# convolution2dLayer

2-D convolutional layer

## Description

A 2-D convolutional layer applies sliding convolutional filters to the input. The layer convolves the input by moving the filters along the input vertically and horizontally and computing the dot product of the weights and the input, and then adding a bias term.

## Creation

### Syntax

``layer = convolution2dLayer(filterSize,numFilters)``
``layer = convolution2dLayer(filterSize,numFilters,Name,Value)``

### Description

````layer = convolution2dLayer(filterSize,numFilters)` creates a 2-D convolutional layer and sets the `FilterSize` and `NumFilters` properties.```

example

````layer = convolution2dLayer(filterSize,numFilters,Name,Value)` sets the optional `Stride`, `DilationFactor`, `NumChannels`, Parameters and Initialization, Learn Rate and Regularization, and `Name` properties using name-value pairs. To specify input padding, use the `'Padding'` name-value pair argument. For example, `convolution2dLayer(11,96,'Stride',4,'Padding',1)` creates a 2-D convolutional layer with 96 filters of size ```[11 11]```, a stride of `[4 4]`, and padding of size 1 along all edges of the layer input. You can specify multiple name-value pairs. Enclose each property name in single quotes.```

### Input Arguments

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Name-Value Pair Arguments

Use comma-separated name-value pair arguments to specify the size of the padding to add along the edges of the layer input or to set the `Stride`, `DilationFactor`, `NumChannels`, Parameters and Initialization, Learn Rate and Regularization, and `Name` properties. Enclose names in single quotes.

Example: `convolution2dLayer(3,16,'Padding','same')` creates a 2-D convolutional layer with 16 filters of size `[3 3]` and `'same'` padding. At training time, the software calculates and sets the size of the padding so that the layer output has the same size as the input.

Input edge padding, specified as the comma-separated pair consisting of `'Padding'` and one of these values:

• `'same'` — Add padding of size calculated by the software at training or prediction time so that the output has the same size as the input when the stride equals 1. If the stride is larger than 1, then the output size is `ceil(inputSize/stride)`, where `inputSize` is the height or width of the input and `stride` is the stride in the corresponding dimension. The software adds the same amount of padding to the top and bottom, and to the left and right, if possible. If the padding that must be added vertically has an odd value, then the software adds extra padding to the bottom. If the padding that must be added horizontally has an odd value, then the software adds extra padding to the right.

• Nonnegative integer `p` — Add padding of size `p` to all the edges of the input.

• Vector `[a b]` of nonnegative integers — Add padding of size `a` to the top and bottom of the input and padding of size `b` to the left and right.

• Vector `[t b l r]` of nonnegative integers — Add padding of size `t` to the top, `b` to the bottom, `l` to the left, and `r` to the right of the input.

Example: `'Padding',1` adds one row of padding to the top and bottom, and one column of padding to the left and right of the input.

Example: `'Padding','same'` adds padding so that the output has the same size as the input (if the stride equals 1).

## Properties

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### Convolution

Height and width of the filters, specified as a vector `[h w]` of two positive integers, where `h` is the height and `w` is the width. `FilterSize` defines the size of the local regions to which the neurons connect in the input.

When creating the layer, you can specify `FilterSize` as a scalar to use the same value for the height and width.

Example: `[5 5]` specifies filters with a height of 5 and a width of 5.

Number of filters, specified as a positive integer. This number corresponds to the number of neurons in the convolutional layer that connect to the same region in the input. This parameter determines the number of channels (feature maps) in the output of the convolutional layer.

Example: `96`

Step size for traversing the input vertically and horizontally, specified as a vector `[a b]` of two positive integers, where `a` is the vertical step size and `b` is the horizontal step size. When creating the layer, you can specify `Stride` as a scalar to use the same value for both step sizes.

Example: `[2 3]` specifies a vertical step size of 2 and a horizontal step size of 3.

