Pushing Cuckoo Eggs under Inverse Radon Transforms












4












$begingroup$


Essentially the inverse of the Radon transforms $Rf(L)=int_L{f(x)|dx|}$ has the ability to reconstruct $f(x)$ from the integrals over all lines; or, expressed differently to (re)construct $f(x)$ from a functional, that maps lines to real values.



Essentially this question is aimed at cuckoo eggs the inverse Radon transform will hatch if pushed under it:






Question:



what are examples of other functionals, that map lines to real values, to which the inverse Radon transform can be applied beneficially?



I am especially interested in examples that have been mentioned in publications.
Two examples of such line functionals that would be possible candidates, are




  • mapping lines to the geodesic distance between the pair of points, in which they intersect the boundary of a compact, convex subset of $mathbb{R}^n$ (lines that miss the boundary or are tangent to it are mapped to $0$)


  • mapping lines to their moments of inertia w.r.t. the model of a compact physical object; lines missing a sufficiently small compact containing volume will be mapped to $0$







fa.functional-analysis integral-transforms







share|cite|improve this question











$endgroup$

















    4












    $begingroup$


    Essentially the inverse of the Radon transforms $Rf(L)=int_L{f(x)|dx|}$ has the ability to reconstruct $f(x)$ from the integrals over all lines; or, expressed differently to (re)construct $f(x)$ from a functional, that maps lines to real values.



    Essentially this question is aimed at cuckoo eggs the inverse Radon transform will hatch if pushed under it:






    Question:



    what are examples of other functionals, that map lines to real values, to which the inverse Radon transform can be applied beneficially?



    I am especially interested in examples that have been mentioned in publications.
    Two examples of such line functionals that would be possible candidates, are




    • mapping lines to the geodesic distance between the pair of points, in which they intersect the boundary of a compact, convex subset of $mathbb{R}^n$ (lines that miss the boundary or are tangent to it are mapped to $0$)


    • mapping lines to their moments of inertia w.r.t. the model of a compact physical object; lines missing a sufficiently small compact containing volume will be mapped to $0$







    fa.functional-analysis integral-transforms







    share|cite|improve this question











    $endgroup$















      4












      4








      4


      1



      $begingroup$


      Essentially the inverse of the Radon transforms $Rf(L)=int_L{f(x)|dx|}$ has the ability to reconstruct $f(x)$ from the integrals over all lines; or, expressed differently to (re)construct $f(x)$ from a functional, that maps lines to real values.



      Essentially this question is aimed at cuckoo eggs the inverse Radon transform will hatch if pushed under it:






      Question:



      what are examples of other functionals, that map lines to real values, to which the inverse Radon transform can be applied beneficially?



      I am especially interested in examples that have been mentioned in publications.
      Two examples of such line functionals that would be possible candidates, are




      • mapping lines to the geodesic distance between the pair of points, in which they intersect the boundary of a compact, convex subset of $mathbb{R}^n$ (lines that miss the boundary or are tangent to it are mapped to $0$)


      • mapping lines to their moments of inertia w.r.t. the model of a compact physical object; lines missing a sufficiently small compact containing volume will be mapped to $0$







      fa.functional-analysis integral-transforms







      share|cite|improve this question











      $endgroup$




      Essentially the inverse of the Radon transforms $Rf(L)=int_L{f(x)|dx|}$ has the ability to reconstruct $f(x)$ from the integrals over all lines; or, expressed differently to (re)construct $f(x)$ from a functional, that maps lines to real values.



      Essentially this question is aimed at cuckoo eggs the inverse Radon transform will hatch if pushed under it:






      Question:



      what are examples of other functionals, that map lines to real values, to which the inverse Radon transform can be applied beneficially?



      I am especially interested in examples that have been mentioned in publications.
      Two examples of such line functionals that would be possible candidates, are




      • mapping lines to the geodesic distance between the pair of points, in which they intersect the boundary of a compact, convex subset of $mathbb{R}^n$ (lines that miss the boundary or are tangent to it are mapped to $0$)


      • mapping lines to their moments of inertia w.r.t. the model of a compact physical object; lines missing a sufficiently small compact containing volume will be mapped to $0$







      fa.functional-analysis integral-transforms




      fa.functional-analysis integral-transforms






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago









      YCor

      27.3k480132




      27.3k480132










      asked 2 days ago









      Manfred WeisManfred Weis

      4,48221340




      4,48221340






















          1 Answer
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          Suppose that $newcommand{bR}{mathbb{R}}$ $Ssubset bR^n$ is a compact semialgebraic set. Denote by $AL_n$ the Grassmannian of affine lines in $bR^n$. We define a "motivic" Radon transform that associates to each line $Lin AL_n$ the Euler characteristic $R_S(L):=chi(Lcap S)$. Pierre Schapira showed that the set $S$ is completely determined by its motivic Radon transform $R_S: AL_nto mathbb{Z}$.



