Compute an optimal wavelet basis for vector data:

The best basis is stored in the resulting DiscreteWaveletData object:

Show the coefficient array corresponding to each index in the basis:

Wavelet packet transforms give a DiscreteWaveletData object that includes a default basis:

The basis includes all coefficients at the highest level of refinement:

WaveletBestBasis gives a new DiscreteWaveletData object with a different, optimal basis:

Compute best basis for packet transforms with any number of levels of refinement:

Compute wavelet basis, minimizing log-energy for an image:

Compare default basis with best basis in a hierarchical grid layout of coefficient images:

Compute best basis with custom cost function:

Choose basis with fewest coefficients lying between -1 and 1:

Count values lying between -1 and 1 in default basis:

Best basis:

WaveletBestBasis gives the optimal basis among all possible wavelet bases:

Recursively enumerate all possible wavelet bases for levels of refinement:

Show the 5 possible wavelet bases for 2 levels of refinement:

The number of possible bases grows extremely quickly as a function of refinement level:

Cost value of optimal basis:

Distribution of cost values of a random sample of 1000 bases:

Apply cost functions to coefficients with fixed total energy:

Plot cost as a function of first component energy for "ShannonEntropy":

"LogEnergy":

{"Norm",p}:

{"Threshold",δ}:

In general, each cost function can lead to a different best basis:

Plot distribution of energy among best basis coefficient values for each cost function:

Different cost functions often lead to similar best bases:

Define a custom cost function that favors coefficients that are nearer to integers:

Best basis for list data using custom cost function:

The computed cost value for the {} coefficient is the custom cost value of the original data:

Cost value of the best basis coefficients:

The best basis coefficients are clustered around integers:

Reproduce the built-in cost functions as custom ones:

"ShannonEntropy":

"LogEnergy":

"Norm":

"Threshold":

Use WaveletBestBasis to choose a wavelet basis for threshold-based data compression:

Use a hard threshold of 0.1:

Count nonzero values in thresholded coefficients as a measure of compression:

Compare with compression by thresholding original data:

Plot original and reconstructed data:

Compare threshold-based compression for various threshold values :

Use the same threshold value for best basis selection and thresholding:

Show reconstructed data along with the number of nonzero values as a measure of compression:

Length of uncompressed original data:

Simple lossy image compression for various threshold values :

Show reconstructed data along with the number of nonzero values as a measure of compression:

Total number of pixel values in all channels of original image: