# kornia.losses¶

## Reconstruction¶

ssim(img1, img2, window_size, max_val=1.0, eps=1e-12)[source]

Function that computes the Structural Similarity (SSIM) index map between two images.

Measures the (SSIM) index between each element in the input x and target y.

The index can be described as:

$\text{SSIM}(x, y) = \frac{(2\mu_x\mu_y+c_1)(2\sigma_{xy}+c_2)} {(\mu_x^2+\mu_y^2+c_1)(\sigma_x^2+\sigma_y^2+c_2)}$
where:
• $$c_1=(k_1 L)^2$$ and $$c_2=(k_2 L)^2$$ are two variables to stabilize the division with weak denominator.

• $$L$$ is the dynamic range of the pixel-values (typically this is $$2^{\#\text{bits per pixel}}-1$$).

Parameters
• img1 (Tensor) – the first input image with shape $$(B, C, H, W)$$.

• img2 (Tensor) – the second input image with shape $$(B, C, H, W)$$.

• window_size (int) – the size of the gaussian kernel to smooth the images.

• max_val (float, optional) – the dynamic range of the images. Default: 1.0

• eps (float, optional) – Small value for numerically stability when dividing. Default: 1e-12

Return type

Tensor

Returns

The ssim index map with shape $$(B, C, H, W)$$.

Examples

>>> input1 = torch.rand(1, 4, 5, 5)
>>> input2 = torch.rand(1, 4, 5, 5)
>>> ssim_map = ssim(input1, input2, 5)  # 1x4x5x5

ssim_loss(img1, img2, window_size, max_val=1.0, eps=1e-12, reduction='mean')[source]

Function that computes a loss based on the SSIM measurement.

The loss, or the Structural dissimilarity (DSSIM) is described as:

$\text{loss}(x, y) = \frac{1 - \text{SSIM}(x, y)}{2}$

See ssim() for details about SSIM.

Parameters
• img1 (Tensor) – the first input image with shape $$(B, C, H, W)$$.

• img2 (Tensor) – the second input image with shape $$(B, C, H, W)$$.

• window_size (int) – the size of the gaussian kernel to smooth the images.

• max_val (float, optional) – the dynamic range of the images. Default: 1.0

• eps (float, optional) – Small value for numerically stability when dividing. Default: 1e-12

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'mean'

Return type

Tensor

Returns

The loss based on the ssim index.

Examples

>>> input1 = torch.rand(1, 4, 5, 5)
>>> input2 = torch.rand(1, 4, 5, 5)
>>> loss = ssim_loss(input1, input2, 5)

psnr(input, target, max_val)[source]

Creates a function that calculates the PSNR between 2 images.

PSNR is Peek Signal to Noise Ratio, which is similar to mean squared error. Given an m x n image, the PSNR is:

$\text{PSNR} = 10 \log_{10} \bigg(\frac{\text{MAX}_I^2}{MSE(I,T)}\bigg)$

where

$\text{MSE}(I,T) = \frac{1}{mn}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1} [I(i,j) - T(i,j)]^2$

and $$\text{MAX}_I$$ is the maximum possible input value (e.g for floating point images $$\text{MAX}_I=1$$).

Parameters
• input (Tensor) – the input image with arbitrary shape $$(*)$$.

• labels – the labels image with arbitrary shape $$(*)$$.

• max_val (float) – The maximum value in the input tensor.

Return type

Tensor

Returns

the computed loss as a scalar.

Examples

>>> ones = torch.ones(1)
>>> psnr(ones, 1.2 * ones, 2.) # 10 * log(4/((1.2-1)**2)) / log(10)
tensor(20.0000)

Reference:

https://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio#Definition

psnr_loss(input, target, max_val)[source]

Function that computes the PSNR loss.

The loss is computed as follows:

$\text{loss} = -\text{psnr(x, y)}$

See psnr() for details abut PSNR.

Parameters
Return type

Tensor

Returns

the computed loss as a scalar.

Examples

>>> ones = torch.ones(1)
>>> psnr_loss(ones, 1.2 * ones, 2.) # 10 * log(4/((1.2-1)**2)) / log(10)
tensor(-20.0000)

total_variation(img)[source]

Function that computes Total Variation according to [1].

Parameters

img (Tensor) – the input image with shape $$(N, C, H, W)$$ or $$(C, H, W)$$.

Return type

Tensor

Returns

a scalar with the computer loss.

Examples

>>> total_variation(torch.ones(3, 4, 4))
tensor(0.)


Note

See a working example here.

Reference:
inverse_depth_smoothness_loss(idepth, image)[source]

Criterion that computes image-aware inverse depth smoothness loss.

