching operations.

Exploring the potential of 3D Zernike descriptors and SVM for protein\u2013protein interface prediction

BackgroundThe correct determination of protein–protein interaction interfaces is important for understanding disease mechanisms and for rational drug design. To date, several computational methods for the prediction of protein interfaces have been developed, but the interface prediction problem is still not fully understood. Experimental evidence suggests that the location of binding sites is imprinted in the protein structure, but there are major differences among the interfaces of the various protein types: the characterising properties can vary a lot depending on the interaction type and function. The selection of an optimal set of features characterising the protein interface and the development of an effective method to represent and capture the complex protein recognition patterns are of paramount importance for this task.ResultsIn this work we investigate the potential of a novel local surface descriptor based on 3D Zernike moments for the interface prediction task. Descriptors invariant to roto-translations are extracted from circular patches of the protein surface enriched with physico-chemical properties from the HQI8 amino acid index set, and are used as samples for a binary classification problem. Support Vector Machines are used as a classifier to distinguish interface local surface patches from non-interface ones. The proposed method was validated on 16 classes of proteins extracted from the Protein–Protein Docking Benchmark 5.0 and compared to other state-of-the-art protein interface predictors (SPPIDER, PrISE and NPS-HomPPI).ConclusionsThe 3D Zernike descriptors are able to capture the similarity among patterns of physico-chemical and biochemical properties mapped on the protein surface arising from the various spatial arrangements of the underlying residues, and their usage can be easily extended to other sets of amino acid properties. The results suggest that the choice of a proper set of features characterising the protein interface is crucial for the interface prediction task, and that optimality strongly depends on the class of proteins whose interface we want to characterise. We postulate that different protein classes should be treated separately and that it is necessary to identify an optimal set of features for each protein class.

Unified framework for human behaviour recognition: An approach using 3D Zernike moments

In this paper, we present a unified framework for the analysis of video databases by using Markov spatio-temporal random walks on graph. The proposed framework provides an efficient approach for clustering, data organization, dimension reduction and recognition. The aim of our work is to develop a vision-based approach for human behaviour recognition. Our contribution lies in three aspects. First, we employ 3D Zernike moments to encode the object of interest in a video clip. Then, we propose a new method to represent the video database as a weighted undirected graph where each vertex is a video clip. The weight of an edge between two video clips is defined by a Gaussian kernel on their 3D Zernike moments and their respective neighbourhoods in the feature space. Our objective is to obtain a robust low-dimensional space through spectral graph embedding which provides efficient keypoints transcription into an euclidean manifold, and allows to achieve higher classification accuracy through agglomerative categorization. Finally, we describe a variational framework for manifold denoising based on p-Laplacian, thereby lessening the negative impact of outliers, enhancing keypoints classification and thus, boosting the recognition accuracy. The proposed method is tested on the Weizmann and KTH human action datasets and on a hand gesture dataset. The retrieved results using the 3D Zernike moments prove that the proposed method can effectively capture the form of the behaviours with low order moments. Moreover, our framework allows to classify various behaviours and achieves a significant recognition rate.

Novel binding pocket descriptors based on DrugScore potential fields encoded by 3D Zernike descriptors

Proteins interact with other molecules, e.g. ligands or other proteins, in specific binding sites. Key factors for these interactions are the shape, size, and buriedness of the binding site, as well as its physicochemical composition. Since all these properties usually significantly vary among different proteins, up to now there is no standard definition what constitutes a binding site [1]. Thus, novel pocket descriptors allowing an in-depth characterization of binding sites are highly desired. Hence, we developed novel binding pocket descriptors based on 3D molecular interaction fields. The binding pocket of a protein is characterized using the distance dependent, knowledge-based pair potentials of the DrugScore scoring function [2] in combination with multiple ligand atom probes. To allow an efficient comparison of the resulting potential fields, the 3D grids are encoded using 3D Zernike descriptors. The 3D Zernike polynomials Znlm are orthonormal basis functions on the unit sphere. Thus, any 3D object can be represented as: using a 3D Zernike function expansion. We utilized the resulting function expansion coefficients cnlm, i.e. the 3D Zernike moments, to describe the 3D molecular potential fields characterizing a protein's binding pocket. The resulting descriptors are invariant under rotation, scaling, and translation and enable a fast comparison and an efficient characterization of protein binding pockets. Thus, these novel pocket descriptors can be used to predict target druggability or to calculate similarities between binding pockets, e.g. to predict potential off-targets or to perform protein function de-orphanization.

An Algorithm for Fast Computation of 3D Zernike Moments for Volumetric Images

An algorithm was proposed for very fast and low‐complexity computation of three‐dimensional Zernike moments. The 3D Zernike moments were expressed in terms of exact 3D geometric moments where the later are computed exactly through the mathematical integration of the monomial terms over the digital image/object voxels. A new symmetry‐based method was proposed to compute 3D Zernike moments with 87% reduction in the computational complexity. A fast 1D cascade algorithm was also employed to add more complexity reduction. The comparison with existing methods was performed, where the numerical experiments and the complexity analysis ensured the efficiency of the proposed method especially with image and objects of large sizes.

