# dltranspconv

Deep learning transposed convolution

## Syntax

## Description

The transposed convolution operation upsamples feature maps.

The `dltranspconv`

function applies the deep learning transposed
convolution operation to `dlarray`

data.
Using `dlarray`

objects makes working with high
dimensional data easier by allowing you to label the dimensions. For example, you can label
which dimensions correspond to spatial, time, channel, and batch dimensions using the
`"S"`

, `"T"`

, `"C"`

, and
`"B"`

labels, respectively. For unspecified and other dimensions, use the
`"U"`

label. For `dlarray`

object functions that operate
over particular dimensions, you can specify the dimension labels by formatting the
`dlarray`

object directly, or by using the `DataFormat`

option.

computes the deep learning transposed convolution of the input `Y`

= dltranspconv(`X`

,`weights`

,`bias`

)`X`

using the filters defined by `weights`

, and adds the constant
`bias`

. The input `X`

must be a formatted
`dlarray`

. The output `Y`

is a formatted
`dlarray`

with the same dimension format as
`X`

.

The function, by default, convolves over up to three dimensions
of `X`

labeled `"S"`

(spatial). To convolve over dimensions
labeled `"T"`

(time), specify `weights`

with a
`"T"`

dimension using a formatted `dlarray`

object or by
using the `WeightsFormat`

option.

For unformatted input data, use the `DataFormat`

option.

specifies options using one or more name-value pair arguments in addition to the
input arguments in previous syntaxes. For example, `Y`

= dltranspconv(___`Name=Value`

)`Stride=3`

sets
the stride of the convolution operation.

## Examples

### Perform 2-D Transposed Convolution

Create a formatted `dlarray`

object containing a batch of 128 28-by-28 images with 3 channels. Specify the format `"SSCB"`

(spatial, spatial, channel, batch).

```
miniBatchSize = 128;
inputSize = [28 28];
numChannels = 3;
X = rand(inputSize(1),inputSize(2),numChannels,miniBatchSize);
X = dlarray(X,"SSCB");
```

View the size and format of the input data.

size(X)

`ans = `*1×4*
28 28 3 128

dims(X)

ans = 'SSCB'

Initialize the weights and bias for 2-D transposed convolution. For the weights, specify 64 3-by-3 filters. For the bias, specify a vector of zeros.

filterSize = [3 3]; numFilters = 64; weights = rand(filterSize(1),filterSize(2),numFilters,numChannels); bias = zeros(1,numFilters);

Apply 2-D transposed convolution using the `dltranspconv`

function.

Y = dltranspconv(X,weights,bias);

View the size and format of the output.

size(Y)

`ans = `*1×4*
30 30 64 128

dims(Y)

ans = 'SSCB'

### Perform Grouped Transposed Convolution

Create a formatted `dlarray`

object containing a batch of 128 28-by-28 images with 16 channels. Specify the format `"SSCB"`

(spatial, spatial, channel, batch).

```
miniBatchSize = 128;
inputSize = [28 28];
numChannels = 16;
X = rand(inputSize(1),inputSize(2),numChannels,miniBatchSize);
X = dlarray(X,"SSCB");
```

View the size and format of the input data.

size(X)

`ans = `*1×4*
28 28 16 128

dims(X)

ans = 'SSCB'

Initialize the weights and bias for 2-D grouped transposed convolution. For the weights, specify two groups of 64 3-by-3 filters. For the bias, specify a vector of zeros.

The number of channels per group is given by the number of channels of the input data divided by the number of groups. The size of the bias vector is the number of filters per group multiplied by the number of groups.

filterSize = [3 3]; numFiltersPerGroup = 64; numGroups = 2; numChannelsPerGroup = numChannels / numGroups; weights = rand(filterSize(1),filterSize(2),numFiltersPerGroup,numChannelsPerGroup,numGroups); bias = zeros(1,numFiltersPerGroup*numGroups);

Apply 2-D grouped transposed convolution using the `dltranspconv`

function.

Y = dltranspconv(X,weights,bias);

View the size and format of the output.

size(Y)

`ans = `*1×4*
30 30 128 128

dims(Y)

ans = 'SSCB'

## Input Arguments

`X`

— Input data

`dlarray`

| numeric array

Input data, specified as a formatted `dlarray`

, an
unformatted `dlarray`

, or a numeric array.

If `X`

is an unformatted `dlarray`

or a
numeric array, then you must specify the format using the `DataFormat`

option. If `X`

is a numeric
array, then either `weights`

or `bias`

must be a `dlarray`

object.

The function, by default, convolves over up to three dimensions
of `X`

labeled `"S"`

(spatial). To convolve over dimensions
labeled `"T"`

(time), specify `weights`

with a
`"T"`

dimension using a formatted `dlarray`

object or by
using the `WeightsFormat`

option.

`weights`

— Filters

`dlarray`

| numeric array

Filters, specified as a formatted `dlarray`

, an unformatted
`dlarray`

, or a numeric array.

