<a name="nn.convlayers.dok"></a>
# Convolutional layers #

A convolution is an integral that expresses the amount of overlap of one function `g` as it is shifted over another function `f`. It therefore "blends" one function with another. The neural network package supports convolution, pooling, subsampling and other relevant facilities. These are divided base on the dimensionality of the input and output [Tensors](https://github.com/torch/torch7/blob/master/doc/tensor.md#tensor):

  * [Temporal Modules](#nn.TemporalModules) apply to sequences with a one-dimensional relationship
(e.g. sequences of words, phonemes and letters. Strings of some kind).
    * [TemporalConvolution](#nn.TemporalConvolution) : a 1D convolution over an input sequence ;
    * [TemporalSubSampling](#nn.TemporalSubSampling) : a 1D sub-sampling over an input sequence ;
    * [TemporalMaxPooling](#nn.TemporalMaxPooling) : a 1D max-pooling operation over an input sequence ;
    * [LookupTable](#nn.LookupTable) : a convolution of width `1`, commonly used for word embeddings ;
  * [Spatial Modules](#nn.SpatialModules) apply to inputs with two-dimensional relationships (e.g. images):
    * [SpatialConvolution](#nn.SpatialConvolution) : a 2D convolution over an input image ;
    * [SpatialFullConvolution](#nn.SpatialFullConvolution) : a 2D full convolution over an input image ;
    * [SpatialConvolutionLocal](#nn.SpatialConvolutionLocal) : a 2D locally-connected layer over an input image ;
    * [SpatialSubSampling](#nn.SpatialSubSampling) : a 2D sub-sampling over an input image ;
    * [SpatialMaxPooling](#nn.SpatialMaxPooling) : a 2D max-pooling operation over an input image ;
    * [SpatialFractionalMaxPooling](#nn.SpatialFractionalMaxPooling) : a 2D fractional max-pooling operation over an input image ;
    * [SpatialAveragePooling](#nn.SpatialAveragePooling) : a 2D average-pooling operation over an input image ;
    * [SpatialAdaptiveMaxPooling](#nn.SpatialAdaptiveMaxPooling) : a 2D max-pooling operation which adapts its parameters dynamically such that the output is of fixed size ;
    * [SpatialMaxUnpooling](#nn.SpatialMaxUnpooling) : a 2D max-unpooling operation ;
    * [SpatialLPPooling](#nn.SpatialLPPooling) : computes the `p` norm in a convolutional manner on a set of input images ;
    * [SpatialConvolutionMap](#nn.SpatialConvolutionMap) : a 2D convolution that uses a generic connection table ;
    * [SpatialZeroPadding](#nn.SpatialZeroPadding) : padds a feature map with specified number of zeros ;
    * [SpatialReflectionPadding](#nn.SpatialReflectionPadding) : padds a feature map with the reflection of the input ;
    * [SpatialReplicationPadding](#nn.SpatialReplicationPadding) : padds a feature map with the value at the edge of the input borders ;
    * [SpatialSubtractiveNormalization](#nn.SpatialSubtractiveNormalization) : a spatial subtraction operation on a series of 2D inputs using
    * [SpatialCrossMapLRN](#nn.SpatialCrossMapLRN) : a spatial local response normalization between feature maps ;
    * [SpatialBatchNormalization](#nn.SpatialBatchNormalization): mean/std normalization over the mini-batch inputs and pixels, with an optional affine transform that follows
a kernel for computing the weighted average in a neighborhood ;
    * [SpatialUpsamplingNearest](#nn.SpatialUpSamplingNearest): A simple upsampler applied to every channel of the feature map.
  * [Volumetric Modules](#nn.VolumetricModules) apply to inputs with three-dimensional relationships (e.g. videos) :
    * [VolumetricConvolution](#nn.VolumetricConvolution) : a 3D convolution over an input video (a sequence of images) ;
    * [VolumetricFullConvolution](#nn.VolumetricFullConvolution) : a 3D full convolution over an input video (a sequence of images) ;
    * [VolumetricMaxPooling](#nn.VolumetricMaxPooling) : a 3D max-pooling operation over an input video.
    * [VolumetricAveragePooling](#nn.VolumetricAveragePooling) : a 3D average-pooling operation over an input video.
    * [VolumetricMaxUnpooling](#nn.VolumetricMaxUnpooling) : a 3D max-unpooling operation ;


<a name="nn.TemporalModules"></a>
## Temporal Modules ##
Excluding an optional first batch dimension, temporal layers expect a 2D Tensor as input. The
first dimension is the number of frames in the sequence (e.g. `nInputFrame`), the last dimension
is the number of features per frame (e.g. `inputFrameSize`). The output will normally have the same number
of dimensions, although the size of each dimension may change. These are commonly used for processing acoustic signals or sequences of words, i.e. in Natural Language Processing.

