Padding arrays

Introduction

The padarray() function generates a padded image from an array img and a specification border of the boundary conditions and amount of padding to add.

padarray([T], img, border)

The function returns a new image that is an expansion of the input image, in which additional pixels are derived from the border of the input image using the extrapolation scheme specified by border.

The function supports one, two or multi-dimensional images. You can specify the element type T of the output image.

The `Pad` type

The type Pad designates the form of padding which should be used to extrapolate pixels beyond the boundary of an image. Instances must set style, a Symbol specifying the boundary conditions of the image.

The symbol must be one of:

:replicate (repeat edge values to infinity),
:circular (image edges "wrap around"),
:symmetric (the image reflects relative to a position between pixels),
:reflect (the image reflects relative to the edge itself).

Refer to the documentation of Pad for more details and examples for each option.

The `Fill` type

The type Fill designates a particular value which will be used to extrapolate pixels beyond the boundary of an image.

Refer to the documentation of Fill for more details and illustrations.

2D Examples

The following examples show the effects of modifying the input array:

\[\mathbf{A} = \boxed{ \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 4 & 6 & 8 & 10 & 12 \\ 3 & 6 & 9 & 12 & 15 & 18 \\ 4 & 8 & 12 & 16 & 20 & 24 \\ 5 & 10 & 15 & 20 & 25 & 30 \\ 6 & 12 & 18 & 24 & 30 & 36 \end{matrix}}.\]

Examples with `Pad`

The command padarray(A, Pad(:replicate, 4, 4)) yields

\[\boxed{ \begin{array}{ccccccccccccc} 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ 1 & 1 & 1 & 1 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 6 & 6 & 6 & 6 \\ 2 & 2 & 2 & 2 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 12 & 12 & 12 & 12 \\ 3 & 3 & 3 & 3 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 18 & 18 & 18 & 18 \\ 4 & 4 & 4 & 4 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 24 & 24 & 24 & 24 \\ 5 & 5 & 5 & 5 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 30 & 30 & 30 & 30 \\ 6 & 6 & 6 & 6 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \end{array} }.\]

The command padarray(A, Pad(:circular,4,4)) yields

\[\boxed{ \begin{array}{ccccccccccccc} 9 & 12 & 15 & 18 & 3 & 6 & 9 & 12 & 15 & 18 & 3 & 6 & 9 & 12 \\ 12 & 16 & 20 & 24 & 4 & 8 & 12 & 16 & 20 & 24 & 4 & 8 & 12 & 16 \\ 15 & 20 & 25 & 30 & 5 & 10 & 15 & 20 & 25 & 30 & 5 & 10 & 15 & 20 \\ 18 & 24 & 30 & 36 & 6 & 12 & 18 & 24 & 30 & 36 & 6 & 12 & 18 & 24 \\ 3 & 4 & 5 & 6 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 1 & 2 & 3 & 4 \\ 6 & 8 & 10 & 12 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 2 & 4 & 6 & 8 \\ 9 & 12 & 15 & 18 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 3 & 6 & 9 & 12 \\ 12 & 16 & 20 & 24 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 4 & 8 & 12 & 16 \\ 15 & 20 & 25 & 30 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 5 & 10 & 15 & 20 \\ 18 & 24 & 30 & 36 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 6 & 12 & 18 & 24 \\ 3 & 4 & 5 & 6 & 1 & 2 & 3 & 4 & 5 & 6 & 1 & 2 & 3 & 4 \\ 6 & 8 & 10 & 12 & 2 & 4 & 6 & 8 & 10 & 12 & 2 & 4 & 6 & 8 \\ 9 & 12 & 15 & 18 & 3 & 6 & 9 & 12 & 15 & 18 & 3 & 6 & 9 & 12 \\ 12 & 16 & 20 & 24 & 4 & 8 & 12 & 16 & 20 & 24 & 4 & 8 & 12 & 16 \end{array} }.\]

