Mathematics of imaging the black hole












45












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The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 .



It has been claimed that state-of-the-art imaging algorithms were an enabler for this historic success. Does anybody care to describe the difficulties, and (quite certainly non-trivial) mathematics that went into this effort ?










share|cite|improve this question











$endgroup$








  • 7




    $begingroup$
    For reference here is the paper where the collaboration described data processing: iopscience.iop.org/article/10.3847/2041-8213/ab0c57/meta
    $endgroup$
    – Neal
    2 days ago










  • $begingroup$
    Katie Bouman's work might shed some light here!
    $endgroup$
    – yousuf soliman
    2 days ago






  • 3




    $begingroup$
    But, @yousuf, will that light be able to escape from the black hole?
    $endgroup$
    – Gerry Myerson
    yesterday






  • 5




    $begingroup$
    The actual image reconstruction is described in paper IV: iopscience.iop.org/article/10.3847/2041-8213/ab0e85. In a nutshell, they formulated a baker's dozen of data misfit (for all the different acquired data as described in paper III) and regularization (including total variation and sparsity) functionals, and then minimized their weighted sum using L-BFGS. Not quite state-of-the-art from a mathematical point of view (especially using BFGS for a non-differentiable functional), but close enough.
    $endgroup$
    – Christian Clason
    yesterday








  • 4




    $begingroup$
    Great post and answer. For those also interested on the physics side of the discussions, here's a recent post: physics.stackexchange.com/questions/471972/…
    $endgroup$
    – user929304
    yesterday
















45












$begingroup$


The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 .



It has been claimed that state-of-the-art imaging algorithms were an enabler for this historic success. Does anybody care to describe the difficulties, and (quite certainly non-trivial) mathematics that went into this effort ?










share|cite|improve this question











$endgroup$








  • 7




    $begingroup$
    For reference here is the paper where the collaboration described data processing: iopscience.iop.org/article/10.3847/2041-8213/ab0c57/meta
    $endgroup$
    – Neal
    2 days ago










  • $begingroup$
    Katie Bouman's work might shed some light here!
    $endgroup$
    – yousuf soliman
    2 days ago






  • 3




    $begingroup$
    But, @yousuf, will that light be able to escape from the black hole?
    $endgroup$
    – Gerry Myerson
    yesterday






  • 5




    $begingroup$
    The actual image reconstruction is described in paper IV: iopscience.iop.org/article/10.3847/2041-8213/ab0e85. In a nutshell, they formulated a baker's dozen of data misfit (for all the different acquired data as described in paper III) and regularization (including total variation and sparsity) functionals, and then minimized their weighted sum using L-BFGS. Not quite state-of-the-art from a mathematical point of view (especially using BFGS for a non-differentiable functional), but close enough.
    $endgroup$
    – Christian Clason
    yesterday








  • 4




    $begingroup$
    Great post and answer. For those also interested on the physics side of the discussions, here's a recent post: physics.stackexchange.com/questions/471972/…
    $endgroup$
    – user929304
    yesterday














45












45








45


5



$begingroup$


The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 .



It has been claimed that state-of-the-art imaging algorithms were an enabler for this historic success. Does anybody care to describe the difficulties, and (quite certainly non-trivial) mathematics that went into this effort ?










share|cite|improve this question











$endgroup$




The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 .



It has been claimed that state-of-the-art imaging algorithms were an enabler for this historic success. Does anybody care to describe the difficulties, and (quite certainly non-trivial) mathematics that went into this effort ?







signal-analysis inverse-problems machine-learning image-processing popularization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









Rodrigo de Azevedo

1,8422820




1,8422820










asked 2 days ago









Piyush GroverPiyush Grover

1,63511328




1,63511328








  • 7




    $begingroup$
    For reference here is the paper where the collaboration described data processing: iopscience.iop.org/article/10.3847/2041-8213/ab0c57/meta
    $endgroup$
    – Neal
    2 days ago










  • $begingroup$
    Katie Bouman's work might shed some light here!
    $endgroup$
    – yousuf soliman
    2 days ago






  • 3




    $begingroup$
    But, @yousuf, will that light be able to escape from the black hole?
    $endgroup$
    – Gerry Myerson
    yesterday






  • 5




    $begingroup$
    The actual image reconstruction is described in paper IV: iopscience.iop.org/article/10.3847/2041-8213/ab0e85. In a nutshell, they formulated a baker's dozen of data misfit (for all the different acquired data as described in paper III) and regularization (including total variation and sparsity) functionals, and then minimized their weighted sum using L-BFGS. Not quite state-of-the-art from a mathematical point of view (especially using BFGS for a non-differentiable functional), but close enough.
    $endgroup$
    – Christian Clason
    yesterday








  • 4




    $begingroup$
    Great post and answer. For those also interested on the physics side of the discussions, here's a recent post: physics.stackexchange.com/questions/471972/…
    $endgroup$
    – user929304
    yesterday














