Dimensional Singularities Research Articles

MEDICAL IMAGE FUSION BASED ON RIPPLET TRANSFORM TYPE-I

The motivation behind fusing multimodality, multi- resolution images is to create a single image with improved interpretability. In this paper, we propose a novel multimodality Medical Image Fusion (MIF) method, based on Ripplet Transform Type-I (RT) for spatially registered, multi-sensor, multi-resolution medical images. RT is a new Multi-scale Geometric Analysis (MGA) tool, capable of resolving two dimensional (2D) singularities and representing image edges more e-ciently. The source medical images are flrst transformed by discrete RT (DRT). Difierent fusion rules are applied to the difierent subbands of the transformed images. Then inverse DRT (IDRT) is applied to the fused coe-cients to get the fused image. The performance of the proposed scheme is evaluated by various quantitative measures like Mutual Information (MI), Spatial Frequency (SF), and Entropy (EN) etc. Visual and quantitative analysis shows, that the proposed technique performs better compared to fusion scheme based on Contourlet Transform (CNT).

Six-dimensional exceptional quotient singularities

We classify six-dimensional exceptional quotient singularities and show that seven- dimensional exceptional quotient singularities do not exist. Inter alia we prove that the irre- ducible six-dimensional projective representation of the sporadic simple Hall-Janko group gives rise to an exceptional quotient singularity. where Ei is a ξ-exceptional divisor, and bi ∈ Q. Let B be an effective Q-divisor on V. Put B = m X i=1 aiBi, where Bi is a prime Weil divisor on V , and ai ∈ Q>0. Suppose that B is a Q-Cartier divisor. Then m X i=1 ai ¯ Bi ∼Q ξ ∗ m X i=1 aiBi !

The classification of special Cohen–Macaulay modules

In this paper we completely classify all the special Cohen–Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras.

On symplectic fillings of lens spaces

Let ξ ¯ st \overline \xi _\textrm {st} be the contact structure naturally induced on the lens space L ( p , q ) = S 3 / Z / p Z L(p,q)=S^3/\mathbb {Z}/p\mathbb {Z} by the standard contact structure ξ st \xi _\textrm {st} on the three–sphere S 3 S^3 . We obtain a complete classification of the symplectic fillings of ( L ( p , q ) , ξ ¯ st ) (L(p,q),\overline \xi _\textrm {st}) up to orientation–preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two–dimensional cyclic quotient singularities.

Coupling leads to chaos

The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension 4. On the other hand, we will prove that in any generic unfolding X μ of an n-dimensional nilpotent singularity of codimension n there are bifurcation curves of ( n − 1 ) -dimensional nilpotent singularities of codimension n − 1 which are in turn generically unfolded by X μ . Arguments conclude recalling that any generic unfolding of the 3-dimensional nilpotent singularity of codimension 3 exhibits strange attractors.

Linear feature extraction based on complex ridgelet transform

In our previous work, a new dual-tree complex wavelet transform (DT-CWT) model for surface analysis has been built, which solved the problem of the lack of shift-invariance that existed in the first and second generation wavelet models. Unfortunately, the DT-CWT model still has the same problem as the previous wavelet models in the lack of ability to detect line singularities or higher dimensional singularities, which causes the edges not to be smooth when extracting the directional features from engineering surfaces. In this paper, a complex finite ridgelet transform (CFRIT), which provides approximate shift invariance and analysis of line singularities, is proposed by taking the DT-CWT on the projections of the finite Radon transform (FRAT). The Numerical experiments show the remarkable potential of the methodology to analyse engineering and bioengineering surfaces with linear scratches in comparison to wavelet-based methods developed in our pervious work.

A new digital implementation of ridgelet transform for images of dyadic length

Ridgelet transform is a new directional multi-resolution transform and it is more suitable for describing the signals with high dimensional singularities. Finite ridgelet transform is a discrete version of ridgelet transform, which is as numerical precision as the continuous ridgelet transform and has low computational complexity. However, finite ridgelet transform is only suitable for images of prime-pixels length, which is a limitation of its applications in image processing. In this paper, a new digital implementation of ridgelet transform that is suitable for images of dyadic length is proposed. This method not only expands the applications of finite ridgelet transform, but also simplifies the algorithm. First, we introduce the concept of ridgelet transform in the continuous domain. Then, we illustrate finite ridgelet transform and the new method. Finally, we compare the new method with finite ridgelet transform by applying both digital ridgelet transforms to the denoising of images embedded in additive white Gaussian noise and the new method gets a better performance in image denoising. The new algorithm can also be used as the important building block in curvelet transform and get surprising visual performances in denoising for natural image.

Improved epsilon expansion for three-dimensional turbulence: summation of nearest dimensional singularities.

An improved epsilon expansion in the d-dimensional (d>2) stochastic theory of turbulence is constructed by taking into account pole singularities at d-->2 in coefficients of the epsilon expansion of universal quantities. Effectiveness of the method is illustrated by a two-loop calculation of the Kolmogorov constant in three dimensions.

The invariant -k2 and continued fractions for 2- dimensional cyclic quotient singularities

Every rational number greater than 1 corresponds to a 2-dimensional cyclic quotient singularity and therefore corresponds to a rational number-K 2,an invariant of this singularity.

