kornia.losses

Reconstruction

kornia.losses.ssim_loss(img1, img2, window_size, max_val=1.0, eps=1e-12, reduction='mean', padding='same')

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'

• padding (str, optional) – 'same' | 'valid'. Whether to only use the “valid” convolution area to compute SSIM to match the MATLAB implementation of original SSIM paper. Default: 'same'

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)

kornia.losses.ssim3d_loss(img1, img2, window_size, max_val=1.0, eps=1e-12, reduction='mean', padding='same')

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 \((B, C, D, H, W)\).

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

• 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'

• padding (str, optional) – 'same' | 'valid'. Whether to only use the “valid” convolution area to compute SSIM to match the MATLAB implementation of original SSIM paper. Default: 'same'

Return type:

Tensor

Returns:

The loss based on the ssim index.

Examples

>>> input1 = torch.rand(1, 4, 5, 5, 5)
>>> input2 = torch.rand(1, 4, 5, 5, 5)
>>> loss = ssim3d_loss(input1, input2, 5)

kornia.losses.psnr_loss(input, target, max_val)

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:

• input (Tensor) – the input image with shape \((*)\).

• labels – the labels image with 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_loss(ones, 1.2 * ones, 2.) # 10 * log(4/((1.2-1)**2)) / log(10)
tensor(-20.0000)

kornia.losses.total_variation(img, reduction='sum')

Function that computes Total Variation according to [1].

Parameters:

• img (Tensor) – the input image with shape \((*, H, W)\).

• reduction (str, optional) – Specifies the reduction to apply to the output: 'mean' | 'sum'. 'mean': the sum of the output will be divided by the number of elements in the output, 'sum': the output will be summed. Default: 'sum'

Return type:

Tensor

Returns:

a tensor with shape \((*,)\).

Examples

>>> total_variation(torch.ones(4, 4))
tensor(0.)
>>> total_variation(torch.ones(2, 5, 3, 4, 4)).shape
torch.Size([2, 5, 3])

Note

See a working example here. Total Variation is formulated with summation, however this is not resolution invariant. Thus, reduction=’mean’ was added as an optional reduction method.

Reference:

[1] https://en.wikipedia.org/wiki/Total_variation

kornia.losses.inverse_depth_smoothness_loss(idepth, image)

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:

• idepth (Tensor) – tensor with the inverse depth with shape \((N, 1, H, W)\).

• image (Tensor) – tensor with the input image with shape \((N, 3, H, W)\).

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)

kornia.losses.charbonnier_loss(img1, img2, reduction='none')

Criterion that computes the Charbonnier [2] (aka. L1-L2 [3]) loss.

According to [1], we compute the Charbonnier loss as follows:

\[\text{WL}(x, y) = \sqrt{(x - y)^{2} + 1} - 1\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] https://ieeexplore.ieee.org/document/413553 [3] https://hal.inria.fr/inria-00074015/document [4] https://arxiv.org/pdf/1712.05927.pdf

. note::

This implementation follows the formulation by Barron [1]. Other works utilize a slightly different implementation (see [4]).

Parameters:

• img1 (Tensor) – the predicted tensor with shape \((*)\).

• img2 (Tensor) – the target tensor with the same shape as img1.

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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'

Return type:

Tensor

Returns:

a scalar with the computed loss.

Example

>>> img1 = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 32)
>>> output = charbonnier_loss(img1, img2, reduction="sum")
>>> output.backward()

kornia.losses.welsch_loss(img1, img2, reduction='none')

Criterion that computes the Welsch [2] (aka. Leclerc [3]) loss.

According to [1], we compute the Welsch loss as follows:

\[\text{WL}(x, y) = 1 - exp(-\frac{1}{2} (x - y)^{2})\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] https://www.tandfonline.com/doi/abs/10.1080/03610917808812083 [3] https://link.springer.com/article/10.1007/BF00054839

Parameters:

• img1 (Tensor) – the predicted tensor with shape \((*)\).

