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Expand Up @@ -60,11 +60,10 @@
"which is a linear system $Hx = b$ that needs to be solved for $x$. This is done using the conjugate gradient method.\n",
"\n",
"```{note}\n",
"In Pruessmann, K.P., et al. MRM 2001 (https://doi.org/10.1002/mrm.1241) the k-space density is used to reweight the\n",
"loss to achieve faster convergence. This increases reconstruction error, see Ong F., Uecker M., Lustig M. TMI 2020\n",
"(https://doi.org/10.1109/TMI.2019.2954121). We follow a recommendation by Fessler and Noll\n",
"(https://ece-classes.usc.edu/ee591/library/Fessler-Iterative%20Reconstruction.pdf) and use the DCF to obtain a good\n",
"starting point.\n",
"In [Pruessmann et al. 2001](https://doi.org/10.1002/mrm.1241) the k-space density is used to reweight the\n",
"loss to achieve faster convergence. This increases reconstruction error, see [Ong F., Uecker M., Lustig M. 2020](https://doi.org/10.1109/TMI.2019.2954121).\n",
"We follow a recommendation by [Fessler and Noll](https://ece-classes.usc.edu/ee591/library/Fessler-Iterative%20Reconstruction.pdf)\n",
"and use the DCF to obtain a good starting point.\n",
"```"
]
},
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Expand Up @@ -95,19 +95,18 @@
"One important question of course is, what to use as $x_\\mathrm{{reg}}$ and $B$. For dynamic images (e.g. cine MRI)\n",
"low-resolution dynamic images or high-quality static images have been proposed.\n",
"In recent years, the output of neural networks has also been used, i.e. $x_{\\mathrm{reg}} = u_{\\theta}(x_0)$\n",
"$B=\\mathrm{Id}$ for a pre-trained network $u_{\\theta}$ and initial image $x_0$ [Kofler et al., IOP PMB 2020].\n",
"$B=\\mathrm{Id}$ for a pre-trained network $u_{\\theta}$ and initial image $x_0$ [[Kofler et al. 2020](https://doi.org/10.1088/1361-6560/ab990e)].\n",
"\n",
"In this example we are going to use a high-quality image to regularize the reconstruction of an undersampled image.\n",
"Both images are obtained from the same data acquisition - one using all the acquired data ($x_{\\mathrm{reg}}$),\n",
"and one using only parts of it ($x$). This is, of course, an unrealistic case but it will allow us to demonstrate\n",
"the effect of the regularization.\n",
"\n",
"```{note}\n",
"In Pruessmann, K.P., et al. MRM 2001 (https://doi.org/10.1002/mrm.1241) the k-space density is used to reweight the\n",
"loss to achieve faster convergence. This increases reconstruction error, see Ong F., Uecker M., Lustig M. TMI 2020\n",
"(https://doi.org/10.1109/TMI.2019.2954121). We follow a recommendation by Fessler and Noll\n",
"(https://ece-classes.usc.edu/ee591/library/Fessler-Iterative%20Reconstruction.pdf) and use the DCF to obtain a good\n",
"starting point.\n",
"In [Pruessmann et al. 2001](https://doi.org/10.1002/mrm.1241) the k-space density is used to reweight the\n",
"loss to achieve faster convergence. This increases reconstruction error, see [Ong F., Uecker M., Lustig M. 2020](https://doi.org/10.1109/TMI.2019.2954121).\n",
"We follow a recommendation by [Fessler and Noll](https://ece-classes.usc.edu/ee591/library/Fessler-Iterative%20Reconstruction.pdf)\n",
"and use the DCF to obtain a good starting point.\n",
"```"
]
},
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