diff --git a/Article/Expe_auto/bsNew.png b/Article/Expe_auto/bsNew.png
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index 8311feb0685ab1d59f68f7f1db80d4371e6c574f..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/bsOld.png b/Article/Expe_auto/bsOld.png
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index 95f8a8d80f7a304c476fc54346520715de6385a5..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/buro.png b/Article/Expe_auto/buro.png
deleted file mode 100755
index b3aa23a4ff700d29efcabcbbb50a917c9fdb5ea0..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/buroDetail.png b/Article/Expe_auto/buroDetail.png
deleted file mode 100755
index a73b1a1656de95efb0a570af8c14c7676d50169a..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/cannyTable.tex b/Article/Expe_auto/cannyTable.tex
deleted file mode 100644
index bb2f214ae13ffe3c50c551b35f5371fc9c068d3b..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/cannyTable.tex
+++ /dev/null
@@ -1,14 +0,0 @@
-\begin{tabular}{|l||r|r|r|r|r|}
-\hline
-Measure $M$ & \multicolumn{1}{c|}{$T$ (ms)} & \multicolumn{1}{c|}{$C$}
-& \multicolumn{1}{c|}{$N$}
-& \multicolumn{1}{c|}{$L$} & \multicolumn{1}{c|}{$L/N$} \\
-\hline
-Canny
-& 75.4 $\pm$ 11.7 & 60.6 $\pm$ 10.6
-& 466 $\pm$ 138 & 17678 $\pm$ 4419 & 39.5 $\pm$ 10.2 \\
-Ours
-& 83.2 $\pm$ 20.1 & 61.5 $\pm$ 10.8
-& 613 $\pm$ 140 & 20769 $\pm$ 4000 & 34.6 $\pm$ 5.4 \\
-\hline
-\end{tabular}
diff --git a/Article/Expe_auto/coloredNew.png b/Article/Expe_auto/coloredNew.png
deleted file mode 100644
index 76e163c45f9a4411ce309a24a0d1ce24c300b59f..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/coloredOld.png b/Article/Expe_auto/coloredOld.png
deleted file mode 100644
index afedf2807e2064a97870fe63031151f04c3e54f9..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/dssDetailNew.png b/Article/Expe_auto/dssDetailNew.png
deleted file mode 100644
index 2313b7fc473548728e6c26c69086384d2f78bb27..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/dssDetailOld.png b/Article/Expe_auto/dssDetailOld.png
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index 56349908259b4ba57287220f5182119ea04153c1..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/dssNew.png b/Article/Expe_auto/dssNew.png
deleted file mode 100644
index 9cd53b790d1513e289413b7bfb9437e37787e579..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/dssOld.png b/Article/Expe_auto/dssOld.png
deleted file mode 100644
index 9cfe50a24cf1c84b3918212c26cb435706e88ff3..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_auto/perfTable.tex b/Article/Expe_auto/perfTable.tex
deleted file mode 100644
index decd46a58d983bcc5e24cc0ddc7da2f7b9593be9..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perfTable.tex
+++ /dev/null
@@ -1,20 +0,0 @@
-\begin{tabular}{|l||r|r|r|r|}
-\hline
-Measure $M$ & \multicolumn{1}{c|}{$T$} & \multicolumn{1}{c|}{$N$}
-& \multicolumn{1}{c|}{$L$} & \multicolumn{1}{c|}{$W$} \\
-\hline
-$M_{old}$ on image of \RefFig{fig:auto}
-& 29.51 ms & 306 segments & 38.58 pixels & 2.47 pixels \\
-$M_{new}$ on image of \RefFig{fig:auto}
-& 25.85 ms & 352 segments & 33.25 pixels & 2.17 pixels \\
-\hline \hline
-$M_{new}/M_{old}$ (\%) & & & & \\
-\hspace{0.4cm} on image of \RefFig{fig:auto}
-& \multicolumn{1}{l|}{87.60} & \multicolumn{1}{l|}{115.03}
-& \multicolumn{1}{l|}{86.18} & \multicolumn{1}{l|}{87.85} \\
-\hspace{0.4cm} on the set of test images
-& 85.47 $\pm$ 2.16 & 108.19 $\pm$ 4.86 & 91.23 $\pm$ 3.63 & 86.18 $\pm$ 3.88 \\
-\hspace{0.4cm} on CannyLines images
-& 86.02 $\pm$ 2.44 & 110.15 $\pm$ 6.51 & 89.23 $\pm$ 5.11 & 84.70 $\pm$ 2.98 \\
-\hline
-\end{tabular}
diff --git a/Article/Expe_auto/perf_buro.txt b/Article/Expe_auto/perf_buro.txt
deleted file mode 100644
index 52280913a4b139f22a1bbec586912f708e2eef3f..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_buro.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-800
-533
-2.58545
-914
-352
-94
-33.2509
-2.17087
-2.21914
-2.95147
-851
-306
-85
-38.5787
-2.47154
-2.46201
diff --git a/Article/Expe_auto/perf_indoor01.txt b/Article/Expe_auto/perf_indoor01.txt
deleted file mode 100644
index 3e91f64fe58143a91e16c4ea76cc3db166f2b74e..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor01.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-5.55119
-2323
-834
-178
-28.8666
-2.18444
-2.32847
-6.5528
-2279
-786
-176
-30.9693
-2.60664
-2.705
diff --git a/Article/Expe_auto/perf_indoor02.txt b/Article/Expe_auto/perf_indoor02.txt
deleted file mode 100644
index 045b632e7f6091486fe39f83102b3fb6005d410f..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor02.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.80259
-1498
-540
-154
-37.068
-1.84411
-1.88157
-4.4721
-1455
-460
-157
-43.7546
-2.25229
-2.27455
diff --git a/Article/Expe_auto/perf_indoor03.txt b/Article/Expe_auto/perf_indoor03.txt
deleted file mode 100644
index feb842c3622a7585cd76cc95b041f4f5fea6d70f..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor03.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.61507
-1490
-544
-89
-28.1896
-2.15282
-2.19388
-4.21664
-1427
-497
-99
-30.912
-2.56824
-2.61858
diff --git a/Article/Expe_auto/perf_indoor04.txt b/Article/Expe_auto/perf_indoor04.txt
deleted file mode 100644
index dac2c2c5fb24953c47ba18faf63a0e837d1d8fe8..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor04.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-1.96869
-734
-251
-85
-43.1107
-2.02305
-2.07427
-2.30199
-730
-205
-83
-54.239
-2.32858
-2.3503
diff --git a/Article/Expe_auto/perf_indoor05.txt b/Article/Expe_auto/perf_indoor05.txt
deleted file mode 100644
index 21678296ff433af6fdd33e79bf8957b368a395e4..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor05.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.90785
-1594
-649
-110
-25.5561
-2.16213
-2.17938
-4.70566