• img2 (Tensor) – the target tensor with the same shape as img1.

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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'

Return type:

Tensor

Returns:

a scalar with the computed loss.

Example

>>> img1 = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 32)
>>> output = welsch_loss(img1, img2, reduction="mean")
>>> output.backward()

kornia.losses.cauchy_loss(img1, img2, reduction='none')

Criterion that computes the Cauchy [2] (aka. Lorentzian) loss.

According to [1], we compute the Cauchy loss as follows:

\[\text{WL}(x, y) = log(\frac{1}{2} (x - y)^{2} + 1)\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] https://files.is.tue.mpg.de/black/papers/cviu.63.1.1996.pdf

Parameters:

• img1 (Tensor) – the predicted tensor with shape \((*)\).

• img2 (Tensor) – the target tensor with the same shape as img1.

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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'

Return type:

Tensor

Returns:

a scalar with the computed loss.

Example

>>> img1 = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 32)
>>> output = cauchy_loss(img1, img2, reduction="mean")
>>> output.backward()

kornia.losses.geman_mcclure_loss(img1, img2, reduction='none')

Criterion that computes the Geman-McClure loss [2].

According to [1], we compute the Geman-McClure loss as follows:

\[\text{WL}(x, y) = \frac{2 (x - y)^{2}}{(x - y)^{2} + 4}\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] Bayesian image analysis: An application to single photon emission tomography, Geman and McClure, 1985

Parameters:

• img1 (Tensor) – the predicted tensor with shape \((*)\).

• img2 (Tensor) – the target tensor with the same shape as img1.

• reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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'

Return type:

Tensor

Returns:

a scalar with the computed loss.

Example

>>> img1 = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 32)
>>> output = geman_mcclure_loss(img1, img2, reduction="mean")
>>> output.backward()

class kornia.losses.SSIMLoss(window_size, max_val=1.0, eps=1e-12, reduction='mean', padding='same')

Create 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'

• padding (str, optional) – 'same' | 'valid'. Whether to only use the “valid” convolution area to compute SSIM to match the MATLAB implementation of original SSIM paper. Default: 'same'

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 kornia.losses.SSIM3DLoss(window_size, max_val=1.0, eps=1e-12, reduction='mean', padding='same')

Create 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'

• padding (str, optional) – 'same' | 'valid'. Whether to only use the “valid” convolution area to compute SSIM to match the MATLAB implementation of original SSIM paper. Default: 'same'

Returns:

The loss based on the ssim index.

Examples

>>> input1 = torch.rand(1, 4, 5, 5, 5)
>>> input2 = torch.rand(1, 4, 5, 5, 5)
>>> criterion = SSIM3DLoss(5)
>>> loss = criterion(input1, input2)

class kornia.losses.MS_SSIMLoss(sigmas=[0.5, 1.0, 2.0, 4.0, 8.0], data_range=1.0, K=(0.01, 0.03), alpha=0.025, compensation=200.0, reduction='mean')

Creates a criterion that computes MSSIM + L1 loss.

According to [1], we compute the MS_SSIM + L1 loss as follows:

\[\text{loss}(x, y) = \alpha \cdot \mathcal{L_{MSSIM}}(x,y)+(1 - \alpha) \cdot G_\alpha \cdot \mathcal{L_1}(x,y)\]

Where:

• \(\alpha\) is the weight parameter.

• \(x\) and \(y\) are the reconstructed and true reference images.

• \(\mathcal{L_{MSSIM}}\) is the MS-SSIM loss.

• \(G_\alpha\) is the sigma values for computing multi-scale SSIM.

• \(\mathcal{L_1}\) is the L1 loss.

