# dltranspconv

Deep learning transposed convolution

## Syntax

``Y = dltranspconv(X,weights,bias)``
``Y = dltranspconv(X,weights,bias,DataFormat=FMT)``
``Y = dltranspconv(___Name=Value)``

## 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.

Note

This function applies the deep learning transposed convolution operation to `dlarray` data. If you want to apply transposed convolution within a `layerGraph` object or `Layer` array, use one of the following layers:

example

````Y = dltranspconv(X,weights,bias)` computes the deep learning transposed convolution of the input `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.```
````Y = dltranspconv(X,weights,bias,DataFormat=FMT)` applies the deep learning transposed convolution operation to the unformatted `dlarray` object `X` with format specified by `FMT`. The output `Y` is an unformatted `dlarray` object with dimensions in the same order as `X`.```

example

````Y = dltranspconv(___Name=Value)` specifies options using one or more name-value pair arguments in addition to the input arguments in previous syntaxes. For example, `Stride=3` sets the stride of the convolution operation. ```

## Examples

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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' ```

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

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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.

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.

WeightsFormat
1-D transposed convolution`"S"` (spatial) or `"T"` (time)Filter size

`filterSize`-by-`numFilters`-by-`numChannels` array, where `filterSize` is the size of the 1-D filters, `numFilters` is the number of filters, and `numChannels` is the number of channels of the input data.

`"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

`filterSize`-by-`numFiltersPerGroup`-by-`numChannelsPerGroup`-by-`numGroups` array, where `filterSize` is the size of the 1-D filters, `numFiltersPerGroup` is the number of filters per group, `numChannelsPerGroup` is the number of channels per group of the input data, and `numGroups` is the number groups.

`numChannelsPerGroup` must equal the number of the channels of the input data divided by `numGroups`.

`"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 convolutionFirst `"S"` (spatial)Filter height

`filterSize(1)`-by-`filterSize(2)`-by-`numFilters`-by-`numChannels` array, where `filterSize(1)` and `filterSize(2)` are the height and width of the 2-D filters, respectively, `numFilters` is the number of filters, and `numChannels` is the number of channels of the input data.

`"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 convolutionFirst `"S"` (spatial)Filter height

`filterSize(1)`-by-`filterSize(2)`-by-`numFiltersPerGroup`-by-`numChannelsPerGroup`-by-`numGroups` array, where `filterSize(1)` and `filterSize(2)` are the height and width of the 2-D filters, respectively, `numFiltersPerGroup` is the number of filters per group, `numChannelsPerGroup` is the number of channels per group of the input data, and `numGroups` is the number of groups.

`numChannelsPerGroup` must equal the number of the channels of the input data divided by `numGroups`.

`"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 convolutionFirst `"S"` (spatial)Filter height

`filterSize(1)`-by-`filterSize(2)`-by-`filterSize(3)`-by-`numFilters`-by-`numChannels` array, where `filterSize(1)`, `filterSize(2)`, and `filterSize(3)` are the height, width, and depth of the 3-D filters, respectively, `numFilters` is the number of filters, and `numChannels` is the number of channels of the input data.

`"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 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 .

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.

Dimension order of unformatted input data, specified as a character vector or string scalar `FMT` that provides a label for each dimension of the data.

When you specify the format of a `dlarray` object, each character provides a label for each dimension of the data and must be one of the following:

• `"S"` — Spatial

• `"C"` — Channel

• `"B"` — Batch (for example, samples and observations)

• `"T"` — Time (for example, time steps of sequences)

• `"U"` — Unspecified

You can specify multiple dimensions labeled `"S"` or `"U"`. You can use the labels `"C"`, `"B"`, and `"T"` at most once.

You must specify `DataFormat` when the input data is not a formatted `dlarray`.

Data Types: `char` | `string`

Dimension order of the weights, specified as a character vector or string scalar that provides a label for each dimension of the weights.

The default value of `WeightsFormat` depends on the task:

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.

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.

Data Types: `char` | `string`

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`

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 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

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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

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### 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 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 and bias matrices for standard convolution derived from the layer weights and biases, respectively. This operation is equivalent to the backward function of a standard convolution layer.

## Version History

Introduced in R2019b