Note: The [LookupTable](#nn.LookupTable) is special in that while it does output a temporal Tensor of size `nOutputFrame x outputFrameSize`,
its input is a 1D Tensor of indices of size `nIndices`. Again, this is excluding the option first batch dimension.

<a name="nn.TemporalConvolution"></a>
## TemporalConvolution ##

```lua
module = nn.TemporalConvolution(inputFrameSize, outputFrameSize, kW, [dW])
```

Applies a 1D convolution over an input sequence composed of `nInputFrame` frames. The `input` tensor in
`forward(input)` is expected to be a 2D tensor (`nInputFrame x inputFrameSize`) or a 3D tensor (`nBatchFrame x nInputFrame x inputFrameSize`).

The parameters are the following:
  * `inputFrameSize`: The input frame size expected in sequences given into `forward()`.
  * `outputFrameSize`: The output frame size the convolution layer will produce.
  * `kW`: The kernel width of the convolution
  * `dW`: The step of the convolution. Default is `1`.

Note that depending of the size of your kernel, several (of the last)
frames of the sequence might be lost. It is up to the user to add proper padding frames in the input
sequences.

If the input sequence is a 2D tensor of dimension `nInputFrame x inputFrameSize`, the output sequence will be
`nOutputFrame x outputFrameSize` where
```lua
nOutputFrame = (nInputFrame - kW) / dW + 1
```

If the input sequence is a 3D tensor of dimension `nBatchFrame x nInputFrame x inputFrameSize`, the output sequence will be
`nBatchFrame x nOutputFrame x outputFrameSize`.

The parameters of the convolution can be found in `self.weight` (Tensor of
size `outputFrameSize x (inputFrameSize x kW) `) and `self.bias` (Tensor of
size `outputFrameSize`). The corresponding gradients can be found in
`self.gradWeight` and `self.gradBias`.

For a 2D input, the output value of the layer can be precisely described as:
```lua
output[t][i] = bias[i]
  + sum_j sum_{k=1}^kW weight[i][j][k]
                                * input[dW*(t-1)+k)][j]
```

Here is a simple example:

```lua
inp=5;  -- dimensionality of one sequence element
outp=1; -- number of derived features for one sequence element
kw=1;   -- kernel only operates on one sequence element per step
dw=1;   -- we step once and go on to the next sequence element

mlp=nn.TemporalConvolution(inp,outp,kw,dw)

x=torch.rand(7,inp) -- a sequence of 7 elements
print(mlp:forward(x))
```
which gives:
```lua
-0.9109
-0.9872
-0.6808
-0.9403
-0.9680
-0.6901
-0.6387
[torch.Tensor of dimension 7x1]
```

This is equivalent to:
```lua
weights=torch.reshape(mlp.weight,inp) -- weights applied to all
bias= mlp.bias[1];
for i=1,x:size(1) do -- for each sequence element
   element= x[i]; -- features of ith sequence element
   print(element:dot(weights) + bias)
end
```
which gives:
```lua
-0.91094998687717
-0.98721705771773
-0.68075004276185
-0.94030132495887
-0.96798754116609
-0.69008470895581
-0.63871422284166
```

<a name="nn.TemporalMaxPooling"></a>
## TemporalMaxPooling ##

```lua
module = nn.TemporalMaxPooling(kW, [dW])
```

Applies 1D max-pooling operation in `kW` regions by step size
`dW` steps. Input sequence composed of `nInputFrame` frames. The `input` tensor in
`forward(input)` is expected to be a 2D tensor (`nInputFrame x inputFrameSize`)
or a 3D tensor (`nBatchFrame x nInputFrame x inputFrameSize`).

If the input sequence is a 2D tensor of dimension `nInputFrame x inputFrameSize`, the output sequence will be
`nOutputFrame x inputFrameSize` where
```lua
nOutputFrame = (nInputFrame - kW) / dW + 1
```

<a name="nn.TemporalSubSampling"></a>
## TemporalSubSampling ##

```lua
module = nn.TemporalSubSampling(inputFrameSize, kW, [dW])
```

Applies a 1D sub-sampling over an input sequence composed of `nInputFrame` frames. The `input` tensor in
`forward(input)` is expected to be a 2D tensor (`nInputFrame x inputFrameSize`). The output frame size
will be the same as the input one (`inputFrameSize`).

The parameters are the following:
  * `inputFrameSize`: The input frame size expected in sequences given into `forward()`.
  * `kW`: The kernel width of the sub-sampling
  * `dW`: The step of the sub-sampling. Default is `1`.