Factor for dilated convolution (also known as atrous convolution), specified as a vector `[h w]` of two positive integers, where `h` is the vertical dilation and `w` is the horizontal dilation. When creating the layer, you can specify `DilationFactor` as a scalar to use the same value for both horizontal and vertical dilations.

Use dilated convolutions to increase the receptive field (the area of the input which the layer can see) of the layer without increasing the number of parameters or computation.

The layer expands the filters by inserting zeros between each filter element. The dilation factor determines the step size for sampling the input or equivalently the upsampling factor of the filter. It corresponds to an effective filter size of (Filter Size – 1) .* Dilation Factor + 1. For example, a 3-by-3 filter with the dilation factor `[2 2]` is equivalent to a 5-by-5 filter with zeros between the elements.

Example: `[2 3]`

Size of padding to apply to input borders, specified as a vector `[t b l r]` of four nonnegative integers, where `t` is the padding applied to the top, `b` is the padding applied to the bottom, `l` is the padding applied to the left, and `r` is the padding applied to the right.

When you create a layer, use the `'Padding'` name-value pair argument to specify the padding size.

Example: `[1 1 2 2]` adds one row of padding to the top and bottom, and two columns of padding to the left and right of the input.

Method to determine padding size, specified as `'manual'` or `'same'`.

The software automatically sets the value of `PaddingMode` based on the `'Padding'` value you specify when creating a layer.

• If you set the `'Padding'` option to a scalar or a vector of nonnegative integers, then the software automatically sets `PaddingMode` to `'manual'`.

• If you set the `'Padding'` option to `'same'`, then the software automatically sets `PaddingMode` to `'same'` and calculates the size of the padding at training time so that the output has the same size as the input when the stride equals 1. If the stride is larger than 1, then the output size is `ceil(inputSize/stride)`, where `inputSize` is the height or width of the input and `stride` is the stride in the corresponding dimension. The software adds the same amount of padding to the top and bottom, and to the left and right, if possible. If the padding that must be added vertically has an odd value, then the software adds extra padding to the bottom. If the padding that must be added horizontally has an odd value, then the software adds extra padding to the right.

Note

`Padding` property will be removed in a future release. Use `PaddingSize` instead. When creating a layer, use the `'Padding'` name-value pair argument to specify the padding size.

Size of padding to apply to input borders vertically and horizontally, specified as a vector `[a b]` of two nonnegative integers, where `a` is the padding applied to the top and bottom of the input data and `b` is the padding applied to the left and right.

Example: `[1 1]` adds one row of padding to the top and bottom, and one column of padding to the left and right of the input.

Value to pad data, specified as one of the following:

`PaddingValue`DescriptionExample
ScalarPad with the specified scalar value.
`$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 3& 1& 4& 0& 0\\ 0& 0& 1& 5& 9& 0& 0\\ 0& 0& 2& 6& 5& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$`
`'symmetric-include-edge'`Pad using mirrored values of the input, including the edge values.
`$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 1& 1& 5& 9& 9& 5\\ 1& 3& 3& 1& 4& 4& 1\\ 1& 3& 3& 1& 4& 4& 1\\ 5& 1& 1& 5& 9& 9& 5\\ 6& 2& 2& 6& 5& 5& 6\\ 6& 2& 2& 6& 5& 5& 6\\ 5& 1& 1& 5& 9& 9& 5\end{array}\right]$`
`'symmetric-exclude-edge'`Pad using mirrored values of the input, excluding the edge values.
`$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\\ 9& 5& 1& 5& 9& 5& 1\\ 5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\end{array}\right]$`
`'replicate'`Pad using repeated border elements of the input
`$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 1& 1& 1& 5& 9& 9& 9\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\end{array}\right]$`

Number of channels for each filter, specified as `'auto'` or a positive integer.