Reference:

[1]: https://research.nvidia.com/sites/default/files/pubs/2017-03_Loss-Functions-for/NN_ImgProc.pdf#page11

Parameters:

• sigmas (optional) – gaussian sigma values. Default: [0.5, 1.0, 2.0, 4.0, 8.0]

• data_range (optional) – the range of the images. Default: 1.0

• K (optional) – k values. Default: (0.01, 0.03)

• alpha (optional) – specifies the alpha value Default: 0.025

• compensation (optional) – specifies the scaling coefficient. Default: 200.0

• reduction (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 computed loss.

Shape:

• Input1: \((N, C, H, W)\).

• Input2: \((N, C, H, W)\).

• Output: \((N, H, W)\) or scalar if reduction is set to 'mean' or 'sum'.

Examples

>>> input1 = torch.rand(1, 3, 5, 5)
>>> input2 = torch.rand(1, 3, 5, 5)
>>> criterion = kornia.losses.MS_SSIMLoss()
>>> loss = criterion(input1, input2)

class kornia.losses.TotalVariation

Compute the Total Variation according to [1].

Shape:

• Input: \((*, H, W)\).

• Output: \((*,)\).

Examples

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

Reference:

[1] https://en.wikipedia.org/wiki/Total_variation

class kornia.losses.PSNRLoss(max_val)

Create 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 kornia.losses.InverseDepthSmoothnessLoss

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 kornia.losses.CharbonnierLoss(reduction='none')

Criterion that computes the Charbonnier [2] (aka. L1-L2 [3]) loss.

According to [1], we compute the Charbonnier loss as follows:

\[\text{WL}(x, y) = \sqrt{(x - y)^{2} + 1} - 1\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] https://ieeexplore.ieee.org/document/413553 [3] https://hal.inria.fr/inria-00074015/document [4] https://arxiv.org/pdf/1712.05927.pdf

. note::

This implementation follows the formulation by Barron [1]. Other works utilize a slightly different implementation (see [4]).

Parameters:

reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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:

• img1: the predicted tensor with shape \((*)\).

• img2: the target tensor with the same shape as img1.

Example

>>> criterion = CharbonnierLoss(reduction="mean")
>>> img1 = torch.randn(2, 3, 32, 2107, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 2107)
>>> output = criterion(img1, img2)
>>> output.backward()

class kornia.losses.WelschLoss(reduction='none')

Criterion that computes the Welsch [2] (aka. Leclerc [3]) loss.

According to [1], we compute the Welsch loss as follows:

\[\text{WL}(x, y) = 1 - exp(-\frac{1}{2} (x - y)^{2})\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] https://www.tandfonline.com/doi/abs/10.1080/03610917808812083 [3] https://link.springer.com/article/10.1007/BF00054839

Parameters:

reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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:

• img1: the predicted tensor with shape \((*)\).

• img2: the target tensor with the same shape as img1.

Example

>>> criterion = WelschLoss(reduction="mean")
>>> img1 = torch.randn(2, 3, 32, 1904, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 1904)
>>> output = criterion(img1, img2)
>>> output.backward()

class kornia.losses.CauchyLoss(reduction='none')

Criterion that computes the Cauchy [2] (aka. Lorentzian) loss.

According to [1], we compute the Cauchy loss as follows:

\[\text{WL}(x, y) = log(\frac{1}{2} (x - y)^{2} + 1)\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] https://files.is.tue.mpg.de/black/papers/cviu.63.1.1996.pdf

Parameters:

reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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:

• img1: the predicted tensor with shape \((*)\).

• img2: the target tensor with the same shape as img1.

Example

>>> criterion = CauchyLoss(reduction="mean")
>>> img1 = torch.randn(2, 3, 32, 2107, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 2107)
>>> output = criterion(img1, img2)
>>> output.backward()

class kornia.losses.GemanMcclureLoss(reduction='none')

Criterion that computes the Geman-McClure loss [2].

According to [1], we compute the Geman-McClure loss as follows:

\[\text{WL}(x, y) = \frac{2 (x - y)^{2}}{(x - y)^{2} + 4}\]

Where:

• \(x\) is the prediction.