Note that depending of the size of your kernel, several (of the last)
frames of the sequence might be lost. It is up to the user to add proper padding frames in the input
sequences.

If the input sequence is a 2D tensor `nInputFrame x inputFrameSize`, the output sequence will be
`inputFrameSize x nOutputFrame` where
```lua
nOutputFrame = (nInputFrame - kW) / dW + 1
```

The parameters of the sub-sampling can be found in `self.weight` (Tensor of
size `inputFrameSize`) and `self.bias` (Tensor of
size `inputFrameSize`). The corresponding gradients can be found in
`self.gradWeight` and `self.gradBias`.

The output value of the layer can be precisely described as:
```lua
output[i][t] = bias[i] + weight[i] * sum_{k=1}^kW input[i][dW*(t-1)+k)]
```

<a name="nn.LookupTable"></a>
## LookupTable ##

```lua
module = nn.LookupTable(nIndex, size, [paddingValue])
```

This layer is a particular case of a convolution, where the width of the convolution would be `1`.
When calling `forward(input)`, it assumes `input` is a 1D or 2D tensor filled with indices.
If the input is a matrix, then each row is assumed to be an input sample of given batch. Indices start
at `1` and can go up to `nIndex`. For each index, it outputs a corresponding `Tensor` of size
specified by `size`.

LookupTable can be very slow if a certain input occurs frequently compared to other inputs;
this is often the case for input padding. During the backward step, there is a separate thread
for each input symbol which results in a bottleneck for frequent inputs.
generating a `n x size1 x size2 x ... x sizeN` tensor, where `n`
is the size of a 1D `input` tensor.

Again with a 1D input, when only `size1` is provided, the `forward(input)` is equivalent to
performing the following matrix-matrix multiplication in an efficient manner:
```lua
M P
```
where `M` is a 2D matrix `size x nIndex` containing the parameters of the lookup-table and
`P` is a 2D matrix, where each column vector `i` is a zero vector except at index `input[i]` where it is `1`.

1D example:
```lua
 -- a lookup table containing 10 tensors of size 3
 module = nn.LookupTable(10, 3)

 input = torch.Tensor{1,2,1,10}
 print(module:forward(input))
```

Outputs something like:
```lua
-1.4415 -0.1001 -0.1708
-0.6945 -0.4350  0.7977
-1.4415 -0.1001 -0.1708
-0.0745  1.9275  1.0915
[torch.DoubleTensor of dimension 4x3]
```
Note that the first row vector is the same as the 3rd one!

Given a 2D input tensor of size `m x n`, the output is a `m x n x size`
tensor, where `m` is the number of samples in
the batch and `n` is the number of indices per sample.

2D example:
```lua
 -- a lookup table containing 10 tensors of size 3
 module = nn.LookupTable(10, 3)

 -- a batch of 2 samples of 4 indices each
 input = torch.Tensor({{1,2,4,5},{4,3,2,10}})
 print(module:forward(input))
```

Outputs something like:
```lua
(1,.,.) =
 -0.0570 -1.5354  1.8555
 -0.9067  1.3392  0.6275
  1.9662  0.4645 -0.8111
  0.1103  1.7811  1.5969

(2,.,.) =
  1.9662  0.4645 -0.8111
  0.0026 -1.4547 -0.5154
 -0.9067  1.3392  0.6275
 -0.0193 -0.8641  0.7396
[torch.DoubleTensor of dimension 2x4x3]
```

<a name="nn.SpatialModules"></a>
## Spatial Modules ##
Excluding an optional batch dimension, spatial layers expect a 3D Tensor as input. The
first dimension is the number of features (e.g. `frameSize`), the last two dimensions
are spatial (e.g. `height x width`). These are commonly used for processing images.

<a name="nn.SpatialConvolution"></a>
### SpatialConvolution ###

```lua
module = nn.SpatialConvolution(nInputPlane, nOutputPlane, kW, kH, [dW], [dH], [padW], [padH])
```

Applies a 2D convolution over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 3D tensor (`nInputPlane x height x width`).

The parameters are the following:
  * `nInputPlane`: The number of expected input planes in the image given into `forward()`.
  * `nOutputPlane`: The number of output planes the convolution layer will produce.
  * `kW`: The kernel width of the convolution
  * `kH`: The kernel height of the convolution
  * `dW`: The step of the convolution in the width dimension. Default is `1`.
  * `dH`: The step of the convolution in the height dimension. Default is `1`.
  * `padW`: The additional zeros added per width to the input planes. Default is `0`, a good number is `(kW-1)/2`.
  * `padH`: The additional zeros added per height to the input planes. Default is `padW`, a good number is `(kH-1)/2`.

