# convolution3dLayer

3-D convolutional layer

## Description

A 3-D convolutional layer applies sliding cuboidal convolution filters to 3-D input. The layer convolves the input by moving the filters along the input vertically, horizontally, and along the depth, computing the dot product of the weights and the input, and then adding a bias term.

## Creation

### Description

layer = convolution3dLayer(filterSize,numFilters) creates a 3-D convolutional layer and sets the FilterSize and NumFilters properties.

example

layer = convolution3dLayer(filterSize,numFilters,Name,Value) sets the optional Stride, DilationFactor, NumChannels, Parameters and Initialization, Learning Rate and Regularization, and Name properties using name-value pairs. To specify input padding, use the 'Padding' name-value pair argument. For example, convolution3dLayer(11,96,'Stride',4,'Padding',1) creates a 3-D convolutional layer with 96 filters of size [11 11 11], a stride of [4 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, Learning Rate and Regularization, and Name properties. Enclose names in single quotes.

Example: convolution3dLayer(3,16,'Padding','same') creates a 3-D convolutional layer with 16 filters of size [3 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, width, or depth 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, to the left and right, and to the front and back, if possible. If the padding in a given dimension has an odd value, then the software adds the extra padding to the input as postpadding. In other words, the software adds extra vertical padding to the bottom, extra horizontal padding to the right, and extra depth padding to the back of the input.

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

• Three-element vector [a b c] of nonnegative integers — Add padding of size a to the top and bottom, padding of size b to the left and right, and padding of size c to the front and back of the input.

• 2-by-3 matrix [t l f;b r k] of nonnegative integers — Add padding of size t to the top, b to the bottom, l to the left, r to the right, f to the front, and k to the back of the input. In other words, the top row specifies the prepadding and the second row defines the postpadding in the three dimensions.

Example: 'Padding',1 adds one row of padding to the top and bottom, one column of padding to the left and right, and one plane of padding to the front and back 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, width, and depth of the filters, specified as a vector [h w d] of three positive integers, where h is the height, w is the width, and d is the depth. 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, width, and depth.

Example: [5 5 5] specifies filters with a height, width, and depth 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.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Step size for traversing the input in three dimensions, specified as a vector [a b c] of three positive integers, where a is the vertical step size, b is the horizontal step size, and c is the step size along the depth. When creating the layer, you can specify Stride as a scalar to use the same value for step sizes in all three directions.

Example: [2 3 1] specifies a vertical step size of 2, a horizontal step size of 3, and a step size along the depth of 1.

Factor for dilated convolution (also known as atrous convolution), specified as a vector [h w d] of three positive integers, where h is the vertical dilation, w is the horizontal dilation, and d is the dilation along the depth. When creating the layer, you can specify DilationFactor as a scalar to use the same value for dilation in all three directions.

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-by-3 filter with the dilation factor [2 2 2] is equivalent to a 5-by-5-by-5 filter with zeros between the elements.

Example: [2 3 1] dilates the filter vertically by a factor of 2, horizontally by a factor of 3, and along the depth by a factor of 1.

Size of padding to apply to input borders, specified as 2-by-3 matrix [t l f;b r k] of nonnegative integers, where t and b are the padding applied to the top and bottom in the vertical direction, l and r are the padding applied to the left and right in the horizontal direction, and f and k are the padding applied to the front and back along the depth. In other words, the top row specifies the prepadding and the second row defines the postpadding in the three dimensions.

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

Example: [1 2 4;1 2 4] adds one row of padding to the top and bottom, two columns of padding to the left and right, and four planes of padding to the front and back 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, width, or depth 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, to the left and right, and to the front and back, if possible. If the padding in a given dimension has an odd value, then the software adds the extra padding to the input as postpadding. In other words, the software adds extra vertical padding to the bottom, extra horizontal padding to the right, and extra depth padding to the back of the input.

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

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

$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 1& 1& 1& 5& 9& 9& 9\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\end{array}\right]$

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | char | string

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 [1] (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)*FilterSize(3)*NumChannels and numOut = FilterSize(1)*FilterSize(2)*FilterSize(3)*NumFilters.

• 'he' – Initialize the weights with the He initializer [2]. The He initializer samples from a normal distribution with zero mean and variance 2/numIn, where numIn = FilterSize(1)*FilterSize(2)*FilterSize(3)*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 you train 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-FilterSize(3)-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 you train 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-1-by-NumFilters array.

Data Types: single | double

### Learning 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 you specify using the trainingOptions function.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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 you specify using the trainingOptions function.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

### Layer

Layer name, specified as a character vector or a string scalar. For Layer array input, the trainNetwork, assembleNetwork, layerGraph, and dlnetwork functions automatically assign names to layers with Name set to ''.

