From 3d4de8feb231afc36c93dbeb8286877976d754b6 Mon Sep 17 00:00:00 2001
From: even <philippe.even@loria.fr>
Date: Wed, 1 May 2019 13:07:09 +0200
Subject: [PATCH] Article : intro and last comparisons
---
Article/Expe_auto/compTable.tex | 20 ++++++++--------
Article/Fig_synth/statsTable.tex | 4 ++--
Article/abstract.tex | 8 +++----
Article/expeV2.tex | 3 ++-
Article/introV2.tex | 17 +++++++-------
Article/methodV2.tex | 2 +-
Article/notionsV2.tex | 15 ++++++------
Code/FBSD/BSTools/bsdetectionwidget.cpp | 8 +++++++
Code/FBSD/BSTools/extlines.cpp | 23 ++++++++++++-------
.../BlurredSegment/blurredsegmentproto.cpp | 9 ++++----
.../FBSD/BlurredSegment/blurredsegmentproto.h | 6 +++++
Code/FBSD/BlurredSegment/bsdetector.cpp | 2 +-
Code/FBSD/BlurredSegment/bstracker.cpp | 12 ++++++----
Code/FBSD/ConvexHull/antipodal.cpp | 11 +++++----
Code/FBSD/ConvexHull/antipodal.h | 16 +++++++++----
Code/FBSD/ConvexHull/convexhull.cpp | 13 +++++------
Code/FBSD/ConvexHull/convexhull.h | 9 ++++----
17 files changed, 107 insertions(+), 71 deletions(-)
diff --git a/Article/Expe_auto/compTable.tex b/Article/Expe_auto/compTable.tex
index 10ccfd0..619e13c 100644
--- a/Article/Expe_auto/compTable.tex
+++ b/Article/Expe_auto/compTable.tex
@@ -1,20 +1,20 @@
-\begin{tabular}{|l||r@{$\pm$}r|r@{$\pm$}r|r@{$\pm$}r|r@{$\pm$}r|r@{$\pm$}r|}
+\begin{tabular}{|l||r@{~$\pm~$}r|r@{~$\pm$~}r|r@{~$\pm$~}r|r@{~$\pm$~}r|r@{~$\pm$~}r|}
\hline
Measure $M$ & \multicolumn{2}{c|}{$T$ (ms)} & \multicolumn{2}{c|}{$C$ (\%)}
& \multicolumn{2}{c|}{$N$}
& \multicolumn{2}{c|}{$L$ (pixels)} & \multicolumn{2}{c|}{$L/N$} \\
\hline
LSD
-& 63.2 & 14.0 & 60.9 & 11.2
-& 561 & 202 & 17956 & 5387 & 33.5 & 7.7 \\
+& 63.5 & 13.6~ & 60.9 & 11.2~
+& 536 & 193~ & 17745 & 5337 & 34.6 & 7.9~ \\
ED-Lines
-& 55.5 & 10.0 & 64.0 & 11.2
-& 617 & 228 & 19775 & 5790 & 33.8 & 8.3 \\
+& 55.5 & 9.9~ & 64.0 & 11.2~
+& 570 & 210~ & 19351 & 5669 & 35.8 & 8.9~ \\
CannyLines
-& 69.8 & 10.5 & 60.5 & 10.6
-& 478 & 142 & 17779 & 4407 & 38.8 & 10.1 \\
-Our detector
-& 68.7 & 15.8 & \textbf{67.9} & \textbf{9.6}
-& 449 & 104 & 19175 & 3912 & \textbf{43.7} & 7.7 \\
+& 69.6 & 10.5~ & 60.5 & 10.6~
+& 467 & 138~ & 17679 & 4398 & 39.5 & 10.1~ \\
+Our detector ~~
+& ~66.9 & 15.6~ & ~\textbf{67.9} & \textbf{9.6}~
+& ~478 & 110~ & 19472 & 3914 & ~\textbf{41.7} & \textbf{7.5}~ \\
\hline
\end{tabular}
diff --git a/Article/Fig_synth/statsTable.tex b/Article/Fig_synth/statsTable.tex
index d93b912..e74ea9a 100644
--- a/Article/Fig_synth/statsTable.tex
+++ b/Article/Fig_synth/statsTable.tex
@@ -14,9 +14,9 @@ Recall (ratio of true detection, \%): $R = \#(D\cap S)/\#S$
& 89.20 & $\pm$ & 3.94 & \textbf{90.08} & $\pm$ & \textbf{2.77} \\
F-measure (harmonic mean,\%): $F = 2\times P\times R/(P+R)$
& 79.85 & $\pm$ & 6.78 & \textbf{84.17} & $\pm$ & \textbf{4.17} \\
-Thickness difference (in pixels) to matched input segment
+Thickness difference (pixels) to matched input segment
& 0.92 & $\pm$ & 0.31 & \textbf{0.76} & $\pm$ & \textbf{0.23} \\
-Angle difference (in degrees) to matched input segment
+Angle difference (degrees) to matched input segment
& 1.48 & $\pm$ & 1.42 & \textbf{1.05} & $\pm$ & \textbf{0.80} \\
\hline
\end{tabular}
diff --git a/Article/abstract.tex b/Article/abstract.tex
index 74c13ef..5cd7860 100755
--- a/Article/abstract.tex