• \(y\) is the target to be regressed to.

Reference:

[1] https://arxiv.org/pdf/1701.03077.pdf [2] Bayesian image analysis: An application to single photon emission tomography, Geman and McClure, 1985

Parameters:

reduction (str, optional) – Specifies the reduction to apply to the output: 'none' | 'mean' | 'sum'. 'none': no reduction will be applied (default), '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:

• img1: the predicted tensor with shape \((*)\).

• img2: the target tensor with the same shape as img1.

Example

>>> criterion = GemanMcclureLoss(reduction="mean")
>>> img1 = torch.randn(2, 3, 32, 2107, requires_grad=True)
>>> img2 = torch.randn(2, 3, 32, 2107)
>>> output = criterion(img1, img2)
>>> output.backward()

Semantic Segmentation

kornia.losses.binary_focal_loss_with_logits(input, target, alpha=0.25, gamma=2.0, reduction='none', eps=None, pos_weight=None)

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 of arbitrary shape.

• target (Tensor) – the target tensor with shape matching input.

• 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 | None, optional) – Deprecated: scalar for numerically stability when dividing. This is no longer used. Default: None

• pos_weight (Tensor | None, optional) – a weight of positive examples. It’s possible to trade off recall and precision by adding weights to positive examples. Must be a vector with length equal to the number of classes. Default: None

Return type:

Tensor

Returns:

the computed loss.

Examples

>>> 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(21.8725)

kornia.losses.focal_loss(input, target, alpha, gamma=2.0, reduction='none', eps=None)

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 | None, optional) – Deprecated: scalar to enforce numerical stabiliy. This is no longer used. Default: None

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()

kornia.losses.dice_loss(input, target, average='micro', eps=1e-08)

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:

[1] https://en.wikipedia.org/wiki/S%C3%B8rensen%E2%80%93Dice_coefficient

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 \(0 ≤ targets[i] ≤ C−1\).

• average (str, optional) – Reduction applied in multi-class scenario: - 'micro' [default]: Calculate the loss across all classes. - 'macro': Calculate the loss for each class separately and average the metrics across classes. Default: 'micro'

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

Return type:

Tensor

Returns:

One-element tensor of 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()

kornia.losses.tversky_loss(input, target, alpha, beta, eps=1e-08)

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:

• input (Tensor) – logits tensor with shape \((N, C, H, W)\) where C = number of classes.

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

• alpha (float) – the first coefficient in the denominator.

• beta (float) – the second coefficient in the denominator.

• eps (float, optional) – scalar for numerical stability. 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 = tversky_loss(input, target, alpha=0.5, beta=0.5)
>>> output.backward()

kornia.losses.lovasz_hinge_loss(pred, target)

Criterion that computes a surrogate binary intersection-over-union (IoU) loss.

According to [2], we compute the IoU as follows:

\[\text{IoU}(x, class) = \frac{|X \cap Y|}{|X \cup Y|}\]

[1] approximates this fomular with a surrogate, which is fully differentable.

Where:

• \(X\) expects to be the scores of each class.

• \(Y\) expects to be the binary tensor with the class labels.

the loss, is finally computed as:

\[\text{loss}(x, class) = 1 - \text{IoU}(x, class)\]

Reference:

[1] http://proceedings.mlr.press/v37/yub15.pdf [2] https://arxiv.org/pdf/1705.08790.pdf

. note::

This loss function only supports binary labels. For multi-class labels please use the Lovasz-Softmax loss.

Parameters:

• pred (Tensor) – logits tensor with shape \((N, 1, H, W)\).

• labels – labels tensor with shape \((N, H, W)\) with binary values.

Return type:

Tensor

Returns:

a scalar with the computed loss.

Example

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

kornia.losses.lovasz_softmax_loss(pred, target)

Criterion that computes a surrogate multi-class intersection-over-union (IoU) loss.