-1554
-604
-134
-28.002
-2.4826
-2.50369
diff --git a/Article/Expe_auto/perf_indoor06.txt b/Article/Expe_auto/perf_indoor06.txt
deleted file mode 100644
index b85532b1345585d5c92cc9f6ace1c4f1de36ebc1..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor06.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.64825
-1234
-500
-168
-43.8536
-1.78707
-1.7937
-4.36547
-1248
-394
-174
-55.1621
-2.09328
-2.01921
diff --git a/Article/Expe_auto/perf_indoor07.txt b/Article/Expe_auto/perf_indoor07.txt
deleted file mode 100644
index 8202eee8529805662d9c5fe4e17cda972a85827a..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor07.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-4.05357
-1470
-520
-142
-38.5894
-2.07342
-2.11269
-4.6441
-1333
-440
-134
-47.63
-2.49047
-2.55303
diff --git a/Article/Expe_auto/perf_indoor08.txt b/Article/Expe_auto/perf_indoor08.txt
deleted file mode 100644
index 3e88308b9f68678e118cdbc1ce320a84445087a2..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor08.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-4.87889
-2141
-764
-106
-24.1643
-2.3129
-2.33964
-5.74342
-2146
-715
-117
-25.6504
-2.74824
-2.84597
diff --git a/Article/Expe_auto/perf_indoor09.txt b/Article/Expe_auto/perf_indoor09.txt
deleted file mode 100644
index 1221db9407eceb90c63876fb9be39dca54bec62c..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor09.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-5.55145
-2335
-865
-149
-26.4754
-2.24392
-2.24859
-6.52401
-2295
-788
-146
-29.8719
-2.61334
-2.56096
diff --git a/Article/Expe_auto/perf_indoor10.txt b/Article/Expe_auto/perf_indoor10.txt
deleted file mode 100644
index 70999a38648c26aab21a166fcf7082b24e9159e9..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_indoor10.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-2.06293
-772
-270
-93
-42.3207
-1.96
-1.95388
-2.58662
-777
-258
-88
-45.3709
-2.53972
-2.57716
diff --git a/Article/Expe_auto/perf_outdoor01.txt b/Article/Expe_auto/perf_outdoor01.txt
deleted file mode 100644
index f3c8e82e77eb19aa578c34c44fd43f00fd1e551e..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor01.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-4.07407
-1619
-590
-120
-28.621
-2.13309
-2.1309
-4.87129
-1538
-540
-130
-31.9144
-2.57515
-2.59164
diff --git a/Article/Expe_auto/perf_outdoor02.txt b/Article/Expe_auto/perf_outdoor02.txt
deleted file mode 100644
index 2a813a519c0feb7a04c06644e852dd5780c6ff2f..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor02.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-5.944
-3000
-964
-99
-17.5359
-2.50626
-2.48442
-6.6842
-2917
-904
-112
-19.1165
-2.78864
-2.82444
diff --git a/Article/Expe_auto/perf_outdoor03.txt b/Article/Expe_auto/perf_outdoor03.txt
deleted file mode 100644
index 0504c6d469f59f9d922e8fde93b0e6e23a0fbe80..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor03.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.98861
-2018
-615
-56
-18.6491
-2.38418
-2.35207
-4.49043
-1951
-591
-54
-19.9451
-2.81475
-2.83261
diff --git a/Article/Expe_auto/perf_outdoor04.txt b/Article/Expe_auto/perf_outdoor04.txt
deleted file mode 100644
index 13bbc2038df5435e41a77122c7ec49d9518937ce..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor04.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-4.86978
-2471
-767
-88
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-2.38793
-2.39488
-5.46851
-2356
-726
-99
-19.9373
-2.80044
-2.89292
diff --git a/Article/Expe_auto/perf_outdoor05.txt b/Article/Expe_auto/perf_outdoor05.txt
deleted file mode 100644
index 6d475c93fe378cc43fe7428a01703127188c5377..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor05.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.09307
-1404
-496
-78
-23.7321
-2.33562
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-1349
-436
-83
-27.5756
-2.71522
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diff --git a/Article/Expe_auto/perf_outdoor06.txt b/Article/Expe_auto/perf_outdoor06.txt
deleted file mode 100644
index e84f583fea14383958229088a1fee30f21c158b7..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor06.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-6.80802
-3549
-1044
-75
-17.3937
-2.42717
-2.32804
-7.609
-3441
-1054
-94
-17.5495
-2.78552
-2.85436
diff --git a/Article/Expe_auto/perf_outdoor07.txt b/Article/Expe_auto/perf_outdoor07.txt
deleted file mode 100644
index ace6ab181a990b84007a89e15e0a0b0b1086c0db..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor07.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-6.41568
-2790
-1029
-150
-24.6812
-2.30869
-2.64853
-7.34472
-2529
-916
-180
-29.1043
-2.67184
-2.81002
diff --git a/Article/Expe_auto/perf_outdoor08.txt b/Article/Expe_auto/perf_outdoor08.txt
deleted file mode 100644
index eb3cb8ba9d8e488d98a68bba07af14430bfffdf2..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor08.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-6.90651
-3248
-1037
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-2.37173
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-7.78974
-3115
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-187
-25.699
-2.77951
-2.91742
diff --git a/Article/Expe_auto/perf_outdoor09.txt b/Article/Expe_auto/perf_outdoor09.txt
deleted file mode 100644
index f337357ede81efaa5ffa659ac16423a68b8c834d..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor09.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.77171
-1734
-571
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-1.93375
-1.87111
-4.37652
-1648
-523
-114
-31.0221
-2.46606
-2.43659
diff --git a/Article/Expe_auto/perf_outdoor10.txt b/Article/Expe_auto/perf_outdoor10.txt
deleted file mode 100644
index 3733e88c6ea237faa4e3005b4740255cfd2fdae3..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/perf_outdoor10.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-100