+++ b/Article/abstract.tex
@@ -2,9 +2,9 @@
detector in gray-level images,
where line segments are enriched with a thickness parameter
intended to provide a quality criterion on the extracted feature.
-This study firstly enhances previous works on interactive
-line detection with a better estimation of the segment thickness and
-orientation through two main improvements: adaptive directional scans and
-the control of assigned thickness to the detection algorithm.
+This study is based on a previous work on interactive line detection
+in gray level images. At first, a better estimation of the segment thickness
+and orientation is achieved through two main improvements:
+adaptive directional scans and control of assigned thickness.
Then, a new contribution to the automatic detection of all the segments in
a single image is also proposed and left available in an online demonstration.
diff --git a/Article/expeV2.tex b/Article/expeV2.tex
index 942de10..1fddd26 100755
--- a/Article/expeV2.tex
+++ b/Article/expeV2.tex
@@ -74,7 +74,8 @@ that the new detector is faster and finds more edges than the previous one.
\caption{Measured performance of both versions of the detector on a set of
synthesized images.
Old refers to the previous version \cite{KerautretEven09}, whereas new is
-the present detector augmented with ADS and CAW concepts.
+the present detector (with adaptive directional scans and control of
+assigned width).
$S$ is the set of all the input segments,
$D$ the set of all the detected blurred segments.}
\label{tab:synth}
diff --git a/Article/introV2.tex b/Article/introV2.tex
index 52688b0..b317741 100755
--- a/Article/introV2.tex
+++ b/Article/introV2.tex
@@ -20,7 +20,7 @@ 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
+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.
@@ -48,15 +48,16 @@ with optimal thickness 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 thickness was initially fixed by the user and not estimated,
+%A first attempt was already made in a previous work \cite{KerautretEven09}
+An interactive tool was already designed for live line extractions in
+gray level images \cite{KerautretEven09}.
+But the segment thickness 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
+Here, the limitations of this first detector are solved
by the introduction of two new concepts:
-(i) adaptive directional scan designed to get some
-compliance to the unpredictable orientation problem;
-(ii) control of assigned thickness to the blurred segment
-recognition algorithm.
+(i) adaptive directional scans (ADS) designed to better track the detected line;
+%get some compliance to the unpredictable orientation problem;
+(ii) control of assigned thickness (CAT) to bound its scattering.
% 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
diff --git a/Article/methodV2.tex b/Article/methodV2.tex
index 23bac4e..573f496 100755
--- a/Article/methodV2.tex
+++ b/Article/methodV2.tex
@@ -83,7 +83,7 @@ following sections \ref{subsec:ads} and \ref{subsec:caw}.
The output segment $\mathcal{B}'$ is finally accepted according to the
application needs.
For the present work, only a size test is performed so that blurred
-segments with less than 12 pixels are rejected.
+segments with less than 10 pixels are rejected.