$\text{loss} = \left | \partial_x d_{ij} \right | e^{-\left \| \partial_x I_{ij} \right \|} + \left | \partial_y d_{ij} \right | e^{-\left \| \partial_y I_{ij} \right \|}$
Parameters
Return type

Tensor

Returns

a scalar with the computed loss.

Examples

>>> idepth = torch.rand(1, 1, 4, 5)
>>> image = torch.rand(1, 3, 4, 5)
>>> loss = inverse_depth_smoothness_loss(idepth, image)


## Semantic Segmentation¶

binary_focal_loss_with_logits(input, target, alpha=0.25, gamma=2.0, reduction='none', eps=1e-08)[source]

Function that computes Binary Focal loss.

$\text{FL}(p_t) = -\alpha_t (1 - p_t)^{\gamma} \, \text{log}(p_t)$
where:
• $$p_t$$ is the model’s estimated probability for each class.

Parameters
• input (Tensor) – input data tensor with shape $$(N, 1, *)$$.

• target (Tensor) – the target tensor with shape $$(N, 1, *)$$.

• alpha (float, optional) – Weighting factor for the rare class $$\alpha \in [0, 1]$$. Default: 0.25

• gamma (float, optional) – Focusing parameter $$\gamma >= 0$$. Default: 2.0

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'none'

• eps (float, optional) – for numerically stability when dividing. Default: 1e-08

Return type

Tensor

Returns

the computed loss.

Examples

>>> num_classes = 1
>>> kwargs = {"alpha": 0.25, "gamma": 2.0, "reduction": 'mean'}
>>> logits = torch.tensor([[[[6.325]]],[[[5.26]]],[[[87.49]]]])
>>> labels = torch.tensor([[[1.]],[[1.]],[[0.]]])
>>> binary_focal_loss_with_logits(logits, labels, **kwargs)
tensor(4.6052)

focal_loss(input, target, alpha, gamma=2.0, reduction='none', eps=1e-08)[source]

Criterion that computes Focal loss.

According to [LGG+18], the Focal loss is computed as follows:

$\text{FL}(p_t) = -\alpha_t (1 - p_t)^{\gamma} \, \text{log}(p_t)$
Where:
• $$p_t$$ is the model’s estimated probability for each class.

Parameters
• input (Tensor) – logits tensor with shape $$(N, C, *)$$ where C = number of classes.

• target (Tensor) – labels tensor with shape $$(N, *)$$ where each value is $$0 ≤ targets[i] ≤ C−1$$.

• alpha (float) – Weighting factor $$\alpha \in [0, 1]$$.

• gamma (float, optional) – Focusing parameter $$\gamma >= 0$$. Default: 2.0

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'none'

• eps (float, optional) – Scalar to enforce numerical stabiliy. Default: 1e-08

Return type

Tensor

Returns

the computed loss.

Example

>>> N = 5  # num_classes
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = focal_loss(input, target, alpha=0.5, gamma=2.0, reduction='mean')
>>> output.backward()

dice_loss(input, target, eps=1e-08)[source]

Criterion that computes Sørensen-Dice Coefficient loss.

According to [1], we compute the Sørensen-Dice Coefficient as follows:

$\text{Dice}(x, class) = \frac{2 |X| \cap |Y|}{|X| + |Y|}$
Where:
• $$X$$ expects to be the scores of each class.

• $$Y$$ expects to be the one-hot tensor with the class labels.

the loss, is finally computed as:

$\text{loss}(x, class) = 1 - \text{Dice}(x, class)$
Reference:
Parameters
• input (Tensor) – logits tensor with shape $$(N, C, H, W)$$ where C = number of classes.

• labels – labels tensor with shape $$(N, H, W)$$ where each value is $$0 ≤ targets[i] ≤ C−1$$.

• eps (float, optional) – Scalar to enforce numerical stabiliy. Default: 1e-08

Return type

Tensor

Returns

the computed loss.

Example

>>> N = 5  # num_classes
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = dice_loss(input, target)
>>> output.backward()

tversky_loss(input, target, alpha, beta, eps=1e-08)[source]

Criterion that computes Tversky Coefficient loss.

According to [SEG17], we compute the Tversky Coefficient as follows:

$\text{S}(P, G, \alpha; \beta) = \frac{|PG|}{|PG| + \alpha |P \setminus G| + \beta |G \setminus P|}$
Where:
• $$P$$ and $$G$$ are the predicted and ground truth binary labels.

• $$\alpha$$ and $$\beta$$ control the magnitude of the penalties for FPs and FNs, respectively.

Note

• $$\alpha = \beta = 0.5$$ => dice coeff

• $$\alpha = \beta = 1$$ => tanimoto coeff

• $$\alpha + \beta = 1$$ => F beta coeff

Parameters
Return type

Tensor

Returns

the computed loss.

