Open Curve Detection via Minimum Path Keypoints Detection

This partial implementation of a research paper focus on the tools and applications for open curve detection.
Given an input image of a simple or complex and unparametrized curve, we would like to automaticaaly derive a discrete parametrisation
that fit the 2D curve. The applications are for example in medical image analysis where we would like to derive for example
the vessels or aortic valves.

Fast-Marching and 2D distance map

First, we use the Fast-Marching Algorithm to create a distance map on the image. For example, starting from the following:\
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the Fast-Marching, derives automatically the potential map shown below:\
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The Fast-Marching algorithm might be implemented via two techniques:

• Djisktra (L1 norm neighborhood)
• Eikonal equation (L2 norm neighborhood)

Automatic path specification

Using this algorithm, on a open curve:\
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We derive the chart distance:\
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And using back propagation to find the geodesic, we find a potential path on the image:\

Complex curve

On more complex curves, this technique is not robust, see the following figures:\
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Minimal Path Keypoint Detection

To avoid this, the authors introduce a method based on intermediary computation of keypoints
(kind of step by step geodesic), that will be more robust than the Fast-Marching applied directly.

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Using some conditions specifications on the distance between three following keypoints, we can even
come up with a method where we don’t need to specify starting and ending points. See the figures below:\
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The program proposed in the notebooks does not go as far as the research paper but the computations are
very interesting anyway.