### Abstract

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) self-similar. For computational reasons the set is also (e) orthogonal and lends itself to (f) rapid computation. The systems are derived from concepts in matrix algebra, but are closely connected to decompositions based on quadrature mirror filters. Our computations take place hierarchically, leading to a pyramid representation in which all of the basis functions have the same basic shape, and appear at many scales. By placing the high-pass and low-pass kernels on staggered grids, we can derived odd-tap QMF kernels that are quite compact. We have developed pyramids using separable, quincunx, and hexagonal kernels. Image data compression with the pyramids gives excellent results, both in terms of MSE and visual appearance. A non-orthogonal variant allows good performance with 3-tap basis kernels and the appropriate inverse sampling kernels.

Original language | English (US) |
---|---|

Pages (from-to) | 50-58 |

Number of pages | 9 |

Journal | Proceedings of SPIE - The International Society for Optical Engineering |

Volume | 845 |

DOIs | |

State | Published - Oct 13 1987 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics
- Electrical and Electronic Engineering
- Computer Science Applications

### Cite this

**Orthogonal Pyramid Transforms for Image Coding.** / Adelson, Edward H.; Simoncelli, Eero.

Research output: Contribution to journal › Article

*Proceedings of SPIE - The International Society for Optical Engineering*, vol. 845, pp. 50-58. https://doi.org/10.1117/12.976485

}

TY - JOUR

T1 - Orthogonal Pyramid Transforms for Image Coding

AU - Adelson, Edward H.

AU - Simoncelli, Eero

PY - 1987/10/13

Y1 - 1987/10/13

N2 - 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) self-similar. For computational reasons the set is also (e) orthogonal and lends itself to (f) rapid computation. The systems are derived from concepts in matrix algebra, but are closely connected to decompositions based on quadrature mirror filters. Our computations take place hierarchically, leading to a pyramid representation in which all of the basis functions have the same basic shape, and appear at many scales. By placing the high-pass and low-pass kernels on staggered grids, we can derived odd-tap QMF kernels that are quite compact. We have developed pyramids using separable, quincunx, and hexagonal kernels. Image data compression with the pyramids gives excellent results, both in terms of MSE and visual appearance. A non-orthogonal variant allows good performance with 3-tap basis kernels and the appropriate inverse sampling kernels.

AB - 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) self-similar. For computational reasons the set is also (e) orthogonal and lends itself to (f) rapid computation. The systems are derived from concepts in matrix algebra, but are closely connected to decompositions based on quadrature mirror filters. Our computations take place hierarchically, leading to a pyramid representation in which all of the basis functions have the same basic shape, and appear at many scales. By placing the high-pass and low-pass kernels on staggered grids, we can derived odd-tap QMF kernels that are quite compact. We have developed pyramids using separable, quincunx, and hexagonal kernels. Image data compression with the pyramids gives excellent results, both in terms of MSE and visual appearance. A non-orthogonal variant allows good performance with 3-tap basis kernels and the appropriate inverse sampling kernels.

UR - http://www.scopus.com/inward/record.url?scp=84957527628&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957527628&partnerID=8YFLogxK

U2 - 10.1117/12.976485

DO - 10.1117/12.976485

M3 - Article

AN - SCOPUS:84957527628

VL - 845

SP - 50

EP - 58

JO - Proceedings of SPIE - The International Society for Optical Engineering

JF - Proceedings of SPIE - The International Society for Optical Engineering

SN - 0277-786X

ER -