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

BACKGROUND: Direct comparison of 2D images is computationally inefficient due to the need for translation, rotation, and scaling of the images to evaluate their similarity. In many biological applications, such as digital pathology and cryo-EM, often identifying specific local regions of images is of particular interest. Therefore, finding invariant descriptors that can efficiently retrieve local image patches or subimages becomes necessary. RESULTS: We present a software package called Two-Dimensional Krawtchouk Descriptors that allows to perform local subimage search in 2D images. The new toolkit uses only a small number of invariant descriptors per image for efficient local image retrieval. This enables querying an image and comparing similar patterns locally across a potentially large database. We show that these descriptors appear to be useful for searching local patterns or small particles in images and demonstrate some test cases that can be helpful for both assembly software developers and their users. CONCLUSIONS: Local image comparison and subimage search can prove cumbersome in both computational complexity and runtime, due to factors such as the rotation, scaling, and translation of the object in question. By using the 2DKD toolkit, relatively few descriptors are developed to describe a given image, and this can be achieved with minimal memory usage.

A global map of the protein shape universe.

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.

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

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 (3DKDs) 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 3DKDs and apply it to the local shape comparison of protein surfaces in order to predict ligand molecules that bind to query proteins.

Comparison of image patches using local moment invariants.

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.

Solving a generalized distance geometry problem for protein structure determination.

We propose a new approach to the problem of determining an ensemble of protein structures with a set of interatomic distance bounds in NMR protein modeling. Similarly to X-ray crystallography, we assume that the protein has an equilibrium structure and the atoms fluctuate around their equilibrium positions. Then, the problem can be formulated as a generalized distance geometry problem, to find the equilibrium positions and maximal possible fluctuation radii for the atoms in the protein, subject to the condition that the fluctuations should be within the given distance bounds. We describe the scientific background of the work, the motivation of the new approach and the formulation of the problem. We develop a geometric buildup algorithm for an approximate solution to the problem and present some preliminary test results as a first step concept proofing. We also discuss related theoretical and computational issues and potential impacts of this work in NMR protein modeling.

A geometric buildup algorithm for the solution of the distance geometry problem using least-squares approximation.

We propose a new geometric buildup algorithm for the solution of the distance geometry problem in protein modeling, which can prevent the accumulation of the rounding errors in the buildup calculations successfully and also tolerate small errors in given distances. In this algorithm, we use all instead of a subset of available distances for the determination of each unknown atom and obtain the position of the atom by using a least-squares approximation instead of an exact solution to the system of distance equations. We show that the least-squares approximation can be obtained by using a special singular value decomposition method, which not only tolerates and minimizes small distance errors, but also prevents the rounding errors from propagation effectively, especially when the distance data is sparse. We describe the least-squares formulations and their solution methods, and present the test results from applying the new algorithm for the determination of a set of protein structures with varying degrees of availability and accuracy of the distances. We show that the new development of the algorithm increases the modeling ability, and improves stability and robustness of the geometric buildup approach significantly from both theoretical and practical points of view.