The command padarray(A, Pad(:symmetric,4,4)) yields

\[\boxed{ \begin{array}{ccccccccccccc} 16 & 12 & 8 & 4 & 4 & 8 & 12 & 16 & 20 & 24 & 24 & 20 & 16 & 12 \\ 12 & 9 & 6 & 3 & 3 & 6 & 9 & 12 & 15 & 18 & 18 & 15 & 12 & 9 \\ 8 & 6 & 4 & 2 & 2 & 4 & 6 & 8 & 10 & 12 & 12 & 10 & 8 & 6 \\ 4 & 3 & 2 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 5 & 4 & 3 \\ 4 & 3 & 2 & 1 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 6 & 5 & 4 & 3 \\ 8 & 6 & 4 & 2 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 12 & 10 & 8 & 6 \\ 12 & 9 & 6 & 3 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 18 & 15 & 12 & 9 \\ 16 & 12 & 8 & 4 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 24 & 20 & 16 & 12 \\ 20 & 15 & 10 & 5 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 30 & 25 & 20 & 15 \\ 24 & 18 & 12 & 6 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 36 & 30 & 24 & 18 \\ 24 & 18 & 12 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 30 & 24 & 18 \\ 20 & 15 & 10 & 5 & 5 & 10 & 15 & 20 & 25 & 30 & 30 & 25 & 20 & 15 \\ 16 & 12 & 8 & 4 & 4 & 8 & 12 & 16 & 20 & 24 & 24 & 20 & 16 & 12 \\ 12 & 9 & 6 & 3 & 3 & 6 & 9 & 12 & 15 & 18 & 18 & 15 & 12 & 9 \end{array} }.\]

The command padarray(A, Pad(:reflect,4,4)) yields

\[\boxed{ \begin{array}{ccccccccccccc} 25 & 20 & 15 & 10 & 5 & 10 & 15 & 20 & 25 & 30 & 25 & 20 & 15 & 10 \\ 20 & 16 & 12 & 8 & 4 & 8 & 12 & 16 & 20 & 24 & 20 & 16 & 12 & 8 \\ 15 & 12 & 9 & 6 & 3 & 6 & 9 & 12 & 15 & 18 & 15 & 12 & 9 & 6 \\ 10 & 8 & 6 & 4 & 2 & 4 & 6 & 8 & 10 & 12 & 10 & 8 & 6 & 4 \\ 5 & 4 & 3 & 2 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 5 & 4 & 3 & 2 \\ 10 & 8 & 6 & 4 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 10 & 8 & 6 & 4 \\ 15 & 12 & 9 & 6 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 15 & 12 & 9 & 6 \\ 20 & 16 & 12 & 8 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 20 & 16 & 12 & 8 \\ 25 & 20 & 15 & 10 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 25 & 20 & 15 & 10 \\ 30 & 24 & 18 & 12 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 30 & 24 & 18 & 12 \\ 25 & 20 & 15 & 10 & 5 & 10 & 15 & 20 & 25 & 30 & 25 & 20 & 15 & 10 \\ 20 & 16 & 12 & 8 & 4 & 8 & 12 & 16 & 20 & 24 & 20 & 16 & 12 & 8 \\ 15 & 12 & 9 & 6 & 3 & 6 & 9 & 12 & 15 & 18 & 15 & 12 & 9 & 6 \\ 10 & 8 & 6 & 4 & 2 & 4 & 6 & 8 & 10 & 12 & 10 & 8 & 6 & 4 \end{array} }.\]

Examples with `Fill`

The command padarray(A, Fill(0,(4,4),(4,4))) yields

\[\boxed{ \begin{array}{ccccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} }.\]

3D Examples

Each example is based on a multi-dimensional array $\mathsf{A} \in\mathbb{R}^{2 \times 2 \times 2}$ given by:

\[\mathsf{A}(:,:,1) = \boxed{ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}} \quad \text{and} \quad \mathsf{A}(:,:,2) = \boxed{ \begin{array}{cc} 5 & 6 \\ 7 & 8 \end{array}}.\]

Note that each example will yield a new multi-dimensional array $\mathsf{A}' \in \mathbb{R}^{4 \times 4 \times 4}$ of type OffsetArray, where prepended dimensions may be negative or start from zero.

