## Formulation

$$y_{g t}^{b}=\operatorname{pool}\left(1-y_{g t}, \theta_{0}\right)-\left(1-y_{g t}\right), y_{p d}^{b}=\operatorname{pool}\left(1-y_{p d}, \theta_{0}\right)-\left(1-y_{p d}\right)$$

$$y_{g t}^{b, e x t}=\operatorname{pool}\left(y_{g t}^{b}, \theta\right), y_{p d}^{b, e x t}=\operatorname{pool}\left(y_{p d}^{b}, \theta\right)$$

$$P^{c}=\frac{\operatorname{sum}\left(y_{p d}^{b} \circ y_{g t}^{b, e x t}\right)}{\operatorname{sum}\left(y_{p d}^{b}\right)}, R^{c}=\frac{\operatorname{sum}\left(y_{g t}^{b} \circ y_{p d}^{b, e x t}\right)}{\operatorname{sum}\left(y_{g t}^{b}\right)}$$

$$B F_{1}^{c}=\frac{2 P^{c} R^{c}}{P^{c}+R^{c}}, L_{B F_{1}^{c}}=1-B F_{1}^{c}$$

## Hyperparameters

Acccording to the paper [1], “Hyperparameter $\theta_0$ must be as small as possible to extract vicious boundary; usually we set $\theta_0=3$”.

And, “The value of hyperparameter $\theta$ can be determined as not greater than the minimum distance between neighboring segments of the binary ground truth map”.

## Results

The loss calculation of a class with $\theta=\theta_0=9$; (b) ground truth segment (gt); (c) predicted segment (pred); (d) boundary of gt; (e) boundary of pred; (f) expanded boundary of gt; (g) expanded boundary of pred; (h) pixel-wise multiplication of masks (d) and (g); (i) pixel-wise multiplication of masks (e) and (f)

The boundary F1 score of the same label and prediction in the figure with different combinations of the hyperparameters:

$\theta=3$ $\theta=5$ $\theta=7$ $\theta=9$ $\theta=11$ $\theta=13$ $\theta=15$ $\theta=17$ $\theta=19$
$\theta_0=3$ 0.06 0.07 0.12 0.20 0.29 0.47 0.60 0.80 0.88
$\theta_0=5$ 0.06 0.09 0.16 0.24 0.38 0.53 0.70 0.84 0.91
$\theta_0=7$ 0.08 0.12 0.20 0.32 0.45 0.62 0.76 0.87 0.94
$\theta_0=9$ 0.10 0.16 0.27 0.38 0.54 0.68 0.81 0.92 0.97
$\theta_0=11$ 0.14 0.22 0.33 0.46 0.61 0.76 0.88 0.96 0.98
$\theta_0=13$ 0.19 0.28 0.41 0.55 0.70 0.83 0.92 0.96 0.98
$\theta_0=15$ 0.25 0.37 0.50 0.65 0.79 0.89 0.94 0.97 0.98
$\theta_0=17$ 0.34 0.47 0.60 0.74 0.85 0.91 0.94 0.97 0.98
$\theta_0=19$ 0.44 0.57 0.71 0.82 0.88 0.92 0.95 0.97 0.98

The IoU score on this example is 0.18:

Heatmap of F1 scores same as the table Ground Truth and Prediction

### Medium Example

The IoU score on this example is 0.56:

Heatmap of F1 scores same as the table Ground Truth and Prediction

### Good Example

The IoU score on this example is 0.74:

Heatmap of F1 scores same as the table Ground Truth and Prediction

## Implementation

class BoundaryLoss(nn.Module):
"""Boundary Loss proposed in:
Alexey Bokhovkin et al., Boundary Loss for Remote Sensing Imagery Semantic Segmentation
https://arxiv.org/abs/1905.07852
"""

def __init__(self, theta0=5, theta=11):
super().__init__()

self.theta0 = theta0
self.theta = theta

def forward(self, pred, gt):
"""
Input:
- pred: the output from model (before softmax)
shape (N, C, H, W)
- gt: ground truth map
shape (N, H, w)
Return:
- boundary loss, averaged over mini-bathc
"""
n, c, _, _ = pred.shape

# softmax so that predicted map can be distributed in [0, 1]
pred = torch.softmax(pred, dim=1)

# one-hot vector of ground truth
one_hot_gt = one_hot(gt, c)

# boundary map
gt_b = F.max_pool2d(
1 - one_hot_gt, kernel_size=self.theta0, stride=1, padding=(self.theta0 - 1) // 2)
gt_b -= 1 - one_hot_gt

pred_b = F.max_pool2d(
1 - pred, kernel_size=self.theta0, stride=1, padding=(self.theta0 - 1) // 2)
pred_b -= 1 - pred

# extended boundary map
gt_b_ext = F.max_pool2d(
gt_b, kernel_size=self.theta, stride=1, padding=(self.theta - 1) // 2)

pred_b_ext = F.max_pool2d(
pred_b, kernel_size=self.theta, stride=1, padding=(self.theta - 1) // 2)

# reshape
gt_b = gt_b.view(n, c, -1)
pred_b = pred_b.view(n, c, -1)
gt_b_ext = gt_b_ext.view(n, c, -1)
pred_b_ext = pred_b_ext.view(n, c, -1)

# Precision, Recall
eps = 1e-7
P = (torch.sum(pred_b * gt_b_ext, dim=2) + eps) / (torch.sum(pred_b, dim=2) + eps)
R = (torch.sum(pred_b_ext * gt_b, dim=2) + eps) / (torch.sum(gt_b, dim=2) + eps)

# Boundary F1 Score
BF1 = (2 * P * R + eps) / (P + R + eps)

# summing BF1 Score for each class and average over mini-batch
loss = torch.mean(1-BF1)

return loss


## References

[1] Bokhovkin, A., & Burnaev, E. (2019, July). Boundary Loss for Remote Sensing Imagery Semantic Segmentation. In International Symposium on Neural Networks (pp. 388-401). Springer, Cham.