haart2

2-D Haar wavelet transform

Syntax

[a,h,v,d] = haart2(x)

[a,h,v,d] = haart2(x,level)

[a,h,v,d] = haart2(___,integerflag)

Description

[a,h,v,d] = haart2(x) performs the 2-D Haar discrete wavelet transform (DWT) of the matrix, x. x is a 2-D, 3-D, or 4-D matrix with even length row and column dimensions. If x is 4-D, the dimensions are Spatial-by-Spatial-by-Channel-by-Batch. The Haar transform is always computed along the row and column dimensions of the input. If the row and column dimensions of x are powers of two, the Haar transform is obtained down to level log2(min(size(x,[1 2]))). If the row or column dimension of x is even, but not a power of two, the Haar transform is obtained down to level floor(log2(min(size(x,[1 2])/2))).

haart2 returns the approximation coefficients, a, at the coarsest level. haart2 also returns cell arrays of matrices containing the horizontal, vertical, and diagonal detail coefficients by level. If the 2-D Haar transform is computed only at one level coarser in resolution, then h, v, and d are matrices. The default level depends on the number of rows of x.

[a,h,v,d] = haart2(x,level) performs the 2-D Haar transform down to the specified level.

[a,h,v,d] = haart2(___,integerflag) specifies how the 2-D Haar transform handles integer-valued data, using any of the previous syntaxes.

Examples

Haar Transform and First Level Details of 2-D Data

Obtain the 2-D Haar transform of 2-D data and plot its diagonal and horizontal level 1 details.

load xbox;
[a,h,v,d] = haart2(xbox);
imagesc(xbox)
title('Original Image')

figure
subplot(2,1,1)
imagesc(d{1})
title('Diagonal Level 1 Details')
subplot(2,1,2)
imagesc(h{1})
title('Horizontal Level 1 Details')

Haar Transform of Image Down to a Specified Level

Show the effect of limiting the maximum level of the 2-D Haar transform on an image.

Load and display the image of a cameraman.

im = imread('cameraman.tif');
imagesc(im)

Obtain the 2-D Haar transform to level 2 and view the level 2 approximation.

[a2,h2,v2,d2] = haart2(im,2);
imagesc(a2)

Haar Transform Using Integer Image Data

Compare 2-D Haar transform results using the default 'noninteger' flag and the 'integer' flag. The cameraman image is uint8 data, so its maximum value is 255.

Obtain the default Haar transform. The approximation detail coefficient is outside the range 0 to 255.

im = imread('cameraman.tif');
[a,h,v,d] = haart2(im);
a

a = 
3.0393e+04

Obtain the Haar transform, limiting it to integer values. The approximation detail is an integer and is within the range of the original image data.

[a,h,v,d] = haart2(im,'integer');
a

a = 
119

Input Arguments

x — Input signal

matrix

Input signal, specified as a 2-D, 3-D, or 4-D real-valued matrix. If x is 4-D, the dimensions are Spatial-by-Spatial-by-Channel-by-Batch. The row and column sizes of x must be even length.

Data Types: single | double

level — Maximum level

positive integer

Maximum level to which to perform the 2-D Haar transform, specified as a positive integer. The default value depends on the length of the input signal, x.

• If both the row and column sizes of x are powers of two, the 2-D Haar transform is obtained down to level log2(min(size(x,[1 2]))).

• If both the row and column sizes of x are even, but at least one is not a power of two, level is equal to floor(log2(min(size(x,[1 2])/2))).

If level is greater than 1, then h, v, and d are cell arrays. If level is equal to 1, then h, v, and d are matrices.

integerflag — Integer-valued data handling

'noninteger' (default) | 'integer'

Integer-valued data handling, specified as either 'noninteger' or 'integer'. 'noninteger' does not preserve integer-valued data in the 2-D Haar transform, and 'integer' preserves it. The 'integer' option applies only if all elements of the input, x, are integers. For integer-valued input, haart2 returns integer-valued wavelet coefficients. For both 'noninteger' and 'integer', however, the 2-D Haar transform algorithm uses floating-point arithmetic. If x is a single-precision input, the numeric type of the Haar transform coefficients is single precision. For all other numeric types, the numeric type of the coefficients is double precision.

Output Arguments

a — Approximation coefficients

scalar | matrix

Approximation coefficients at the coarsest scale, returned as a scalar or matrix of coefficients, depending on the level to which the transform is calculated. Approximation, or scaling, coefficients are a lowpass representation of the input. At each level, the approximation coefficients are divided into coarser approximation and detail coefficients.

Data Types: single | double

h — Horizontal detail coefficients

matrix | cell array

Horizontal detail coefficients by level, returned as a matrix or cell array of matrices. If level is greater than 1, h is a cell array. If level is equal to 1, the 2-D Haar transform is computed at only one level coarser in resolution and h is a matrix.

Note: Generated C and C++ code always returns the horizontal detail coefficients h in a cell array.

Data Types: single | double

v — Vertical detail coefficients

matrix | cell array

Vertical detail coefficients by level, returned as a matrix or cell array of matrices. If level is greater than 1, v is a cell array. If level is equal to 1, the 2-D Haar transform is computed at only one level coarser in resolution and v is a matrix.

Note: Generated C and C++ code always returns the vertical detail coefficients v in a cell array.

Data Types: single | double

d — Diagonal detail coefficients

matrix | cell array

Diagonal detail coefficients by level, returned as a matrix or cell array of matrices. If level is greater than 1, d is a cell array. If level is equal to 1, the 2-D Haar transform is computed at only one level coarser in resolution and d is a matrix.

Note: Generated C and C++ code always returns the diagonal detail coefficients d in a cell array.

Data Types: single | double

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Version History

Introduced in R2016b