This parameter is always equal to the number of channels of the input to the convolutional layer. For example, if the input is a color image, then the number of channels for the input is 3. If the number of filters for the convolutional layer prior to the current layer is 16, then the number of channels for the current layer is 16.

If `NumChannels` is `'auto'`, then the software determines the number of channels at training time.

Example: `256`

### Parameters and Initialization

Function to initialize the weights, specified as one of the following:

• `'glorot'` – Initialize the weights with the Glorot initializer [4] (also known as Xavier initializer). The Glorot initializer independently samples from a uniform distribution with zero mean and variance ```2/(numIn + numOut)```, where ```numIn = FilterSize(1)*FilterSize(2)*NumChannels``` and ```numOut = FilterSize(1)*FilterSize(2)*NumFilters```.

• `'he'` – Initialize the weights with the He initializer [5]. The He initializer samples from a normal distribution with zero mean and variance `2/numIn`, where ```numIn = FilterSize(1)*FilterSize(2)*NumChannels```.

• `'narrow-normal'` – Initialize the weights by independently sampling from a normal distribution with zero mean and standard deviation 0.01.

• `'zeros'` – Initialize the weights with zeros.

• `'ones'` – Initialize the weights with ones.

• Function handle – Initialize the weights with a custom function. If you specify a function handle, then the function must be of the form `weights = func(sz)`, where `sz` is the size of the weights. For an example, see Specify Custom Weight Initialization Function.

The layer only initializes the weights when the `Weights` property is empty.

Data Types: `char` | `string` | `function_handle`

Function to initialize the bias, specified as one of the following:

• `'zeros'` – Initialize the bias with zeros.

• `'ones'` – Initialize the bias with ones.

• `'narrow-normal'` – Initialize the bias by independently sampling from a normal distribution with zero mean and standard deviation 0.01.

• Function handle – Initialize the bias with a custom function. If you specify a function handle, then the function must be of the form `bias = func(sz)`, where `sz` is the size of the bias.

The layer only initializes the bias when the `Bias` property is empty.

Data Types: `char` | `string` | `function_handle`

Layer weights for the convolutional layer, specified as a numeric array.

The layer weights are learnable parameters. You can specify the initial value for the weights directly using the `Weights` property of the layer. When training a network, if the `Weights` property of the layer is nonempty, then `trainNetwork` uses the `Weights` property as the initial value. If the `Weights` property is empty, then `trainNetwork` uses the initializer specified by the `WeightsInitializer` property of the layer.

At training time, `Weights` is a `FilterSize(1)`-by-`FilterSize(2)`-by-`NumChannels`-by-`NumFilters` array.

Data Types: `single` | `double`

Layer biases for the convolutional layer, specified as a numeric array.

The layer biases are learnable parameters. When training a network, if `Bias` is nonempty, then `trainNetwork` uses the `Bias` property as the initial value. If `Bias` is empty, then `trainNetwork` uses the initializer specified by `BiasInitializer`.

At training time, `Bias` is a 1-by-1-by-`NumFilters` array.

Data Types: `single` | `double`

### Learn Rate and Regularization

Learning rate factor for the weights, specified as a nonnegative scalar.

The software multiplies this factor by the global learning rate to determine the learning rate for the weights in this layer. For example, if `WeightLearnRateFactor` is 2, then the learning rate for the weights in this layer is twice the current global learning rate. The software determines the global learning rate based on the settings specified with the `trainingOptions` function.

Example: `2`

Learning rate factor for the biases, specified as a nonnegative scalar.

The software multiplies this factor by the global learning rate to determine the learning rate for the biases in this layer. For example, if `BiasLearnRateFactor` is 2, then the learning rate for the biases in the layer is twice the current global learning rate. The software determines the global learning rate based on the settings specified with the `trainingOptions` function.

Example: `2`

L2 regularization factor for the weights, specified as a nonnegative scalar.

