\section{Introduction}

\label{sec:intro}

Straight lines are commonly used as visual features for many image analysis
processes.
For instance in computer vision, they are used to estimate the vanishing
points associated to main directions of the 3D world, thus allowing to compute camera
orientation. They are also used to detect structured features for
3D reconstruction.

Therefore, straight line detection is always an active research topic
centered on the quest of still faster, more accurate or more robust-to-noise
methods \cite{AkinlarTopal12,GioiAl10,LuAl15,MatasAl00}.
Most of the times, they rely on the extraction of an edge map based
on gradient magnitude. Gradient orientation is often used to discriminate
candidates and thus provide better efficiency.
However, they seldom provide an exploitable measure of the output line
quality, based on intrinsic properties such as sharpness, connectivity
or scattering.
This information could be useful to get some confidence level and help to
classify these features for further exploitation.
In computer vision applications, it could also be a base for uncertainty
propagation within 3D interpretation tools, in order to dispose of
complementary measures to reprojection errors for local accuracy evaluation.

%Some information may sometimes be drawn from their specific context,
%for example through an analysis of the peak in a Hough transform accumulator.

In digital geometry, new mathematical definitions of classical
geometric objects, such as lines or circles, have been developed
to better fit to the discrete nature of most of today's data to process.
In particular, the notion of blurred segment \cite{Buzer07,DebledAl05} was
introduced to cope with the image noise or other sources of imperfections
from the real world using a width\footnote{We use equivalently the terms
width and thickness in this work.}
parameter.
Efficient algorithms have already been designed to recognize
these digital objects in binary images \cite{DebledAl06}.
Blurred segments seem well suited to reflect the required line quality
information.
%Its preimage,
%i.e. the space of geometric entities which numerization matches this
%blurred segment, may be used to compute some confidence level in the delivered
%3D interpretations, as a promising extension of former works
%on discrete epipolar geometry \cite{NatsumiAl08}.

The present work aims at designing a flexible tool to detect blurred segments
with optimal width and orientation in gray-level images for as well
supervised as unsupervised contexts.
User-friendly solutions are sought, with ideally no parameter to set,
or at least quite few values with intuitive meaning.
A first attempt was already made in a previous work \cite{KerautretEven09}
but the segment width was initially fixed by the user and not estimated,
leading to erroneous orientations of the detected lines.
In the present work, the limitations of this first detector were solved
by the introduction of two new concepts:
(i) {\bf adaptive directional scan} designed to get some
compliance to the unpredictable orientation problem;
(ii) {\bf control of the assigned width} to the blurred segment
recognition algorithm, intended to derive more reliable information on the
line orientation and quality.
As a side effect, these two major evolutions also led to a noticeable
improvement of the time performance of the detector.
They are also put forward within a global line extraction algorithm
which can be evaluated through an online demonstration at :
\href{http://ipol-geometry.loria.fr/~kerautre/ipol_demo/FBSD_IPOLDemo}{
\small{\url{http://ipol-geometry.loria.fr/~kerautre/ipol_demo/FBSD_IPOLDemo}}}

In the next section, the main theoretical notions used in this work are
introduced.
The new detector workflow, the adaptive directional scan, the control
of the assigned width and their integration into both supervised and
unsupervised contexts are then presented in \RefSec{sec:method}.
Experiments led to assess the achieved performance of this new detector
are decribed in \RefSec{sec:expe}.
Finally, \RefSec{sec:conclusion} gives a short conclusion
followed by some open perspectives for future works.