According to [1], we compute the IoU as follows:

\[\text{IoU}(x, class) = \frac{|X \cap Y|}{|X \cup Y|}\]

[1] approximates this fomular with a surrogate, which is fully differentable.

Where:

• \(X\) expects to be the scores of each class.

• \(Y\) expects to be the binary tensor with the class labels.

the loss, is finally computed as:

\[\text{loss}(x, class) = 1 - \text{IoU}(x, class)\]

Reference:

[1] https://arxiv.org/pdf/1705.08790.pdf

. note::

This loss function only supports multi-class (C > 1) labels. For binary labels please use the Lovasz-Hinge loss.

Parameters:

• pred (Tensor) – logits tensor with shape \((N, C, H, W)\) where C = number of classes > 1.

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

Return type:

Tensor

Returns:

a scalar with the computed loss.

Example

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

class kornia.losses.BinaryFocalLossWithLogits(alpha, gamma=2.0, reduction='none', pos_weight=None)

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 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'

• pos_weight (Tensor | None, optional) – a weight of positive examples. It’s possible to trade off recall and precision by adding weights to positive examples. Must be a vector with length equal to the number of classes. Default: None

Shape:

• Input: \((N, *)\).

• Target: \((N, *)\).

Examples

>>> kwargs = {"alpha": 0.25, "gamma": 2.0, "reduction": 'mean'}
>>> loss = BinaryFocalLossWithLogits(**kwargs)
>>> input = torch.randn(1, 3, 5, requires_grad=True)
>>> target = torch.empty(1, 3, 5, dtype=torch.long).random_(2)
>>> output = loss(input, target)
>>> output.backward()

class kornia.losses.DiceLoss(average='micro', eps=1e-08)

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:

[1] https://en.wikipedia.org/wiki/S%C3%B8rensen%E2%80%93Dice_coefficient

Parameters:

• average (str, optional) – Reduction applied in multi-class scenario: - 'micro' [default]: Calculate the loss across all classes. - 'macro': Calculate the loss for each class separately and average the metrics across classes. Default: 'micro'

• 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 kornia.losses.TverskyLoss(alpha, beta, eps=1e-08)

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:

• alpha (float) – the first coefficient in the denominator.

• beta (float) – the second coefficient in the denominator.

• eps (float, optional) – scalar for numerical stability. Default: 1e-08

Shape:

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

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

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 kornia.losses.FocalLoss(alpha, gamma=2.0, reduction='none', eps=None)

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 | None, optional) – Deprecated: scalar to enforce numerical stability. This is no longer used. Default: None

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 kornia.losses.LovaszHingeLoss

Criterion that computes a surrogate binary intersection-over-union (IoU) loss.

According to [2], we compute the IoU as follows:

\[\text{IoU}(x, class) = \frac{|X \cap Y|}{|X \cup Y|}\]

[1] approximates this fomular with a surrogate, which is fully differentable.

Where:

• \(X\) expects to be the scores of each class.

• \(Y\) expects to be the binary tensor with the class labels.

the loss, is finally computed as:

\[\text{loss}(x, class) = 1 - \text{IoU}(x, class)\]

Reference:

[1] http://proceedings.mlr.press/v37/yub15.pdf [2] https://arxiv.org/pdf/1705.08790.pdf

. note::

This loss function only supports binary labels. For multi-class labels please use the Lovasz-Softmax loss.

Parameters:

• pred – logits tensor with shape \((N, 1, H, W)\).

• labels – labels tensor with shape \((N, H, W)\) with binary values.

Returns:

a scalar with the computed loss.

Example

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

class kornia.losses.LovaszSoftmaxLoss

Criterion that computes a surrogate multi-class intersection-over-union (IoU) loss.

According to [1], we compute the IoU as follows:

\[\text{IoU}(x, class) = \frac{|X \cap Y|}{|X \cup Y|}\]

[1] approximates this fomular with a surrogate, which is fully differentable.