The software multiplies this factor by the global L2 regularization factor to determine the L2 regularization for the weights in this layer. For example, if `WeightL2Factor` is 2, then the L2 regularization for the weights in this layer is twice the global L2 regularization factor. You can specify the global L2 regularization factor using the `trainingOptions` function.

Example: `2`

L2 regularization factor for the biases, specified as a nonnegative scalar.

The software multiplies this factor by the global L2 regularization factor to determine the L2 regularization for the biases in this layer. For example, if `BiasL2Factor` is 2, then the L2 regularization for the biases in this layer is twice the global L2 regularization factor. You can specify the global L2 regularization factor using the `trainingOptions` function.

Example: `2`

### Layer

Layer name, specified as a character vector or a string scalar. To include a layer in a layer graph, you must specify a nonempty, unique layer name. If you train a series network with the layer and `Name` is set to `''`, then the software automatically assigns a name to the layer at training time.

Data Types: `char` | `string`

Number of inputs of the layer. This layer accepts a single input only.

Data Types: `double`

Input names of the layer. This layer accepts a single input only.

Data Types: `cell`

Number of outputs of the layer. This layer has a single output only.

Data Types: `double`

Output names of the layer. This layer has a single output only.

Data Types: `cell`

## Examples

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Create a convolutional layer with 96 filters, each with a height and width of 11. Use a stride (step size) of 4 in the horizontal and vertical directions.

`layer = convolution2dLayer(11,96,'Stride',4)`
```layer = Convolution2DLayer with properties: Name: '' Hyperparameters FilterSize: [11 11] NumChannels: 'auto' NumFilters: 96 Stride: [4 4] DilationFactor: [1 1] PaddingMode: 'manual' PaddingSize: [0 0 0 0] PaddingValue: 0 Learnable Parameters Weights: [] Bias: [] Show all properties ```

Include a convolutional layer in a `Layer` array.

```layers = [ imageInputLayer([28 28 1]) convolution2dLayer(5,20) reluLayer maxPooling2dLayer(2,'Stride',2) fullyConnectedLayer(10) softmaxLayer classificationLayer]```
```layers = 7x1 Layer array with layers: 1 '' Image Input 28x28x1 images with 'zerocenter' normalization 2 '' Convolution 20 5x5 convolutions with stride [1 1] and padding [0 0 0 0] 3 '' ReLU ReLU 4 '' Max Pooling 2x2 max pooling with stride [2 2] and padding [0 0 0 0] 5 '' Fully Connected 10 fully connected layer 6 '' Softmax softmax 7 '' Classification Output crossentropyex ```

To specify the weights and bias initializer functions, use the `WeightsInitializer` and `BiasInitializer` properties respectively. To specify the weights and biases directly, use the `Weights` and `Bias` properties respectively.

Specify Initialization Functions

Create a convolutional layer with 32 filters, each with a height and width of 5 and specify the weights initializer to be the He initializer.

```filterSize = 5; numFilters = 32; layer = convolution2dLayer(filterSize,numFilters, ... 'WeightsInitializer','he')```
```layer = Convolution2DLayer with properties: Name: '' Hyperparameters FilterSize: [5 5] NumChannels: 'auto' NumFilters: 32 Stride: [1 1] DilationFactor: [1 1] PaddingMode: 'manual' PaddingSize: [0 0 0 0] PaddingValue: 0 Learnable Parameters Weights: [] Bias: [] Show all properties ```

Note that the `Weights` and `Bias` properties are empty. At training time, the software initializes these properties using the specified initialization functions.

Specify Custom Initialization Functions

To specify your own initialization function for the weights and biases, set the `WeightsInitializer` and `BiasInitializer` properties to a function handle. For these properties, specify function handles that take the size of the weights and biases as input and output the initialized value.

Create a convolutional layer with 32 filters, each with a height and width of 5 and specify initializers that sample the weights and biases from a Gaussian distribution with a standard deviation of 0.0001.

