1. 3-D RECONSTRUCTION OF A
VISION BASED ON ITS
STEREOSCOPIC MODEL
Tesista
Guillermo Enrique Medina Zegarra
Orientador
PhD. Edgar Lobaton, USA
Co-Orientador
PhD. Nestor Calvo, Argentina
Agradezco a
Arequipa, Per´u
May 07, 2012

2. 1
Index
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

3. 2
Index
1 Introduction
Motivation and context
Definition the problem
General objective
Specific objectives
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

4. 3
Motivation and context
Limitations on pre-Renaissance to create 3D.
(a) Jesus into
Jerusalem

5. 3
Motivation and context
The artists during the Renaissance and the depth.
The vanishing points and three dimensions.
(b) The School of Athens

6. 3
Motivation and context
Limitations on pre-Renaissance to create 3D.
The artists during the Renaissance and the depth.
The vanishing points and three dimensions.
(a) Jesus into
Jerusalem
(b) The School of Athens
Figura: Painting pre-Renaissance and Renaissance [Ma et al., 2004].

7. 4
Definition the problem
Physical architecture, location, distribution and lighting.
(a) One camara
[Cipolla et al., 2010]
(b) Artificial lighting
[VISGRAF., 2012]

8. 5
Definition the problem (cont...)
Figura: How to get the parameters to map an object to the image plane
? [Faugeras, 1993].

9. 6
Definition the problem (cont...)
Figura: How to find corresponding points ? [Szeliski, 2011].

10. 7
Definition the problem (cont...)
Figura: How to find a 3D point of each pair of corresponding points ?
[Szeliski, 2011].

11. 8
Definition the problem (cont...)
Figura: How to reconstruction and smooth a surface from a cloud of
points ? [Hartley and Zisserman, 2004].

12. 9
General objective
Objetivo general
Propose a model for the reconstruction of a 3D image of an object
from two images captured by two cameras located adequately.

13. 10
Specific objectives
Specific objectives
Position two digital cameras on a physical architecture for image
acquisition and calibration.
Rectification of the stereo image pair and calculate the disparity
map through the normalized cross-correlation.
Create the object’s surface from the Delaunay triangulation of the
disparity map.

14. 11
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

15. 12
Pinhole camera model
Help to understand the image formation from geometric point of
view.
Parts of the pinhole camera model: optical center (o), focal
distance (f ) and image plane (I).
x = ¯op ∩ I x ∈ R2
, p ∈ R3
Figura: Pinhole camera model [Ma et al., 2004].

16. 13
Pinhole camera model (cont...)
Figura: Example of the projection of an object on image plane .

17. 14
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

18. 15
Epipolar geometry
Study the geometric relationship and mathematical analysis of a
3-D p point in their image planes.
Figura: Two projections x1, x2 ∈ R2
of a 3-D point p from two vantage
points [Ma et al., 2004].

19. 16
Epipolar geometry (cont...)
Figura: Example of the projection of a cube image on two image planes.

20. 17
Rectification
Figura: Rectification of the stereo image pair [Fusiello et al., 2000].

21. 18
Disparity calculation
(a) (b) Disparity map
(a - b) Tsukuba image pair [Scharstein and Szeliski, 2002].

22. 19
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

23. 20
Pipeline of the proposal

24. 21
Description of the proposed pipeline
Physical architecture
Canon SD1200 Sony DSC-S750

25. 21
Description of the proposed pipeline
Physical architecture
Image acquisition
features Sony DSC-S750 Canon SD1200 IS
Sensor type CCD CCD
Image size 640 × 480 640 × 480
ISO 100 100
Flash off off
Technical characteristics of the two digital cameras

26. 21
Description of the proposed pipeline
Physical architecture
Image acquisition
Calibration
Chessboard (7 × 10)

27. 22
Description of the proposed pipeline
Rectification
Linear search
The correspondence of points is
in the same horizontal line
Original images Rectified images

28. 23
Description of the proposed pipeline
Pre-processing
Manual segmentation
Gaussian filter
Rectified images Pre-processed images

29. 24
Description of the proposed pipeline
Disparity map
Normalized Cross-Correlation
Median filter
Left image pre-processed Right image pre-processed Disparity map

30. 25
Description of the proposed pipeline
3-D mesh
Delaunay triangulation
Intersection of lines
3-D mesh
Disparity map Point Cloud 3-D mesh