%Too short, too sparse or too fragmented segments
%can be rejected. Length, sparsity or fragmentation thresholds are
%intuitive parameters left at the end user disposal.
diff --git a/Article/notionsV2.tex b/Article/notionsV2.tex
index 8933340..42619cb 100755
--- a/Article/notionsV2.tex
+++ b/Article/notionsV2.tex
@@ -22,15 +22,16 @@ its period (i.e. the length of its periodic pattern).
When $\nu = p(\mathcal{L})$, then $\mathcal{L}$ is the narrowest 8-connected
line and is called a {\it naive line}.
-The {\it thickness} $\mu = \frac{\nu}{max(|a|,|b|)}$ of the digital straight
-line $\mathcal{L}(a,b,c,\nu)$ is the minimum of the vertical and horizontal
+The {\it thickness} $\mu = \frac{\nu}{max(|a|,|b|)}$ of
+% the digital straight line
+$\mathcal{L}(a,b,c,\nu)$ is the minimum of the vertical and horizontal
distances between lines $ax + by = c$ and $ax + by = c + \nu$.
\begin{definition}
A \textbf{blurred segment} $\mathcal{B}$ of assigned thickness $\varepsilon$
-is a set of points in $\mathbb{Z}^2$ that all belong to a digital straight
-line $\mathcal{L}$ of thickness $\mu = \varepsilon$.
-The \textbf{optimal line} of the blurred segment is the digital straight
+is a set of points in $\mathbb{Z}^2$ that all belong to a covering digital
+straight line $\mathcal{L}$ of thickness $\mu = \varepsilon$.
+The \textbf{optimal line} of the blurred segment is the covering
line with minimal thickness.
The \textbf{thickness} of the blurred segment is the thickness of its
optimal line.
@@ -89,7 +90,7 @@ $\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}) = 0$
expresses the orthogonality constraint between the scan lines $\mathcal{N}_i$
and the scan strip $\mathcal{D}$.
Then the shift of the period $p(\mathcal{D})$ between successive scans
-guarantees that all points of the scan strip are travelled one and only one
+guarantees that all points of the scan strip are traversed one and only one
time.
The scans $S_i$ are developed on each side of a start scan $S_0$,
@@ -164,7 +165,7 @@ If we note
$p_{\vec{D}} = max (|X_D|,|Y_D|)$,
$\nu_{\vec{D}} = \lceil w\cdot p_{\vec{D}} \rceil$,
$c_3 = x_C\cdot Y_D - y_C\cdot X_D - \frac{\nu_{\vec{D}}}{2}$,
-$c_4 = X_D\cdot x_C + Y_D\cdot y_C - \frac{p_{\vec{D}}}{2}$,
+and $c_4 = x_C\cdot X_D + y_C\cdot Y_D - \frac{p_{\vec{D}}}{2}$,
it is then defined by
the following scan strip $\mathcal{D}^{C,\vec{D},w}$ and scan lines
$\mathcal{N}_i^{C,\vec{D},w}$:
diff --git a/Code/FBSD/BSTools/bsdetectionwidget.cpp b/Code/FBSD/BSTools/bsdetectionwidget.cpp
index cd6969d..ca0ccf1 100755
--- a/Code/FBSD/BSTools/bsdetectionwidget.cpp
+++ b/Code/FBSD/BSTools/bsdetectionwidget.cpp
@@ -899,6 +899,14 @@ void BSDetectionWidget::keyPressEvent (QKeyEvent *event)
yorkview = new BSYorkView (&loadedImage, &detector);
yorkview->show ();
break;
+
+ case Qt::Key_Period :
+ // Switches the thickenning control
+ detector.switchStrict ();
+ cout << "Assigned strict (leaning lines) thickness control "
+ << (detector.isStrictOn () ? "on" : "off") << endl;
+ extract ();
+ break;
// DEV OUT
case Qt::Key_Plus :
diff --git a/Code/FBSD/BSTools/extlines.cpp b/Code/FBSD/BSTools/extlines.cpp
index 3c37dd2..0cb8240 100755
