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docs: Fixed LaTeX syntax in docstrings (#107)
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@frgfm
frgfm authored Oct 31, 2021
1 parent 5f510c3 commit e34fc42
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38 changes: 19 additions & 19 deletions torchcam/methods/activation.py
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Original file line number Diff line number Diff line change
Expand Up @@ -25,7 +25,7 @@ class CAM(_CAM):
of the visual feature extraction block. The localization map is computed as follows:
.. math::
L^{(c)}_{CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)} A_k(x, y)\\Big)
L^{(c)}_{CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)} A_k(x, y)\Big)
where :math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
position :math:`(x, y)`,
Expand Down Expand Up @@ -96,7 +96,7 @@ class ScoreCAM(_CAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{Score-CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)} A_k(x, y)\\Big)
L^{(c)}_{Score-CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)} A_k(x, y)\Big)
with the coefficient :math:`w_k^{(c)}` being defined as:
Expand All @@ -109,10 +109,10 @@ class ScoreCAM(_CAM):
and :math:`M_k` is defined as follows:
.. math::
M_k = \\frac{U(A_k) - \\min\\limits_m U(A_m)}{\\max\\limits_m U(A_m) - \\min\\limits_m U(A_m)})
\\odot X
M_k = \frac{U(A_k) - \min\limits_m U(A_m)}{\max\limits_m U(A_m) - \min\limits_m U(A_m)})
\odot X
where :math:`\\odot` refers to the element-wise multiplication and :math:`U` is the upsampling operation.
where :math:`\odot` refers to the element-wise multiplication and :math:`U` is the upsampling operation.
Example::
>>> from torchvision.models import resnet18
Expand Down Expand Up @@ -216,12 +216,12 @@ class SSCAM(ScoreCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{SS-CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)} A_k(x, y)\\Big)
L^{(c)}_{SS-CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)} A_k(x, y)\Big)
with the coefficient :math:`w_k^{(c)}` being defined as:
.. math::
w_k^{(c)} = \\frac{1}{N} \\sum\\limits_1^N softmax(Y^{(c)}(M_k) - Y^{(c)}(X_b))
w_k^{(c)} = \frac{1}{N} \sum\limits_1^N softmax(Y^{(c)}(M_k) - Y^{(c)}(X_b))
where :math:`N` is the number of samples used to smooth the weights,
:math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
Expand All @@ -230,12 +230,12 @@ class SSCAM(ScoreCAM):
and :math:`M_k` is defined as follows:
.. math::
M_k = \\Bigg(\\frac{U(A_k) - \\min\\limits_m U(A_m)}{\\max\\limits_m U(A_m) - \\min\\limits_m U(A_m)} +
\\delta\\Bigg) \\odot X
M_k = \Bigg(\frac{U(A_k) - \min\limits_m U(A_m)}{\max\limits_m U(A_m) - \min\limits_m U(A_m)} +
\delta\Bigg) \odot X
where :math:`\\odot` refers to the element-wise multiplication, :math:`U` is the upsampling operation,
:math:`\\delta \\sim \\mathcal{N}(0, \\sigma^2)` is the random noise that follows a 0-mean gaussian distribution
with a standard deviation of :math:`\\sigma`.
where :math:`\odot` refers to the element-wise multiplication, :math:`U` is the upsampling operation,
:math:`\delta \sim \mathcal{N}(0, \sigma^2)` is the random noise that follows a 0-mean gaussian distribution
with a standard deviation of :math:`\sigma`.
Example::
>>> from torchvision.models import resnet18
Expand Down Expand Up @@ -306,12 +306,12 @@ class ISCAM(ScoreCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{ISS-CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)} A_k(x, y)\\Big)
L^{(c)}_{ISS-CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)} A_k(x, y)\Big)
with the coefficient :math:`w_k^{(c)}` being defined as:
.. math::
w_k^{(c)} = \\sum\\limits_{i=1}^N \\frac{i}{N} softmax(Y^{(c)}(M_k) - Y^{(c)}(X_b))
w_k^{(c)} = \sum\limits_{i=1}^N \frac{i}{N} softmax(Y^{(c)}(M_k) - Y^{(c)}(X_b))
where :math:`N` is the number of samples used to smooth the weights,
:math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
Expand All @@ -320,12 +320,12 @@ class ISCAM(ScoreCAM):
and :math:`M_k` is defined as follows:
.. math::
M_k = \\Bigg(\\frac{U(A_k) - \\min\\limits_m U(A_m)}{\\max\\limits_m U(A_m) - \\min\\limits_m U(A_m)} +
\\delta\\Bigg) \\odot X
M_k = \Bigg(\frac{U(A_k) - \min\limits_m U(A_m)}{\max\limits_m U(A_m) - \min\limits_m U(A_m)} +
\delta\Bigg) \odot X
where :math:`\\odot` refers to the element-wise multiplication, :math:`U` is the upsampling operation,
:math:`\\delta \\sim \\mathcal{N}(0, \\sigma^2)` is the random noise that follows a 0-mean gaussian distribution
with a standard deviation of :math:`\\sigma`.
where :math:`\odot` refers to the element-wise multiplication, :math:`U` is the upsampling operation,
:math:`\delta \sim \mathcal{N}(0, \sigma^2)` is the random noise that follows a 0-mean gaussian distribution
with a standard deviation of :math:`\sigma`.