Examples with `Pad`

The command padarray(A, Pad(:replicate, 1, 1, 1)) gives:

\[\begin{aligned} \mathsf{A}'(:,:,0) & = \boxed{ \begin{array}{cccc} 1 & 1 & 2 & 2 \\ 1 & 1 & 2 & 2 \\ 3 & 3 & 4 & 4 \\ 3 & 3 & 4 & 4 \end{array}} & \mathsf{A}'(:,:,1) & = \boxed{ \begin{array}{cccc} 1 & 1 & 2 & 2 \\ 1 & \boxed{1} & \boxed{2} & 2 \\ 3 & \boxed{3} & \boxed{4} & 4 \\ 3 & 3 & 4 & 4 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 5 & 5 & 6 & 6 \\ 5 & \boxed{5} & \boxed{6} & 6 \\ 7 & \boxed{7} & \boxed{8} & 8 \\ 7 & 7 & 8 & 8 \end{array}} & \mathsf{A}'(:,:,3) & = \boxed{ \begin{array}{cccc} 5 & 5 & 6 & 6 \\ 5 & 5 & 6 & 6 \\ 7 & 7 & 8 & 8 \\ 7 & 7 & 8 & 8 \end{array}} \end{aligned} .\]

The command padarray(A, Pad(:circular, 1, 1, 1)) gives:

\[\begin{aligned} \mathsf{A}'(:,:,0) & = \boxed{ \begin{array}{cccc} 8 & 7 & 8 & 7 \\ 6 & 5 & 6 & 5 \\ 8 & 7 & 8 & 7 \\ 6 & 5 & 6 & 5 \end{array}} & \mathsf{A}'(:,:,1) & = \boxed{ \begin{array}{cccc} 4 & 3 & 4 & 3 \\ 2 & \boxed{1} & \boxed{2} & 1 \\ 4 & \boxed{3} & \boxed{4} & 3 \\ 2 & 1 & 2 & 1 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 8 & 7 & 8 & 7 \\ 6 & \boxed{5} & \boxed{6} & 5 \\ 8 & \boxed{7} & \boxed{8} & 7 \\ 6 & 5 & 6 & 5 \end{array}} & \mathsf{A}'(:,:,3) & = \boxed{ \begin{array}{cccc} 4 & 3 & 4 & 3 \\ 2 & 1 & 2 & 1 \\ 4 & 3 & 4 & 3 \\ 2 & 1 & 2 & 1 \end{array}} \end{aligned} .\]

The command padarray(A,Pad(:symmetric, 1, 1, 1)) gives:

\[\begin{aligned} \mathsf{A}'(:,:,0) & = \boxed{ \begin{array}{cccc} 1 & 1 & 2 & 2 \\ 1 & 1 & 2 & 2 \\ 3 & 3 & 4 & 4 \\ 3 & 3 & 4 & 4 \end{array}} & \mathsf{A}'(:,:,1) & = \boxed{ \begin{array}{cccc} 1 & 1 & 2 & 2 \\ 1 & \boxed{1} & \boxed{2} & 2 \\ 2 & \boxed{3} & \boxed{4} & 4 \\ 2 & 3 & 4 & 4 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 5 & 5 & 6 & 6 \\ 5 & \boxed{5} & \boxed{6} & 6 \\ 7 & \boxed{7} & \boxed{8} & 8 \\ 7 & 7 & 8 & 8 \end{array}} & \mathsf{A}'(:,:,3) & = \boxed{ \begin{array}{cccc} 5 & 5 & 6 & 6 \\ 5 & 5 & 6 & 6 \\ 7 & 7 & 8 & 8 \\ 7 & 7 & 8 & 8 \end{array}} \end{aligned} .\]

The command padarray(A, Pad(:reflect, 1, 1, 1)) gives:

Examples with `Fill`

The command padarray(A, Fill(0, (1, 1, 1))) gives:

\[\begin{aligned} \mathsf{A}'(:,:,0) & = \boxed{ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}} & \mathsf{A}'(:,:,1) & = \boxed{ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & \boxed{1} & \boxed{2} & 0 \\ 0 & \boxed{3} & \boxed{4} & 0 \\ 0 & 0 & 0 & 0 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & \boxed{5} & \boxed{6} & 0 \\ 0 & \boxed{7} & \boxed{8} & 0 \\ 0 & 0 & 0 & 0 \end{array}} & \mathsf{A}'(:,:,3) & = \boxed{ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}} \end{aligned} .\]

BorderArray

BorderArray(inner::AbstractArray, border::AbstractBorder) <: AbstractArray

Construct a thin wrapper around the array inner, with given border. No data is copied in the constructor. Instead, border values are computed on the fly in getindex calls.

Useful for stencil computations.