--- a/Code/FBSD/BSTools/extlines.cpp
+++ b/Code/FBSD/BSTools/extlines.cpp
@@ -176,7 +176,7 @@ double ExtLines::covering (const char *name, int nbinfo,
cl.ey = height - 1 - val[3];
clLength = sqrt ((cl.sx - cl.ex) * (cl.sx - cl.ex)
+ (cl.sy - cl.ey) * (cl.sy - cl.ey));
- if (clLength > minl)
+ if (clLength >= minl)
{
cannys.push_back (cl);
lg += clLength;
@@ -245,6 +245,7 @@ double ExtLines::covering (const char *name, int nbinfo,
int ExtLines::count (const char *name, int nbinfo,
double &lg, double minl) const
{
+cout << "COUNTING" << endl;
vector<ExtLine> cannys;
double val[nbinfo];
double clLength = 0.;
@@ -314,21 +315,27 @@ void ExtLines::stats (const QImage &im)
outf.close ();
}
- double lbs = 0.;
- int nbs = (int) bss.size ();
+ double mylength, mylg = 0.;
+ int mynb = 0;
+ // int nbs = (int) bss.size ();
vector<BlurredSegment *>::iterator it = bss.begin ();
while (it != bss.end ())
{
Pt2i lastleft = (*it)->getLastLeft ();
Pt2i lastright = (*it)->getLastRight ();
- lbs += sqrt ((lastleft.x () - lastright.x ())
- * (lastleft.x () - lastright.x ())
- + (lastleft.y () - lastright.y ())
- * (lastleft.y () - lastright.y ()));
+ mylength = sqrt ((lastleft.x () - lastright.x ())
+ * (lastleft.x () - lastright.x ())
+ + (lastleft.y () - lastright.y ())
+ * (lastleft.y () - lastright.y ()));
+ if (mylength >= MIN_LENGTH)
+ {
+ mylg += mylength;
+ mynb++;
+ }
it ++;
}
ofstream outl ("Data/fbsdln.txt", ios::out);
- outl << nbs << " " << lbs << " " << (lbs / nbs) << endl;
+ outl << mynb << " " << mylg << " " << (mylg / mynb) << endl;
outl.close ();
for (int i = 0; i < height; i++) delete [] tabImage[i];
diff --git a/Code/FBSD/BlurredSegment/blurredsegmentproto.cpp b/Code/FBSD/BlurredSegment/blurredsegmentproto.cpp
index 925ae8a..b357866 100755
--- a/Code/FBSD/BlurredSegment/blurredsegmentproto.cpp
+++ b/Code/FBSD/BlurredSegment/blurredsegmentproto.cpp
@@ -58,7 +58,7 @@ BlurredSegmentProto::~BlurredSegmentProto ()
AbsRat BlurredSegmentProto::strictThickness () const
{
- return (convexhull != NULL ? convexhull->strictThickness () : AbsRat (0, 1));
+ return (convexhull != NULL ? convexhull->thickness () : AbsRat (0, 1));
}
@@ -83,9 +83,10 @@ DigitalStraightLine *BlurredSegmentProto::getLine () const
convexhull->antipodalEdgeAndVertex (s, e, v);
return (new DigitalStraightLine (s, e, v));
}
- if (bsFlat) return (new DigitalStraightLine (getLastLeft (), getLastRight (),
- DigitalStraightLine::DSL_THIN));
- return (NULL);
+ if (bsFlat || leftOK || rightOK)
+ return (new DigitalStraightLine (getLastLeft (), getLastRight (),
+ DigitalStraightLine::DSL_THIN));
+ return (NULL); // No line if only one point
}
diff --git a/Code/FBSD/BlurredSegment/blurredsegmentproto.h b/Code/FBSD/BlurredSegment/blurredsegmentproto.h
index bad3be9..28589ce 100755
--- a/Code/FBSD/BlurredSegment/blurredsegmentproto.h
+++ b/Code/FBSD/BlurredSegment/blurredsegmentproto.h
@@ -40,6 +40,12 @@ public:
*/
~BlurredSegmentProto ();
+ /**
+ * \brief Checks if the blurred segment has at least two points.
+ */
+ inline bool isLineable () const {
+ return (bsOK || bsFlat || leftOK || rightOK); }
+
/**
* \brief Returns the built-in blurred segment strict thickness.