Example::
>>> from torchvision.models import resnet18
Expand Down
66 changes: 33 additions & 33 deletions torchcam/methods/gradient.py
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Original file line number Diff line number Diff line change
Expand Up @@ -72,13 +72,13 @@ class GradCAM(_GradCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{Grad-CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)} A_k(x, y)\\Big)
L^{(c)}_{Grad-CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)} A_k(x, y)\Big)
with the coefficient :math:`w_k^{(c)}` being defined as:
.. math::
w_k^{(c)} = \\frac{1}{H \\cdot W} \\sum\\limits_{i=1}^H \\sum\\limits_{j=1}^W
\\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)}
w_k^{(c)} = \frac{1}{H \cdot W} \sum\limits_{i=1}^H \sum\limits_{j=1}^W
\frac{\partial Y^{(c)}}{\partial A_k(i, j)}
where :math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
position :math:`(x, y)`,
Expand Down Expand Up @@ -114,26 +114,26 @@ class GradCAMpp(_GradCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{Grad-CAM++}(x, y) = \\sum\\limits_k w_k^{(c)} A_k(x, y)
L^{(c)}_{Grad-CAM++}(x, y) = \sum\limits_k w_k^{(c)} A_k(x, y)
with the coefficient :math:`w_k^{(c)}` being defined as:
.. math::
w_k^{(c)} = \\sum\\limits_{i=1}^H \\sum\\limits_{j=1}^W \\alpha_k^{(c)}(i, j) \\cdot
ReLU\\Big(\\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)}\\Big)
w_k^{(c)} = \sum\limits_{i=1}^H \sum\limits_{j=1}^W \alpha_k^{(c)}(i, j) \cdot
ReLU\Big(\frac{\partial Y^{(c)}}{\partial A_k(i, j)}\Big)
where :math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
position :math:`(x, y)`,
:math:`Y^{(c)}` is the model output score for class :math:`c` before softmax,
and :math:`\\alpha_k^{(c)}(i, j)` being defined as:
and :math:`\alpha_k^{(c)}(i, j)` being defined as:
.. math::
\\alpha_k^{(c)}(i, j) = \\frac{1}{\\sum\\limits_{i, j} \\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)}}
= \\frac{\\frac{\\partial^2 Y^{(c)}}{(\\partial A_k(i,j))^2}}{2 \\cdot
\\frac{\\partial^2 Y^{(c)}}{(\\partial A_k(i,j))^2} + \\sum\\limits_{a,b} A_k (a,b) \\cdot
\\frac{\\partial^3 Y^{(c)}}{(\\partial A_k(i,j))^3}}
\alpha_k^{(c)}(i, j) = \frac{1}{\sum\limits_{i, j} \frac{\partial Y^{(c)}}{\partial A_k(i, j)}}
= \frac{\frac{\partial^2 Y^{(c)}}{(\partial A_k(i,j))^2}}{2 \cdot
\frac{\partial^2 Y^{(c)}}{(\partial A_k(i,j))^2} + \sum\limits_{a,b} A_k (a,b) \cdot
\frac{\partial^3 Y^{(c)}}{(\partial A_k(i,j))^3}}
if :math:`\\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)} = 1` else :math:`0`.
if :math:`\frac{\partial Y^{(c)}}{\partial A_k(i, j)} = 1` else :math:`0`.