```filterSize = 5; numFilters = 32; layer = convolution2dLayer(filterSize,numFilters, ... 'WeightsInitializer', @(sz) rand(sz) * 0.0001, ... 'BiasInitializer', @(sz) rand(sz) * 0.0001)```
```layer = Convolution2DLayer with properties: Name: '' Hyperparameters FilterSize: [5 5] NumChannels: 'auto' NumFilters: 32 Stride: [1 1] DilationFactor: [1 1] PaddingMode: 'manual' PaddingSize: [0 0 0 0] PaddingValue: 0 Learnable Parameters Weights: [] Bias: [] Show all properties ```

Again, the `Weights` and `Bias` properties are empty. At training time, the software initializes these properties using the specified initialization functions.

Specify Weights and Bias Directly

Create a fully connected layer with an output size of 10 and set the weights and bias to `W` and `b` in the MAT file `Conv2dWeights.mat` respectively.

```filterSize = 5; numFilters = 32; load Conv2dWeights layer = convolution2dLayer(filterSize,numFilters, ... 'Weights',W, ... 'Bias',b)```
```layer = Convolution2DLayer with properties: Name: '' Hyperparameters FilterSize: [5 5] NumChannels: 3 NumFilters: 32 Stride: [1 1] DilationFactor: [1 1] PaddingMode: 'manual' PaddingSize: [0 0 0 0] PaddingValue: 0 Learnable Parameters Weights: [5x5x3x32 double] Bias: [1x1x32 double] Show all properties ```

Here, the `Weights` and `Bias` properties contain the specified values. At training time, if these properties are non-empty, then the software uses the specified values as the initial weights and biases. In this case, the software does not use the initializer functions.

Suppose the size of the input is 28-by-28-by-1. Create a convolutional layer with 16 filters, each with a height of 6 and a width of 4. Set the horizontal and vertical stride to 4.

Make sure the convolution covers the input completely. For the convolution to fully cover the input, both the horizontal and vertical output dimensions must be integer numbers. For the horizontal output dimension to be an integer, one row of padding is required on the top and bottom of the image: (28 – 6+ 2 * 1)/4 + 1 = 7. For the vertical output dimension to be an integer, no zero padding is required: (28 – 4+ 2 * 0)/4 + 1 = 7.

Construct the convolutional layer.

`layer = convolution2dLayer([6 4],16,'Stride',4,'Padding',[1 0])`
```layer = Convolution2DLayer with properties: Name: '' Hyperparameters FilterSize: [6 4] NumChannels: 'auto' NumFilters: 16 Stride: [4 4] DilationFactor: [1 1] PaddingMode: 'manual' PaddingSize: [1 1 0 0] PaddingValue: 0 Learnable Parameters Weights: [] Bias: [] Show all properties ```

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## Compatibility Considerations

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Behavior changed in R2019a

## References

[1] LeCun, Y., B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel. "Handwritten Digit Recognition with a Back-Propagation Network." In Advances in Neural Information Processing Systems 2 (D. Touretzky, ed.). San Francisco: Morgan Kaufmann, 1990.

[2] LeCun, Y., L. Bottou, Y. Bengio, and P. Haffner. ''Gradient-Based Learning Applied to Document Recognition.'' Proceedings of the IEEE. Vol. 86, Number 11, 1998, pp. 2278–2324.

[3] Murphy, K. P. Machine Learning: A Probabilistic Perspective. Cambridge, MA: MIT Press, 2012.

[4] Glorot, Xavier, and Yoshua Bengio. "Understanding the Difficulty of Training Deep Feedforward Neural Networks." In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 249–356. Sardinia, Italy: AISTATS, 2010.

[5] He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification." In Proceedings of the 2015 IEEE International Conference on Computer Vision, 1026–1034. Washington, DC: IEEE Computer Vision Society, 2015.

## Extended Capabilities

Introduced in R2016a

[1] Image credit: Convolution arithmetic (License)