          This problem can be formulated cleverly to express $1_S$, the indicator function of $S$, as an inverse Radon transform of $R_S$. The story is very rich and a bit too complex to expose here. Instead I refer to section 4.1 and 4.2 of my notes where you will find detailed explanations and precise references. Also, here is a link to Schapira's original paper.



          The Crofton formulae allow you to compute certain invariants of $S$ given the knowledge of other invariants of the intersection of $S$ with affine planes.






          share|cite|improve this answer











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            1 Answer
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            $begingroup$

            Suppose that $newcommand{bR}{mathbb{R}}$ $Ssubset bR^n$ is a compact semialgebraic set. Denote by $AL_n$ the Grassmannian of affine lines in $bR^n$. We define a "motivic" Radon transform that associates to each line $Lin AL_n$ the Euler characteristic $R_S(L):=chi(Lcap S)$. Pierre Schapira showed that the set $S$ is completely determined by its motivic Radon transform $R_S: AL_nto mathbb{Z}$.



            This problem can be formulated cleverly to express $1_S$, the indicator function of $S$, as an inverse Radon transform of $R_S$. The story is very rich and a bit too complex to expose here. Instead I refer to section 4.1 and 4.2 of my notes where you will find detailed explanations and precise references. Also, here is a link to Schapira's original paper.



            The Crofton formulae allow you to compute certain invariants of $S$ given the knowledge of other invariants of the intersection of $S$ with affine planes.






            share|cite|improve this answer











            $endgroup$


















              3












              $begingroup$

              Suppose that $newcommand{bR}{mathbb{R}}$ $Ssubset bR^n$ is a compact semialgebraic set. Denote by $AL_n$ the Grassmannian of affine lines in $bR^n$. We define a "motivic" Radon transform that associates to each line $Lin AL_n$ the Euler characteristic $R_S(L):=chi(Lcap S)$. Pierre Schapira showed that the set $S$ is completely determined by its motivic Radon transform $R_S: AL_nto mathbb{Z}$.



              This problem can be formulated cleverly to express $1_S$, the indicator function of $S$, as an inverse Radon transform of $R_S$. The story is very rich and a bit too complex to expose here. Instead I refer to section 4.1 and 4.2 of my notes where you will find detailed explanations and precise references. Also, here is a link to Schapira's original paper.



              The Crofton formulae allow you to compute certain invariants of $S$ given the knowledge of other invariants of the intersection of $S$ with affine planes.






              share|cite|improve this answer











              $endgroup$
















                3












                3








                3





                $begingroup$

                Suppose that $newcommand{bR}{mathbb{R}}$ $Ssubset bR^n$ is a compact semialgebraic set. Denote by $AL_n$ the Grassmannian of affine lines in $bR^n$. We define a "motivic" Radon transform that associates to each line $Lin AL_n$ the Euler characteristic $R_S(L):=chi(Lcap S)$. Pierre Schapira showed that the set $S$ is completely determined by its motivic Radon transform $R_S: AL_nto mathbb{Z}$.



                This problem can be formulated cleverly to express $1_S$, the indicator function of $S$, as an inverse Radon transform of $R_S$. The story is very rich and a bit too complex to expose here. Instead I refer to section 4.1 and 4.2 of my notes where you will find detailed explanations and precise references. Also, here is a link to Schapira's original paper.



                The Crofton formulae allow you to compute certain invariants of $S$ given the knowledge of other invariants of the intersection of $S$ with affine planes.






                share|cite|improve this answer











                $endgroup$



                Suppose that $newcommand{bR}{mathbb{R}}$ $Ssubset bR^n$ is a compact semialgebraic set. Denote by $AL_n$ the Grassmannian of affine lines in $bR^n$. We define a "motivic" Radon transform that associates to each line $Lin AL_n$ the Euler characteristic $R_S(L):=chi(Lcap S)$. Pierre Schapira showed that the set $S$ is completely determined by its motivic Radon transform $R_S: AL_nto mathbb{Z}$.



                This problem can be formulated cleverly to express $1_S$, the indicator function of $S$, as an inverse Radon transform of $R_S$. The story is very rich and a bit too complex to expose here. Instead I refer to section 4.1 and 4.2 of my notes where you will find detailed explanations and precise references. Also, here is a link to Schapira's original paper.



                The Crofton formulae allow you to compute certain invariants of $S$ given the knowledge of other invariants of the intersection of $S$ with affine planes.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 2 days ago

























                answered 2 days ago









                Liviu NicolaescuLiviu Nicolaescu

                25.5k259110




                25.5k259110






























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