MATLAB - Multi-Dimensional Array

MATLAB - Multi-dimensional Arrays

While vectors (1-D) and matrices (2-D) are the most common arrays in MATLAB, the language natively supports arrays with any number of dimensions. A multi-dimensional (or N-D) array is a powerful way to store and organize complex data. You can think of a 3-D array as a book of spreadsheets, where each page is a 2-D matrix.

Practical Use Cases

Multi-dimensional arrays are essential for many real-world applications:

• Image Processing: A standard color (RGB) image is an M-by-N-by-3 array, where M is the height, N is the width, and the 3 “pages” represent the Red, Green, and Blue color channels.
• Time-Series Analysis: Storing data from multiple sensors over time can be represented as a num_samples-by-num_sensors-by-num_trials array.
• Numerical Simulation: Storing the results of a parameter sweep, like the temperature on a plate over time, could be an x_position-by-y_position-by_time_step array.

Creating Multi-dimensional Arrays

You can create N-D arrays using initialization functions or by concatenating existing arrays.

% 1. Using initialization functions (recommended)
% Create a 2x4x3 array of random numbers
A = rand(2, 4, 3);

% Create a 5x2x4 array of all zeros
Z = zeros(5, 2, 4);

% 2. Concatenating existing matrices
% The cat() function is the robust way to build N-D arrays
mat1 = [1 2; 3 4]; % 2x2 matrix
mat2 = [5 6; 7 8]; % Another 2x2 matrix

% Concatenate along the 3rd dimension (pages) to create a 2x2x2 array
C = cat(3, mat1, mat2);

The size() function works for N-D arrays, returning the length of each dimension.

size(C)
ans =
     2     2     2

Accessing Elements

To access elements, you provide an index for each dimension, separated by commas. The colon : operator selects all elements in a dimension, and the end keyword refers to the last element.

% Let's use the 2x2x2 array C from the previous example
% C(:,:,1) is [1 2; 3 4], C(:,:,2) is [5 6; 7 8]

% Get the element in row 1, col 2, page 2
elem = C(1, 2, 2);
% Result: 6

% Get the entire first row from the first page
row_vec = C(1, :, 1);
% Result: [1 2]

% Get the entire second column from all pages
% This results in a 2x1x2 array
col_matrix = C(:, 2, :);

% Use squeeze() to remove singleton dimensions (like the middle '1')
squeezed_cols = squeeze(col_matrix);
% Result: a 2x2 matrix [2 6; 4 8]

Computation with Multi-dimensional Arrays

Most MATLAB mathematical functions can operate along a specified dimension of an N-D array. This is extremely powerful for data analysis.

% Create a 3x4x2 sample array
data = randi(10, [3, 4, 2]);

% Find the mean of each column (i.e., operate along dimension 1)
mean_cols = mean(data, 1);
% Result will be a 1x4x2 array

% Find the sum of each page (i.e., operate along dimension 3)
sum_pages = sum(data, 3);
% Result will be a 3x4 matrix

Manipulating Array Dimensions

Sometimes you need to rearrange the dimensions of an array for computation or interfacing with other tools. MATLAB provides several key functions for this.

• reshape(A, sz): Changes the size of an array without changing the order of its elements in memory. The total number of elements must remain the same.
• permute(A, order): Rearranges the dimensions of the array. This is useful for tasks like transposing an RGB image from MxNx3 to 3xMxN.
• squeeze(A): Removes dimensions of length 1. Useful for simplifying arrays after slicing.
• pagetranspose(A) or A.': Transposes each 2-D page of a multi-dimensional array (new in R2020b).

Example: Permuting Image Channels

% Create a pseudo-image: 100 pixels high, 80 pixels wide, 3 color channels
rgbImage = rand(100, 80, 3);

% Some libraries might expect the channel dimension first (channel x height x width)
% The original order is [1, 2, 3] for (height, width, channel)
% We want the new order to be [3, 1, 2]
permutedImage = permute(rgbImage, [3, 1, 2]);

size(rgbImage)
ans =
   100    80     3

size(permutedImage)
ans =
     3   100    80