Directional total variation

The total variation regularization (i.e. minimizing the norm of the gradient) is not well adapted to curvilinear structures since it penalizes the object contours. Indeed, curvilinear structures are essentially composed of edges and so, are highly penalized and tend to be removed by the total variation.

We propose a more suitable regularization term for curvilinear structure filtering, by considering both intensity and directional features of RORPO. We regularize inside the curvilinear structures solely along their local axis, and we keep an isotropic regularization outside the curvilinear structures.

Initial image image_comparison_synthetic
Regularization weight λ=0.1 λ=0.6 λ=1


segmentation_classique_0-10 segmentation_classique_0-60 segmentation_classique_1-00
Our directional


segmentation_directions_0-10 segmentation_directions_0-60 segmentation_directions_1-00

This directional regularization tends to reconnect noisy curvilinear structures segments and better preserves their extremities. We used our regularization in a segmentation framework, but it could also be used in other variational problems such as denoising, debluring, inpainting, etc


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