-640
-480
-3.82884
-1576
-648
-82
-22.4578
-2.41069
-2.47638
-4.49443
-1535
-610
-94
-24.359
-2.71668
-2.86795
diff --git a/Article/Expe_auto/readme.txt b/Article/Expe_auto/readme.txt
deleted file mode 100644
index 058065687e348f1595e4426a7df011b774c13865..0000000000000000000000000000000000000000
--- a/Article/Expe_auto/readme.txt
+++ /dev/null
@@ -1,36 +0,0 @@
-Seg ../buro.png
-20 y pour obtenir un niveau de fond d'ecran a 100
-m pour une detection m
-! pour un affichage sombre
-m pour une detection multiple avec le nouveau detecteur
-p pour recuperer la capture (capture.png -> coloredNew.png)
-6 pour une detection multiple avec l'ancien detecteur
-p pour recuperer la capture (capture.png -> coloredOld.png)
-= pour remplacer les couleurs aleatoires par du noir
-p pour recuperer la capture (capture.png -> bsOld.png)
-6 pour une detection multiple avec le nouveau detecteur
-p pour recuperer la capture (capture.png -> bsNew.png)
-Ctrl y - Ctrl u pour afficher les segments englobants
-p pour recuperer la capture (capture.png -> dssNew.png)
-6 pour une detection multiple avec l'ancien detecteur
-p pour recuperer la capture (capture.png -> dssOld.png)
-8 pour un test de performance (perf.txt -> perf_buro.txt)
-cp dssNew.png dssDetailNew.png
-Selectionner la zone (544, 512) (742, 448)
-cp dssOld.png dssDetailOld.png
-Selectionner la zone (544, 512) (742, 448)
-
-Contenu de perf_buro.txt
-- Nombre de detections multiples
-- Largeur d'image
-- Hauteur d'image
-puis avec le nouveau detecteur
-- Temps d'execution
-- Nombre de minimum locaux testes
-- Nombre de segments flous detectes
-- Nombre de segments d'extension > 40 pixels
-- Extension moyenne des segments flous detectes
-- Largeur moyenne des segments flous detectes
-- Largeur moyenne des segments flous longs (> 40 pixels) detectes
-puis pareil avec l'ancien detecteur
-
diff --git a/Article/Expe_hard/hardDetailNew.png b/Article/Expe_hard/hardDetailNew.png
deleted file mode 100644
index 4ee88ca837b9a94a69421daca4fb654b9cdcbcd9..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_hard/hardDetailOld.png b/Article/Expe_hard/hardDetailOld.png
deleted file mode 100644
index 2cb5c40a315583485993761372f7c536a02eb605..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_hard/hardNew.png b/Article/Expe_hard/hardNew.png
deleted file mode 100644
index 1f40affbb154e2abfe7676be82cec496d763cd33..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_hard/hardOld.png b/Article/Expe_hard/hardOld.png
deleted file mode 100644
index 65f20b0d4a9f2ca176e05d95107279c2a0048661..0000000000000000000000000000000000000000
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diff --git a/Article/Expe_hard/readme.txt b/Article/Expe_hard/readme.txt
deleted file mode 100644
index 3ac1d6b5e8281ba85229bbf0930a4722060cda9c..0000000000000000000000000000000000000000
--- a/Article/Expe_hard/readme.txt
+++ /dev/null
@@ -1,13 +0,0 @@
-Seg ../parpaings.gif
-30 y : Met le fond d'ecran au niveau 150
-7 l : Out89put BS min size -> 12 (note qu'avec l = 16 c'est pas mal non plus)
-m : detection auto avec le nouveau detecteur
-! : affiche les segments en sombre
-= : supprime les couleurs aleatoires (-> segments noirs)
-p : sauve l'image capture.png -> ismm/Expe_hard/hardNew.png
-6 : detection auto avec l'ancien detecteur
-p : sauve l'image capture.png -> ismm/Expe_hard/hardOld.png
-
-Pour les images hardDetailNew.png et hardDetailOld.png,
-selectionner le coin haut gauche de l'image depuis le
-point (346, 140).
diff --git a/Article/Fig_method/parpaings.png b/Article/Fig_method/parpaings.png
deleted file mode 100644
index ac6716017d8dd3a3b6d1bde446140a7032de1386..0000000000000000000000000000000000000000
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diff --git a/Article/Fig_method/parpaings2.png b/Article/Fig_method/parpaings2.png
deleted file mode 100644
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diff --git a/Article/Fig_method/parpaings3.png b/Article/Fig_method/parpaings3.png
deleted file mode 100644
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diff --git a/Article/Fig_method/plafondAuto.png b/Article/Fig_method/plafondAuto.png
deleted file mode 100644
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diff --git a/Article/Fig_method/plafond_petit.png b/Article/Fig_method/plafond_petit.png
deleted file mode 100755
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diff --git a/Article/Fig_method/vcercle.png b/Article/Fig_method/vcercle.png
deleted file mode 100644
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diff --git a/Article/Fig_method/vcercleAuto.png b/Article/Fig_method/vcercleAuto.png
deleted file mode 100644
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diff --git a/Article/expe.tex b/Article/expe.tex
deleted file mode 100755
index 6b3fcd358d9a46be773bbf173e9b2cd20e45f6a0..0000000000000000000000000000000000000000
--- a/Article/expe.tex
+++ /dev/null
@@ -1,70 +0,0 @@
-\section{Experimental validation}
-
-\label{sec:expe}
-
-The main goal of this work is to provide straight segments with a quality
-indication through the associated width parameter.
-In lack of available reference tool, the evaluation stage mostly aims
-at quantifying the advantages of the new detector compared to the previous
-one in unsupervised context.
-The process flow of the former method (initial detection followed by two
-refinement steps) is integrated as an option into the code of the new
-detector, so that both methods rely on the same optimized basic routines.
-During all these experiments, only the blurred segment size and its
-orientation compared to the initial stroke are tested at the end of
-the initial detection, and only the segment size is tested at the end
-of the fine tracking stage.
