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@benchmark fobj($pinit)

# Now we benchmark the gradient
@benchmark gfobj($pinit)</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Chael A, et al. <em>Inteferometric Imaging Directly with Closure Phases</em> 2018 ApJ 857 1 arXiv:1803/07088</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../vlbi_imaging_problem/">Introduction to the VLBI Imaging Problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 24 August 2023 13:10">Thursday 24 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
@benchmark gfobj($pinit)</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Chael A, et al. <em>Inteferometric Imaging Directly with Closure Phases</em> 2018 ApJ 857 1 arXiv:1803/07088</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../vlbi_imaging_problem/">Introduction to the VLBI Imaging Problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 31 August 2023 03:02">Thursday 31 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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\begin{pmatrix}
\tilde{I} + \tilde{Q} &amp; \tilde{U} + i\tilde{V}\\
\tilde{U} - i\tilde{V} &amp; \tilde{I} - \tilde{Q}
\end{pmatrix}.\]</p><p>where e.g., <span>$\left&lt;XY^*\right&gt; = 2\left&lt;v_{pX}v^*_{pY}\right&gt;$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../vlbi_imaging_problem/">« Introduction to the VLBI Imaging Problem</a><a class="docs-footer-nextpage" href="../examples/data/">Loading Data into Comrade »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 24 August 2023 13:10">Thursday 24 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
\end{pmatrix}.\]</p><p>where e.g., <span>$\left&lt;XY^*\right&gt; = 2\left&lt;v_{pX}v^*_{pY}\right&gt;$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../vlbi_imaging_problem/">« Introduction to the VLBI Imaging Problem</a><a class="docs-footer-nextpage" href="../examples/data/">Loading Data into Comrade »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 31 August 2023 03:02">Thursday 31 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Stokes I Simultaneous Image and Instrument Modeling · Comrade.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">Comrade.jl</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../benchmarks/">Benchmarks</a></li><li><a class="tocitem" href="../../vlbi_imaging_problem/">Introduction to the VLBI Imaging Problem</a></li><li><a class="tocitem" href="../../conventions/">Conventions</a></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="../data/">Loading Data into Comrade</a></li><li><a class="tocitem" href="../geometric_modeling/">Geometric Modeling of EHT Data</a></li><li><a class="tocitem" href="../imaging_closures/">Imaging a Black Hole using only Closure Quantities</a></li><li class="is-active"><a class="tocitem" href>Stokes I Simultaneous Image and Instrument Modeling</a><ul class="internal"><li><a class="tocitem" href="#Load-the-Data"><span>Load the Data</span></a></li><li><a class="tocitem" href="#Reconstructing-the-Image-and-Instrument-Effects"><span>Reconstructing the Image and Instrument Effects</span></a></li></ul></li><li><a class="tocitem" href="../imaging_pol/">Polarized Image and Instrumental Modeling</a></li><li><a class="tocitem" href="../hybrid_imaging/">Hybrid Imaging of a Black Hole</a></li></ul></li><li><span class="tocitem">Libraries</span><ul><li><a class="tocitem" href="../../libs/optimization/">ComradeOptimization</a></li><li><a class="tocitem" href="../../libs/ahmc/">ComradeAHMC</a></li><li><a class="tocitem" href="../../libs/nested/">ComradeNested</a></li><li><a class="tocitem" href="../../libs/dynesty/">ComradeDynesty</a></li><li><a class="tocitem" href="../../libs/adaptmcmc/">ComradeAdaptMCMC</a></li></ul></li><li><a class="tocitem" href="../../base_api/">ComradeBase API</a></li><li><a class="tocitem" href="../../api/">Comrade API</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Stokes I Simultaneous Image and Instrument Modeling</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Stokes I Simultaneous Image and Instrument Modeling</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/ptiede/Comrade.jl/blob/main/examples/imaging_vis.jl#" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Stokes-I-Simultaneous-Image-and-Instrument-Modeling"><a class="docs-heading-anchor" href="#Stokes-I-Simultaneous-Image-and-Instrument-Modeling">Stokes I Simultaneous Image and Instrument Modeling</a><a id="Stokes-I-Simultaneous-Image-and-Instrument-Modeling-1"></a><a class="docs-heading-anchor-permalink" href="#Stokes-I-Simultaneous-Image-and-Instrument-Modeling" title="Permalink"></a></h1><p>In this tutorial, we will create a preliminary reconstruction of the 2017 M87 data on April 6 by simultaneously creating an image and model for the instrument. By instrument model, we mean something akin to self-calibration in traditional VLBI imaging terminology. However, unlike traditional self-cal, we will at each point in our parameter space effectively explore the possible self-cal solutions. This will allow us to constrain and marginalize over the instrument effects, such as time variable gains.</p><p>To get started we load Comrade.</p><pre><code class="language-julia hljs">using Comrade</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi"> Activating project at `~/work/Comrade.jl/Comrade.jl/examples`</code></pre><pre><code class="language-julia hljs">using Pyehtim
