## Abstract

In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.

Original language | English (US) |
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Pages (from-to) | 45-77 |

Number of pages | 33 |

Journal | Acta Applicandae Mathematicae |

Volume | 59 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1999 |

### Bibliographical note

Funding Information:★ This work was partially supported by the National Science Foundation ECS 91-22106, DMS 92-04192, DMS 95-00931, by the National Science Foundation Learning and Intelligent Systems Program (LIS), by the Air Force Office of Scientific Research AF/F49620-94-1-0058DEF, by the Army Research Office DAAH04-94-G-0054, DAAH04-93-G-0332, AFOSR-MURI, by the Office of Naval Research ONR-N00014-97-1-0509, by the Office of Naval Research Young Investigator Award, and by the Presidential Early Career Awards for Scientists and Engineers (PECASE).

## Keywords

- Active contours
- Affine gradient
- Affine invariant detection
- Affine scale-space
- Denoising
- Geodesics
- Gradient flows
- Invariant metrics
- Segmentation