-All other tests, sparsity or fragmentation, are disabled.
-The segment minimal size is set to 5 pixels, except where precised.
-
-The first test compares the performance of both
-detectors on a set of 1000 synthesized images containing 10 randomly
-placed input segments with random width between 2 and 5 pixels.
-The absolute value of the difference of each found segment to its
-matched input segment is measured.
-On these ground-truth image, the numerical error on the gradient extraction
-biases the line width measures. This bias was first estimated using 1000
-images containing only one input segment (no possible interaction)
-and the found value (1.4 pixel) was taken into account in the test.
-\RefTab{tab:synth} shows
-slightly better width and angle measurements for the new detector.
-The new detector shows more precise, with a smaller amount of false
-detections and succeeds in finding most of the input segments.
-
-%\begin{figure}[h]
-%\center
-% \begin{tabular}{
-% c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c}
-% \includegraphics[width=0.19\textwidth]{Fig_synth/statsExample.png} &
-% \includegraphics[width=0.19\textwidth]{Fig_synth/statsoldPoints.png} &
-% \includegraphics[width=0.19\textwidth]{Fig_synth/statsoldBounds.png} &
-% \includegraphics[width=0.19\textwidth]{Fig_synth/statsnewPoints.png} &
-% \includegraphics[width=0.19\textwidth]{Fig_synth/statsnewBounds.png}
-% \begin{picture}(1,1)
-% \put(-310,0){a)}
-% \put(-240,0){b)}
-% \put(-170,0){c)}
-% \put(-100,0){d)}
-% \put(-30,0){e)}
-% \end{picture}
-% \end{tabular}
-% \caption{Evaluation on synthesized images:
-% a) one of the test images,
-% b) output blurred segments from the old detector and
-% c) their enclosing digital segments,
-% d) output blurred segments from the new detector and
-% e) their enclosing digital segments.}
-% \label{fig:synth}
-%\end{figure}
-\begin{table}
-\centering
-\input{Fig_synth/statsTable}
-\caption{Measured performance of both detectors on a set of synthesized images.
-$S$ is the set of all the input segments,
-$D$ the set of all the detected blurred segments.}
-\label{tab:synth}
-\end{table}
-
-\input{expeAuto}
-
-%\input{expeHard}
diff --git a/Article/expeAuto.tex b/Article/expeAuto.tex
deleted file mode 100755
index 48c6175c5714114d329acb260acafa11e8eea71c..0000000000000000000000000000000000000000
--- a/Article/expeAuto.tex
+++ /dev/null
@@ -1,81 +0,0 @@
-The next experiments aim at evaluating the performance of the new
-detector with respect to the previous one on a test set composed of a
-selection of 20 real images.
-One of them is displayed on \RefFig{fig:auto}.
-Compared measures $M$ are the execution time $T$, the amount $N$ of detected
-blurred segments, their mean length $L$ and their mean width $W$.
-For the sake of objectivity, these results are also compared to the same
-measurements made on the 20 images data base used for the CannyLine line
-segment detector \cite{LuAl15}.
-\RefTab{tab:auto} gives the achieved results.
-\begin{figure}[h]
-%\center
- \begin{tabular}{
- c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}}
- \includegraphics[width=0.32\textwidth]{Expe_auto/buro.png} &
-% \includegraphics[width=0.32\textwidth]{Expe_auto/coloredOld.png} &
-% \includegraphics[width=0.32\textwidth]{Expe_auto/coloredNew.png} \\
- \includegraphics[width=0.32\textwidth]{Expe_auto/bsOld.png} &
- \includegraphics[width=0.32\textwidth]{Expe_auto/bsNew.png} \\
- \includegraphics[width=0.22\textwidth]{Expe_auto/buroDetail.png} &
- \includegraphics[width=0.22\textwidth]{Expe_auto/dssDetailOld.png} &
- \includegraphics[width=0.22\textwidth]{Expe_auto/dssDetailNew.png}
- \begin{picture}(1,1)
- {\color{red}{
- \put(-19.5,31){\framebox(28,9)}
- \put(-5.5,31){\vector(-2,-1){20}}
- \put(-133.5,31){\framebox(28,9)}
- \put(-117.5,31){\vector(-2,-1){20}}
- \put(-247.5,31){\framebox(28,9)}
- \put(-231.5,31){\vector(-2,-1){20}}
- }}
- {\color{dwhite}{
- \put(-291,32.5){\circle*{8}}
- \put(-177,32.5){\circle*{8}}
- \put(-63,32.5){\circle*{8}}
- \put(-302,4.5){\circle*{8}}
- \put(-188,4.5){\circle*{8}}
- \put(-75,4.5){\circle*{8}}
- }}
- \put(-293.5,30){a}
- \put(-179.5,30){b}
- \put(-65.5,30){c}
- \put(-305,2){d}
- \put(-191,2){e}
- \put(-77.5,2){f}
- \end{picture}
- \end{tabular}
- \caption{Automatic detection on real images:
- an input image (a), the segments found by the old detector (b)
- and those found by the new detector (c), and a detail of the
- image (d) and the enclosing digital segments for both old (e)
- and new (f) detectors.}
- \label{fig:auto}
-\end{figure}
-\begin{table}
-\centering
-\input{Expe_auto/perfTable}
-\caption{Measured performance of both detectors on standard images.
- $M_{old}$ (resp. $M_{new}$) denotes the measure obtained with
- the previous (resp. new) detector.}
-\label{tab:auto}
-\end{table}
-
-The new detector is faster and finds more edges than the previous one.
-Details of \RefFig{fig:auto} d) and e) illustrate achieved
-accuracy improvements.
-Output segments are thinner but also shorter.
-The control of the assigned width to fit detected segment width
-has the side effect of blocking the segment expansion when remote parts
-are noisier.
-Found edges are thus more fragmented.
-The relevance of this behavior depends strongly on application requirements.
-Therefore the control of the assigned width is left as an option, the user
-can let or cancel it.
-%In both case, it could be useful to combine the detector with a tool
-%to merge aligned segments.
