### Abstract

Recent papers in multiscale morphological filtering, particularly, have renovated the interest in signal representation via multiscale openings. Although most of the analysis was done with flat structuring elements, extensions to grayscale structuring elements (GSE) are certainly possible. In fact, we have shown that opening a signal with convex and symmetric GSE does not introduce additional zero-crossings as the filter moves to a coarser scales. However, the issue of finding an optimal GSE is still an open problem. In this paper, we present a procedure to find an optimal GSE under the least mean square (LMS) algorithm subject to three constraints: The GSE must be convex, symmetric, and non-negative. The use of the basis functions simplifies the problem formulation. In fact, we show that the basis functions for convex and symmetric GSE are concave and symmetric, thus alternative constraints are developed. The results of this algorithm are compared with our previous work.

Original language | English (US) |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Editors | Edward R. Dougherty, Jaakko T. Astola, Harold G. Longbotham |

Pages | 129-141 |

Number of pages | 13 |

State | Published - Jan 1 1996 |

Event | Nonlinear Image Processing VII - San Jose, CA, USA Duration: Jan 29 1996 → Jan 30 1996 |

### Publication series

Name | Proceedings of SPIE - The International Society for Optical Engineering |
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Volume | 2662 |

ISSN (Print) | 0277-786X |

### Other

Other | Nonlinear Image Processing VII |
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City | San Jose, CA, USA |

Period | 1/29/96 → 1/30/96 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(pp. 129-141). (Proceedings of SPIE - The International Society for Optical Engineering; Vol. 2662).

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*Proceedings of SPIE - The International Society for Optical Engineering.*Proceedings of SPIE - The International Society for Optical Engineering, vol. 2662, pp. 129-141, Nonlinear Image Processing VII, San Jose, CA, USA, 1/29/96.

**Finding optimal convex gray-scale structuring elements for morphological multiscale representation.** / Morales, Aldo W.; Ko, Sung Jea.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Finding optimal convex gray-scale structuring elements for morphological multiscale representation

AU - Morales, Aldo W.

AU - Ko, Sung Jea

PY - 1996/1/1

Y1 - 1996/1/1

N2 - Recent papers in multiscale morphological filtering, particularly, have renovated the interest in signal representation via multiscale openings. Although most of the analysis was done with flat structuring elements, extensions to grayscale structuring elements (GSE) are certainly possible. In fact, we have shown that opening a signal with convex and symmetric GSE does not introduce additional zero-crossings as the filter moves to a coarser scales. However, the issue of finding an optimal GSE is still an open problem. In this paper, we present a procedure to find an optimal GSE under the least mean square (LMS) algorithm subject to three constraints: The GSE must be convex, symmetric, and non-negative. The use of the basis functions simplifies the problem formulation. In fact, we show that the basis functions for convex and symmetric GSE are concave and symmetric, thus alternative constraints are developed. The results of this algorithm are compared with our previous work.

AB - Recent papers in multiscale morphological filtering, particularly, have renovated the interest in signal representation via multiscale openings. Although most of the analysis was done with flat structuring elements, extensions to grayscale structuring elements (GSE) are certainly possible. In fact, we have shown that opening a signal with convex and symmetric GSE does not introduce additional zero-crossings as the filter moves to a coarser scales. However, the issue of finding an optimal GSE is still an open problem. In this paper, we present a procedure to find an optimal GSE under the least mean square (LMS) algorithm subject to three constraints: The GSE must be convex, symmetric, and non-negative. The use of the basis functions simplifies the problem formulation. In fact, we show that the basis functions for convex and symmetric GSE are concave and symmetric, thus alternative constraints are developed. The results of this algorithm are compared with our previous work.

UR - http://www.scopus.com/inward/record.url?scp=0029708890&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029708890&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0029708890

SN - 0819420360

SN - 9780819420367

T3 - Proceedings of SPIE - The International Society for Optical Engineering

SP - 129

EP - 141

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

A2 - Dougherty, Edward R.

A2 - Astola, Jaakko T.

A2 - Longbotham, Harold G.

ER -