Where:

• \(X\) expects to be the scores of each class.

• \(Y\) expects to be the binary tensor with the class labels.

the loss, is finally computed as:

\[\text{loss}(x, class) = 1 - \text{IoU}(x, class)\]

Reference:

[1] https://arxiv.org/pdf/1705.08790.pdf

. note::

This loss function only supports multi-class (C > 1) labels. For binary labels please use the Lovasz-Hinge loss.

Parameters:

• pred – logits tensor with shape \((N, C, H, W)\) where C = number of classes > 1.

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

Returns:

a scalar with the computed loss.

Example

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

Distributions

kornia.losses.js_div_loss_2d(input, target, reduction='mean')

Calculate 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'

Return type:

Tensor

Examples

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

kornia.losses.kl_div_loss_2d(input, target, reduction='mean')

Calculate 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'

Return type:

Tensor

Examples

>>> input = torch.full((1, 1, 2, 4), 0.125)
>>> loss = kl_div_loss_2d(input, input)
>>> loss.item()
0.0

Morphology

class kornia.losses.HausdorffERLoss(alpha=2.0, k=10, reduction='mean')

Binary Hausdorff loss based on morphological erosion.

Hausdorff Distance loss measures the maximum distance of a predicted segmentation boundary to the nearest ground-truth edge pixel. For two segmentation point sets X and Y , the one-sided HD from X to Y is defined as:

\[hd(X,Y) = \max_{x \in X} \min_{y \in Y}||x - y||_2\]

Furthermore, the bidirectional HD is:

\[HD(X,Y) = max(hd(X, Y), hd(Y, X))\]

This is an Hausdorff Distance (HD) Loss that based on morphological erosion, which provided a differentiable approximation of Hausdorff distance as stated in [KS19]. The code is refactored on top of here.

Parameters:

• alpha (float, optional) – controls the erosion rate in each iteration. Default: 2.0

• k (int, optional) – the number of iterations of erosion. Default: 10

• reduction (str, optional) – Specifies the reduction to apply to the output: ‘none’ | ‘mean’ | ‘sum’. ‘none’: no reduction will be applied, ‘mean’: the weighted mean of the output is taken, ‘sum’: the output will be summed. Default: 'mean'

Examples

>>> hdloss = HausdorffERLoss()
>>> input = torch.randn(5, 3, 20, 20)
>>> target = (torch.rand(5, 1, 20, 20) * 2).long()
>>> res = hdloss(input, target)

class kornia.losses.HausdorffERLoss3D(alpha=2.0, k=10, reduction='mean')

Binary 3D Hausdorff loss based on morphological erosion.

Hausdorff Distance loss measures the maximum distance of a predicted segmentation boundary to the nearest ground-truth edge pixel. For two segmentation point sets X and Y , the one-sided HD from X to Y is defined as:

\[hd(X,Y) = \max_{x \in X} \min_{y \in Y}||x - y||_2\]

Furthermore, the bidirectional HD is:

\[HD(X,Y) = max(hd(X, Y), hd(Y, X))\]

This is a 3D Hausdorff Distance (HD) Loss that based on morphological erosion, which provided a differentiable approximation of Hausdorff distance as stated in [KS19]. The code is refactored on top of here.

Parameters:

• alpha (float, optional) – controls the erosion rate in each iteration. Default: 2.0

• k (int, optional) – the number of iterations of erosion. Default: 10

• reduction (str, optional) – Specifies the reduction to apply to the output: ‘none’ | ‘mean’ | ‘sum’. ‘none’: no reduction will be applied, ‘mean’: the weighted mean of the output is taken, ‘sum’: the output will be summed. Default: 'mean'

Examples

>>> hdloss = HausdorffERLoss3D()
>>> input = torch.randn(5, 3, 20, 20, 20)
>>> target = (torch.rand(5, 1, 20, 20, 20) * 2).long()
>>> res = hdloss(input, target)