arXiv subjects
Publications and source records attributed to Victor Palamodov.
A non-iterative method of reconstruction is proposed from data of MRI system and of a harmonic electro-magnetic field at Larmor frequency. The method is based on the exact analytic formula for the contrast source function. A geometric method for acquisition of the full inductive field is discussed.
New lensless diffractive X-ray technic for micro-scale imaging of biological tissue is based on quantitative phase retrieval schemes. By incorporating refraction, this method yields improved contrast compared to purely absorption-based radiography but involves a phase retrieval problem since of physical limitation of detectors. A general method is proposed in this paper for one step reconstruction of the ray integral of complex refractive index of an optically weak object from intensity distribution of the hologram.
Any even function defined on 2-sphere is reconstructed from its integrals over big circles by means of the classical Funk formula. For the non-geodesic Funk transform on the sphere of arbitrary dimension, there is the explicit inversion formula similar to that for the geodesic transform. A function defined on the sphere of radius one is integrated over traces of hyperplanes tangent to a sphere contained in the unit ball. This reconstruction is generalized in the paper for Riemannian hypersurfaces in an affine space.
The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.
A reconstruction of a function from integrals over the family of confocal paraboloids is given by a FBP formula.
A class of photoacoustic acquisition geometries in n-space is considered such that the spherical mean transform admits an exact filtered back projection reconstruction formula. The reconstruction is interpreted as a time reversion mirror that reproduces exactly an arbitrary source distribution in the cavity. A series of examples of non-uniqueness of the inverse potential problem is constructed basing on the same geometrical technique.
Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is stated which implies uniqueness and stability for this problem. If the conformal class is not known a unique reconstruction is not possible since of shortage of information. It is proved that the list of all geodesic lengths is sufficient for unique determination of a Riemannian metric in a compact surface with boundary up to an automorphism which fix the boundary. Some related problems of integral geometry are studied. Key words: Geodesic curve, Travel-time, Conjugate point, Geodesic flow, Hodograph, Geodesic integral transform.
A general method for analytic inversion of geometric integral transforms is proposed
The phenomenon of removable singularity is studied for overedetermined systems of differential equations. We show that the dimension of the characteristic variety plays a key role in the problem.
We study properties of the general integral transform defined for a family of hypersurfaces in a smooth manifold. Estimates of Sobolev norms, range conditions and approximation theorem for the kernel of the integral transform are stated. Applications to the spherical mean transform that appears in thermo/opto/photoacoustic tomography are discussed.
Global constructions of quantization deformation and obstructions are discussed for an arbitrary complex analytic space in terms of adapted (analytic) Hochschild cohomology. For K3-surfaces an explicit global construction of a Poisson bracket is given. It is shown that the analytic Hochschild (co)homology on a complex space has structure of coherent analytic sheaf in each degree.
An integral stability estimate is proved for refraction coefficients of two conformal metrics in a plane domain in terms of its travel times. No assumption on absence of conjugate points of geodesics is made.