The size and format of the weights depends on the type of task. If
`weights`

is an unformatted `dlarray`

or a numeric array, then the size and shape of `weights`

depends on the `WeightsFormat`

option.

The following table describes the size and format of the weights for
various tasks. You can specify an array with the dimensions in any order
using formatted `dlarray`

objects or by using the
`WeightsFormat`

option. When the weights has multiple
dimensions with the same label (for example, multiple dimensions labeled
`"S"`

), then those dimensions must be in ordered as
described in this table.

Task | Required Dimensions | Size | Example | |
---|---|---|---|---|

Weights | Format | |||

1-D transposed convolution | `"S"` (spatial) or
`"T"` (time) | Filter size |
| `"SCU"` (spatial,
channel, unspecified) |

`"C"` (channel) | Number of channels | |||

`"U"` (unspecified) | Number of filters | |||

1-D grouped transposed convolution | `"S"` (spatial) or
`"T"` (time) | Filter size |
| `"SCUU"` (spatial,
channel, unspecified, unspecified) |

`"C"` (channel) | Number of channels per group | |||

First `"U"` (unspecified) | Number of filters per group | |||

Second `"U"` (unspecified) | Number of groups | |||

2-D transposed convolution | First `"S"` (spatial) | Filter height |
| `"SSCU"` (spatial,
spatial, channel, unspecified) |

Second `"S"` (spatial) or
`"T"` (time) | Filter width | |||

`"C"` (channel) | Number of channels | |||

`"U"` (unspecified) | Number of filters | |||

2-D grouped transposed convolution | First `"S"` (spatial) | Filter height |
| `"SSCUU"` (spatial,
spatial, channel, unspecified, unspecified) |

Second `"S"` (spatial) or
`"T"` (time) | Filter width | |||

`"C"` (channel) | Number of channels per group | |||

First `"U"` (unspecified) | Number of filters per group | |||

Second `"U"` (unspecified) | Number of groups | |||

3-D transposed convolution | First `"S"` (spatial) | Filter height |
| `"SSSCU"` (spatial,
spatial, spatial, channel, unspecified) |

Second `"S"` (spatial) | Filter width | |||

Third `"S"` (spatial) or
`"T"` (time) | Filter depth | |||

`"C"` (channel) | Number of channels | |||

`"U"` (unspecified) | Number of filters |

**Tip**

The function, by default, convolves over up to three dimensions
of `X`

labeled `"S"`

(spatial). To convolve over dimensions
labeled `"T"`

(time), specify `weights`

with a
`"T"`

dimension using a formatted `dlarray`

object or by
using the `WeightsFormat`

option.

`bias`

— Bias constant

`dlarray`

vector | `dlarray`

scalar | numeric vector | numeric scalar

Bias constant, specified as a formatted or unformatted
`dlarray`

vector or `dlarray`

scalar,
a numeric vector, or a numeric scalar.

If

`bias`

is a scalar or has only singleton dimensions, the same bias is applied to each entry of the output.If

`bias`

has a nonsingleton dimension, each element of`bias`

is the bias applied to the corresponding convolutional filter specified by`weights`

. The number of elements of`bias`

must match the number of filters specified by`weights`

.

If `bias`

is a formatted `dlarray`

, the
nonsingleton dimension must be a channel dimension labeled
`"C"`

.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`Stride=2`

sets the stride of each filter to
2.

`DataFormat`

— Description of data dimensions

character vector | string scalar

Description of the data dimensions, specified as a character vector or string scalar.

A data format is a string of characters, where each character describes the type of the corresponding data dimension.

The characters are:

`"S"`

— Spatial`"C"`

— Channel`"B"`

— Batch`"T"`

— Time`"U"`

— Unspecified

For example, consider an array containing a batch of sequences where the first, second,
and third dimensions correspond to channels, observations, and time steps, respectively. You
can specify that this array has the format `"CBT"`

(channel, batch,
time).

You can specify multiple dimensions labeled `"S"`

or `"U"`

.
You can use the labels `"C"`

, `"B"`

, and
`"T"`

once each, at most. The software ignores singleton trailing
`"U"`

dimensions after the second dimension.

If the input data is not a formatted `dlarray`

object, then you must
specify the `DataFormat`

option.

For more information, see Deep Learning Data Formats.

**Data Types: **`char`

| `string`

`WeightsFormat`

— Description of weights dimensions

character vector | string scalar

Description of weights dimensions, specified as a character vector or string scalar.

A data format is a string of characters, where each character describes the type of the corresponding dimension of the data.