-
-%Although these observations in unsupervised context should be reproduced
-%in supervised context, similar experiments require an application context
-%to dispose of a ground truth and of real users to assess the detector
-%relevance through ergonomy evaluations.
diff --git a/Article/expeHard.tex b/Article/expeHard.tex
deleted file mode 100755
index 4304047ccbb2fb9715f51ebde2592280d6418092..0000000000000000000000000000000000000000
--- a/Article/expeHard.tex
+++ /dev/null
@@ -1,33 +0,0 @@
-The last test visually compares the results of both detectors on very textured
-images, also difficult to process for other detectors from the literature.
-The minimal size parameter was raised to 12 pixels to reject small segments
-considered as outliers.
-On the example of \RefFig{fig:hard}, the new detector provides less residual
-outliers and misaligned segments, and globally more relevant informations
-to infere the structure of the brick wall.
-\begin{figure}
- \center
- \begin{tabular}{
- c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}}
- \includegraphics[width=0.205\textwidth]{Fig_method/parpaings.png} &
- \includegraphics[width=0.32\textwidth]{Expe_hard/hardDetailOld.png} &
- \includegraphics[width=0.32\textwidth]{Expe_hard/hardDetailNew.png}
- \begin{picture}(1,1)
- {\color{red}{
- \put(-302,34){\framebox(29,11.5)}
- }}
- {\color{dwhite}{
- \put(-266,4.5){\circle*{8}}
- \put(-171,4.5){\circle*{8}}
- \put(-58,4.5){\circle*{8}}
- }}
- \put(-268.5,2){a}
- \put(-173.5,2){b}
- \put(-60.5,2){c}
- \end{picture}
- \end{tabular}
- \caption{Results on very textured images: test image (a),
- detail (top left corner) on the segments found by the old
- detector (b) and on those found by the new detector (c).}
- \label{fig:hard}
-\end{figure}
diff --git a/Article/intro.tex b/Article/intro.tex
deleted file mode 100755
index 4499fd50d086c666b2990870c9f717f835f689b1..0000000000000000000000000000000000000000
--- a/Article/intro.tex
+++ /dev/null
@@ -1,99 +0,0 @@
-\section{Introduction}
-
-\label{sec:intro}
-
-\subsection{Motivations}
-
-Straight line 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{AkinlarTopal12,GioiAl10,LuAl15,MatasAl00}.
-However, they seldom provide an exploitable measure of the output line
-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 research field 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{Buzer07,DebledAl05} was
-introduced to cope with the image noise or other sources of imperfections
-from the real world by the mean of a width parameter.
-Efficient algorithms have already been designed to recognize
-these digital objects in binary images \cite{DebledAl06}.
-Straight edges are rich visual features for 3D scene reconstruction from 2D
-images.
-A blurred segment seems well suited to reflect the required edge 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.
-
-\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 image area
-defined by the user. The goal is to disclose the presence of a straight edge.
-Therefore as simple a 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 performed.
-On the 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.
-Because the numerization rounding fixes a limit on this estimated orientation
-accuracy, more steps are inevitably necessary to process larger images.
-
-\subsection{Main contributions}
-
-The present work aims at solving both former mentioned
-drawbacks through two main contributions:
-(i) the concept of {\bf adaptive directional scan} designed to get some
-compliance to the unpredictable orientation problem;
-(ii) the {\bf 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.
-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 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 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.
diff --git a/Article/method.tex b/Article/method.tex
deleted file mode 100755
index efa15402d4950edff6eb57ddc675f669ca940d0b..0000000000000000000000000000000000000000
--- a/Article/method.tex
+++ /dev/null
@@ -1,280 +0,0 @@
-\section{The detection method}
-
-\label{sec:method}
-
-\subsection{Workflow of the detection process}
-
-The workflow of the detection process is summerized in the following figure.
-
-\begin{figure}[h]
-\center
- \input{Fig_method/workflow}
- \caption{The detection process main workflow.}
- \label{fig:workflow}
-\end{figure}
-
-The initial detection consists in building and extending a blurred segment
-$\mathcal{B}$ based on points with highest norm gradient found in each scan
-of a static directional scan defined by an input segment $AB$.
-
-Validity tests are then applied to decide of the detection pursuit.
-They aim at rejecting too short or too sparse blurred segments, or
-those with a close orientation to $AB$.
-In case of positive response, the position $C$ and direction $\vec{D}$
-of this initial blurred segment are extracted.
-
-In the fine tracking step, another blurred segment $\mathcal{B}'$ is built
-and extended with points that correspond to local maxima of the
-image gradient, ranked by magnitude order, and with gradient direction
-close to start point gradient direction.
-At this refinement step, a control of the assigned width is applied
-and an adaptive directional scan based on the found position $C$ and
-direction $\vec{D}$ is used in order to extends the segment in the
-appropriate direction. These two improvements are described in the
-following sections.
-
-The output segment $\mathcal{B}'$ is finally tested according to the
-application needs. 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.
-%None of these tests are activated for the experimental stage in order
-%to put forward achievable performance.
-
-\subsection{Adaptive directional scan}
-
-The blurred segment is searched within a directional scan with a position
-and an orientation approximately provided by the user, or blindly defined
-in unsupervised mode.
-Most of the time, the detection stops where the segment escapes sideways
-from the scan strip (\RefFig{fig:escape} a).
-A second search is then run using another directional scan aligned
-on the detected segment (\RefFig{fig:escape} b).
-In the given example, an outlier added to the initial segment leads to a
-wrong orientation value.
-But even in case of a correct detection, this estimated orientation is
-subject to the numerization rounding, and the longer the real segment is,
-the higher the probability gets to fail again on an escape from the scan strip.
-
-\begin{figure}[h]
-\center
- \begin{tabular}{c@{\hspace{0.2cm}}c@{\hspace{0.2cm}}c}
- \includegraphics[width=0.24\textwidth]{Fig_notions/escapeLightFirst_half.png} &
- \includegraphics[width=0.24\textwidth]{Fig_notions/escapeLightSecond_half.png} &
- \includegraphics[width=0.48\textwidth]{Fig_notions/escapeLightThird_zoom.png}
- \begin{picture}(1,1)(0,0)
- {\color{dwhite}{
- \put(-307,4.5){\circle*{8}}
- \put(-216,4.5){\circle*{8}}
- \put(-127,4.5){\circle*{8}}
- }}
- \put(-309.5,2){a}
- \put(-219,1.5){b}
- \put(-129.5,2){c}
- \end{picture}
- \end{tabular}
- \caption{Aborted detections on side escapes of static directional scans
- and successful detection using an adaptive directional scan.
