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31.3.2 Wavelet transforms in two dimension

The library provides functions to perform two-dimensional discrete wavelet transforms on square matrices. The matrix dimensions must be an integer power of two. There are two possible orderings of the rows and columns in the two-dimensional wavelet transform, referred to as the “standard” and “non-standard” forms.

The “standard” transform performs a complete discrete wavelet transform on the rows of the matrix, followed by a separate complete discrete wavelet transform on the columns of the resulting row-transformed matrix. This procedure uses the same ordering as a two-dimensional Fourier transform.

The “non-standard” transform is performed in interleaved passes on the rows and columns of the matrix for each level of the transform. The first level of the transform is applied to the matrix rows, and then to the matrix columns. This procedure is then repeated across the rows and columns of the data for the subsequent levels of the transform, until the full discrete wavelet transform is complete. The non-standard form of the discrete wavelet transform is typically used in image analysis.

The functions described in this section are declared in the header file gsl_wavelet2d.h.

— Function: int gsl_wavelet2d_transform (const gsl_wavelet * w, double * data, size_t tda, size_t size1, size_t size2, gsl_wavelet_direction dir, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_transform_forward (const gsl_wavelet * w, double * data, size_t tda, size_t size1, size_t size2, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_transform_inverse (const gsl_wavelet * w, double * data, size_t tda, size_t size1, size_t size2, gsl_wavelet_workspace * work)

These functions compute two-dimensional in-place forward and inverse discrete wavelet transforms in standard form on the array data stored in row-major form with dimensions size1 and size2 and physical row length tda. The dimensions must be equal (square matrix) and are restricted to powers of two. For the transform version of the function the argument dir can be either forward (+1) or backward (-1). A workspace work of the appropriate size must be provided. On exit, the appropriate elements of the array data are replaced by their two-dimensional wavelet transform.

The functions return a status of GSL_SUCCESS upon successful completion. GSL_EINVAL is returned if size1 and size2 are not equal and integer powers of 2, or if insufficient workspace is provided.

— Function: int gsl_wavelet2d_transform_matrix (const gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_direction dir, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_transform_matrix_forward (const gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_transform_matrix_inverse (const gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)

These functions compute the two-dimensional in-place wavelet transform on a matrix a.

— Function: int gsl_wavelet2d_nstransform (const gsl_wavelet * w, double * data, size_t tda, size_t size1, size_t size2, gsl_wavelet_direction dir, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_nstransform_forward (const gsl_wavelet * w, double * data, size_t tda, size_t size1, size_t size2, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_nstransform_inverse (const gsl_wavelet * w, double * data, size_t tda, size_t size1, size_t size2, gsl_wavelet_workspace * work)

These functions compute the two-dimensional wavelet transform in non-standard form.

— Function: int gsl_wavelet2d_nstransform_matrix (const gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_direction dir, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_nstransform_matrix_forward (const gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)
— Function: int gsl_wavelet2d_nstransform_matrix_inverse (const gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)

These functions compute the non-standard form of the two-dimensional in-place wavelet transform on a matrix a.