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Depending on the size of the signal and the filter, any one of the following

- **Non overlapping Batches** - All batched filters are applied to all batched signals. The batch

axis of Signal and Filter **should not** be the same.

\note All non-overlapping(interleaved) convolutions default to frequency domain

\ref AF_CONV_FREQ irrespective of the provided convolution mode argument.

signal and filter respectively. This formulation only decides the dimensionality of convolution.

Given below are some examples on how convolution dimensionality is computed.

| Signal Size | Filter Size | Input Rank | Filter Rank | Convolve Dimensionality |

|:--------------:|:--------------:|:----------:|:-----------:|:-------------------------:|

| \dims{m,n,1,1} | \dims{m,1,1,1} | 2 | 1 | \f$ min(2, 1) => \f$ 1D |

| \dims{m,1,1,1} | \dims{m,n,1,1} | 1 | 2 | \f$ min(1, 2) => \f$ 1D |

| \dims{m,n,1,1} | \dims{m,n,1,1} | 2 | 2 | \f$ min(2, 2) => \f$ 2D |

| \dims{m,n,1,1} | \dims{m,n,p,1} | 2 | 3 | \f$ min(2, 3) => \f$ 2D |

| \dims{m,n,1,p} | \dims{m,n,1,q} | 4 | 4 | 3D |

| \dims{m,n,p,1} | \dims{m,n,q,1} | 3 | 3 | \f$ min(3, 3) => \f$ 3D |

\note In the cases similar to the fifth row of the above table,

signal and filter are of rank 4, the function delegates the

operation to three dimensional convolution \ref signal_func_convolve3

If the operation you intend to perform doesn't align with what this

function does, please check the rank specific convolve functions (hyperlinked below)

documentation to find out more.

For one dimensional signals (lets say m is size of 0th axis), below batch operations are possible.

| Signal Size | Filter Size | Output Size | Batch Mode | Description |

| :------------: | :------------: | :------------: | :----------------------: | :----------------------------------------------------------------- |

| \dims{m,1,1,1} | \dims{m,1,1,1} | \dims{m,1,1,1} | No Batch | Output will be a single convolved array |

| \dims{m,1,1,1} | \dims{m,n,1,1} | \dims{m,n,1,1} | Filter is Batched | n filters applied to same input |

| \dims{m,n,1,1} | \dims{m,1,1,1} | \dims{m,n,1,1} | Signal is Batched | 1 filter applied to n inputs |

| \dims{m,n,p,q} | \dims{m,n,p,q} | \dims{m,n,p,q} | Identical Batches | n*p*q filters applied to n*p*q inputs in one-to-one correspondence |

| \dims{m,n,1,1} | \dims{m,1,p,q} | \dims{m,n,p,q} | Non-overlapping batches | p*q filters applied to n inputs to produce n x p x q results |

There are various other permutations of signal and filter sizes that fall under

the category of non-overlapping batch mode that are not listed in the above

table. For any signal and filter size combination to fall under the

non-overlapping batch mode, they should satisfy one of the following conditions.

- Signal and filter size along a given batch axis (\f$ > 1 \f$) should be same.

- Either signal size or filter size along a given batch axis (\f$ > 1 \f$) should be equal to one.

\note For the above tabular illustrations, we assumed \ref af_conv_mode is \ref AF_CONV_DEFAULT.

A detailed explanation of each batch mode for 2D convolutions is provided below.

Given below are definitions of variables and constants that are used to

facilitate easy illustration of the operations.

- \f$[M\quad N]\f$, \f$[A\quad B]\f$ are signal, filter sizes along

\f$0^{th}\f$ & \f$1^{st}\f$ axes respectively.

- \f$P\f$ and \f$Q\f$ are two constants, integers greater than one.

- \f$ p \f$ is an integer variable with range \f$ \ 0 \leq p < P \f$.

- \f$ q \f$ is an integer variable with range \f$ \ 0 \leq q < Q \f$.

- O, S and F are notations for Output, Signal and Filter respectively.

We have also used images to showcase some examples which follow the

below notation.

- Each blue line is a two dimensional matrix.

- Each orange line indicates a full 2d convolution operation.

- Suffix of each letter indicates indices along \f$ 3^{rd}\f$ and \f$ 4^{th}\f$

axes in the order of appearance from left to right in the suffix.

- O, S and F are notations for Output, Signal and Filter respectively.

### No Batch

Given below is an example of no batch mode.

\image html "conv_docs_images/basic.png" "Single 2d convolution with 2d filter"

For input size \dims{M,N,1,1} and filter size \dims{A,B,1,1}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,1,1} = \convolve_eq{\shape_t{S,M,N,1,1},\shape_t{F,A,B,1,1}}

### Batched Filter

Given below is an example of filter batch mode.

\image html "conv_docs_images/filter.png" "Single signal convolved with many filters independently"

For input size \dims{M,N,1,1} and filter size \dims{A,B,P,1}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,P,1} = \set_eq{\convolve_t{\shape_t{S,M,N,1,1},\shape_t{f,A,B,p,1}}, \forall \shape_t{f,A,B,p,1} \in \shape_t{F,A,B,P,1}}

### Batched Signal

Given below is an example of signal batch mode.

\image html "conv_docs_images/signal.png" "Single filter convolved with many signals independently"

For input size \dims{M,N,P,1} and filter size \dims{A,B,1,1}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,P,1} = \set_eq{\convolve_t{\shape_t{s,M,N,p,1},\shape_t{F,A,B,1,1}}, \forall \shape_t{s,M,N,p,1} \in \shape_t{S,M,N,P,1}}

### Identical Batch Sizes

Given below is an example of identical batch mode.