Note that depending of the size of your kernel, several (of the last)
columns or rows of the input image might be lost. It is up to the user to
add proper padding in images.

If the input image is a 3D tensor `nInputPlane x height x width`, the output image size
will be `nOutputPlane x oheight x owidth` where
```lua
owidth  = floor((width  + 2*padW - kW) / dW + 1)
oheight = floor((height + 2*padH - kH) / dH + 1)
```

The parameters of the convolution can be found in `self.weight` (Tensor of
size `nOutputPlane x nInputPlane x kH x kW`) and `self.bias` (Tensor of
size `nOutputPlane`). The corresponding gradients can be found in
`self.gradWeight` and `self.gradBias`.

The output value of the layer can be precisely described as:
```lua
output[i][j][k] = bias[k]
  + sum_l sum_{s=1}^kW sum_{t=1}^kH weight[s][t][l][k]
                                    * input[dW*(i-1)+s)][dH*(j-1)+t][l]
```


<a name="nn.SpatialConvolutionMap"></a>
### SpatialConvolutionMap ###

```lua
module = nn.SpatialConvolutionMap(connectionMatrix, kW, kH, [dW], [dH])
```

This class is a generalization of
[nn.SpatialConvolution](#nn.SpatialConvolution). It uses a generic
connection table between input and output features. The
[nn.SpatialConvolution](#nn.SpatialConvolution) is equivalent to
using a [full connection table](#nn.tables.full). One can specify
different types of connection tables.

<a name="nn.tables.full"></a>
#### Full Connection Table ####

```lua
table = nn.tables.full(nin,nout)
```

This is a precomputed table that specifies connections between every
input and output node.

<a name="nn.tables.onetoone"></a>
#### One to One Connection Table ####

```lua
table = nn.tables.oneToOne(n)
```

This is a precomputed table that specifies a single connection to each
output node from corresponding input node.

<a name="nn.tables.random"></a>
#### Random Connection Table ####

```lua
table = nn.tables.random(nin,nout, nto)
```

This table is randomly populated such that each output unit has
`nto` incoming connections. The algorithm tries to assign uniform
number of outgoing connections to each input node if possible.

<a name="nn.SpatialFullConvolution"></a>
### SpatialFullConvolution ###

```lua
module = nn.SpatialFullConvolution(nInputPlane, nOutputPlane, kW, kH, [dW], [dH], [padW], [padH], [adjW], [adjH])
```

Applies a 2D full convolution over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 3D or 4D tensor.

Other frameworks call this operation "In-network Upsampling", "Fractionally-strided convolution", "Backwards Convolution," "Deconvolution", or "Upconvolution."

The parameters are the following:
  * `nInputPlane`: The number of expected input planes in the image given into `forward()`.
  * `nOutputPlane`: The number of output planes the convolution layer will produce.
  * `kW`: The kernel width of the convolution
  * `kH`: The kernel height of the convolution
  * `dW`: The step of the convolution in the width dimension. Default is `1`.
  * `dH`: The step of the convolution in the height dimension. Default is `1`.
  * `padW`: The additional zeros added per width to the input planes. Default is `0`, a good number is `(kW-1)/2`.
  * `padH`: The additional zeros added per height to the input planes. Default is `0`, a good number is `(kH-1)/2`.
  * `adjW`: Extra width to add to the output image. Default is `0`. Cannot be greater than dW-1.
  * `adjH`: Extra height to add to the output image. Default is `0`. Cannot be greater than dH-1.

If the input image is a 3D tensor `nInputPlane x height x width`, the output image size
will be `nOutputPlane x oheight x owidth` where
```lua
owidth  = (width  - 1) * dW - 2*padW + kW + adjW
oheight = (height - 1) * dH - 2*padH + kH + adjH
```

Further information about the full convolution can be found in the following paper: [Fully Convolutional Networks for Semantic Segmentation](http://www.cs.berkeley.edu/~jonlong/long_shelhamer_fcn.pdf).

<a name="nn.SpatialConvolutionLocal"></a>
### SpatialConvolutionLocal ###

```lua
module = nn.SpatialConvolutionLocal(nInputPlane, nOutputPlane, iW, iH, kW, kH, [dW], [dH], [padW], [padH])
```

Applies a 2D locally-connected layer over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 3D or 4D tensor.

A locally-connected layer is similar to a convolution layer but without weight-sharing.