31. 26
Description of the proposed pipeline
Reconstructed model
Smoothing the surface
Texturing of the right image
Creation of the surface Smoothing of the surface Texturing of the model

32. 27
“Cubo m´agico”
Original images Rectified images Pre-processed images

33. 28
“Cubo m´agico” (cont...)
Disparity map Point Cloud 3-D mesh
Creation of the surface Smoothing of the surface Texturing of the model

34. 29
Multiple views of the “Magic Cube”

35. 29
Multiple views of the “Magic Cube”

36. 29
Multiple views of the “Magic Cube”

37. 29
Multiple views of the “Magic Cube”

38. 29
Multiple views of the “Magic Cube”

39. 29
Multiple views of the “Magic Cube”

40. 29
Multiple views of the “Magic Cube”

41. 29
Multiple views of the “Magic Cube”

42. 30
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
Teddy bear
Human face
6 Limitations and problems founded
7 Conclusions and future work

43. 31
Teddy bear
Original images Rectified imagenes Pre-processed imagenes

44. 32
Teddy bear (cont...)
Disparity map Cloud of points 3-D mesh
Creation of the surface Smoothing of the surface Texturing of the model

45. 33
Human face
Original images Rectified imagenes Pre-processed imagenes

46. 34
Human face (cont...)
Cloud of points Model without smoothing Smoothing model “Transformed” model
3-D mesh Model without smoothing Smoothing model “Transformed” model

47. 35
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

48. 36
Limitations and problems founded
neighbourhood size Imperfections of the created model

49. 37
Limitations and problems founded (cont...)
Imperfect original image Imperfect original image
Wrong disparity map “Amorphous” 3-D reconstruction

50. 38
Index
1 Introduction
2 Image formation
3 Geometry from two views
4 Proposal
5 Results
6 Limitations and problems founded
7 Conclusions and future work

51. 39
Conclusions
Physic architecture was designed simple and profit.
Lighting conditions must be adequate.
A pipeline was proposed with a sequence of steps needed to get a
3-D reconstruction of a stereo image pair.
The method for the disparity calculation is simple and no robust.
There is a strong dependency between each step of the
reconstruction.

52. 40
Future work
Create an environment with
appropriate conditions for calibration,
lighting and image acquisition.
Physical architecture and artificial lighting [Bradley et al., 2008]

53. 40
Future work
Create an environment with
appropriate conditions for calibration,
lighting and image acquisition.
Physical architecture and artificial lighting [Bradley et al., 2008]
Use multiple cameras.
Multiple views [Hartley and Zisserman, 2004]

54. 41
Future work (cont...)
Use robust methods.

55. 42
Publicated on
Symposium article
“3-D visual reconstruction : a system perspective.”
G. Medina-Zegarra y E. Lobaton
2nd International Symposium on Innovation and Techno-
logy (2011)
pag. 102-107, November 28-30, Lima, Peru
ISBN: 978-612-45917-1-6
Place: Technological University of Peru
Editor: International Institute of Innovation and Techno-
logy (IIITEC)
Chair: Mario Chauca Saavedra

56. 43
Acknowledgements
PhD. Alfedro Miranda
Mag. Alfedro Paz
PhD. Carlos Leyton
PhD(c). Christian L´opez del Alamo
PhD. Edgar Lobaton
PhD. Eduardo Tejada
PhD. Jes´us Mena
PhD. Jos´e Corrales-Nieves
PhD(c). Juan Carlos Gutierrez
Lic. Lu´ıs Pareja
PhD. Nestor Calvo
PhD(c). Regina Ticona
Family Barrios Neyra

57. 44
References
Bradley, D., Popa, T., Sheffer, A., Heidrich, W., and Boubekeur, T. (2008).
Markerless garment capture.
ACM Transactions on Graphics (TOG), 27:99:1–99:9.
Cipolla, R., Battiato, S., and Farinella, G. M. (2010).
Computer Vision: Detection, Recognition and Reconstruction.
Springer.
Faugeras, O. (1993).
Three-dimensional Computer Vision: A Geometric Viewpoint.
The MIT Press. ISBN: 0262061589.
Fusiello, A., Trucco, E., and Verri, A. (2000).
A compact algorithm for rectification of stereo pairs.
Machine Vision and Applications, 12:16–22.
Hartley, R. and Zisserman, A. (2004).
Multiple View Geometry in Computer Vision. Second Edition.
Cambridge University Press. ISBN: 0521540518.
Ma, Y., Soatto, S., Koˇseck´a, J., and Sastry, S. S. (2004).
An Invitation to 3D Vision from Images to Geometric Models.
Springer. ISBN: 0387008934.
Scharstein, D. and Szeliski, R. (2002).
A taxonomy and evaluation of dense two-frame stereo correspondence algorithms.
International Journal of Computer Vision, 47:7–42.
Szeliski, R. (2011).
Computer Vision: Algorithms and Applications.
Springer. ISBN: 9781848829343.