- The last points added to the left of the blurred segment during
- initial detection (a) lead to a bad estimation of its
- orientation, and thus to an incomplete fine tracking with a
- classical directional scan (b). An adaptive directional scan
- instead of the static one allows to continue the segment
- expansion as far as necessary (c).
- Input selection is drawn in red color, scan strip bounds
- in blue and detected blurred segments in green.}
- \label{fig:escape}
-\end{figure}
-
-To overcome this issue, in the former work, an additional refinement step is
-run in the direction estimated from this longer segment.
-It is enough to completely detect most of the tested edges, but certainly
-not all, especially if big size images with much longer edges are processed.
-As a solution, this operation could be itered as long as the blurred segment
-escapes from the directional scan using as any fine detection steps as
-necessary.
-But at each iteration, already tested points are processed again,
-thus producing a useless computational cost.
-
-Here the proposed solution is to dynamically align the scan direction on
-the blurred segment all along the expansion stage.
-At each iteration $i$ of the expansion, the scan strip is aligned on the
-direction of the blurred segment $\mathcal{B}_{i-1}$ computed at previous
-iteration $i-1$.
-More generally, an adaptive directional scan $ADS$ is defined by:
-\begin{equation}
-ADS = \left\{
-S_i = \mathcal{D}_i \cap \mathcal{N}_i \cap \mathcal{I}
-\left| \begin{array}{l}
-\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}_0) = 0 \\
-\wedge~ h(\mathcal{N}_i) = h(\mathcal{N}_{i-1}) + p(\mathcal{D}_0) \\
-\wedge~ \mathcal{D}_{i} = \mathcal{D} (C_{i-1}, \vec{D}_{i-1}, w_{i-1}),
-i > \lambda
-\end{array} \right. \right\}
-\end{equation}
-where $C_{i}$, $\vec{D}_{i}$ and $w_{i}$ are respectively a position,
-a director vector and a width observed at iteration $i$.
-In the scope of the present detector, $C_{i-1}$ is the intersection of
-the input selection and the central line of $\mathcal{B}_{i-1}$,
-$\vec{D}_{i-1}$ the support vector of the enclosing digital segment
-$E(\mathcal{B}_{i-1})$, and $w_{i-1}$ a value slightly greater than the
-minimal width of $\mathcal{B}_{i-1}$.
-So the last clause expresses the update of the scan bounds at iteration $i$.
-Compared to static directional scans where the scan strip remains fixed to
-the initial line $\mathcal{D}_0$, here the scan strip moves while
-scan lines remain fixed.
-This behavior ensures a complete detection of the blurred segment even when
-the orientation of $\mathcal{D}_0$ is badly estimated (\RefFig{fig:escape} c).
-In practice, it is started after $\lambda = 20$ iterations when the observed
-direction becomes more stable.
-
-\subsection{Control of the assigned width}
-
-The assigned width $\varepsilon$ to the blurred segment recognition algorithm
-is initially set to a large value $\varepsilon_0$ in order to allow the
-detection of large blurred segments.
-Then, when no more augmentation of the minimal width is observed after
-$\tau$ iterations ($\mu_{i+\tau} = \mu_i$), it is set to a much
-stricter value able to circumscribe the possible interpretations of the
-segment, that take into account the digitization margins:
-\begin{equation}
-\varepsilon = \mu_{i+\tau} + \frac{\textstyle 1}{\textstyle 2}
-\end{equation}
-This strategy aims at preventing the incorporation of spurious outliers in
-further parts of the segment.
-Setting the observation distance to a constant value $\tau = 20$ seems
-appropriate in most experimented situations.
-
-\subsection{Supervised blurred segments detection}
-
-In supervised context, the user draws an input stroke across the specific
-edge that he wants to extract from the image.
-The detection method previously described is continuously run during mouse
-dragging and the output blurred segment is displayed on-the-fly.
-
-%The method is quite sensitive to the local conditions of the initial detection
-%so that the output blurred segment may be quite unstable.
-%In order to temper this undesirable behavior for interactive applications,
-%the initial detection can be optionally run twice, the second fast scan being
-%aligned on the first detection output.
-%This strategy provides a first quick analysis of the local context before
-%extracting the segment and contributes to notably stabilize the overall
-%process.
-%
-%When selecting candidates for the fine detection stage, an option, called
-%{\it edge selection mode}, is left to also filter the points according to
-%their gradient direction.
-%In {\it main edge selection mode}, only the points with a gradient vector
-%in the same direction as the start point gradient vector are added to the
-%blurred segment.
-%In {\it opposite edge selection mode}, only the points with an opposite
-%gradient vector direction are kept.
-%In {\it line selection mode} this direction-based filter is not applied,
-%and all the candidate points are aggregated into a same blurred segment,
-%whatever the direction of their gradient vector.
-%As illustrated on \RefFig{fig:edgeDir}, this mode allows the detection of
-%the two opposite edges of a thin straight object.
-%
-%\begin{figure}[h]
-%\center
-% \begin{tabular}{c@{\hspace{0.2cm}}c}
-% \includegraphics[width=0.4\textwidth]{Fig_method/selectLine_zoom.png} &
-% \includegraphics[width=0.4\textwidth]{Fig_method/selectEdges_zoom.png}
-% \end{tabular}
-% \begin{picture}(1,1)(0,0)
-% {\color{dwhite}{
-% \put(-220,-14.5){\circle*{8}}
-% \put(-74,-14.5){\circle*{8}}
-% }}
-% \put(-222.5,-17){a}
-% \put(-76.5,-17){b}
-% \end{picture}
-% \caption{Blurred segments obtained in \textit{line} or \textit{edge
-% selection mode} as a result of the gradient direction filtering
-% when adding points.
-% In \textit{line selection mode} (a), a thick blurred segment is
-% built and extended all along the brick join.
-% In \textit{edge selection mode} (b), a thin blurred segment is
-% built along one of the two join edges.
-% Both join edges are detected with the \textit{multi-selection}
-% option.
-% On that very textured image, they are much shorter than the whole
-% join detected in line selection mode.