  • 7




    $begingroup$
    For reference here is the paper where the collaboration described data processing: iopscience.iop.org/article/10.3847/2041-8213/ab0c57/meta
    $endgroup$
    – Neal
    2 days ago










  • $begingroup$
    Katie Bouman's work might shed some light here!
    $endgroup$
    – yousuf soliman
    2 days ago






  • 3




    $begingroup$
    But, @yousuf, will that light be able to escape from the black hole?
    $endgroup$
    – Gerry Myerson
    yesterday






  • 5




    $begingroup$
    The actual image reconstruction is described in paper IV: iopscience.iop.org/article/10.3847/2041-8213/ab0e85. In a nutshell, they formulated a baker's dozen of data misfit (for all the different acquired data as described in paper III) and regularization (including total variation and sparsity) functionals, and then minimized their weighted sum using L-BFGS. Not quite state-of-the-art from a mathematical point of view (especially using BFGS for a non-differentiable functional), but close enough.
    $endgroup$
    – Christian Clason
    yesterday








  • 4




    $begingroup$
    Great post and answer. For those also interested on the physics side of the discussions, here's a recent post: physics.stackexchange.com/questions/471972/…
    $endgroup$
    – user929304
    yesterday








7




7




$begingroup$
For reference here is the paper where the collaboration described data processing: iopscience.iop.org/article/10.3847/2041-8213/ab0c57/meta
$endgroup$
– Neal
2 days ago




$begingroup$
For reference here is the paper where the collaboration described data processing: iopscience.iop.org/article/10.3847/2041-8213/ab0c57/meta
$endgroup$
– Neal
2 days ago












$begingroup$
Katie Bouman's work might shed some light here!
$endgroup$
– yousuf soliman
2 days ago




$begingroup$
Katie Bouman's work might shed some light here!
$endgroup$
– yousuf soliman
2 days ago




3




3




$begingroup$
But, @yousuf, will that light be able to escape from the black hole?
$endgroup$
– Gerry Myerson
yesterday




$begingroup$
But, @yousuf, will that light be able to escape from the black hole?
$endgroup$
– Gerry Myerson
yesterday




5




5




$begingroup$
The actual image reconstruction is described in paper IV: iopscience.iop.org/article/10.3847/2041-8213/ab0e85. In a nutshell, they formulated a baker's dozen of data misfit (for all the different acquired data as described in paper III) and regularization (including total variation and sparsity) functionals, and then minimized their weighted sum using L-BFGS. Not quite state-of-the-art from a mathematical point of view (especially using BFGS for a non-differentiable functional), but close enough.
$endgroup$
– Christian Clason
yesterday






$begingroup$
The actual image reconstruction is described in paper IV: iopscience.iop.org/article/10.3847/2041-8213/ab0e85. In a nutshell, they formulated a baker's dozen of data misfit (for all the different acquired data as described in paper III) and regularization (including total variation and sparsity) functionals, and then minimized their weighted sum using L-BFGS. Not quite state-of-the-art from a mathematical point of view (especially using BFGS for a non-differentiable functional), but close enough.
$endgroup$
– Christian Clason
yesterday






4




4




$begingroup$
Great post and answer. For those also interested on the physics side of the discussions, here's a recent post: physics.stackexchange.com/questions/471972/…
$endgroup$
– user929304
yesterday




$begingroup$
Great post and answer. For those also interested on the physics side of the discussions, here's a recent post: physics.stackexchange.com/questions/471972/…
$endgroup$
– user929304
yesterday










1 Answer
1






active

oldest

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35












$begingroup$

Essential elements of the reconstruction algorithm were developed at MIT under the name CHIRP = Continuous High-resolution Image Reconstruction using Patch priors, as described in Computational Imaging for VLBI Image Reconstruction (2015).



The difficulty of VLBI (Very Long Baseline Interferometry Image) reconstruction is that the inversion problem is highly-ill posed, there are many images that explain the data. The challenge is to find an explanation that respects our prior assumptions about the “visual” universe while still satisfying the observed data. Bayesian approaches are generally employed for that purpose, in CHIRP machine learning is used to automatically identify visual patterns --- obviating the need for hand training of the algorithm.



A key technical innovation is a way to correct for the delays in the signal received from the various telescopes. The delays are difficult to predict, since they depend the local variations in the speed of the radio waves through the noisy atmosphere. CHIRP adopts an algebraic solution known as phase closure to this problem: If the measurements from three telescopes are multiplied, the extra delays caused by atmospheric noise cancel each other out.



One test case that shows the resolving power of CHIRP, compared to a competing algorithm (BU) is shown below (taken from the MIT paper). Notice how CHIRP is able to resolve 2 separate, previously unresolved, bright emissions in the blazar OJ287.








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    35












    $begingroup$

    Essential elements of the reconstruction algorithm were developed at MIT under the name CHIRP = Continuous High-resolution Image Reconstruction using Patch priors, as described in Computational Imaging for VLBI Image Reconstruction (2015).