The parameters are the following:
  * `nInputPlane`: The number of expected input planes in the image given into `forward()`.
  * `nOutputPlane`: The number of output planes the locally-connected layer will produce.
  * `iW`: The input width.
  * `iH`: The input height.
  * `kW`: The kernel width.
  * `kH`: The kernel height.
  * `dW`: The step in the width dimension. Default is `1`.
  * `dH`: The step in the height dimension. Default is `1`.
  * `padW`: The additional zeros added per width to the input planes. Default is `0`, a good number is `(kW-1)/2`.
  * `padH`: The additional zeros added per height to the input planes. Default is `0`, a good number is `(kH-1)/2`.

If the input image is a 3D tensor `nInputPlane x iH x iW`, the output image size
will be `nOutputPlane x oH x oW` where
```lua
oW  = floor((iW  + 2*padW - kW) / dW + 1)
oH = floor((iH + 2*padH - kH) / dH + 1)
```

<a name="nn.SpatialLPPooling"></a>
### SpatialLPPooling ###

```lua
module = nn.SpatialLPPooling(nInputPlane, pnorm, kW, kH, [dW], [dH])
```

Computes the `p` norm in a convolutional manner on a set of 2D input planes.

<a name="nn.SpatialMaxPooling"></a>
### SpatialMaxPooling ###

```lua
module = nn.SpatialMaxPooling(kW, kH [, dW, dH, padW, padH])
```

Applies 2D max-pooling operation in `kWxkH` regions by step size
`dWxdH` steps. The number of output features is equal to the number of
input planes.

If the input image is a 3D tensor `nInputPlane x height x width`, the output
image size will be `nOutputPlane x oheight x owidth` where

```lua
owidth  = op((width  + 2*padW - kW) / dW + 1)
oheight = op((height + 2*padH - kH) / dH + 1)
```

`op` is a rounding operator. By default, it is `floor`. It can be changed
by calling `:ceil()` or `:floor()` methods.

<a name="nn.SpatialFractionalMaxPooling"></a>
### SpatialFractionalMaxPooling ###

```lua
module = nn.SpatialFractionalMaxPooling(kW, kH, outW, outH)
--   the output should be the exact size (outH x outW)
OR
module = nn.SpatialFractionalMaxPooling(kW, kH, ratioW, ratioH)
--   the output should be the size (floor(inH x ratioH) x floor(inW x ratioW))
--   ratios are numbers between (0, 1) exclusive
```

Applies 2D Fractional max-pooling operation as described in the
paper ["Fractional Max Pooling" by Ben Graham](http://arxiv.org/abs/1412.6071) in the "pseudorandom" mode.

The max-pooling operation is applied in `kWxkH` regions by a stochastic step size determined by the target output size.
The number of output features is equal to the number of input planes.

There are two constructors available.

Constructor 1:
```lua
module = nn.SpatialFractionalMaxPooling(kW, kH, outW, outH)
```

Constructor 2:
```lua
module = nn.SpatialFractionalMaxPooling(kW, kH, ratioW, ratioH)
```
If the input image is a 3D tensor `nInputPlane x height x width`, the output
image size will be `nOutputPlane x oheight x owidth`

 where

```lua
owidth  = floor(width * ratioW)
oheight = floor(height * ratioH)
```
ratios are numbers between (0, 1) exclusive


<a name="nn.SpatialAveragePooling"></a>
### SpatialAveragePooling ###

```lua
module = nn.SpatialAveragePooling(kW, kH [, dW, dH, padW, padH])
```

Applies 2D average-pooling operation in `kWxkH` regions by step size
`dWxdH` steps. The number of output features is equal to the number of
input planes.

If the input image is a 3D tensor `nInputPlane x height x width`, the output
image size will be `nOutputPlane x oheight x owidth` where

```lua
owidth  = op((width  + 2*padW - kW) / dW + 1)
oheight = op((height + 2*padH - kH) / dH + 1)
```

`op` is a rounding operator. By default, it is `floor`. It can be changed
by calling `:ceil()` or `:floor()` methods.

By default, the output of each pooling region is divided by the number of
elements inside the padded image (which is usually `kW*kH`, except in some
corner cases in which it can be smaller). You can also divide by the number
of elements inside the original non-padded image. To switch between different
division factors, call `:setCountIncludePad()` or `:setCountExcludePad()`. If
`padW=padH=0`, both options give the same results.

<a name="nn.SpatialAdaptiveMaxPooling"></a>
### SpatialAdaptiveMaxPooling ###

```lua
module = nn.SpatialAdaptiveMaxPooling(W, H)
```

Applies 2D max-pooling operation in an image such that the output is of
size `WxH`, for any input size. The number of output features is equal
to the number of input planes.