Precipitate shape fitting and reconstruction by means of 3D Zernike functions

3D Zernike functions are defined and used for the reconstruction of precipitate shapes. These functions are orthogonal over the unit ball and allow for an arbitrary shape, scaled to fit inside an embedding sphere, to be decomposed into 3D harmonics. Explicit expressions are given for the general Zernike moments, correcting typographical errors in the literature. Explicit expressions of the Zernike moments for the ellipsoid and the cube are given. The 3D Zernike functions and moments are applied to the reconstruction of γ′ precipitate shapes in two Ni-based superalloys, one with nearly cuboidal precipitate shapes, and one with more complex dendritic shapes.

Efficient 3D Geometric and Zernike Moments Computation from Unstructured Surface Meshes

This paper introduces and evaluates a fast exact algorithm and a series of faster approximate algorithms for the computation of 3D geometric moments from an unstructured surface mesh of triangles. Being based on the object surface reduces the computational complexity of these algorithms with respect to volumetric grid-based algorithms. In contrast, it can only be applied for the computation of geometric moments of homogeneous objects. This advantage and restriction is shared with other proposed algorithms based on the object boundary. The proposed exact algorithm reduces the computational complexity for computing geometric moments up to order N with respect to previously proposed exact algorithms, from N(9) to N(6). The approximate series algorithm appears as a power series on the rate between triangle size and object size, which can be truncated at any desired degree. The higher the number and quality of the triangles, the better the approximation. This approximate algorithm reduces the computational complexity to N(3). In addition, the paper introduces a fast algorithm for the computation of 3D Zernike moments from the computed geometric moments, with a computational complexity N(4), while the previously proposed algorithm is of order N(6). The error introduced by the proposed approximate algorithms is evaluated in different shapes and the cost-benefit ratio in terms of error, and computational time is analyzed for different moment orders.

Fingerprint‐based structure retrieval using electron density

We present a computational approach that can quickly search a large protein structural database to identify structures that fit a given electron density, such as determined by cryo-electron microscopy. We use geometric invariants (fingerprints) constructed using 3D Zernike moments to describe the electron density, and reduce the problem of fitting of the structure to the electron density to simple fingerprint comparison. Using this approach, we are able to screen the entire Protein Data Bank and identify structures that fit two experimental electron densities determined by cryo-electron microscopy.

The application of 3D Zernike moments for the description of "model-free" molecular structure, functional motion, and structural reliability.

Protein structures are not static entities consisting of equally well-determined atomic coordinates. Proteins undergo continuous motion, and as catalytic machines, these movements can be of high relevance for understanding function. In addition to this strong biological motivation for considering shape changes is the necessity to correctly capture different levels of detail and error in protein structures. Some parts of a structural model are often poorly defined, and the atomic displacement parameters provide an excellent means to characterize the confidence in an atom's spatial coordinates. A mathematical framework for studying these shape changes, and handling positional variance is therefore of high importance. We present an approach for capturing various protein structure properties in a concise mathematical framework that allows us to compare features in a highly efficient manner. We demonstrate how three-dimensional Zernike moments can be employed to describe functions, not only on the surface of a protein but throughout the entire molecule. A number of proof-of-principle examples are given which demonstrate how this approach may be used in practice for the representation of movement and uncertainty.

DISCRIMINATIVE SPHERICAL WAVELET FEATURES FOR CONTENT-BASED 3D MODEL RETRIEVAL

The description of 3D shapes using features that possess descriptive power and are invariant under similarity transformations is one of the most challenging issues in content-based 3D model retrieval. Spherical harmonics-based descriptors have been proposed for obtaining rotation invariant representations. However, spherical harmonic analysis is based on a latitude-longitude parameterization of the sphere which has singularities at each pole, and therefore, variations of the north pole affect significantly the shape function. In this paper we discuss these issues and propose the usage of spherical wavelet transforms as a tool for the analysis of 3D shapes represented by functions on the unit sphere. We introduce three new descriptors extracted from the wavelet coefficients, namely: (1) a subset of the spherical wavelet coefficients, (2) the L1 and, (3) the L2 energies of the spherical wavelet sub-bands. The advantage of this tool is threefold; First, it takes into account feature localization and local orientations. Second, the energies of the wavelet transform are rotation invariant. Third, shape features are uniformly represented which makes the descriptors more efficient. Spherical wavelet descriptors are natural extensions of spherical harmonics and 3D Zernike moments. We evaluate, on the Princeton Shape Benchmark, the proposed descriptors regarding computational aspects and shape retrieval performance.