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 3-D convolution layer with 16 filters, each with a height, width, and depth of 5. Use a stride (step size) of 4 in all three directions.

layer = convolution3dLayer(5,16,'Stride',4)
layer =
Convolution3DLayer with properties:

Name: ''

Hyperparameters
FilterSize: [5 5 5]
NumChannels: 'auto'
NumFilters: 16
Stride: [4 4 4]
DilationFactor: [1 1 1]

Learnable Parameters
Weights: []
Bias: []

Show all properties

Include a 3-D convolution layer in a Layer array.

layers = [ ...
image3dInputLayer([28 28 28 3])
convolution3dLayer(5,16,'Stride',4)
reluLayer
maxPooling3dLayer(2,'Stride',4)
fullyConnectedLayer(10)
softmaxLayer
classificationLayer]
layers =
7x1 Layer array with layers:

1   ''   3-D Image Input         28x28x28x3 images with 'zerocenter' normalization
2   ''   Convolution             16 5x5x5 convolutions with stride [4  4  4] and padding [0  0  0; 0  0  0]
3   ''   ReLU                    ReLU
4   ''   3-D Max Pooling         2x2x2 max pooling with stride [4  4  4] and padding [0  0  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 3-D convolutional layer with 32 filters, each with a height, width, and depth of 5. Specify the weights initializer to be the He initializer.

filterSize = 5;
numFilters = 32;
layer = convolution3dLayer(filterSize,numFilters, ...
'WeightsInitializer','he')
layer =
Convolution3DLayer with properties:

Name: ''

Hyperparameters
FilterSize: [5 5 5]
NumChannels: 'auto'
NumFilters: 32
Stride: [1 1 1]
DilationFactor: [1 1 1]

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, width, and depth of 5. Specify initializers that sample the weights and biases from a Gaussian distribution with a standard deviation of 0.0001.

filterSize = 5;
numFilters = 32;

layer = convolution3dLayer(filterSize,numFilters, ...
'WeightsInitializer', @(sz) rand(sz) * 0.0001, ...
'BiasInitializer', @(sz) rand(sz) * 0.0001)
layer =
Convolution3DLayer with properties:

Name: ''

Hyperparameters
FilterSize: [5 5 5]
NumChannels: 'auto'
NumFilters: 32
Stride: [1 1 1]
DilationFactor: [1 1 1]

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 3-D convolutional layer compatible with color images. Set the weights and bias to W and b in the MAT file Conv3dWeights.mat respectively.

filterSize = 5;
numFilters = 32;

layer = convolution3dLayer(filterSize,numFilters, ...
'Weights',W, ...
'Bias',b)
layer =
Convolution3DLayer with properties:

Name: ''

Hyperparameters
FilterSize: [5 5 5]
NumChannels: 3
NumFilters: 32
Stride: [1 1 1]
DilationFactor: [1 1 1]

Learnable Parameters
Weights: [5-D double]
Bias: [1x1x1x32 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-28-by-1. Create a 3-D convolutional layer with 16 filters, each with a height of 6, a width of 4, and a depth of 5. Set the stride in all dimensions to 4.

Make sure the convolution covers the input completely. For the convolution to fully cover the input, the output dimensions must be integer numbers. When there is no dilation, the i-th output dimension is calculated as (imageSize(i) - filterSize(i) + padding(i)) / stride(i) + 1.

• For the horizontal output dimension to be an integer, two rows of padding are required: (28 – 6 + 2)/4 + 1 = 7. Distribute the padding symmetrically by adding one row of padding at the top and bottom of the image.

• For the vertical output dimension to be an integer, no padding is required: (28 – 4+ 0)/4 + 1 = 7.

• For the depth output dimension to be an integer, one plane of padding is required: (28 – 5 + 1)/4 + 1 = 7. You must distribute the padding asymmetrically across the front and back of the image. This example adds one plane of padding to the back of the image.

Construct the convolutional layer. Specify 'Padding' as a 2-by-3 matrix. The first row specifies prepadding and the second row specifies postpadding in the three dimensions.

layer = convolution3dLayer([6 4 5],16,'Stride',4,'Padding',[1 0 0;1 0 1])
layer =
Convolution3DLayer with properties:

Name: ''

Hyperparameters
FilterSize: [6 4 5]
NumChannels: 'auto'
NumFilters: 16
Stride: [4 4 4]
DilationFactor: [1 1 1]

Learnable Parameters
Weights: []
Bias: []

Show all properties

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## References

[1] 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.

[2] 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.

Introduced in R2019a