Boundary reconstruction for the broken ray transform (bibtex)

by Joonas Ilmavirta

Abstract:

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary.
For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form.
The problem studied here is related to Calder\'on's problem with partial data.

Reference:

Boundary reconstruction for the broken ray transform (Joonas Ilmavirta), Annales Academiae Scientiarum Fennicae Mathematica, volume 39, number 2, pp. 485–502, 2014.
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We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary.
For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form.
The problem studied here is related to Calderón's problem with partial data.
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Bibtex Entry:

@article{bdy-det, author = {Joonas Ilmavirta}, title = {{Boundary reconstruction for the broken ray transform}}, journal={Annales Academiae Scientiarum Fennicae Mathematica}, month = jun, year = {2014}, volume = {39}, number = {2}, pages = {485--502}, arxiv = {1310.2025}, MRNUMBER = {3237032}, doi = {10.5186/aasfm.2014.3935}, url={http://users.jyu.fi/~jojapeil/pub/brt-boundary-reconstruction.pdf}, gsid = {16787522604397989183}, eprint = {https://www.mittag-leffler.se/preprints/list.php?program_code=1213s}, abstract = {We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form. The problem studied here is related to Calder\'on's problem with partial data.} }

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