By V. P. Golubyatnikov
The 1st a part of this new quantity within the Inverse and Ill-Posed difficulties sequence reports distinctiveness questions for recuperating the shapes of the convex and extra complex our bodies from shapes in their projections onto planes of low size. a few balance estimates of the recommendations to those inverse difficulties are given. the second one half offers with inverse issues of projection information at once attached to tomography, particularly to obvious contours of delicate surfaces, that have sensible interpretations resembling skinny cracks in non-stop media that are studied in wave optics, and so forth. New effects on reconstruction of delicate surfaces from observations of the wave fronts iteration through those surfaces are bought. This booklet will be of curiosity to researchers within the fields of inverse difficulties, crucial geometry and tomography.
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Golubyatnikov. Uniqueness questions of tomography the pairs of antipodal points with outward normals ±фк obtained from the vectors ±φο by rotations through the angles kn-α. If φ^ £ (Ε(ω2)\Α π ), then the radii of curvature of dV{{uj2) and öV^o^) at the points with outward normals ±фк coincide. If фь € (Ε(ω2) \ Ao), then the sum of the curvature radii equals w at such points. If Ao Π Απ is nowhere dense, then by rotating it through the angles kir • a we obtain nowhere dense sets (Ло Π The countable union of these intersections, according to Baire's theorem, has dense complement D С Ε(ω2), and the sequence of the unit vectors can be constructed in this complement D.
In view of the previous lemma, it is sufficient to examine the case when <^_1(0) lies on some great circle. By the assumptions, the preimage _1(0) is not contained in φ~1(π). 4 it follows that for all ω G ψ~ι(0) \ (
P. Golubyatnikov. Uniqueness questions of tomography Here is the norm of this vector, and Л is a symmetric η χ η matrix whose eigenvalues λ ι , . . , λ„ satisfy the condition max|Ai-Aj-| < 1/2. 2) This inequality provides the positivity of the curvature radius of the boundd2h ary of the projection + h for all directions tangent to the unit sphere αφζ for both support functions hi and /12, which implies the convexity of these bodies V\ and V2 (see, for example, Pogorelov, 1973). Direct computations show that the projections of these bodies onto twodimensional plane P(u, v) spanned on the orthogonal pair of unit vectors u, ν e Sn~1 are similar with the ratio of the similitude ехр((Лг;, v) + (Au, u)), and after rotation by the angle π/2 these projections become directly homothetic.
