## 2 Materials and Methods
### Proposed Loss Functions
#### 2.3.2 Dissimilarity from False Positive  
Among the predicted pixels, there are false positive pixels. While increasing the intersection with the ground truth, the model should also know about the false positive pixels to minimize the prediction of those pixels in the future. The relation of the number of the true positive pixels with the number of the false positive pixels is also inversely proportional. Reduction in the false positive pixels will help to focus or give information to increase the intersection of the ground truth and the predicted region, resulting in the improvement of the DSC.  
\[DSC\propto\frac{1}{{{{P}_{FP}}}} \tag{8}\]  
where \({{P}_{FP}}\) is the number of pixels of the false positive in the predicted image. Figure 6 shows the process of getting false positive pixels from the predicted output. To get false negative pixels, we first invert the ground truth image. To do so, we subtract each pixel of the ground truth image from a integer value one. Then the inverted image is multiplied by the predicted image. As both images (ground truth and inverted predicted output) are binary images, only the intersection between them will not be zero. This remaining non-zero area is the false positive pixel as this area is non-zero in the output image. From this pixel value, we can calculate a dissimilarity coefficient for false positive. The dissimilarity coefficient for false-positive pixels prediction is calculated as,  
\[DC_{FP}=\frac{{\left|{{inv}(I)\ \cap\ \hat{I}}\right|}}{{\left|{\hat{I}}\right|}} \tag{9}\]  
where \({\left|{{inv}\ (I)}\right|}\) is the number of pixels of the ROI from the inverted ground truth, and \({\left|{\hat{I}}\right|}\) is the number of pixels of the ROI from the predicted output image. \(DC_{FP}\) is a metric of area overlap between the inverted ground truth and the predicted image. \({\left|{{inv}\ (I)\ \cap\ \hat{I}}\right|}\) represents the number of pixels that are predicted as foreground but are not foreground in the ground truth image. Thus, we consider these pixels as a false positive. The process of calculating dissimilarity coefficient for false positive values is summarized in Algorithm 1. From Equation (9) we can say that the \(DC_{FP}\) is the ratio of number of false positive pixels, or false foreground pixels, to the number of total foreground pixels, or ROI predicted. This ratio tells us how many of the predicted ROI pixels do not match with the actual ground truth ROI pixels.  
```
Input: ground truth image \(I\), predicted image \(\hat{I}\) Output: \(DC_{FN}\), \(DC_{FP}\)
1. Inverse_\(\hat{I}\gets 1-\hat{I}\)
2. Intersection_FN \(\hat{I}\times\) inverse_\(\hat{I}\)
3. union_FN \(\hat{\leftarrow}\) sum (inverse_\(\hat{I}\))
4. \(DC_{FN}\gets 2*\) Intersection_FN/union_FN
5. Inverse_\(\hat{I}\gets 1-\hat{I}\)
6. Intersection_FP \(\hat{\leftarrow}\) inverse_\(\hat{I}*\hat{I}\)
7. union_FP \(\hat{\leftarrow}\) sum (\(\hat{I}\))
8. \(DC_{FP}\gets 2\times\) Intersection_FP/union_FP
9. return \(DC_{FP}\), \(DC_{FN}\)
```  
**Algorithm 1** Computation of dissimilarity coefficient for false negative \(DC_{FN}\) and false positive  
Figure 5: Binary maps of: (**a**) the ground truth, (**b**) the predicted output, (**c**) the inverted predicted output, and (**d**) the false negative area obtained by multiplying (**a**,**c**).