Fractal functions and wavelet expansions based on several scaling functions
Journal of Approximation Theory
Classification Using Adaptive Wavelets for Feature Extraction
IEEE Transactions on Pattern Analysis and Machine Intelligence
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Algorithms for designing wavelets to match a specified signal
IEEE Transactions on Signal Processing
A new approach for estimation of statistically matched wavelet
IEEE Transactions on Signal Processing
-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
IEEE Transactions on Signal Processing
Sparse signal reconstruction from limited data using FOCUSS: are-weighted minimum norm algorithm
IEEE Transactions on Signal Processing
Algebraic theory of optimal filterbanks
IEEE Transactions on Signal Processing
Matching pursuits with time-frequency dictionaries
IEEE Transactions on Signal Processing
Results on principal component filter banks: colored noise suppression and existence issues
IEEE Transactions on Information Theory
Singularity detection and processing with wavelets
IEEE Transactions on Information Theory - Part 2
On the optimal choice of a wavelet for signal representation
IEEE Transactions on Information Theory - Part 2
Signal analysis using a multiresolution form of the singular value decomposition
IEEE Transactions on Image Processing
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We propose a new approach to construct adaptive multiscale orthonormal (AMO) bases of R^N that provide highly sparse signal representations. Our new multilayer AMO basis design produces a high proportion of small scale vectors. The basis vectors are built from small scale to large scales, layer by layer. For each layer, the basis vector maximizes a p-norm measure of sparsity. We compare the sparsity ratios SR (i.e. the percentage of negligibly small coefficients) obtained with AMO and Daubechies wavelet bases for seven families of piecewise smooth signals with randomly located discontinuities. The signals are composed of polynomial, sinusoidal and exponential pieces. In all cases, AMO bases produce a SR increase ranging from 6% to 37%. AMO bases have three main advantages over wavelets. First, they are found automatically by solving a sequence of optimization problems, which eliminates the problem of selecting a wavelet for a given signal. Second, they can provide a significantly sparser representation. Finally, they have the ability to produce zero coefficients for a larger family of piecewise smooth signals. The drawbacks of AMO bases are computational: the basis computation is more expensive, the basis vectors require storage space and no fast transform is known.