Domains of Holomorphy with Edges and Lower Dimensional Boundary Singularities

It is shown that a domain in C N with piecewise smooth boundary (and also of some more general shape) is a domain of holomorphy, provided the Levi form at every regular point is positively semidefinite and the tangent cone is convex at every point outside a boundary subset of zero Hausdorff (2N-2)-dimensional measure.

Pinkham-Demazure construction for two dimensional cyclic quotient singularities

The affine ring of any cyclic quotient singularity has countably many natural graded ring structures. In two dimensional case, we construct every such graded ring in terms of Q-coefficient divisor on $P^{1}$ which is naturally associated with a resolution of the singularity.

Development of FEM Program for Stress Singularity Analysis and Database of Three Dimensional Interfacial Corner Singularities.

Development of FEM Program for Stress Singularity Analysis and Database of Three Dimensional Interfacial Corner Singularities.

Fuzzy cosets and their gravity duals

Dp-branes placed in a certain external RR (p+4)-form field expand into a transverse fuzzy two-sphere, as shown by Myers. We find that by changing the (p+4)-form background other fuzzy cosets can be obtained. Three new examples, S^2 X S^2, CP^2 and SU(3)/(U(1) X U(1)) are constructed. The first two are four-dimensional while the last is six-dimensional. The dipole and quadrupole moments which result in these configurations are discussed. Finally, the gravity backgrounds dual to these vacua are examined in a leading order approximation. These are multi-centered solutions containing (p+4)- or (p+6)-dimensional brane singularities.

Electronic transport in k-component Fibonacci quantumwaveguides

We present a study of electronic behaviours in the k-componentFibonacci (KCF) quantum waveguides, in which k differentincommensurate intervals are arranged according to a substitutionrule. On the basis of the transfer matrix method, the quantum transmissionproperties of the KCF stub structures are obtained. It is shownthat the transmission coefficient depends on the wavevector ofthe electron and the number of different incommensurate intervals k.For the KCF waveguides with the same k, on increasing thenumber of stubs, the minima in transmission extend gradually intothe band gap over which the transmission is blocked. Meanwhile moretransmission peaks appear. For finite KCF stub structures, onincreasing the number of different incommensurate intervals k, thetotal transmission over the spectral region of interest decreasesgradually and the width of the electronic band gap is enlarged.Moreover, when the value of k is large enough, the transmission isbasically shut off, except at a few energies where resonanttunnelling takes place. These properties make it possible to usethis kind of KCF waveguide as a switching device for digitalapplications. On the other hand, the charge-density distributionsin these structures are singularly continuous. We propose that theycan be analysed using a multifractal concept. A dimensional spectrumof singularities associated with the charge density, f (α),demonstrates that the electronic transport in the KCF waveguidepresents scaling properties; hence the charge-density distributionshows a genuine multifractality.

Vanishing of differentials along ideals and non-archimedean approximation

One of the major new developments in commutative algebra over the last decade or so was the introduction of the theory of tight closure of an ideal by Hochster and Huneke. It proved to be an extremely useful technique to study ideals, and it also turned out to be closely related to many geometric questions. Fedder [F] used derivations in positive characteristic to obtain characterizations of 2 dimensional graded rational singularities in terms of F-purity and F-injectivity. In an attempt to generalize these techniques and to relate rational singularities with F-rationality, Craig Huneke raised the following problem (c.f [FHH]): Let R be a regular local ring, containing a perfect field k, over which R is essentially of finite type, and let C(R/k) be the subring of derivationally constant elements of R/k (i.e. C(R/k) = {x ∈ R : δ(x) = 0 for all δ ∈ Derk(R)}). Then Huneke asked: (1) If I ⊆ R is an ideal, does there exist a constant l = l(R, I) ∈ N with the following property: If x ∈ R with δ(x) ∈ I then there exists a c ∈ C(R/k) with x− c ∈ I. (2) If an l as in (1) exists, is it possible to bound it in a way useful for reduction mod p techniques, i.e. if char(k) = 0, does there exist a model R/A, I ⊆ R of R/k, I with A/Z of finite type and a constant l(I) such that l(R/mR, I+m/m) ≤ l(I) for all m ∈ Max(A).

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn.

For each pair (n, k) with 1 </= k < n, we construct a tight frame (rholambda : lambda in Lambda) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle chin,k-the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to chin,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n-k) x Rn-k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61-83] to map this tight frame isometrically to a tight frame for L2(Rn)-the k-plane ridgelets. This construction makes analysis of a function f in L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for chin,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n - 1.

Geometry of singularities for the steady Boussinesq equations

Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.

A note on irregularity of two dimensional simple hyperbolic singularities

Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singularities. In this note we give an optimal upper bound for a class of surface singularities by the computation of cohomology. Moreover a sufficient condition is given for the positivity of irregularity of some simple hyperbolic surface singularities. Therefore a class of surface singularities with non-rigid deformation is constructed.

ON c1 = 0 RESOLUTION OF QUOTIENT SINGULARITY

An explicit c1 = 0 resolution of the 3-dimensional quotient singularity by the icosahedral group is constructed. Moreover, the Euler number of this resolution is identified with the orbifold Euler characteristic of the group action. Examples of c1 = 0 resolutions of higher dimensional quotient singularities are also obtained.

Perturbing away higher dimensional singularities from area minimizing hypersurfaces

Perturbing away higher dimensional singularities from area minimizing hypersurfaces