-% Blurred segment points are drawn in black color, and the enclosing
-% straight segments in blue.}
-% \label{fig:edgeDir}
-%\end{figure}
-
-%\subsection{Multiple blurred segments detection}
-
-An option, called {\it multi-detection} (Algorithm 1), allows the
-detection of all the segments crossed by the input stroke $AB$.
-In order to avoid multiple detections of the same edge, an occupancy mask,
-initially empty, collects the dilated points of all the blurred segments,
-so that these points can not be added to another segment.
-\input{Fig_method/algoMulti}
-
-First the positions $M_j$ of the prominent local maxima of the gradient
-magnitude found under the stroke are sorted from the highest to the lowest.
-For each of them the main detection process is run with three modifications:
-\begin{enumerate}
-\item the initial detection takes $M_j$ and the orthogonal direction
-$\vec{AB}_\perp$ to the stroke as input to build a static scan of fixed width
-$2~\varepsilon_{ini}$, and $M_j$ is used as start point of the blurred
-segment;
-\item the occupancy mask is filled in with the points of the dilated blurred
-segments $\mathcal{B}_j'$ at the end of each successful detection
-(a 21 pixels neighborhood is used);
-\item points marked as occupied are rejected when selecting candidates for the
-blurred segment extension in the fine tracking step.
-\end{enumerate}
-
-%In edge selection mode (\RefFig{fig:edgeDir} b), the multi-detection
-%algorithm is executed twice, first in main edge selection mode, then
-%in opposite edge selection mode.
-
-\subsection{Automatic blurred segment detection}
-
-An unsupervised mode is also proposed to automatically detect all the
-straight edges in the image. The principle of this automatic detection
-is described in Algorithm 2. A stroke that crosses the whole image, is
-swept in both directions, vertical then horizontal, from the center to
-the borders. At each position, the multi-detection algorithm is run
-to collect all the segments found under the stroke.
-In the present work, the stroke sweeping step $\delta$ is set to 10 pixels.
-
-The automatic detection of blurred segments in a whole image is available
-for testing from an online demonstration
-and from a \textit{GitHub} source code repository: \\
-\href{https://github.com/evenp/FBSD}{
-\small{\url{https://github.com/evenp/FBSD}}}
-
-\input{Fig_method/algoAuto}
-
-%The behavior of the unsupervised detection is depicted through the two
-%examples of \RefFig{fig:auto}.
-%The example on the left shows the detection of thin straight objects on a
-%circle with variable width.
-%On the left half of the circumference, the distance between both edges
-%exceeds the initial assigned width and a thick blurred segment is build
-%for each of them. Of course, on a curve, a continuous thickenning is
-%observed untill the blurred segment minimal width reaches the initial
-%assigned width.
-%On the right half, both edges are encompassed in a common blurred segment,
-%and at the extreme right part of the circle, the few distant residual points
-%are grouped to form a thick segment.
-%
-%The example on the right shows the limits of the edge detector on a picture
-%with quite dense repartition of gradient.
-%All the salient edges are well detected but they are surrounded by a lot
-%of false detections, that rely on the presence of many local maxima of
-%the gradient magnitude with similar orientations.
-%
-%\begin{figure}[h]
-%\center
-% \begin{tabular}{c@{\hspace{0.2cm}}c}
-% \includegraphics[width=0.37\textwidth]{Fig_method/vcercleAuto.png} &
-% \includegraphics[width=0.58\textwidth]{Fig_method/plafondAuto.png}
-% \end{tabular}
-% \caption{Automatic detection of blurred segments.}
-% \label{fig:auto}
-%\end{figure}
diff --git a/Article/notions.tex b/Article/notions.tex
deleted file mode 100755
index f2e631180848faf9ee7a0320a02ba2bfee95dbe3..0000000000000000000000000000000000000000
--- a/Article/notions.tex
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@@ -1,168 +0,0 @@
-\section{Theoretical background}
-
-\label{sec:notions}
-
-\subsection{Blurred segment}
-
-This work relies on the notion of digital straight line as classically
-defined in the digital geometry literature \cite{KletteRosenfeld04}.
-Only the 2D case is considered here.
-
-\begin{definition}
-A \textbf{digital straight line} $\mathcal{L}(a,b,c,\nu)$,
-with $(a,b,c,\nu) \in \mathbb{Z}^4$,
-is the set of points $P(x,y)$ of $\mathbb{Z}^2$ that satisfy :
-$0 \leq ax + by - c < \nu$.
-\end{definition}
-
-In the following, we note $\vec{V}(\mathcal{L}) = (a,b)$ the director vector
-of digital line $\mathcal{L}$, $w(\mathcal{L}) = \nu$ its arithmetical width,
-$h(\mathcal{L}) = c$ its shift to origin, and $p(\mathcal{L}) = max(|a|,|b|)$
-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}.
-
-\begin{definition}
-A \textbf{blurred segment} $\mathcal{B}$ of assigned width $\varepsilon$ is
-a set of points in $\mathbb{Z}^2$ that all belong to a digital straight line
-$\mathcal{L}$ of arithmetical width $w(\mathcal{L}) = \varepsilon$.
-\end{definition}
-
-A linear-time algorithm to recognize a blurred segment of assigned width
-$\varepsilon$ \cite{DebledAl05} is used in this work.
-It is based on an incremental growth of the convex hull of the blurred
-segment when adding each point $P_i$ successively.
-The minimal width $\mu$ of the blurred segment $\mathcal{B}$ is the
-arithmetical width of the narrowest digital straight line that contains
-$\mathcal{B}$.
-%It is also the minimal width of the convex hull of $\mathcal{B}$,
-%that can be computed by Melkman's algorithm \cite{Melkman87}.
-The enclosing digital segment $E(\mathcal{B})$ is the section of this
-optimal digital straight line bounded by the end points of $\mathcal{B}$.
-As depicted on \RefFig{fig:bs},
-the extension of the blurred segment $\mathcal{B}_{i-1}$ of assigned width
-$\varepsilon$ and minimal width $\mu_{i-1}$ at step $i-1$ with a new input
-point $P_i$ is thus controlled by the recognition test $\mu_i < \varepsilon$.