Example

>>> N = 5  # num_classes
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = tversky_loss(input, target, alpha=0.5, beta=0.5)
>>> output.backward()


## Distributions¶

js_div_loss_2d(input, target, reduction='mean')[source]

Calculates the Jensen-Shannon divergence loss between heatmaps.

Parameters
• input (Tensor) – the input tensor with shape $$(B, N, H, W)$$.

• target (Tensor) – the target tensor with shape $$(B, N, H, W)$$.

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'mean'

Examples

>>> input = torch.full((1, 1, 2, 4), 0.125)
>>> loss = js_div_loss_2d(input, input)
>>> loss.item()
0.0

kl_div_loss_2d(input, target, reduction='mean')[source]

Calculates the Kullback-Leibler divergence loss between heatmaps.

Parameters
• input (Tensor) – the input tensor with shape $$(B, N, H, W)$$.

• target (Tensor) – the target tensor with shape $$(B, N, H, W)$$.

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'mean'

Examples

>>> input = torch.full((1, 1, 2, 4), 0.125)
>>> loss = js_div_loss_2d(input, input)
>>> loss.item()
0.0


## Module¶

class DiceLoss(eps=1e-08)[source]

Criterion that computes Sørensen-Dice Coefficient loss.

According to [1], we compute the Sørensen-Dice Coefficient as follows:

$\text{Dice}(x, class) = \frac{2 |X| \cap |Y|}{|X| + |Y|}$
Where:
• $$X$$ expects to be the scores of each class.

• $$Y$$ expects to be the one-hot tensor with the class labels.

the loss, is finally computed as:

$\text{loss}(x, class) = 1 - \text{Dice}(x, class)$
Reference:
Parameters

eps (float, optional) – Scalar to enforce numerical stabiliy. Default: 1e-08

Shape:
• Input: $$(N, C, H, W)$$ where C = number of classes.

• Target: $$(N, H, W)$$ where each value is $$0 ≤ targets[i] ≤ C−1$$.

Example

>>> N = 5  # num_classes
>>> criterion = DiceLoss()
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = criterion(input, target)
>>> output.backward()

class TverskyLoss(alpha, beta, eps=1e-08)[source]

Criterion that computes Tversky Coefficient loss.

According to [SEG17], we compute the Tversky Coefficient as follows:

$\text{S}(P, G, \alpha; \beta) = \frac{|PG|}{|PG| + \alpha |P \setminus G| + \beta |G \setminus P|}$
Where:
• $$P$$ and $$G$$ are the predicted and ground truth binary labels.

• $$\alpha$$ and $$\beta$$ control the magnitude of the penalties for FPs and FNs, respectively.

Note

• $$\alpha = \beta = 0.5$$ => dice coeff

• $$\alpha = \beta = 1$$ => tanimoto coeff

• $$\alpha + \beta = 1$$ => F beta coeff

Parameters
Shape:
• Input: $$(N, C, H, W)$$ where C = number of classes.

• Target: $$(N, H, W)$$ where each value is $$0 ≤ targets[i] ≤ C−1$$.

Examples

>>> N = 5  # num_classes
>>> criterion = TverskyLoss(alpha=0.5, beta=0.5)
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = criterion(input, target)
>>> output.backward()

class FocalLoss(alpha, gamma=2.0, reduction='none', eps=1e-08)[source]

Criterion that computes Focal loss.

According to [LGG+18], the Focal loss is computed as follows:

$\text{FL}(p_t) = -\alpha_t (1 - p_t)^{\gamma} \, \text{log}(p_t)$
Where:
• $$p_t$$ is the model’s estimated probability for each class.

Parameters
• alpha (float) – Weighting factor $$\alpha \in [0, 1]$$.

• gamma (float, optional) – Focusing parameter $$\gamma >= 0$$. Default: 2.0

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'none'

• eps (float, optional) – Scalar to enforce numerical stabiliy. Default: 1e-08

Shape:
• Input: $$(N, C, *)$$ where C = number of classes.

• Target: $$(N, *)$$ where each value is $$0 ≤ targets[i] ≤ C−1$$.

Example

>>> N = 5  # num_classes
>>> kwargs = {"alpha": 0.5, "gamma": 2.0, "reduction": 'mean'}
>>> criterion = FocalLoss(**kwargs)
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = criterion(input, target)
>>> output.backward()

class SSIM(window_size, max_val=1.0, eps=1e-12)[source]

Creates a module that computes the Structural Similarity (SSIM) index between two images.

Measures the (SSIM) index between each element in the input x and target y.

The index can be described as:

$\text{SSIM}(x, y) = \frac{(2\mu_x\mu_y+c_1)(2\sigma_{xy}+c_2)} {(\mu_x^2+\mu_y^2+c_1)(\sigma_x^2+\sigma_y^2+c_2)}$
where:
• $$c_1=(k_1 L)^2$$ and $$c_2=(k_2 L)^2$$ are two variables to stabilize the division with weak denominator.

