# A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

@article{Mallat1989ATF, title={A Theory for Multiresolution Signal Decomposition: The Wavelet Representation}, author={St{\'e}phane Mallat}, journal={IEEE Trans. Pattern Anal. Mach. Intell.}, year={1989}, volume={11}, pages={674-693} }

Multiresolution representations are effective for analyzing the information content of images. [...] Key Method It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. > Expand

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#### References

SHOWING 1-10 OF 86 REFERENCES

Multiresolution approximations and wavelet orthonormal bases of L^2(R)

- Mathematics
- 1989

A multiresolution approximation is a sequence of embedded vector spaces V j jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove… Expand

Orthogonal Pyramid Transforms For Image Coding.

- Mathematics, Engineering
- Other Conferences
- 1987

We describe a set of pyramid transforms that decompose an image into a set of basis functions that are (a) spatial frequency tuned, (b) orientation tuned, (c) spatially localized, and (d)… Expand

DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE

- Mathematics
- 1984

An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant… Expand

The Laplacian Pyramid as a Compact Image Code

- Mathematics, Computer Science
- IEEE Trans. Commun.
- 1983

A technique for image encoding in which local operators of many scales but identical shape serve as the basis functions, which tends to enhance salient image features and is well suited for many image analysis tasks as well as for image compression. Expand

Subband coding of images

- Mathematics, Computer Science
- IEEE Trans. Acoust. Speech Signal Process.
- 1986

A simple yet efficient extension of this concept to the source coding of images by specifying the constraints for a set of two-dimensional quadrature mirror filters for a particular frequency-domain partition and showing that these constraints are satisfied by a separable combination of one-dimensional QMF's. Expand

Scale-Space Filtering

- Mathematics, Computer Science
- IJCAI
- 1983

Scale-space filtering is a method that describes signals qualitatively, managing the ambiguity of scale in an organized and natural way. Expand

Fractal-Based Description of Natural Scenes

- Computer Science, Medicine
- IEEE Transactions on Pattern Analysis and Machine Intelligence
- 1984

The3-D fractal model provides a characterization of 3-D surfaces and their images for which the appropriateness of the model is verifiable and this characterization is stable over transformations of scale and linear transforms of intensity. Expand

Efficiency of a model human image code.

- Computer Science, Medicine
- Journal of the Optical Society of America. A, Optics and image science
- 1987

A code modeled on the simple cells of the primate striate cortex is explored, which maps a digital image into a set of subimages (layers) that are bandpass in spatial frequency and orientation and which is reconstructed from the code. Expand

Cycle-octave and related transforms in seismic signal analysis

- Geology
- 1984

High-resolution seismic methods are needed especially in oil and gas field development. They involve the use of backscattered energy rather than that of reflected signals, and make it interesting to… Expand

Scaling Theorems for Zero Crossings

- Computer Science, Mathematics
- IEEE Transactions on Pattern Analysis and Machine Intelligence
- 1986

It is proved that in any dimension the only filter that does not create generic zero crossings as the scale increases is the Gaussian and this result can be generalized to apply to level crossings of any linear differential operator. Expand