anchor.diff:  
   == Machine learning for inverse problems
   
  -If the forward model maps images to images, such as deblurring, finding the unknown parameters from data using machine learning is at least on a conceptual level straightforward, and any standard convolutional architecture could be used to learn a transformation from data to parameters. Simply use known pairs stem:[(g, f_{\text{true}})] and train a neural network to map stem:[g] to stem:[f_{\text{true}}].
  +If the forward model is local, such as in deblurring where each datapoint is affected only by a small neighborhood of the parameters, finding the unknown parameters from data using machine learning is at least on a conceptual level straightforward. Simply use any standard convolutional neural network and known pairs stem:[(g, f_{\text{true}})] and train the neural network to map stem:[g] to stem:[f_{\text{true}}].
   
   But what happens when stem:[\mathcal{T}] is more complicated? 
   In Computed Tomography, the parameters stem:[f] is an image of the interior of the patient and the forward model is given by the https://en.wikipedia.org/wiki/Radon_transform[Radon transform] which computes all line integrals of the image:
  
positive.code: 
  == Introduction
  
  These problems are called the deblurring and radon inversion problems respectively, and are both examples of _inverse problems_. In inverse problem we seek to find (reconstruct) some parameters given indirect observations. Mathematically this can be stated as finding some parameters stem:[f] that are connected to measured data stem:[g] through the _forward model_ stem:[\mathcal{T}]:
  
  [stem]
  +++++++++++++++++
  
negative.code: 
  == Results
  
  If you ask me, the results look quite good and I especially appreciate that the method is able to avoid some artifacts (some examples shown with red arrows) that the denoiser just wasn't able to remove:
  
  image:compare_results_anotated.png[compare_results_anotated, height=700, width=800]
  
  The quantitative results are also quite good and we outperform learned denoising w.r.t both Peak Signal to Noise Ratio https://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio[(PSNR)] and structural similarity index https://en.wikipedia.org/wiki/Structural_similarity[(SSIM)]. The runtime is not too shabby either and we manage to do all of this using only stem:[2\%] of the trainable parameters used for the denoiser.
  
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  |======================
  