Alexander V. Tuzikov

Evaluation of the symmetry plane in 3D MR brain images

Although the brain is not perfectly symmetrical with respect to the mid-sagittal plane, the automatic detection of this plane and of the degree of symmetry is of interest for many anatomical and functional studies. We propose a method for detecting the best symmetry plane in 3D MR brain images. We express this problem as a registration problem and compute a degree of similarity between the image and its reflection with respect to a plane. The best plane is then obtained by maximizing the similarity measure. This optimization is performed using the downhill simplex method and is initialized by a plane obtained from principal inertia axes, which proves to be close to the global optimum. This is demonstrated on several MR brain images. The proposed algorithm is then successfully tested on simulated and real 3D MR brain images. We also investigated the influence of the optimization procedure control parameters on the computation speed and result precision. Preliminary results obtained on CT and SPECT images suggest that the method can be extended to other modalities.

Similarity and symmetry measures for convex shapes using Minkowski addition

This paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on Minkowski addition and the Brunn-Minkowski inequality. This means, in particular, that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they can be chosen to be invariant under rotations, multiplications, reflections, or the class of affine transformations. It is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle /spl theta/ is a piecewise concave function of /spl theta/. This and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures. Various results obtained in this paper are illustrated by experimental data. Although the paper deals exclusively with the two-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces.

Similarity measures for convex polyhedra based on Minkowski addition

Abstract In this paper we introduce and investigate similarity measures for convex polyhedra based on Minkowski addition and inequalities for the mixed volume and volume related to the Brunn–Minkowski theory. All measures considered are invariant under translations; furthermore, some of them are also invariant under subgroups of the affine transformation group. For the case of rotation and scale invariance, we prove that to obtain the measures based on (mixed) volume, it is sufficient to compute certain functionals only for a finite number of critical rotations. The paper presents a theoretical framework for comparing convex shapes and contains a complexity analysis of the solution. Numerical implementations of the proposed approach are not discussed.

Approximate reflectional symmetries of fuzzy objects with an application in model-based object recognition

This paper is devoted to the study of reflectional symmetries of fuzzy objects. We introduce a symmetry measure which defines the degree of symmetry of an object with respect to a given plane. It is computed by measuring the similarity between the original object and its reflection. The choice of an appropriate measure of comparison is based on the desired properties of the symmetry measure. Then, an algorithm for computing the symmetry plane of fuzzy objects is proposed. This is done using an optimization technique in the space of plane parameters. Finally, we illustrate our approach with an application where the symmetry measure is used as an attribute in graph matching for model-based object recognition.

Computation of volume and surface body moments

Abstract The paper presents explicit formulae for calculation of surface moments for polytopes in R n . We describe also efficient algorithms for calculation of 3D body volume and surface moments. The algorithms are based on explicit formulae for moment calculation and take advantages of shape polygonal representation. They use only coordinates of the body vertices and faces orientation.

Side-Chain Conformational Changes upon Protein–Protein Association

Conformational changes upon protein-protein association are the key element of the binding mechanism. The study presents a systematic large-scale analysis of such conformational changes in the side chains. The results indicate that short and long side chains have different propensities for the conformational changes. Long side chains with three or more dihedral angles are often subject to large conformational transition. Shorter residues with one or two dihedral angles typically undergo local conformational changes not leading to a conformational transition. A relationship between the local readjustments and the equilibrium fluctuations of a side chain around its unbound conformation is suggested. Most of the side chains undergo larger changes in the dihedral angle most distant from the backbone. The frequencies of the core-to-surface interface transitions of six nonpolar residues and Tyr are larger than the frequencies of the opposite surface-to-core transitions. The binding increases both polar and nonpolar interface areas. However, the increase of the nonpolar area is larger for all considered classes of protein complexes, suggesting that the protein association perturbs the unbound interfaces to increase the hydrophobic contribution to the binding free energy. To test modeling approaches to side-chain flexibility in protein docking, conformational changes in the X-ray set were compared with those in the docking decoy sets. The results lead to a better understanding of the conformational changes in proteins and suggest directions for efficient conformational sampling in docking protocols.

Explicit formulae for polyhedra moments

Abstract The paper describes an approach and presents explicit formulae for calculation of polyhedron moments via coordinates of polyhedron vertices. A general formula for calculation of arbitrary finite order moments for polytopes in R n is also obtained.

Brain symmetry plane computation in MR images using inertia axes and optimization

Detection of the best symmetry plane in 3D images can be treated as a registration problem between the original and the reflected images. The registration is performed in a 3D space of parameters defining orientation and shift of a reflection plane. We use the normalized l/sub 2/ metric as the similarity measure between original and reflected images and investigate an algorithm for computation of the best symmetry plane. The algorithm computes first an initial position of the plane by analyzing principal inertia axes. We demonstrate on several MR brain images that the initial position is in the neighborhood of the global maximum. Therefore the downhill simplex method is further used for the computation of the best symmetry plane. The proposed algorithm was tested on simulated and real MR brain images.

Moment computation for objects with spline curve boundary

A new approach is proposed for computation of area and geometric moments for a plane object with a spline curve boundary. The explicit formulae are obtained for area and low order moment calculation. The complexity of calculation depends on the moment order, spline degree, and the number of control points used in spline representation. The formulae proposed use the advantage that the sequence of spline control points is cyclic. It allowed us to reduce substantially the number of summands in them. The formulae might be useful in different applications where it is necessary to perform measurements for shapes with a smooth boundary.

Symmetry Measure Computation for Convex Polyhedra

The paper discusses measures and indices of different kind of symmetry (central, reflection, rotation, mirror rotation) for 3D convex shapes. The measures are based on Minkowski addition and inequalities for volume and mixed volume and are valid for convex shapes only. Symmetry index computation in 3D case is a complicated optimization problem. Taking advantage of convex polyhedra we investigate the situations when these indices can be computed efficiently.