-
-\begin{figure}[h]
-\center
- \input{Fig_notions/bswidth}
- \caption{A growing blurred segment $\mathcal{B}_i$ :
-when adding the new point $P_i$, the blurred segment minimal width
-augments from $\mu_{i-1}$ to $\mu_i$; if the new width $\mu_i$ exceeds
-the assigned width $\varepsilon$, then the new input point is rejected
-and $\mathcal{B}_i = \mathcal{B}_{i-1}$.}
- \label{fig:bs}
-\end{figure}
-
-Associated to this primitive, the following definition of a directional scan
- is an important point the proposed method.
-
-\subsection{Directional scan}
-
-\begin{definition}
-A directional scan $DS$ is an ordered partition restricted to the image
-domain $\mathcal{I}$ of a digital straight line $\mathcal{D}$, called the
-\textbf{scan strip}, into scans $S_i$, each of them being a segment of a
-naive line $\mathcal{N}_i$, called a \textbf{scan line}, orthogonal to
-$\mathcal{D}$.
-\end{definition}
-
-\begin{equation}
-DS = \left\{ S_i = \mathcal{D} \cap \mathcal{N}_i \cap \mathcal{I}
-\left| \begin{array}{l}
-\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}) = 0 \\
-\wedge~ h(\mathcal{N}_i) = h(\mathcal{N}_{i-1}) + p(\mathcal{D})
-\end{array} \right. \right\}
-%S_i = \mathcal{D} \cap \mathcal{N}_i, \mathcal{N}_i \perp \mathcal{D}
-\end{equation}
-In this definition, the clause
-$\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}) = 0$
-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
-time.
-
-The scans $S_i$ are developed on each side of a start scan $S_0$,
-and ordered by their distance to the start line $\mathcal{N}_0$ with
-a positive (resp. negative) sign if they are on the left (resp. right)
-side of $\mathcal{N}_0$ (\RefFig{fig:ds}).
-The directional scan is iteratively parsed from the start scan to both ends.
-At each iteration $i$, the scans $S_i$ and $S_{-i}$ are successively processed.
-
-\begin{figure}[h]
-\center
-% \input{Fig_notions/fig}
- \includegraphics[width=0.8\textwidth]{Fig_notions/scanstrip.eps}
- \begin{picture}(1,1)(0,0)
- \thicklines
- \put(-176,112){\vector(2,-1){30}}
- \put(-90,19){\vector(-2,1){30}}
- {\color{dwhite}{
- \put(-181,114.5){\circle*{10}}
- \put(-84,16.5){\circle*{10}}
- \put(-16,102.5){\circle*{10}}
- \put(-132,66.5){\circle*{12}}
- \put(-72,96.5){\circle*{12}}
- \put(-175.5,65.5){\circle*{20}}
- \put(-117,10.5){\circle*{14}}
- \put(-54,32.5){\circle*{14}}
- \put(-161,10.5){\circle*{20}}
- }}
- \put(-88,13.5){$A$}
- \put(-185,111.5){$B$}
- \put(-20,98){$\mathcal{D}$}
- \put(-137,64){\color{blue}{$S_0$}}
- \put(-77,94){\color{red}{$S_8$}}
- \put(-183,64){\color{dgreen}{$S_{-5}$}}
- \put(-123,8){\color{blue}{$\mathcal{N}_0$}}
- \put(-60,30){\color{red}{$\mathcal{N}_8$}}
- \put(-169,8){\color{dgreen}{$\mathcal{N}_{-5}$}}
- \end{picture}
- \caption{A directional scan.
- The start scan $S_0$ is drawn in blue, odd scans in green,
- even scans in red, the bounds of scan lines $\mathcal{N}_i$
- with plain lines and the bounds of scan strip $\mathcal{D}$
- with dotted lines.}
- \label{fig:ds}
-\end{figure}
-
-A directional scan can be defined by its start scan $S_0$.
-If $A(x_A,y_A)$ and $B(x_B,y_B)$ are the end points of $S_0$,
-and if we note $\delta_x = x_B - x_A$, $\delta_y = y_B - y_A$,
-$c_1 = \delta_x\cdot x_A + \delta_y\cdot y_A$,
-$c_2 = \delta_x\cdot x_B + \delta_y\cdot y_B$ and
-$\nu_{AB} = max (|\delta_x|, |\delta_y|)$, it is then defined by
-the following scan strip $\mathcal{D}^{A,B}$ and scan lines
-$\mathcal{N}_i^{A,B}$:
-\begin{equation}
-\left\{ \begin{array}{l}
-\mathcal{D}^{A,B} =
-\mathcal{L}(\delta_x,~ \delta_y,~ min (c1,c2),~ 1 + |c_1-c_2|) \\
-\mathcal{N}_i^{A,B} = \mathcal{L}(\delta_y,~ -\delta_x,~
-\delta_y\cdot x_A - \delta_x\cdot y_A + i\cdot \nu_{AB},~ \nu_{AB})
-\end{array} \right.
-\end{equation}
-
-%The scan lines length is $d_\infty(AB)$ or $d_\infty(AB)-1$, where $d_\infty$
-%is the chessboard distance ($d_\infty = max (|d_x|,|d_y|)$).
-%In practice, this difference of length between scan lines is not a drawback,
-%as the image bounds should also be processed anyway.
-
-A directional scan can also be defined by its central point $C(x_C,y_C)$,
-its direction $\vec{D}(X_D,Y_D)$ and its width $w$. If we note
-$c_3 = x_C\cdot Y_D - y_C\cdot X_D$,
-$c_4 = X_D\cdot x_C + Y_D\cdot y_C$,
-$\nu_{\vec{D}} = max (|X_D|,|Y_D|)$,
-it is then defined by
-the following scan strip $\mathcal{D}^{C,\vec{D},w}$ and scan lines
-$\mathcal{N}_i^{C,\vec{D},w}$:
-\begin{equation}
-\left\{ \begin{array}{l}
-\mathcal{D}^{C,\vec{D},w}
-= \mathcal{L}(Y_D,~ -X_D,~ c_3 - w / 2,~ w) \\
-\mathcal{N}_i^{C,\vec{D},w} = \mathcal{L}(X_D,~ Y_D,~
- c_4 - w / 2 + i\cdot w,~ \nu_{\vec{D}})
-\end{array} \right.
-\end{equation}