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It is common for example in Cryo-electron microscopy of viruses, that the orientations at which the projections are acquired, are totally unknown. We introduce here a moment based algorithm for recovering them in the three-dimensional parallel beam tomography. In this context, there is likely to be also unknown shifts in the projections. They will be estimated simultaneously. Also stability properties of the algorithm are examined. Our considerations rely on recent results that guarantee a solution to be almost always unique. A similar analysis can also be done in the two-dimensional problem.

We consider uniqueness of three-dimensional parallel beam tomography in which both the object being imaged and the projection orientations are unknown. This problem occurs in certain practical applications, for example in cryo electron microscopy of viral particles, where the projection orientations may be completely unknown due to the random orientations of the particles being imaged. We show that only three projections are needed to guarantee unique recovery of the unknown projection orientations (up to a common orthogonal transformation), if the object belongs to a certain generic set of objects. In particular, the uniqueness holds for almost all objects. We also show that, if the object belongs to that generic set, $k+1$ projections at unknown orientations suffice to determine uniquely also the geometric moments of the object of order less or equal to $k$. As a consequence, the object belonging to that generic set, is uniquely determined (up to an orthogonal transformation) by almost any infinitely many projections at unknown orientations. We show that the uniqueness problem is related to some properties of certain homogeneous polynomials that depend on the projection orientations and the geometric moments of objects. Here certain theorems of algebraic geometry turn out to be useful. We also provide a system of equations, that is uniquely solvable and gives the desired projection orientations and object's geometric moments.

We consider uniqueness of two-dimensional parallel beam tomography with unknown view angles. We show that infinitely many projections at unknown view angles of a sufficiently asymmetric object determine the object uniquely. An explicit expression for the required asymmetry is given in terms of the object's geometric moments. We also show that under certain assumptions finitely many projections guarantee uniqueness for the unknown view angles. Compared to previous results about uniqueness of view angles, our result reduces the minimum number of required projections to approximately half and is applicable to a larger set of objects. Our analysis is based on algebraic geometric properties of a certain system of homogeneous polynomials determined by the Helgason-Ludwig consistency conditions.

We introduce the concept of generalized projection operators, i.e., projection integrals over a body in that generalize the usual result of projected area in a given direction by taking into account shadowing and scattering effects as well as additional convolution functions in the integral. Such operators arise naturally in connection with various observation instruments and data types. We review and discuss some properties of these operators and the related inverse problems, particularly in the cases pertaining to photometric and radar data. We also prove an ambiguity theorem for a special observing geometry common in astrophysics, and uniqueness theorems for radar inverse problems of a spherical target. These theorems are obtained by employing the intrinsic rotational properties of the observing geometries and function representations. We then present examples of the mathematical modelling of the shape and rotation state of a body by simultaneously using complementary data sources corresponding to different generalized projection operators. We show that generalized projection operators unify a number of mathematical considerations and physical observation types under the same concept.

We present a numerical procedure for solving the Minkowski problem, i.e., determining the convex set corresponding to a given curvature function. The method is based on Minkowski's isoperimetric inequality concerning convex and compact sets in R 3 . The support function of the target set is approximated in finite function space, so the problem becomes one of constrained optimization in R n , which in turn is solved by Newtonian (or other) iteration. We prove some properties of the optimization function and the constraining set and present some numerical examples.

A method has been developed for the determination of particle and crystal size distributions of rutile titanium dioxide pigments. This is based on a rigorous model for the turbidity (light extinction) spectra of rutile particles in dilute water suspension. The inversion of the rigorous model leads to an ill-posed problem and to a semi-rigorous method, in which the constraints of the solution have been determined empirically. The pigment particles are approximated as spheroids and a two-dimensional size distribution of particle width and ratio of length and width is calculated. Most of the particles are single crystals but a small amount is in aggregate form. A method to separate the distributions of aggregates and single crystals is proposed. The extinction cross section values for spheroids, needed in the model, are calculated by the T-matrix method. A comparison of the crystal size results with the values determined by transmission electron microscopy shows a very good accuracy for mean values of both the volume equivalent size and for the width, whereas the results for the means of the length/width ratio are much less accurate.

We consider uniqueness of two-dimensional parallel beam tomography in which both the object being imaged and the projection directions are unknown. This problem occurs in certain practical applications. For example, in magnetic resonance imaging there may be uncertainty in the projection directions due to the involuntary motion of the patient. The three-dimensional version of this problem occurs in cryo electron microscopy of viral particles, where the projection directions may be completely unknown due to the random orientations of the particles being imaged. We show that the problem is related to some algebraic geometric properties of a certain system of homogeneous polynomials. We also show that for sufficiently asymmetric objects, the object is uniquely determined up to an orthogonal transformation by the projection data from unknown directions.

A simplified stochastic model for the fiber structure of paper is introduced. The packing density and optical thickness of the fiber network are derived analytically, and their dependence on fiber characteristics can be seen. We undertake a Monte Carlo simulation of light scattering that is based on geometrical optics, using a realization of the model, which gives packing densities and optical thicknesses well in accordance with those given by the stochastic model and the scattering quantities as functions of three angles.