using LinearAlgebra</code></pre><p>For reproducibility we use a stable random number genreator</p><pre><code class="language-julia hljs">using StableRNGs
rng = StableRNG(42)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">StableRNGs.LehmerRNG(state=0x00000000000000000000000000000055)</code></pre><h2 id="Load-the-Data"><a class="docs-heading-anchor" href="#Load-the-Data">Load the Data</a><a id="Load-the-Data-1"></a><a class="docs-heading-anchor-permalink" href="#Load-the-Data" title="Permalink"></a></h2><p>To download the data visit https://doi.org/10.25739/g85n-f134 First we will load our data:</p><pre><code class="language-julia hljs">obs = ehtim.obsdata.load_uvfits(joinpath(dirname(pathof(Comrade)), &quot;..&quot;, &quot;examples&quot;, &quot;SR1_M87_2017_096_hi_hops_netcal_StokesI.uvfits&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Python: &lt;ehtim.obsdata.Obsdata object at 0x7ff4f7565780&gt;</code></pre><p>Now we do some minor preprocessing:</p><ul><li>Scan average the data since the data have been preprocessed so that the gain phases coherent.</li><li>Add 1% systematic noise to deal with calibration issues that cause 1% non-closing errors.</li></ul><pre><code class="language-julia hljs">obs = scan_average(obs.add_fractional_noise(0.01))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Python: &lt;ehtim.obsdata.Obsdata object at 0x7ff504b20c40&gt;</code></pre><p>Now we extract our complex visibilities.</p><pre><code class="language-julia hljs">dvis = extract_table(obs, ComplexVisibilities())</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">EHTObservation{Float64,Comrade.EHTVisibilityDatum{Float64}, ...}
rng = StableRNG(42)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">StableRNGs.LehmerRNG(state=0x00000000000000000000000000000055)</code></pre><h2 id="Load-the-Data"><a class="docs-heading-anchor" href="#Load-the-Data">Load the Data</a><a id="Load-the-Data-1"></a><a class="docs-heading-anchor-permalink" href="#Load-the-Data" title="Permalink"></a></h2><p>To download the data visit https://doi.org/10.25739/g85n-f134 First we will load our data:</p><pre><code class="language-julia hljs">obs = ehtim.obsdata.load_uvfits(joinpath(dirname(pathof(Comrade)), &quot;..&quot;, &quot;examples&quot;, &quot;SR1_M87_2017_096_hi_hops_netcal_StokesI.uvfits&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Python: &lt;ehtim.obsdata.Obsdata object at 0x7fcbf9936e60&gt;</code></pre><p>Now we do some minor preprocessing:</p><ul><li>Scan average the data since the data have been preprocessed so that the gain phases coherent.</li><li>Add 1% systematic noise to deal with calibration issues that cause 1% non-closing errors.</li></ul><pre><code class="language-julia hljs">obs = scan_average(obs.add_fractional_noise(0.01))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Python: &lt;ehtim.obsdata.Obsdata object at 0x7fcc03069180&gt;</code></pre><p>Now we extract our complex visibilities.</p><pre><code class="language-julia hljs">dvis = extract_table(obs, ComplexVisibilities())</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">EHTObservation{Float64,Comrade.EHTVisibilityDatum{Float64}, ...}
source: M87
mjd: 57849
frequency: 2.29070703125e11
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for s in sample(chain, 10)
residual!(p, vlbimodel(post, s), dvis)
end
p</code></pre><p>And viola, you have just finished making a preliminary image and instrument model reconstruction. In reality, you should run the <code>sample</code> step for many more MCMC steps to get a reliable estimate for the reconstructed image and instrument model parameters.</p><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../imaging_closures/">« Imaging a Black Hole using only Closure Quantities</a><a class="docs-footer-nextpage" href="../imaging_pol/">Polarized Image and Instrumental Modeling »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 24 August 2023 13:10">Thursday 24 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
p</code></pre><p>And viola, you have just finished making a preliminary image and instrument model reconstruction. In reality, you should run the <code>sample</code> step for many more MCMC steps to get a reliable estimate for the reconstructed image and instrument model parameters.</p><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../imaging_closures/">« Imaging a Black Hole using only Closure Quantities</a><a class="docs-footer-nextpage" href="../imaging_pol/">Polarized Image and Instrumental Modeling »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 31 August 2023 03:02">Thursday 31 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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