anchor.diff:  If stem:[\mathcal{T}] maps images to images, such as the convolution above, 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 stem:[g] to stem:[f]. Simply give the neural network stem:[g] as input and train it to find stem:[f].
   
   But what happens when stem:[\mathcal{T}] is more complicated? 
  -We'll focus on CT, where the forward model is given by the https://en.wikipedia.org/wiki/Radon_transform[Radon transform]:
  +We'll focus on CT, where the forward model is given by the https://en.wikipedia.org/wiki/Radon_transform[Radon transform] which computes all line integrals of the image:
   
   [stem]
   +++++++++++++++++
  
positive.code: 
  If we compare this to the deblurring problem, we see that while the blurring problem has a local relationship between parameters and data, the radon transform has a global relationship. Here two widely separated points can both influence a point in data as long as they lie on the same line. And since the forward model is global, so is the inverse problem. This means that classical local machine-learning approaches based on convolutions will not work.
  
  One way to solve this would be to use fully-connected layers instead of convolutions, but this quickly becomes infeasible. For example, in the example above the data has size stem:[1000 \times 1000 = 10^6] and the signal stem:[512 \times 512 = 2.6 \cdot 10^5]. A single fully connected layer between these would require a staggering stem:[10^6 \cdot 2.6 \cdot 10^5 = 2.6 \cdot 10^{11}] weights. Storing these in single precision would require one terabyte of data. For a single layer. 
  
  This is obviously not a workable solution.
  
  
negative.code: 
  image:compare_results_anotated.png[compare_results_anotated, height=700, width=800]
  
  The quantiative results are also quite cool 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 number of parameters used for the denoiser.
  
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