• $$L$$ is the dynamic range of the pixel-values (typically this is $$2^{\#\text{bits per pixel}}-1$$).

Parameters
• window_size (int) – the size of the gaussian kernel to smooth the images.

• max_val (float, optional) – the dynamic range of the images. Default: 1.0

• eps (float, optional) – Small value for numerically stability when dividing. Default: 1e-12

Shape:
• Input: $$(B, C, H, W)$$.

• Target $$(B, C, H, W)$$.

• Output: $$(B, C, H, W)$$.

Examples

>>> input1 = torch.rand(1, 4, 5, 5)
>>> input2 = torch.rand(1, 4, 5, 5)
>>> ssim = SSIM(5)
>>> ssim_map = ssim(input1, input2)  # 1x4x5x5

class SSIMLoss(window_size, max_val=1.0, eps=1e-12, reduction='mean')[source]

Creates a criterion that computes a loss based on the SSIM measurement.

The loss, or the Structural dissimilarity (DSSIM) is described as:

$\text{loss}(x, y) = \frac{1 - \text{SSIM}(x, y)}{2}$

See ssim_loss() for details about SSIM.

Parameters
• window_size (int) – the size of the gaussian kernel to smooth the images.

• max_val (float, optional) – the dynamic range of the images. Default: 1.0

• eps (float, optional) – Small value for numerically stability when dividing. Default: 1e-12

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'mean'

Returns

The loss based on the ssim index.

Examples

>>> input1 = torch.rand(1, 4, 5, 5)
>>> input2 = torch.rand(1, 4, 5, 5)
>>> criterion = SSIMLoss(5)
>>> loss = criterion(input1, input2)

class InverseDepthSmoothnessLoss[source]

Criterion that computes image-aware inverse depth smoothness loss.

$\text{loss} = \left | \partial_x d_{ij} \right | e^{-\left \| \partial_x I_{ij} \right \|} + \left | \partial_y d_{ij} \right | e^{-\left \| \partial_y I_{ij} \right \|}$
Shape:
• Inverse Depth: $$(N, 1, H, W)$$

• Image: $$(N, 3, H, W)$$

• Output: scalar

Examples

>>> idepth = torch.rand(1, 1, 4, 5)
>>> image = torch.rand(1, 3, 4, 5)
>>> smooth = InverseDepthSmoothnessLoss()
>>> loss = smooth(idepth, image)

class TotalVariation[source]

Computes the Total Variation according to [1].

Shape:
• Input: $$(N, C, H, W)$$ or $$(C, H, W)$$.

• Output: $$(N,)$$ or scalar.

Examples

>>> tv = TotalVariation()
>>> output = tv(torch.ones((2, 3, 4, 4), requires_grad=True))
>>> output.data
tensor([0., 0.])
>>> output.sum().backward()  # grad can be implicitly created only for scalar outputs

Reference:
class PSNRLoss(max_val)[source]

Creates a criterion that calculates the PSNR loss.

The loss is computed as follows:

$\text{loss} = -\text{psnr(x, y)}$

See psnr() for details abut PSNR.

Parameters

max_val (float) – The maximum value in the input tensor.

Shape:
• Input: arbitrary dimensional tensor $$(*)$$.

• Target: arbitrary dimensional tensor $$(*)$$ same shape as input.

• Output: a scalar.

Examples

>>> ones = torch.ones(1)
>>> criterion = PSNRLoss(2.)
>>> criterion(ones, 1.2 * ones) # 10 * log(4/((1.2-1)**2)) / log(10)
tensor(-20.0000)

class BinaryFocalLossWithLogits(alpha, gamma=2.0, reduction='none')[source]

Criterion that computes Focal loss.

According to [LGG+18], the Focal loss is computed as follows:

$\text{FL}(p_t) = -\alpha_t (1 - p_t)^{\gamma} \, \text{log}(p_t)$
where:
• $$p_t$$ is the model’s estimated probability for each class.

Parameters
• alpha) – Weighting factor for the rare class $$\alpha \in [0, 1]$$.

• gamma (float, optional) – Focusing parameter $$\gamma >= 0$$. Default: 2.0

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied, 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'none'

Shape:
• Input: $$(N, 1, *)$$.

• Target: $$(N, 1, *)$$.

Examples

>>> N = 1  # num_classes
>>> kwargs = {"alpha": 0.25, "gamma": 2.0, "reduction": 'mean'}
>>> loss = BinaryFocalLossWithLogits(**kwargs)
>>> input = torch.randn(1, N, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(N)
>>> output = loss(input, target)
>>> output.backward()