3-D convolutional layer

A 3-D convolutional layer applies sliding cuboidal convolution filters to three-dimensional 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.

`layer = convolution3dLayer(filterSize,numFilters)`

`layer = convolution3dLayer(filterSize,numFilters,Name,Value)`

creates a 3-D convolutional layer and sets the `layer`

= convolution3dLayer(`filterSize`

,`numFilters`

)`FilterSize`

and `NumFilters`

properties.

sets the optional `layer`

= convolution3dLayer(`filterSize`

,`numFilters`

,`Name,Value`

)`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,
`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 zero 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.

Use comma-separated name-value pair arguments to specify the size of the zero 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.

`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 zero padding so that the layer output has the same size as the
input.`'Padding'`

— Input edge padding`0`

(default) | array of nonnegative integers | `'same'`

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).

`FilterSize`

— Height, width, and depth of filtersvector of three positive integers

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.

`NumFilters`

— Number of filterspositive integer

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`

`Stride`

— Step size for traversing input`[1 1 1]`

(default) | vector of three positive integersStep 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.

`DilationFactor`

— Factor for dilated convolution`[1 1 1]`

(default) | vector of three positive integersFactor 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.

`PaddingSize`

— Size of padding`[0 0 0;0 0 0]`

(default) | 2-by-3 matrix of nonnegative integersSize 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.

`PaddingMode`

— Method to determine padding size`'manual'`

(default) | `'same'`

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.

`NumChannels`

— Number of channels for each filter`'auto'`

(default) | positive integerNumber 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`

`WeightsInitializer`

— Function to initialize weights`'glorot'`

(default) | `'he'`

| `'narrow-normal'`

| `'zeros'`

| `'ones'`

| function handleFunction 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`

`BiasInitializer`

— Function to initialize bias`'zeros'`

(default) | `'narrow-normal'`

| `'ones'`

| function handleFunction 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`

`Weights`

— Layer weights`[]`

(default) | numeric arrayLayer 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-`FilterSize(3)`

-by-`NumChannels`

-by-`NumFilters`

array.

**Data Types: **`single`

| `double`

`Bias`

— Layer biases`[]`

(default) | numeric arrayLayer 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-1-by-`NumFilters`

array.

**Data Types: **`single`

| `double`

`WeightLearnRateFactor`

— Learning rate factor for weights1 (default) | nonnegative scalar

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`

`BiasLearnRateFactor`

— Learning rate factor for biases1 (default) | nonnegative scalar

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`

`WeightL2Factor`

— L2 regularization factor for weights1 (default) | nonnegative scalar

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`

`BiasL2Factor`

— L2 regularization factor for biases0 (default) | nonnegative scalar

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`

`Name`

— Layer name`''`

(default) | character vector | string scalar
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`

`NumInputs`

— Number of inputs1 (default)

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

**Data Types: **`double`

`InputNames`

— Input names`{'in'}`

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

**Data Types: **`cell`

`NumOutputs`

— Number of outputs1 (default)

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

**Data Types: **`double`

`OutputNames`

— Output names`{'out'}`

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

**Data Types: **`cell`

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] PaddingMode: 'manual' PaddingSize: [2x3 double] 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] PaddingMode: 'manual' PaddingSize: [2x3 double] 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] PaddingMode: 'manual' PaddingSize: [2x3 double] 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; load Conv3dWeights 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] PaddingMode: 'manual' PaddingSize: [2x3 double] 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 zero 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 zero padding is required: (28 – 4+ 0)/4 + 1 = 7.

For the depth output dimension to be an integer, one plane of zero 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 zero 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] PaddingMode: 'manual' PaddingSize: [2x3 double] Learnable Parameters Weights: [] Bias: [] Show all properties

A convolutional layer applies sliding convolutional filters to the input. A 3-D
convolutional layer extends the functionality of a 2-D convolutional layer to a third
dimension, depth. 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. To learn more, see the definition of convolutional layer
on the `convolution2dLayer`

reference page.

[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*, pp. 249-256. 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 IEEE international conference on computer vision*, pp. 1026-1034. 2015.

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