### Abstract

From Communication to Pattern Recognition, from low layer signal processing to high layer cognition, from practice to theory of engineering principles, the question of inherent complexities of entities represented as sets in euclidean space is of fundamental interest. In this paper, we present some fundamental theoretical results pertaining to the question of how many randomly selected labelled example points it takes to reconstruct a set in euclidean space, and thereby propose a morphological sampling theorem in the form of Stochastic Morphological Sampling Theorem. Drawing on results and concepts from Mathematical Morphology and Learnability Theory, we pursue a set-theoretic approach and demonstrate some provable performances pertaining to euclidean-set-reconstruction from stochastic samples. In particular, we demonstrate a result towards the formulation of a stochastic (morphological) version of the Nyquist Sampling Theorem- that, under weak assumptions on the situation under consideration, the number of randomly-drawn (positive) example points needed to reconstruct the target set is at most polynomial in the performance parameters and also the complexity of the target set as loosely captured by size, dimension and surface-area. The reconstruction result of this paper pertaining to the complexity of euclidean sets has natural interpretations for the process of smoothing modelled formally as a dilation morphological operation. Thus, in this paper, we formulate and demonstrate a certain fundamental (distribution-free) well-behaving aspect of smoothing by proving a fundamental result pertaining to the (set-theoretic) complexity of sets in euclidean space.

Original language | English |
---|---|

Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Editors | E.R. Dougherty, J.T. Astola |

Pages | 108-119 |

Number of pages | 12 |

Volume | 3304 |

DOIs | |

Publication status | Published - 1998 |

Event | Nonlinear Image Processing IX - San Jose, CA, United States Duration: 1998 Jan 26 → 1998 Jan 27 |

### Other

Other | Nonlinear Image Processing IX |
---|---|

Country | United States |

City | San Jose, CA |

Period | 98/1/26 → 98/1/27 |

### Fingerprint

### Keywords

- Classification
- Complexity
- Distribution-free
- Morphology
- Recognition
- Smoothing
- Stochastic sampling theorem

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 3304, pp. 108-119) https://doi.org/10.1117/12.304591

**Morphological approach to smoothing : Stochastic morphological sampling theorem.** / Kim, Woon Kyung.

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

*Proceedings of SPIE - The International Society for Optical Engineering.*vol. 3304, pp. 108-119, Nonlinear Image Processing IX, San Jose, CA, United States, 98/1/26. https://doi.org/10.1117/12.304591

}

TY - GEN

T1 - Morphological approach to smoothing

T2 - Stochastic morphological sampling theorem

AU - Kim, Woon Kyung

PY - 1998

Y1 - 1998

N2 - From Communication to Pattern Recognition, from low layer signal processing to high layer cognition, from practice to theory of engineering principles, the question of inherent complexities of entities represented as sets in euclidean space is of fundamental interest. In this paper, we present some fundamental theoretical results pertaining to the question of how many randomly selected labelled example points it takes to reconstruct a set in euclidean space, and thereby propose a morphological sampling theorem in the form of Stochastic Morphological Sampling Theorem. Drawing on results and concepts from Mathematical Morphology and Learnability Theory, we pursue a set-theoretic approach and demonstrate some provable performances pertaining to euclidean-set-reconstruction from stochastic samples. In particular, we demonstrate a result towards the formulation of a stochastic (morphological) version of the Nyquist Sampling Theorem- that, under weak assumptions on the situation under consideration, the number of randomly-drawn (positive) example points needed to reconstruct the target set is at most polynomial in the performance parameters and also the complexity of the target set as loosely captured by size, dimension and surface-area. The reconstruction result of this paper pertaining to the complexity of euclidean sets has natural interpretations for the process of smoothing modelled formally as a dilation morphological operation. Thus, in this paper, we formulate and demonstrate a certain fundamental (distribution-free) well-behaving aspect of smoothing by proving a fundamental result pertaining to the (set-theoretic) complexity of sets in euclidean space.

AB - From Communication to Pattern Recognition, from low layer signal processing to high layer cognition, from practice to theory of engineering principles, the question of inherent complexities of entities represented as sets in euclidean space is of fundamental interest. In this paper, we present some fundamental theoretical results pertaining to the question of how many randomly selected labelled example points it takes to reconstruct a set in euclidean space, and thereby propose a morphological sampling theorem in the form of Stochastic Morphological Sampling Theorem. Drawing on results and concepts from Mathematical Morphology and Learnability Theory, we pursue a set-theoretic approach and demonstrate some provable performances pertaining to euclidean-set-reconstruction from stochastic samples. In particular, we demonstrate a result towards the formulation of a stochastic (morphological) version of the Nyquist Sampling Theorem- that, under weak assumptions on the situation under consideration, the number of randomly-drawn (positive) example points needed to reconstruct the target set is at most polynomial in the performance parameters and also the complexity of the target set as loosely captured by size, dimension and surface-area. The reconstruction result of this paper pertaining to the complexity of euclidean sets has natural interpretations for the process of smoothing modelled formally as a dilation morphological operation. Thus, in this paper, we formulate and demonstrate a certain fundamental (distribution-free) well-behaving aspect of smoothing by proving a fundamental result pertaining to the (set-theoretic) complexity of sets in euclidean space.

KW - Classification

KW - Complexity

KW - Distribution-free

KW - Morphology

KW - Recognition

KW - Smoothing

KW - Stochastic sampling theorem

UR - http://www.scopus.com/inward/record.url?scp=0032374163&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032374163&partnerID=8YFLogxK

U2 - 10.1117/12.304591

DO - 10.1117/12.304591

M3 - Conference contribution

AN - SCOPUS:0032374163

VL - 3304

SP - 108

EP - 119

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

A2 - Dougherty, E.R.

A2 - Astola, J.T.

ER -