`tile` documentation

`tile` determines indices for dividing a matrix into multiple tiles.

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## Contents

## Syntax

[row,col] = tile(A,maxsiz) [row,col] = tile(siz,maxsiz) [row,col] = tile(...,'overlap',overlap) [row,col,h_rect,h_txt] = tile(...,'show')

## Description

`[row,col] = tile(A,maxsiz)` returns subscript indices `row` and `col` that can be used to divide matrix `A` into tiles, each having a maximum size% `maxsiz`. `maxsiz` can be a scalar to describe the side dimenions of a square tile, or `maxsiz` can be a two-element array to limit tile sizes to rectangular dimensions. Outputs `row` and `col` are cell arrays, each cell containing the range of subscript indices of one tile of `A`. `A` can be a 2D or 3D matrix, but if `A` is a 3D matrix, only its first two dimensions are considered in the tiling calculations.

`[row,col] = tile(siz,maxsiz)` tiles a matrix of size `siz` into tiles of maximum dimensions `maxsiz`.

`[row,col] = tile(...,'overlap',overlap)` specifies the number of rows and columns by which the tiles overlap. overlap can be a scalar or a two-element array.

`[row,col,h_rect,h_txt] = tile(...,'show')` displays numbered rectangles on a plot, showing the tiles. Optional outputs `h_rect` and `h_txt` are handles of the plotted rectangles and text labels.

## Examples

Suppose you have this 20x20 matrix:

Y = peaks(20); imagesc(Y) xlabel 'column \rightarrow' ylabel '\leftarrow row'

But 20x20 is too big for whatever processing algorithm you want to apply to `Y`. It would require too much memory to process the entire matrix at once, so you want to break it up into smaller, bite-size pieces no bigger than 5x5 each. Here's how to do that:

[row,col] = tile(Y,5);

Now we have `row` and `col`, which as you can see:

whos row col

Name Size Bytes Class Attributes col 4x4 2432 cell row 4x4 2432 cell

are each 4x4 cell arrays. Each cell contains the row or column indices that describe a single tile of `Y`. So for example, the row indices of the first tile of `Y` are given by

row{1}

ans = 1.00 2.00 3.00 4.00 5.00

Show all the tiles and their numbers atop the plot of the matrix `Y` by including the `'show'` option when you call `tile`:

tile(20,5,'show'); title 'tile numbers'

The figure above shows that as we would expect, breaking the 20x20 matrix `Y` into tiles, each no larger than 5 rows by 5 columns, results in 16 square tiles. Now it's easy to extract a single tile. For example, to show tile number 10, use `Y(row{10},col{10})`, like this:

figure imagesc(col{10},row{10},Y(row{10},col{10})) title 'tile 10' xlabel 'column \rightarrow' ylabel '\leftarrow row'

## Uneven sizes

What if the size of `Y` isn't perfectly divisible by the maximum tile size? In that case, the resulting tiles will have different sizes.

What if the maximum tile size you want isn't square? In that case, specify the maximum tile size in the form `[maxrow maxcol]`.

Here's how to specify a maximum tile size that isn't square, and that doesn't evenly divide the matrix `Y`:

figure imagesc(Y) xlabel 'column \rightarrow' ylabel '\leftarrow row' tile(Y,[9 4],'show');

## Specifying overlap

For some applications it's necessary to have overlapping tiles. Here's how to add an extra row and column to each tile, such that they overlap:

figure imagesc(Y) xlabel 'column \rightarrow' ylabel '\leftarrow row' tile(Y,[9 6],'overlap',1,'show');

## Process a 2D grid

Suppose you have a big 2160x4320 dataset and you want to process it in tiles no larger than 300x300. Load the dataset and get the rows `r` and columns `c` of each tile:

```
load global_topography.mat
[r,c] = tile(Z,300);
```

To process the 2D matrix `Z` in tiles, we'll loop through each of the 8x15=120 tiles, and we'll process each tile individually. In this example, let's say the processing we want to do is flip each tile upside down. Start by preallocating `Zf`, then operate on each tile:

% Preallocate the output grid: Zf = nan(size(Z)); % Loop through each tile: for k = 1:numel(r) Zf(r{k},c{k}) = flipud(Z(r{k},c{k})); end figure imagescn(lon,lat,Zf) cmocean('topo','pivot') xlabel 'longitude' ylabel 'latitude' title 'each tile flipped upside down'

## Operate on a 3D matrix

The `tile` function was designed primarily to help with large 3D gridded time series that can sometimes strain the limits of your computer's memory. To illustrate how to use this function on such datasets, we'll operate on a relatively small gridded time series, the monthly surface pressure dataset that comes with CDT:

sp = ncread('ERA_Interim_2017.nc','sp'); lat = ncread('ERA_Interim_2017.nc','latitude'); lon = ncread('ERA_Interim_2017.nc','longitude');

The surface pressure dataset is 480x241x12. That "12" at the end corresponds to each of the 12 months of 2017.

What did the median surface pressure look like in 2017? For a small dataset like this we can simply do

spm = median(sp,3);

but if the `sp` dataset were much larger we might need to break it into tiles. To break it into tiles, say each no larger than 50x50, get the rows and columns of each tile like this:

[r,c] = tile(sp,50);

Then to compute the median of `sp` along its third dimension, do something similar to the 2D example above, but add an extra `,:` in the indices to indicate "these rows, these columns, and everything along the third dimension".

Begin by preallocating `spm`, then calculate the median along the third dimension of each tile:

% Preallocate: spm = nan(length(lon),length(lat)); % Loop through each tile: for k = 1:numel(r) spm(r{k},c{k}) = median(sp(r{k},c{k},:),3); end

Plot the 2017 median surface pressure, which we just computed in tiles:

figure imagescn(lon,lat,spm') cmocean tempo % sets the colormap

It looks about right, but we can verify that the tiled solution matches the all-at-once solution like this

isequal(spm,median(sp,3))

ans = logical 1

The `1`, or `true` value says yes, the tiled soltuion matches the non-tiled solution.

## Author Info

This function is part of the Climate Data Toolbox for Matlab. The function and supporting documentation were written by Chad A. Greene of the University of Texas at Austin.