# On Detection of Faint Edges in Noisy Images

## TL;DR

This paper introduces a formal framework and efficient multiscale algorithms for detecting faint edges in noisy images, including straight, curved, and fiber-like structures, with demonstrated success in simulations and real microscopy images.

## Contribution

It provides a formalism for understanding faint edge detectability and develops novel hierarchical algorithms for detecting various edge types in noisy conditions.

## Key findings

- Effective detection of faint edges demonstrated in simulations.
- Algorithms successfully applied to real microscopy images.
- Enhanced fiber detection and nerve axon visualization achieved.

## Abstract

A fundamental question for edge detection in noisy images is how faint can an edge be and still be detected. In this paper we offer a formalism to study this question and subsequently introduce computationally efficient multiscale edge detection algorithms designed to detect faint edges in noisy images. In our formalism we view edge detection as a search in a discrete, though potentially large, set of feasible curves. First, we derive approximate expressions for the detection threshold as a function of curve length and the complexity of the search space. We then present two edge detection algorithms, one for straight edges, and the second for curved ones. Both algorithms efficiently search for edges in a large set of candidates by hierarchically constructing difference filters that match the curves traced by the sought edges. We demonstrate the utility of our algorithms in both simulations and applications involving challenging real images. Finally, based on these principles, we develop an algorithm for fiber detection and enhancement. We exemplify its utility to reveal and enhance nerve axons in light microscopy images.