It is shown that in a broad class of linear systems, including general linear shift-invariant systems, the spatial resolution and the noise satisfy a duality relationship, resembling the uncertainty principle in quantum mechanics. The product of the spatial resolution and the standard deviation of output noise in such systems represents a type of phase-space volume that is invariant with respect to linear scaling of the point-spread function, and it cannot be made smaller than a certain positive absolute lower limit. A corresponding intrinsic "quality" characteristic is introduced and then evaluated for the cases of some popular imaging systems, including computed tomography, generic image convolution and phase-contrast imaging. It is shown that in the latter case the spatial resolution and the noise can sometimes be decoupled, potentially leading to a substantial increase in the imaging quality.

We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded Engel CR manifold.

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We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves "almost" like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of "semi-flat" manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection.In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

Abstract. The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the torsion is explained and the chains of dimensions one and two are discussed.There have been many attempts to use some ideas going back up to Cartan, in order to understand the geometry of CR-manifolds. In the codimension one cases, the satisfactory solution had been worked out in the seventies, see [22,23,8], but the higher codimensions have not been understood yet in a comparable extent. In this paper, the recent general theory of the so called parabolic geometries is applied. In particular, we use the approach developed in [4,21], see also [24,27] for earlier results. Relying on recent achievements by the authors, a clean and quite simple construction of the normal Cartan connection is presented. This Cartan connection replaces the absolute parallelisms from [9] by more powerful geometric tools and it enables the detailed study of geometrical and analytical properties of the CR structures. Consequently the resulting geometric picture is much more transparent and surprising new results are obtained.The main advantage of our approach is the fully coordinate-free handling of the normal Cartan connection and its curvature. Thus we are able to translate the cohomological properties of the structure algebras into full geometrical understanding of the curvature obstruction, without writing down the curvature components explicitly. The initial section introduces the CR structures and provides a brief exposition of distinguished second order osculations of the surfaces by quadrics. Then we observe, that this osculation transfers enough data from the quadric to apply the general construction of normal Cartan connections, due to [24,4]. This leads easily to the main Theorems 1.2 and 1.3. In fact, the Cartan connections are constructed also for certain abstract CRmanifolds though the embedded ones have many distinguished properties. The third section is devoted to the exposition of the generalities 1 2 GERD SCHMALZ AND JAN SLOVÁK on parabolic geometries modelled over |2|-graded algebras and provides the proof of the existence of the normal connections.Next we study the local geometry of the hyperbolic points in detail. We recover easily all known facts from [9], but we go much further. In particular, we identify the complete geometric obstructions against the integrability of the almost product structure on the tangent bundle (Theorem 3.5), the integrability of the almost complex structure on the tangent CR space (Theorem 3.6), and the compatibility of the almost product and almost complex structures (3.8). It turns out that the latter two obstructions always vanish on the embedded hyperbolic CR-structures which results in automatic vanishing of several algebraic brackets. In particular, the whole hyperbolic CR-manifold M ⊂ C 4 is a product of two 3-dimensional CR-manifolds if an...