For an output of dimensions `(owidth,oheight)`, the indexes of the pooling
region `(j,i)` in the input image of dimensions `(iwidth,iheight)` are
given by:

```lua
x_j_start = floor((j   /owidth)  * iwidth)
x_j_end   = ceil(((j+1)/owidth)  * iwidth)

y_i_start = floor((i   /oheight) * iheight)
y_i_end   = ceil(((i+1)/oheight) * iheight)
```

<a name="nn.SpatialMaxUnpooling"></a>
### SpatialMaxUnpooling ###

```lua
module = nn.SpatialMaxUnpooling(poolingModule)
```

Applies 2D "max-unpooling" operation using the indices previously computed
by the SpatialMaxPooling module `poolingModule`.

When `B = poolingModule:forward(A)` is called, the indices of the maximal
values (corresponding to their position within each map) are stored:
`B[{n,k,i,j}] = A[{n,k,indices[{n,k,i}],indices[{n,k,j}]}]`.
If `C` is a tensor of same size as `B`, `module:updateOutput(C)` outputs a
tensor `D` of same size as `A` such that:
`D[{n,k,indices[{n,k,i}],indices[{n,k,j}]}] = C[{n,k,i,j}]`.

Module inspired by:
   "Visualizing and understanding convolutional networks" (2014)
                   by Matthew Zeiler, Rob Fergus

<a name="nn.SpatialSubSampling"></a>
### SpatialSubSampling ###

```lua
module = nn.SpatialSubSampling(nInputPlane, kW, kH, [dW], [dH])
```

Applies a 2D sub-sampling over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 3D tensor (`nInputPlane x height x width`). The number of output
planes will be the same as `nInputPlane`.

The parameters are the following:
  * `nInputPlane`: The number of expected input planes in the image given into `forward()`.
  * `kW`: The kernel width of the sub-sampling
  * `kH`: The kernel height of the sub-sampling
  * `dW`: The step of the sub-sampling in the width dimension. Default is `1`.
  * `dH`: The step of the sub-sampling in the height dimension. Default is `1`.

Note that depending of the size of your kernel, several (of the last)
columns or rows of the input image might be lost. It is up to the user to
add proper padding in images.

If the input image is a 3D tensor `nInputPlane x height x width`, the output image size
will be `nInputPlane x oheight x owidth` where

```lua
owidth  = (width  - kW) / dW + 1
oheight = (height - kH) / dH + 1 .
```

The parameters of the sub-sampling can be found in `self.weight` (Tensor of
size `nInputPlane`) and `self.bias` (Tensor of size `nInputPlane`). The
corresponding gradients can be found in `self.gradWeight` and
`self.gradBias`.

The output value of the layer can be precisely described as:
```lua
output[i][j][k] = bias[k]
  + weight[k] sum_{s=1}^kW sum_{t=1}^kH input[dW*(i-1)+s)][dH*(j-1)+t][k]
```

<a name="nn.SpatialUpSamplingNearest"></a>
### SpatialUpSamplingNearest ###

```lua
module = nn.SpatialUpSamplingNearest(scale)
```

Applies a 2D up-sampling over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 3D or 4D tensor (i.e. for 4D: `nBatchPlane x nInputPlane x height x width`). The number of output planes will be the same.  The v dimension is assumed to be the second last dimension (i.e. for 4D it will be the 3rd dim), and the u dimension is assumed to be the last dimension.

The parameters are the following:
  * `scale`: The upscale ratio.  Must be a positive integer

The up-scaling method is simple nearest neighbor, ie:

```lua
output(u,v) = input(floor((u-1)/scale)+1, floor((v-1)/scale)+1)
```

Where `u` and `v` are index from 1 (as per lua convention).  There are no learnable parameters.

<a name="nn.SpatialZeroPadding"></a>
### SpatialZeroPadding ###

```lua
module = nn.SpatialZeroPadding(padLeft, padRight, padTop, padBottom)
```

Each feature map of a given input is padded with specified number of
zeros. If padding values are negative, then input is cropped.

<a name="nn.SpatialReflectionPadding"></a>
### SpatialReflectionPadding ###

```lua
module = nn.SpatialReflectionPadding(padLeft, padRight, padTop, padBottom)
```

Each feature map of a given input is padded with the reflection of the input boundary

<a name="nn.SpatialReplicationPadding"></a>
### SpatialReplicationPadding ###

```lua
module = nn.SpatialReplicationPadding(padLeft, padRight, padTop, padBottom)
```

Each feature map of a given input is padded with the replication of the input boundary

<a name="nn.SpatialSubtractiveNormalization"></a>
### SpatialSubtractiveNormalization ###

```lua
module = nn.SpatialSubtractiveNormalization(ninputplane, kernel)
```

Applies a spatial subtraction operation on a series of 2D inputs using
`kernel` for computing the weighted average in a neighborhood. The
neighborhood is defined for a local spatial region that is the size as
kernel and across all features. For a an input image, since there is
only one feature, the region is only spatial. For an RGB image, the
weighted average is taken over RGB channels and a spatial region.

