# A Systematic Literature Review on Applications of GAN-Synthesized Images for Brain MRI
## 3 Results
### Loss Functions (RQ 2)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Loss Function** & **Description** & **Probability—** **Based (Yes/No)** & **Ref. No.** \\ \hline \multicolumn{4}{c}{Commonly used loss functions} \\ \multicolumn{4}{c}{The adversarial loss function is created in the repeated} \\ \multicolumn{4}{c}{production and classification cycle. The generator minimizes} \\ \multicolumn{4}{c}{the loss function, and the discriminator maximizes it.} \\ \multicolumn{4}{c}{\(\mathcal{L}_{GAN}\) (G,D) = \(E_{x,y}\) [log (D(x,y)] + \(E_{x,y}\) [log(1 - D(x,G(x,z))]\) ]} \\ \multicolumn{4}{c}{where y is the ground truth image, G is the generator} \\ \multicolumn{4}{c}{network, G (\(x,z\)) is generated image, D is the} \\ \multicolumn{4}{c}{discriminator network.} \\ \multicolumn{4}{c}{A cycle consistency loss allows the generator to learn a} \\ \multicolumn{4}{c}{one-to-one mapping from the input image field to the target} \\ \multicolumn{4}{c}{image field. \(\mathcal{L}_{cycle}^{\prime}(\mathrm{G})=E_{x-p_{data}(x)}\) [\(\|\mathrm{G}(\mathrm{G}(x))-x\|_{1}\|\) ]} \\ \multicolumn{4}{c}{The L1 loss also called mean absolute error (MAE), is a} \\ \multicolumn{4}{c}{pixel-wise error that shows over-smoothing in resultant} \\ \multicolumn{4}{c}{L1 loss} \\ \multicolumn{4}{c}{images. \(\mathcal{L}_{L1}(\mathrm{G})=\frac{1}{4}\left[\|\mathrm{y}-\mathrm{G}(xx)\|_{1}\right]\) where \(n\) is the number} \\ \multicolumn{4}{c}{of voxels in an image, \(1\|.1\|_{1}\) is the sum of} \\ \multicolumn{4}{c}{ voxel-wise residuals} \\ \multicolumn{4}{c}{The L2 loss also called Mean Squared Distance (MSD)} \\ \multicolumn{4}{c}{indicates the error between generated and original images} \\ \multicolumn{4}{c}{\(\mathcal{L}_{L2}(\mathrm{G})=\frac{1}{4}\left[\|\mathrm{y}-\mathrm{G}(xz)\|_{2}\right]\) where} \\ \multicolumn{4}{c}{\(\|\mathrm{l}\|_{1}\) is the sum of squared voxel-wise residuals} \\ \multicolumn{4}{c}{intistyle value.} \\ \multicolumn{4}{c}{Pixel-reconstruction losses give blurry effects in the final} \\ \multicolumn{4}{c}{outputs and cannot express the image’s perceptual quality.} \\ \multicolumn{4}{c}{The perceptual loss is the Euclidean distance in feature space} \\ \multicolumn{4}{c}{to extract semantic features from target images.} \\ \multicolumn{4}{c}{\(\mathcal{L}_{ Perceptual}(\mathrm{G})=\frac{1}{\mathrm{grid}}\left[\|\mathcal{O}(\mathrm{G}(x))- \mathcal{O}(y)\|_{2}^{2}\right]\) where \(\mathcal{O}\) is a} \\ \multicolumn{4}{c}{feature extractor, and w, h, and represent the dimensions of} \\ \multicolumn{4}{c}{frame maps.} \\ \multicolumn{4}{c}{WGAN evaluates the Earth Mover's distance by training the} \\ \multicolumn{4}{c}{discriminator network and is bounded by a} \\ \multicolumn{4}{c}{Lipschitz constraint.} \\ \multicolumn{4}{c}{Wasserstein loss} \\ \multicolumn{4}{c}{\(\mathcal{L}_{WGAN}(\mathrm{D})=-E_{x-p_{r}}[\mathrm{D}(\mathrm{y})]+E_{x-p_{r}}[ \mathrm{D}(\mathrm{G}(\mathrm{x}))]+\lambda E_{x-p_{r}}[(\|\mathrm{Y}_{x}\, \mathcal{D}(\mathrm{g})\|_{2}-1)^{2}]\) where \(x\) is sampled from real} \\ \multicolumn{4}{c}{image ’\(r\)’ and noise ’\(n\)’ is the hyperparameter.} \\ \multicolumn{4}{c}{Other loss functions} \\ \multicolumn{4}{c}{Attention regularization loss} \\ \multicolumn{4}{c}{Binary cross entropy (BCE) loss} \\ \multicolumn{4}{c}{Binary cross entropy (BCE) loss} \\ \multicolumn{4}{c}{It is the average cross-entropy value and the discriminator’s} \\ \multicolumn{4}{c}{logistic sigmoid result.} \\ \multicolumn{4}{c}{Cycle-perceptual loss} \\ \multicolumn{4}{c}{This loss captures the high-level perceptual errors between} \\ \multicolumn{4}{c}{original and dummy images.} \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Summary of loss functions used in applications of GAN-synthesized brain MRI.