# Convolution¶

Versioned name: Convolution-1

Category: Convolution

Short description: Computes 1D, 2D or 3D convolution (cross-correlation to be precise) of input and kernel tensors.

Detailed description: Basic building block of convolution is a dot product of input patch and kernel. Whole operation consist of multiple such computations over multiple input patches and kernels. More thorough explanation can be found in Convolutional Neural Networks and Convolution operation .

For the convolutional layer, the number of output features in each dimension is calculated using the formula:

$n_{out} = \left ( \frac{n_{in} + 2p - k}{s} \right ) + 1$

The receptive field in each layer is calculated using the formulas:

• Jump in the output feature map:

$j_{out} = j_{in} \cdot s$
• Size of the receptive field of output feature:

$r_{out} = r_{in} + ( k - 1 ) \cdot j_{in}$
• Center position of the receptive field of the first output feature:

$start_{out} = start_{in} + ( \frac{k - 1}{2} - p ) \cdot j_{in}$
• Output is calculated using the following formula:

$out = \sum_{i = 0}^{n}w_{i}x_{i} + b$

Attributes:

• strides

• Description: strides is a distance (in pixels) to slide the filter on the feature map over the (z, y, x) axes for 3D convolutions and (y, x) axes for 2D convolutions. For example, strides equal 4,2,1 means sliding the filter 4 pixel at a time over depth dimension, 2 over height dimension and 1 over width dimension.

• Range of values: integer values starting from 0

• Type: int[]

• Required: yes

• Description: pads_begin is a number of pixels to add to the beginning along each axis. For example, pads_begin equal 1,2 means adding 1 pixel to the top of the input and 2 to the left of the input.

• Range of values: integer values starting from 0

• Type: int[]

• Required: yes

• Note: the attribute is ignored when auto_pad attribute is specified.

• Description: pads_end is a number of pixels to add to the ending along each axis. For example, pads_end equal 1,2 means adding 1 pixel to the bottom of the input and 2 to the right of the input.

• Range of values: integer values starting from 0

• Type: int[]

• Required: yes

• Note: the attribute is ignored when auto_pad attribute is specified.

• dilations

• Description: dilations denotes the distance in width and height between elements (weights) in the filter. For example, dilation equal 1,1 means that all the elements in the filter are neighbors, so it is the same as for the usual convolution. dilation equal 2,2 means that all the elements in the filter are matched not to adjacent elements in the input matrix, but to those that are adjacent with distance 1.

• Range of values: integer value starting from 0

• Type: int[]

• Required: yes

• same_upper - the input is padded to match the output size. In case of odd padding value an extra padding is added at the end.

• same_lower - the input is padded to match the output size. In case of odd padding value an extra padding is added at the beginning.

• valid - do not use padding.

• Type: string

• Default value: explicit

• Required: no

Inputs:

• 1: Input tensor of type T and rank 3, 4 or 5. Layout is [N, C_IN, Z, Y, X] (number of batches, number of channels, spatial axes Z, Y, X). Required.

• 2: Kernel tensor of type T and rank 3, 4 or 5. Layout is [C_OUT, C_IN, Z, Y, X] (number of output channels, number of input channels, spatial axes Z, Y, X). Required.

• Note: Type of the convolution (1D, 2D or 3D) is derived from the rank of the input tensors and not specified by any attribute:

• 1D convolution (input tensors rank 3) means that there is only one spatial axis X

• 2D convolution (input tensors rank 4) means that there are two spatial axes Y, X

• 3D convolution (input tensors rank 5) means that there are three spatial axes Z, Y, X

Outputs:

• 1: Output tensor of type T and rank 3, 4 or 5. Layout is [N, C_OUT, Z, Y, X] (number of batches, number of kernel output channels, spatial axes Z, Y, X).

Types:

• T: any numeric type.

Example:

1D Convolution

<layer type="Convolution" ...>
<input>
<port id="0">
<dim>1</dim>
<dim>5</dim>
<dim>128</dim>
</port>
<port id="1">
<dim>16</dim>
<dim>5</dim>
<dim>4</dim>
</port>
</input>
<output>
<port id="2" precision="FP32">
<dim>1</dim>
<dim>16</dim>
<dim>63</dim>
</port>
</output>
</layer>


2D Convolution

<layer type="Convolution" ...>
<input>
<port id="0">
<dim>1</dim>
<dim>3</dim>
<dim>224</dim>
<dim>224</dim>
</port>
<port id="1">
<dim>64</dim>
<dim>3</dim>
<dim>5</dim>
<dim>5</dim>
</port>
</input>
<output>
<port id="2" precision="FP32">
<dim>1</dim>
<dim>64</dim>
<dim>224</dim>
<dim>224</dim>
</port>
</output>
</layer>


3D Convolution

<layer type="Convolution" ...>
<input>
<port id="0">
<dim>1</dim>
<dim>7</dim>
<dim>320</dim>
<dim>320</dim>
<dim>320</dim>
</port>
<port id="1">
<dim>32</dim>
<dim>7</dim>
<dim>3</dim>
<dim>3</dim>
<dim>3</dim>
</port>
</input>
<output>
<port id="2" precision="FP32">
<dim>1</dim>
<dim>32</dim>
<dim>106</dim>
<dim>106</dim>
<dim>106</dim>
</port>
</output>
</layer>