\section{Introduction}

\label{sec:intro}

\subsection{Motivations}

Straight edge detection is a preliminary step of many image analysis
processes. Therefore it is always an active research topic centered on
the quest of still faster, more accurate or more robust-to-noise methods
\cite{MatasAl00,GioiAl10,AkinlarTopal12,LuAl15}.
However they seldom provide an exploitable measure of the output edge
quality, based on intrinsic properties such as sharpness, connectivity
or scattering.
%Some information may sometimes be drawn from their specific context,
%for example through an analysis of the peak in a Hough transform accumulator.

Digital geometry is a recent research domain where new mathematical definitions
of quite classical geometric objects, such as lines or circles, are introduced
to better fit to the discrete nature of most of todays data to process.
In particular, the notion of blurred segment \cite{DebledAl05,Buzer07} was
introduced to cope with the image noise or other sources of imperfections
from the real world by the mean of a width parameter.
It is well suited to reflect the required edge quality information.
Moreover efficient algorithms have already been designed to recognize
these digital objects in binary images.

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 to an end user.

\subsection{Previous work}

In a former paper \cite{KerautretEven09}, an efficient tool to detect
blurred segments of fixed width in gray-level images was already introduced.
It is based on a first rough detection in a local area
of the image either defined by the user in supervised context or blindly
explored in automatic mode. The goal is to disclose the presence of an edge.
Therefore a simple test as the gradient maximal value is performed.

In case of success, refinement steps are run through an exploration of
the image in the direction of the detected edge.
In order to prevent local disturbances such as the presence of a sharper
edge nearby, all the local gradient maxima are successively tested
untill a correct candidate with an acceptable gradient orientation is found.

Only the gradient information is processed as it provides a good information
on the image dynamics, and hence the presence of edges.
Trials to also use the intensity signal were made through costly correlation
techniques, but they were mostly successful for detecting shapes with a
stable appearance such as metallic tubular objects \cite{AubryAl17}.

Despite of good performances achieved, several drawbacks remain.
First, the blurred segment width is not measured but initially set by the
user according to the application requirements. The produced information
on the edge quality is rather poor, and especially when the edge is thin,
the risk to incorporate outlier points is quite high, thus producing a
biased estimation of the edge orientation.

Then, two refinement steps are systematically run.
On one hand, this is useless when the first detection is successfull.
On the other hand, there is no guarantee that this approach is able to
process larger images.
The search direction relies on the support vector of the blurred segment
detected at the former step, and the numerization rounding fixes a limit
on this estimated orientation accuracy.
It results that more steps would inevitably be necessary to process higher
resolution images.

\subsection{Main contributions}

The work presented in this paper aims at solving both former mentioned
drawbacks through two main contributions:
the concept of adaptive directional scanner designed to get some compliance
to the unpredictable orientation problem;
the control of the assigned width to the blurred segment recognition algorithm,
intended to derive more reliable information on the edge orientation and
quality.
As a side effect, these two major evolutions also led to a noticeable
improvement of the time performance of the detector.

In the next section, the main theoretical notions this work relies on are
introduced.
Then the new detector workflow, the adaptive directional scanner, the control
of the assigned with and their integration into both supervised
and unsupervised contexts are presented and discussed in \RefSec{sec:method}.
Experiments led to assess the expected increase of performance are decribed
in \RefSec{sec:expe}.
Finally achieved results are summarized in \RefSec{sec:conclusion},
followed by some open perspectives for future works.