Table 3 Selected loss functions used for digital staining (DS) with featured references, not including adversarial losses. O(x, y) represents the output image, \(\hat{O}(x,y)\) represents the target image, \(\mu\) represents the average value, \(\sigma\) represents the standard deviation, \(c_1\) and \(c_2\) are stabilization constants used to prevent division by weak denominator, respectively

Mean Absolute Error (MAE)

\(L_{MAE} = \sum _{x=1,y=1}^{X,Y}|O(x,y)-\hat{O}(x,y)|\)

[34, 71, 82, 86]

Mean Squared Error (MSE)

\(L_{MSE} = \sum _{x=1,y=1}^{X,Y}(O(x,y)-\hat{O}(x,y))^2\)

[20, 22, 46]

Cross-entropy

\(L_{CE} = \sum _{x=1,y=1}^{X,Y}\hat{O}(x,y)log(O(x,y))\)

[21, 39, 87, 91]

Structural Similarity Index Measure (SSIM)

\(L_{SSIM} = \frac{(2\mu _o\mu _{\hat{o}}+c_1)(2\sigma _o\sigma _{\hat{o}}+c_2)}{(\mu _o^2 + \mu _{\hat{o}}^2+c_1)(\sigma _o^2 + \sigma _{\hat{o}}^2+c_2)}\)

[29, 59]