If the `kernel` is 1D, then it will be used for constructing and seperable
2D kernel. The operations will be much more efficient in this case.

The kernel is generally chosen as a gaussian when it is believed that
the correlation of two pixel locations decrease with increasing
distance. On the feature dimension, a uniform average is used since
the weighting across features is not known.

For this example we use an external package
[image](http://www.github.com/clementfarabet/lua---image/)

```lua
require 'image'
require 'nn'
lena = image.rgb2y(image.lena())
ker = torch.ones(11)
m=nn.SpatialSubtractiveNormalization(1,ker)
processed = m:forward(lena)
w1=image.display(lena)
w2=image.display(processed)
```
![](image/lena.jpg)![](image/lenap.jpg)

<a name="nn.SpatialCrossMapLRN"></a>
### SpatialCrossMapLRN ###

```lua
module = nn.SpatialCrossMapLRN(size [,alpha] [,beta] [,k])
```

Applies Spatial Local Response Normalization between different feature maps.
By default, `alpha = 0.0001`, `beta = 0.75` and `k = 1`

The operation implemented is:
```
                          x_f
y_f =  -------------------------------------------------
        (k+(alpha/size)* sum_{l=l1 to l2} (x_l^2))^beta
```
where `x_f` is the input at spatial locations `h,w` (not shown for simplicity) and feature map `f`,
`l1` corresponds to `max(0,f-ceil(size/2))` and `l2` to `min(F, f-ceil(size/2) + size)`. Here, `F`
is the number of feature maps.
More information can be found [here](https://code.google.com/p/cuda-convnet2/wiki/LayerParams#Local_response_normalization_layer_%28across_maps%29).

<a name="nn.SpatialBatchNormalization"></a>
## SpatialBatchNormalization ##

`module` = `nn.SpatialBatchNormalization(N [,eps] [, momentum] [,affine])`
 where N = number of input feature maps
eps is a small value added to the standard-deviation to avoid divide-by-zero. Defaults to 1e-5
`affine` is a boolean. When set to false, the learnable affine transform is disabled.  Defaults to true

Implements Batch Normalization as described in the paper:
   "Batch Normalization: Accelerating Deep Network Training
                         by Reducing Internal Covariate Shift"
                   by Sergey Ioffe, Christian Szegedy

The operation implemented is:
```
   y =     ( x - mean(x) )
        -------------------- * gamma + beta
       standard-deviation(x)
```
where the mean and standard-deviation are calculated per feature-map over the mini-batches and pixels
and where gamma and beta are learnable parameter vectors of size N (where N = number of feature maps).
The learning of gamma and beta is optional.

   In training time, this layer keeps a running estimate of it's computed mean and std.
   The running sum is kept with a default momentup of 0.1 (unless over-ridden)
   In test time, this running mean/std is used to normalize.



The module only accepts 4D inputs.

```lua
-- with learnable parameters
model = nn.SpatialBatchNormalization(m)
A = torch.randn(b, m, h, w)
C = model:forward(A)  -- C will be of size `b x m x h x w`

-- without learnable parameters
model = nn.SpatialBatchNormalization(m, nil, nil, false)
A = torch.randn(b, m, h, w)
C = model:forward(A)  -- C will be of size `b x m x h x w`
```

<a name="nn.VolumetricModules"></a>
## Volumetric Modules ##
Excluding an optional batch dimension, volumetric layers expect a 4D Tensor as input. The
first dimension is the number of features (e.g. `frameSize`), the second is sequential (e.g. `time`) and the
last two dimensions are spatial (e.g. `height x width`). These are commonly used for processing videos (sequences of images).

<a name="nn.VolumetricConvolution"></a>
### VolumetricConvolution ###

```lua
module = nn.VolumetricConvolution(nInputPlane, nOutputPlane, kT, kW, kH [, dT, dW, dH, padT, padW, padH])
```

Applies a 3D convolution over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 4D tensor (`nInputPlane x time x height x width`).