* The strict thickness is the distance between bounding lines, ie (nu-1)/p.
diff --git a/Code/FBSD/BlurredSegment/bsdetector.cpp b/Code/FBSD/BlurredSegment/bsdetector.cpp
index 0ebae63..81fb515 100755
--- a/Code/FBSD/BlurredSegment/bsdetector.cpp
+++ b/Code/FBSD/BlurredSegment/bsdetector.cpp
@@ -69,7 +69,7 @@ BSDetector::BSDetector ()
finalDensityTestOn = false;
finalLengthTestOn = false;
finalSpreadTestOn = true;
- finalSpreadMin = 12;
+ finalSpreadMin = 10;
// nbSmallBS = 0;
multiSelection = false;
autodet = false;
diff --git a/Code/FBSD/BlurredSegment/bstracker.cpp b/Code/FBSD/BlurredSegment/bstracker.cpp
index a5ca5c4..9dce6c6 100755
--- a/Code/FBSD/BlurredSegment/bstracker.cpp
+++ b/Code/FBSD/BlurredSegment/bstracker.cpp
@@ -280,7 +280,7 @@ BlurredSegment *BSTracker::fineTrack (int bsMaxWidth,
if (thickenOn && stableWidthCount >= thickenningLimit)
{
AbsRat dth (sw);
- if (strictOn) dth = bs.digitalThickness ();
+ if (! strictOn) dth = bs.digitalThickness ();
AbsRat finalWidth (dth.sumHalf ());
if (finalWidth.lessThan (bs.getMaxWidth ())) bs.setMaxWidth (finalWidth);
thickenOn = false;
@@ -306,7 +306,7 @@ BlurredSegment *BSTracker::fineTrack (int bsMaxWidth,
}
// Resets the scan stripe
- if (dynamicScans && count > fittingDelay && bs.getLine () != NULL)
+ if (dynamicScans && count > fittingDelay && bs.isLineable ())
{
// Stops the detection if the segment gets crosswise
if (count == fittingDelay + 1)
@@ -324,7 +324,7 @@ BlurredSegment *BSTracker::fineTrack (int bsMaxWidth,
bs.getLine()->getMedialAxis (ppa, ppb, ppc);
ds->bindTo (ppa, ppb, ppc);
}
- else if (trackCrosswise && count > 3 && bs.getLine () != NULL)
+ else if (trackCrosswise && count > 3 && bs.isLineable ())
{
Vr2i dirn = bs.getSupportVector ();
if (4 * dirn.squaredScalarProduct (scandir)
@@ -360,7 +360,8 @@ BlurredSegment *BSTracker::fineTrack (int bsMaxWidth,
stableWidthCount ++;
if (added)
{
- if (sw.lessThan (bs.strictThickness ())) stableWidthCount = 0;
+ if (thickenOn
+ && sw.lessThan (bs.strictThickness ())) stableWidthCount = 0;
rscan = count;
if (rstop == 0) rstart = 0;
else
@@ -408,7 +409,8 @@ BlurredSegment *BSTracker::fineTrack (int bsMaxWidth,
stableWidthCount ++;
if (added)
{
- if (sw.lessThan (bs.strictThickness ())) stableWidthCount = 0;
+ if (thickenOn
+ && sw.lessThan (bs.strictThickness ())) stableWidthCount = 0;
lscan = count;
if (lstop == 0) lstart = 0;
else
diff --git a/Code/FBSD/ConvexHull/antipodal.cpp b/Code/FBSD/ConvexHull/antipodal.cpp
index c9ef9c7..738fbce 100755
--- a/Code/FBSD/ConvexHull/antipodal.cpp
+++ b/Code/FBSD/ConvexHull/antipodal.cpp
@@ -64,7 +64,7 @@ void Antipodal::init (CHVertex *v1, CHVertex *v2, CHVertex *v3)
}
-AbsRat Antipodal::strictThickness () const
+AbsRat Antipodal::thickness () const
{
int den = ept2->get (iy) - ept1->get (iy);
return (AbsRat (((vpt->get (ix) - ept1->get (ix)) * den
@@ -73,14 +73,17 @@ AbsRat Antipodal::strictThickness () const
}
-void Antipodal::width (int &num, int &den) const
+/*
+void Antipodal::thickness (int &num, int &den) const
{
den = ept2->get (iy) - ept1->get (iy);
- if (den < 0) den = -den;
num = (vpt->get (ix) - ept1->get (ix)) * den
- - (vpt->get (iy) - ept1->get (iy)) * (ept2->get (ix) - ept1->get (ix));
+ - (vpt->get (iy) - ept1->get (iy))
+ * (ept2->get (ix) - ept1->get (ix));
if (num < 0) num = -num;
+ if (den < 0) den = -den;
}
+*/
int Antipodal::remainder (CHVertex *v) const
diff --git a/Code/FBSD/ConvexHull/antipodal.h b/Code/FBSD/ConvexHull/antipodal.h
index 36cd073..df2b9a5 100755
--- a/Code/FBSD/ConvexHull/antipodal.h
+++ b/Code/FBSD/ConvexHull/antipodal.h
@@ -44,13 +44,19 @@ public:
/** Returns the leaning edge end of the antipodal pair. */
inline CHVertex *edgeEnd () const { return ept2; }
- /** Returns the antipodal pair strict thickness.