The characters are:

`"S"`

— Spatial`"C"`

— Channel`"B"`

— Batch`"T"`

— Time`"U"`

— Unspecified

The default value of `WeightsFormat`

depends on the
task:

Task | Default |
---|---|

1-D transposed convolution | `"SCU"` (spatial, channel,
unspecified) |

1-D grouped transposed convolution | `"SCUU"` (spatial, channel,
unspecified, unspecified) |

2-D transposed convolution | `"SSCU"` (spatial, spatial,
channel, unspecified) |

2-D grouped transposed convolution | `"SSCUU"` (spatial, spatial,
channel, unspecified, unspecified) |

3-D transposed convolution | `"SSSCU"` (spatial, spatial,
spatial, channel, unspecified) |

The supported combinations of dimension labels depends on the type of
convolution, for more information, see the `weights`

argument.

For more information, see Deep Learning Data Formats.

**Tip**

`X`

labeled `"S"`

(spatial). To convolve over dimensions
labeled `"T"`

(time), specify `weights`

with a
`"T"`

dimension using a formatted `dlarray`

object or by
using the `WeightsFormat`

option.

**Data Types: **`char`

| `string`

`Stride`

— Step size for traversing input data

`1`

(default) | numeric scalar | numeric vector

Step size for traversing the input data, specified as a numeric scalar or numeric vector.

To use the same step size for all convolution dimensions, specify the stride as a scalar. To specify a different value for each convolution dimension, specify the stride as a vector with elements ordered corresponding to the dimensions labels in the data format.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`DilationFactor`

— Filter dilation factor

`1`

(default) | numeric scalar | numeric vector

Filter dilation factor, specified as specified as a numeric scalar or numeric vector.

To use the dilation factor all convolution dimensions, specify the dilation factor as a scalar. To specify a different value for each convolution dimension, specify the dilation factor as a vector with elements ordered corresponding to the dimensions labels in the data format.

Use the dilation factor to increase the receptive field of the filter (the area of the input that the filter can see) on the input data. Using a dilation factor corresponds to an effective filter size of `filterSize + (filterSize-1)*(dilationFactor-1)`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`Cropping`

— Cropping applied to edges of data

0 (default) |
`"same"`

| numeric scalar | numeric vector | numeric matrix

Cropping applied to edges of data, specified as one of the following.

`"same"`

— Cropping is set so that the output size is the same as the input size when the stride is`1`

. More generally, the output size of each spatial dimension is`inputSize*stride`

, where`inputSize`

is the size of the input along the convolution dimension.Numeric scalar — The same cropping value is applied to both ends of the convolution dimensions.

Numeric vector — A different cropping value is applied along each convolution dimension. Use a vector of size

`d`

, where`d`

is the number of convolution dimensions of the input data. The`i`

th element of the vector specifies the cropping applied to the start and the end along the`i`

th convolution dimension.Numeric matrix — A different cropping value is applied to the start and end of each convolution dimension. Use a matrix of size 2-by-

`d`

, where`d`

is the number of convolution dimensions of the input data. The element`(1,d)`

specifies the cropping applied to the start of convolution dimension`d`

. The element`(2,d)`

specifies the cropping applied to the end of convolution dimension`d`

. For example, in 2-D the format is`[top, left; bottom, right]`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

## Output Arguments

`Y`

— Feature map

`dlarray`

Feature map, returned as a `dlarray`

. The output
`Y`

has the same underlying data type as the input
`X`

.

If the input data `X`

is a formatted
`dlarray`

, then `Y`

has the same
format as `X`

. If the input data is not a formatted
`dlarray`

, then `Y`

is an unformatted
`dlarray`

or numeric array with the same dimension
order as the input data.

The size of the `"C"`

(channel) dimension of
`Y`

depends on the size of the
`weights`

input. The size of the `"C"`

(channel) dimension of output `Y`

is the product of the
size of the dimensions `numFiltersPerGroup`

and
`numGroups`

in the `weights`

argument.
If `weights`

is a formatted `dlarray`

,
this product is the same as the product of the size of the
`"C"`

(channel) dimension and the second
`"U"`

(unspecified) dimension.

## Algorithms

### Transposed Convolution

The *standard* convolution operation *downsamples* the
input by applying sliding convolutional filters to the input. By flattening the input and
output, you can express the convolution operation as $$Y=CX+B$$ for the convolution matrix *C* and bias vector
*B* that can be derived from the layer weights and biases.

Similarly, the *transposed* convolution operation
*upsamples* the input by applying sliding convolutional filters to
the input. To upsample the input instead of downsampling using sliding filters, the layer
zero-pads each edge of the input with padding that has the size of the corresponding filter
edge size minus 1.

By flattening the input and output, the transposed convolution operation is equivalent to $$Y={C}^{\top}X+B$$, where *C* and *B* denote the
convolution matrix and bias vector for standard convolution derived from the layer weights
and biases, respectively. This operation is equivalent to the backward function of a
standard convolution layer.

## Extended Capabilities

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The `dltranspconv`

function
supports GPU array input with these usage notes and limitations:

When at least one of the following input arguments is a

`gpuArray`

or a`dlarray`

with underlying data of type`gpuArray`

, this function runs on the GPU.`X`

`weights`

`bias`

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## Version History

**Introduced in R2019b**

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