The parameters are the following:
  * `nInputPlane`: The number of expected input planes in the image given into `forward()`.
  * `nOutputPlane`: The number of output planes the convolution layer will produce.
  * `kT`: The kernel size of the convolution in time
  * `kW`: The kernel width of the convolution
  * `kH`: The kernel height of the convolution
  * `dT`: The step of the convolution in the time dimension. Default is `1`.
  * `dW`: The step of the convolution in the width dimension. Default is `1`.
  * `dH`: The step of the convolution in the height dimension. Default is `1`.
  * `padT`: The additional zeros added per time to the input planes. Default is `0`, a good number is `(kT-1)/2`.
  * `padW`: The additional zeros added per width to the input planes. Default is `0`, a good number is `(kW-1)/2`.
  * `padH`: The additional zeros added per height to the input planes. Default is `0`, a good number is `(kH-1)/2`.


Note that depending of the size of your kernel, several (of the last)
columns or rows of the input image might be lost. It is up to the user to
add proper padding in images.

If the input image is a 4D tensor `nInputPlane x time x height x width`, the output image size
will be `nOutputPlane x otime x owidth x oheight` where
```lua
otime  = floor((time  + 2*padT - kT) / dT + 1)
owidth  = floor((width  + 2*padW - kW) / dW + 1)
oheight  = floor((height  + 2*padH - kH) / dH + 1)
```

The parameters of the convolution can be found in `self.weight` (Tensor of
size `nOutputPlane x nInputPlane x kT x kH x kW`) and `self.bias` (Tensor of
size `nOutputPlane`). The corresponding gradients can be found in
`self.gradWeight` and `self.gradBias`.

<a name="nn.VolumetricFullConvolution"></a>
### VolumetricFullConvolution ###

```lua
module = nn.VolumetricFullConvolution(nInputPlane, nOutputPlane, kT, kW, kH, [dT], [dW], [dH], [padT], [padW], [padH])
```

Applies a 3D full convolution over an input image composed of several input planes. The `input` tensor in
`forward(input)` is expected to be a 4D or 5D tensor.

The parameters are the following:
* `nInputPlane`: The number of expected input planes in the image given into `forward()`.
* `nOutputPlane`: The number of output planes the convolution layer will produce.
* `kT`: The kernel depth of the convolution
* `kW`: The kernel width of the convolution
* `kH`: The kernel height of the convolution
* `dT`: The step of the convolution in the depth dimension. Default is `1`.
* `dW`: The step of the convolution in the width dimension. Default is `1`.
* `dH`: The step of the convolution in the height dimension. Default is `1`.
* `padT`: The additional zeros added per depth to the input planes. Default is `0`, a good number is `(kT-1)/2`.
* `padW`: The additional zeros added per width to the input planes. Default is `0`, a good number is `(kW-1)/2`.
* `padH`: The additional zeros added per height to the input planes. Default is `0`, a good number is `(kH-1)/2`.

If the input image is a 3D tensor `nInputPlane x depth x height x width`, the output image size
will be `nOutputPlane x odepth x oheight x owidth` where
```lua
odepth  = (depth  - 1) * dT - 2*padT + kT
owidth  = (width  - 1) * dW - 2*padW + kW
oheight = (height - 1) * dH - 2*padH + kH
```

<a name="nn.VolumetricMaxPooling"></a>
### VolumetricMaxPooling ###

```lua
module = nn.VolumetricMaxPooling(kT, kW, kH [, dT, dW, dH, padT, padW, padH])
```

Applies 3D max-pooling operation in `kTxkWxkH` regions by step size
`dTxdWxdH` steps. The number of output features is equal to the number of
input planes / dT. The input can optionally be padded with zeros. Padding should be smaller than half of kernel size.  That is, `padT < kT/2`, `padW < kW/2` and `padH < kH/2`.

<a name="nn.VolumetricAveragePooling"></a>
### VolumetricAveragePooling ###

```lua
module = nn.VolumetricAveragePooling(kT, kW, kH [, dT, dW, dH])
```

Applies 3D average-pooling operation in `kTxkWxkH` regions by step size
`dTxdWxdH` steps. The number of output features is equal to the number of
input planes / dT.

<a name="nn.VolumetricMaxUnpooling"></a>
### VolumetricMaxUnpooling ###

```lua
module = nn.VolumetricMaxUnpooling(poolingModule)
```

Applies 3D "max-unpooling" operation using the indices previously computed
by the VolumetricMaxPooling module `poolingModule`.

When `B = poolingModule:forward(A)` is called, the indices of the maximal
values (corresponding to their position within each map) are stored:
`B[{n,k,t,i,j}] = A[{n,k,indices[{n,k,t}],indices[{n,k,i}],indices[{n,k,j}]}]`.
If `C` is a tensor of same size as `B`, `module:updateOutput(C)` outputs a
tensor `D` of same size as `A` such that:
`D[{n,k,indices[{n,k,t}],indices[{n,k,i}],indices[{n,k,j}]}] = C[{n,k,t,i,j}]`.
