Differentiate an energy containing integral in a region to derive curve evolution

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I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/



Here we want to minimize the energy
enter image description here



where $p$ is the radon transform of $f$:
$$
p ( s , theta ) = int _ Omega f ( x , y ) delta left( P _ theta ( x , y ) - s right) d overline x
$$



and $hat p$ is the randon transform of our estimation of image.



The authors say that if we differentiate the energy with respect to time, we obtain the following:



enter image description here



I found a partial derivation in the first author's thesis (Alvino). But I don't understand how we can apply divergence theorem to step into Eq. (117). (Here C is the curve denoting the boundary of region)



enter image description here




divergence variational-analysis




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    up vote
    0
    down vote

    favorite












    I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/



    Here we want to minimize the energy
    enter image description here



    where $p$ is the radon transform of $f$:
    $$
    p ( s , theta ) = int _ Omega f ( x , y ) delta left( P _ theta ( x , y ) - s right) d overline x
    $$



    and $hat p$ is the randon transform of our estimation of image.



    The authors say that if we differentiate the energy with respect to time, we obtain the following:



    enter image description here



    I found a partial derivation in the first author's thesis (Alvino). But I don't understand how we can apply divergence theorem to step into Eq. (117). (Here C is the curve denoting the boundary of region)



    enter image description here




    divergence variational-analysis




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/



      Here we want to minimize the energy
      enter image description here



      where $p$ is the radon transform of $f$:
      $$
      p ( s , theta ) = int _ Omega f ( x , y ) delta left( P _ theta ( x , y ) - s right) d overline x
      $$



      and $hat p$ is the randon transform of our estimation of image.



      The authors say that if we differentiate the energy with respect to time, we obtain the following:



      enter image description here



      I found a partial derivation in the first author's thesis (Alvino). But I don't understand how we can apply divergence theorem to step into Eq. (117). (Here C is the curve denoting the boundary of region)



      enter image description here




      divergence variational-analysis




      share|cite|improve this question











      I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/



      Here we want to minimize the energy
      enter image description here



      where $p$ is the radon transform of $f$:
      $$
      p ( s , theta ) = int _ Omega f ( x , y ) delta left( P _ theta ( x , y ) - s right) d overline x
      $$



      and $hat p$ is the randon transform of our estimation of image.



      The authors say that if we differentiate the energy with respect to time, we obtain the following:



      enter image description here



      I found a partial derivation in the first author's thesis (Alvino). But I don't understand how we can apply divergence theorem to step into Eq. (117). (Here C is the curve denoting the boundary of region)



      enter image description here





      divergence variational-analysis





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 27 at 14:12









      jakeoung

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