\image html "conv_docs_images/identical.png" "Many signals convolved with many filters in one-on-one manner"

For input size \dims{M,N,P,Q} and filter size \dims{A,B,P,Q}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,P,Q} = \set_eq{\convolve_t{\shape_t{s,M,N,p,q},\shape_t{f,A,B,p,q}}, \forall \shape_t{s,M,N,p,q} \in \shape_t{S,M,N,P,Q} \land \forall \shape_t{f,M,N,p,q} \in \shape_t{F,M,N,P,Q}}

### Non-overlapping Batches

Four different kinds of signal and filter size combinations are handled in this batch mode. Each one

of them are explained in respective sections below.

#### Combination 1

For input size \dims{M,N,P,1} and filter size \dims{A,B,1,Q}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,P,Q} = \set_eq{\set_t{\convolve_t{\shape_t{s,M,N,p,1},\shape_t{f,A,B,1,q}}, \forall \shape_t{s,M,N,p,1} \in \shape_t{S,M,N,P,1}}, \forall \shape_t{f,A,B,1,q} \in \shape_t{F,A,B,1,Q}}

Given below is an example of this batch mode.

\image html "conv_docs_images/non-overlapping_1.png"

#### Combination 2

For input size \dims{M,N,P,1} and filter size \dims{A,B,P,Q}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,P,Q} = \set_eq{\set_t{\convolve_t{\shape_t{s,M,N,p,1},\shape_t{f,A,B,p,q}}, \forall \shape_t{f,A,B,p,q} \in \shape_t{F,A,B,P,Q}}, \forall \shape_t{s,M,N,p,1} \in \shape_t{S,M,N,P,1}}

Given below is an example of this batch mode.

\image html "conv_docs_images/non-overlapping_2.png"

#### Combination 3

For input size \dims{M,N,1,P} and filter size \dims{A,B,Q,1}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,Q,P} = \set_eq{\set_t{\convolve_t{\shape_t{s,M,N,1,p},\shape_t{f,A,B,q,1}}, \forall \shape_t{s,M,N,1,p} \in \shape_t{S,M,N,1,P}}, \forall \shape_t{f,A,B,q,1} \in \shape_t{F,A,B,Q,1}}

Given below is an example of this batch mode.

\image html "conv_docs_images/non-overlapping_3.png"

#### Combination 4

For input size \dims{M,N,P,Q} and filter size \dims{A,B,P,1}, the following set-builder

notation gives a formal definition of all convolutions performed in this mode.

\shape_eq{O,M,N,P,Q} = \set_eq{\set_t{\convolve_t{\shape_t{s,M,N,p,q},\shape_t{f,A,B,p,1}}, \forall \shape_t{s,M,N,p,q} \in \shape_t{S,M,N,P,Q}}, \forall \shape_t{f,A,B,p,1} \in \shape_t{F,A,B,P,1}}

Given below is an example of this batch mode.

\image html "conv_docs_images/non-overlapping_4.png"

The batching behavior of convolve2NN functions(\ref af_convolve2_nn() and

\ref convolve2NN() ) is different from convolve2. The new functions can perform 2D

convolution on 3D signals and filters in a way that is more aligned with

convolutional neural networks.

| Signal Size | Filter Size | Output Size | Batch Mode | Description |

| :-----------------: | :-----------------: | :-----------------: | :------------: | :-------------------------------------------------- |

| \dims{M, N, 1, 1} | \dims{M, N, 1, 1} | \dims{M, N, 1, 1} | No Batch | Output will be a single convolved array |

| \dims{M, N, 1, 1} | \dims{M, N, P, 1} | \dims{M, N, P, 1} | *Invalid* | Size along second axis should be same |

| \dims{M, N, P, 1} | \dims{M, N, 1, 1} | \dims{M, N, P, 1} | *Invalid* | Size along second axis should be same |

| \dims{M, N, P, 1} | \dims{M, N, P, 1} | \dims{M, N, 1, 1} | No Batch | 3D Signal and 3D filter convoled to 2D result |

| \dims{M, N, P, Qs} | \dims{M, N, P, Qf} | \dims{M, N, Qf, Qs} | Batch Qs * Qf | Qs signals and Qf filters to create Qs * Qf results |

\note For the above tabular illustrations, we will assume \ref af_conv_mode is \ref AF_CONV_DEFAULT.

For three dimensional inputs with m, n & p sizes along the 0th, 1st & 2nd axes

| Signal Size | Filter Size | Output Size | Batch Mode | Description |

| :----------------: | :----------------: | :----------------: | :----------------: |:-----------------------------------------------------------|

| \dims{m, n, p, 1} | \dims{a, b, c, 1} | \dims{m, n, p, 1} | No Batch | Output will be a single convolve array |

| \dims{m, n, p, 1} | \dims{a, b, c, d} | \dims{m, n, p, d} | Filter is Batched | d filters applied to same input |

| \dims{m, n, p, q} | \dims{a, b, c, 1} | \dims{m, n, p, q} | Signal is Batched | 1 filter applied to q inputs |

| \dims{m, n, p, k} | \dims{a, b, c, k} | \dims{m, n, p, k} | Identical Batches | k filters applied to k inputs in one-to-one correspondence |

\note For the above tabular illustrations, we assumed \ref af_conv_mode is \ref AF_CONV_DEFAULT.