- * Strict thickness is the vertex horizontal distance to the edge.
+ /**
+ * Returns the antipodal pair horizontal thickness.
+ * It is computed as the vertex horizontal distance to the edge.
*/
- AbsRat strictThickness () const;
+ AbsRat thickness () const;
- /** Gets the rational main axis width of the antipodal pair. */
- void width (int &num, int &den) const;
+ /**
+ * Computes the antipodal pair horizontal thickness.
+ * It is the vertex horizontal distance to the edge.
+ * @param num Numerator of the thickness rational value.
+ * @param den Denominator of the thickness rational value.
+ */
+ //void thickness (int &num, int &den) const;
/** Returns the remainder of the edge line equation for given vertex. */
int remainder (CHVertex *v) const;
diff --git a/Code/FBSD/ConvexHull/convexhull.cpp b/Code/FBSD/ConvexHull/convexhull.cpp
index b20d532..7256ab4 100755
--- a/Code/FBSD/ConvexHull/convexhull.cpp
+++ b/Code/FBSD/ConvexHull/convexhull.cpp
@@ -114,20 +114,19 @@ bool ConvexHull::moveLastPoint (const Pt2i &pix)
}
-AbsRat ConvexHull::strictThickness () const
+AbsRat ConvexHull::thickness () const
{
- AbsRat aphw = aph.strictThickness ();
- AbsRat apvw = apv.strictThickness ();
+ AbsRat aphw = aph.thickness ();
+ AbsRat apvw = apv.thickness ();
return (apvw.lessThan (aphw) ? apvw : aphw);
}
void ConvexHull::antipodalEdgeAndVertex (Pt2i &s, Pt2i &e, Pt2i &v) const
{
- int n1, d1, n2, d2;
- aph.width (n1, d1);
- apv.width (n2, d2);
- const Antipodal *ap = ((n2 * d1 < n1 * d2) ? &apv : &aph);
+ AbsRat aphw = aph.thickness ();
+ AbsRat apvw = apv.thickness ();
+ const Antipodal *ap = (apvw.lessThan (aphw) ? &apv : &aph);
s.set (*(ap->edgeStart ()));
e.set (*(ap->edgeEnd ()));
v.set (*(ap->vertex ()));
diff --git a/Code/FBSD/ConvexHull/convexhull.h b/Code/FBSD/ConvexHull/convexhull.h
index 731bfff..4d06a05 100755
--- a/Code/FBSD/ConvexHull/convexhull.h
+++ b/Code/FBSD/ConvexHull/convexhull.h
@@ -70,11 +70,12 @@ public:
*/
void antipodalEdgeAndVertex (Pt2i &s, Pt2i &e, Pt2i &v) const;
- /** Returns the convex hull strict thickness.
- * Strict thickness is defined as the minimal value of both antipodal pairs.
- * It is also the minimal vertical or horizontal width of the convex hull.
+ /**
+ * Returns the convex hull thickness.
+ * The thickness is the minimal vertical or horizontal thickness.
+ * It is computed as the minimal value of both antipodal pairs.
*/
- AbsRat strictThickness () const;
+ AbsRat thickness () const;
/**
* Returns a string that represents the convex hull.
--
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