Example::
>>> from torchvision.models import resnet18
Expand Down Expand Up @@ -183,34 +183,34 @@ class SmoothGradCAMpp(_GradCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{Smooth Grad-CAM++}(x, y) = \\sum\\limits_k w_k^{(c)} A_k(x, y)
L^{(c)}_{Smooth Grad-CAM++}(x, y) = \sum\limits_k w_k^{(c)} A_k(x, y)
with the coefficient :math:`w_k^{(c)}` being defined as:
.. math::
w_k^{(c)} = \\sum\\limits_{i=1}^H \\sum\\limits_{j=1}^W \\alpha_k^{(c)}(i, j) \\cdot
ReLU\\Big(\\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)}\\Big)
w_k^{(c)} = \sum\limits_{i=1}^H \sum\limits_{j=1}^W \alpha_k^{(c)}(i, j) \cdot
ReLU\Big(\frac{\partial Y^{(c)}}{\partial A_k(i, j)}\Big)
where :math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
position :math:`(x, y)`,
:math:`Y^{(c)}` is the model output score for class :math:`c` before softmax,
and :math:`\\alpha_k^{(c)}(i, j)` being defined as:
and :math:`\alpha_k^{(c)}(i, j)` being defined as:
.. math::
\\alpha_k^{(c)}(i, j)
= \\frac{\\frac{\\partial^2 Y^{(c)}}{(\\partial A_k(i,j))^2}}{2 \\cdot
\\frac{\\partial^2 Y^{(c)}}{(\\partial A_k(i,j))^2} + \\sum\\limits_{a,b} A_k (a,b) \\cdot
\\frac{\\partial^3 Y^{(c)}}{(\\partial A_k(i,j))^3}}
= \\frac{\\frac{1}{n} \\sum\\limits_{m=1}^n D^{(c, 2)}_k(i, j)}{
\\frac{2}{n} \\sum\\limits_{m=1}^n D^{(c, 2)}_k(i, j) + \\sum\\limits_{a,b} A_k (a,b) \\cdot
\\frac{1}{n} \\sum\\limits_{m=1}^n D^{(c, 3)}_k(i, j)}
if :math:`\\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)} = 1` else :math:`0`. Here :math:`D^{(c, p)}_k(i, j)`
\alpha_k^{(c)}(i, j)
= \frac{\frac{\partial^2 Y^{(c)}}{(\partial A_k(i,j))^2}}{2 \cdot
\frac{\partial^2 Y^{(c)}}{(\partial A_k(i,j))^2} + \sum\limits_{a,b} A_k (a,b) \cdot
\frac{\partial^3 Y^{(c)}}{(\partial A_k(i,j))^3}}
= \frac{\frac{1}{n} \sum\limits_{m=1}^n D^{(c, 2)}_k(i, j)}{
\frac{2}{n} \sum\limits_{m=1}^n D^{(c, 2)}_k(i, j) + \sum\limits_{a,b} A_k (a,b) \cdot
\frac{1}{n} \sum\limits_{m=1}^n D^{(c, 3)}_k(i, j)}
if :math:`\frac{\partial Y^{(c)}}{\partial A_k(i, j)} = 1` else :math:`0`. Here :math:`D^{(c, p)}_k(i, j)`
refers to the p-th partial derivative of the class score of class :math:`c` relatively to the activation in layer
:math:`k` at position :math:`(i, j)`, and :math:`n` is the number of samples used to get the gradient estimate.
Please note the difference in the numerator of :math:`\\alpha_k^{(c)}(i, j)`,
which is actually :math:`\\frac{1}{n} \\sum\\limits_{k=1}^n D^{(c, 1)}_k(i,j)` in the paper.
Please note the difference in the numerator of :math:`\alpha_k^{(c)}(i, j)`,
which is actually :math:`\frac{1}{n} \sum\limits_{k=1}^n D^{(c, 1)}_k(i,j)` in the paper.
Example::
>>> from torchvision.models import resnet18
Expand Down Expand Up @@ -311,14 +311,14 @@ class XGradCAM(_GradCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{XGrad-CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)} A_k(x, y)\\Big)
L^{(c)}_{XGrad-CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)} A_k(x, y)\Big)
with the coefficient :math:`w_k^{(c)}` being defined as:
.. math::
w_k^{(c)} = \\sum\\limits_{i=1}^H \\sum\\limits_{j=1}^W
\\Big( \\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)} \\cdot
\\frac{A_k(i, j)}{\\sum\\limits_{m=1}^H \\sum\\limits_{n=1}^W A_k(m, n)} \\Big)
w_k^{(c)} = \sum\limits_{i=1}^H \sum\limits_{j=1}^W
\Big( \frac{\partial Y^{(c)}}{\partial A_k(i, j)} \cdot
\frac{A_k(i, j)}{\sum\limits_{m=1}^H \sum\limits_{n=1}^W A_k(m, n)} \Big)
where :math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
position :math:`(x, y)`,
Expand Down Expand Up @@ -357,12 +357,12 @@ class LayerCAM(_GradCAM):
The localization map is computed as follows:
.. math::
L^{(c)}_{Layer-CAM}(x, y) = ReLU\\Big(\\sum\\limits_k w_k^{(c)}(x, y) \\cdot A_k(x, y)\\Big)
L^{(c)}_{Layer-CAM}(x, y) = ReLU\Big(\sum\limits_k w_k^{(c)}(x, y) \cdot A_k(x, y)\Big)
with the coefficient :math:`w_k^{(c)}(x, y)` being defined as:
.. math::
w_k^{(c)}(x, y) = ReLU\\Big(\\frac{\\partial Y^{(c)}}{\\partial A_k(i, j)}(x, y)\\Big)
w_k^{(c)}(x, y) = ReLU\Big(\frac{\partial Y^{(c)}}{\partial A_k(i, j)}(x, y)\Big)
where :math:`A_k(x, y)` is the activation of node :math:`k` in the target layer of the model at
position :math:`(x, y)`,
Expand Down

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