    The difficulty of VLBI (Very Long Baseline Interferometry Image) reconstruction is that the inversion problem is highly-ill posed, there are many images that explain the data. The challenge is to find an explanation that respects our prior assumptions about the “visual” universe while still satisfying the observed data. Bayesian approaches are generally employed for that purpose, in CHIRP machine learning is used to automatically identify visual patterns --- obviating the need for hand training of the algorithm.



    A key technical innovation is a way to correct for the delays in the signal received from the various telescopes. The delays are difficult to predict, since they depend the local variations in the speed of the radio waves through the noisy atmosphere. CHIRP adopts an algebraic solution known as phase closure to this problem: If the measurements from three telescopes are multiplied, the extra delays caused by atmospheric noise cancel each other out.



    One test case that shows the resolving power of CHIRP, compared to a competing algorithm (BU) is shown below (taken from the MIT paper). Notice how CHIRP is able to resolve 2 separate, previously unresolved, bright emissions in the blazar OJ287.








    share|cite|improve this answer











    $endgroup$


















      35












      $begingroup$

      Essential elements of the reconstruction algorithm were developed at MIT under the name CHIRP = Continuous High-resolution Image Reconstruction using Patch priors, as described in Computational Imaging for VLBI Image Reconstruction (2015).



      The difficulty of VLBI (Very Long Baseline Interferometry Image) reconstruction is that the inversion problem is highly-ill posed, there are many images that explain the data. The challenge is to find an explanation that respects our prior assumptions about the “visual” universe while still satisfying the observed data. Bayesian approaches are generally employed for that purpose, in CHIRP machine learning is used to automatically identify visual patterns --- obviating the need for hand training of the algorithm.



      A key technical innovation is a way to correct for the delays in the signal received from the various telescopes. The delays are difficult to predict, since they depend the local variations in the speed of the radio waves through the noisy atmosphere. CHIRP adopts an algebraic solution known as phase closure to this problem: If the measurements from three telescopes are multiplied, the extra delays caused by atmospheric noise cancel each other out.



      One test case that shows the resolving power of CHIRP, compared to a competing algorithm (BU) is shown below (taken from the MIT paper). Notice how CHIRP is able to resolve 2 separate, previously unresolved, bright emissions in the blazar OJ287.








      share|cite|improve this answer











      $endgroup$
















        35












        35








        35





        $begingroup$

        Essential elements of the reconstruction algorithm were developed at MIT under the name CHIRP = Continuous High-resolution Image Reconstruction using Patch priors, as described in Computational Imaging for VLBI Image Reconstruction (2015).



        The difficulty of VLBI (Very Long Baseline Interferometry Image) reconstruction is that the inversion problem is highly-ill posed, there are many images that explain the data. The challenge is to find an explanation that respects our prior assumptions about the “visual” universe while still satisfying the observed data. Bayesian approaches are generally employed for that purpose, in CHIRP machine learning is used to automatically identify visual patterns --- obviating the need for hand training of the algorithm.



        A key technical innovation is a way to correct for the delays in the signal received from the various telescopes. The delays are difficult to predict, since they depend the local variations in the speed of the radio waves through the noisy atmosphere. CHIRP adopts an algebraic solution known as phase closure to this problem: If the measurements from three telescopes are multiplied, the extra delays caused by atmospheric noise cancel each other out.



        One test case that shows the resolving power of CHIRP, compared to a competing algorithm (BU) is shown below (taken from the MIT paper). Notice how CHIRP is able to resolve 2 separate, previously unresolved, bright emissions in the blazar OJ287.








        share|cite|improve this answer











        $endgroup$



        Essential elements of the reconstruction algorithm were developed at MIT under the name CHIRP = Continuous High-resolution Image Reconstruction using Patch priors, as described in Computational Imaging for VLBI Image Reconstruction (2015).



        The difficulty of VLBI (Very Long Baseline Interferometry Image) reconstruction is that the inversion problem is highly-ill posed, there are many images that explain the data. The challenge is to find an explanation that respects our prior assumptions about the “visual” universe while still satisfying the observed data. Bayesian approaches are generally employed for that purpose, in CHIRP machine learning is used to automatically identify visual patterns --- obviating the need for hand training of the algorithm.



        A key technical innovation is a way to correct for the delays in the signal received from the various telescopes. The delays are difficult to predict, since they depend the local variations in the speed of the radio waves through the noisy atmosphere. CHIRP adopts an algebraic solution known as phase closure to this problem: If the measurements from three telescopes are multiplied, the extra delays caused by atmospheric noise cancel each other out.



        One test case that shows the resolving power of CHIRP, compared to a competing algorithm (BU) is shown below (taken from the MIT paper). Notice how CHIRP is able to resolve 2 separate, previously unresolved, bright emissions in the blazar OJ287.









        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited yesterday

























        answered yesterday









        Carlo BeenakkerCarlo Beenakker

        80.4k9193295




        80.4k9193295






























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