58. 3-D RECONSTRUCTION OF A
VISION BASED ON ITS
STEREOSCOPIC MODEL
Tesista
Guillermo Enrique Medina Zegarra
Orientador
PhD. Edgar Lobaton, USA
Co-Orientador
PhD. Nestor Calvo, Argentina
Agradezco a
Arequipa, Per´u
May 07, 2012

59. 45
View points (perception)
(a) The glass is half full
or half empty ?
(b) is it a duck or a rabbit ?

60. 46
Contenido extra
Contenido extra
1 Datos de procesamiento
2 Geometr´ıa de una vista
3 Geometr´ıa de dos vistas
4 Rectificaci´on e intersecci´on de rectas
5 Mapa de disparidad
6 Filtro de Gauss y filtro de la mediana
7 Propiedades de la triangulaci´on de Delaunay

61. 47
Datos de procesamiento
Un procesador Intel (R) Core (TM) 2 CPU 1.66 GHz y una
memoria RAM 2GB.
El costo computacional del algoritmo en el peor caso es O(n3
) y en
el mejor caso es Θ(n2
).
El tiempo del procesamiento del algoritmo fue de 25 minutos.

62. 48
Modelamiento geom´etrico (matriz de mapeamiento)
propuesta
slide
π : R4
→ R3
; p → x


fsx fsθ ox
0 fsy oy
0 0 1


K
=


sx sθ ox
0 sy oy
0 0 1


Ks


f 0 0
0 f 0
0 0 1


Kf
(1)


u
v
1


x
= K


1 0 0 0
0 1 0 0
0 0 1 0


Π0
R t
0 1
g
π




X
Y
Z
1




p
(2)

63. 49
Ecuaciones de la Geometr´ıa Epipolar
slide
Restricci´on epipolar
xT
2 Fx1 = 0

64. 49
Ecuaciones de la Geometr´ıa Epipolar
slide
Restricci´on epipolar
xT
2 Fx1 = 0
Matriz Fundamental
F = K−T
2 EK−1
1

65. 49
Ecuaciones de la Geometr´ıa Epipolar
slide
Restricci´on epipolar
xT
2 Fx1 = 0
Matriz Fundamental
F = K−T
2 EK−1
1
Matriz esencial
E = [t]x R

66. 49
Ecuaciones de la Geometr´ıa Epipolar
slide
Restricci´on epipolar
xT
2 Fx1 = 0
Matriz Fundamental
F = K−T
2 EK−1
1
Matriz esencial
E = [t]x R
Matriz antisim´etrica
[t]x =


0 −c b
c 0 −a
−b a 0

 (3)

67. 50
Sistema lineal para la matriz F
ui , vi , 1 T


F11 F12 F13
F21 F22 F23
F31 F32 F33

 ui , vi , 1 = 0 , i ∈ R
+
(4)
uu F11 + uv F21 + uF31 + vu F12 + vv F22 + vF32 + u F13 + v F23 + F33 = 0 (5)












u1u1 u1v1 u1 v1u1 v1v1 v1 u1 v1 1
u2u2 u2v2 u2 v2u1 v2v2 v2 u2 v2 1
u3u3 u3v3 u3 v3u1 v3v3 v3 u3 v3 1
u4u4 u4v4 u4 v4u1 v4v4 v4 u4 v4 1
u5u5 u5v5 u5 v5u1 v5v5 v5 u5 v5 1
u6u6 u6v6 u6 v6u1 v6v6 v6 u6 v6 1
u7u7 u7v7 u7 v7u1 v7v7 v7 u7 v7 1
u8u8 u8v8 u8 v8u1 v8v8 v8 u8 v8 1












A













F11
F12
F13
F21
F22
F23
F31
F32
F33













F
= 0 (6)

68. 51
Sistema lineal para la matriz F (cont...)
Minimizar:
A F
2
=
8
i=1
(u
T
i Fui )
2
(7)
Sujeto a:
F 2
= 1 (8)
Por lo tanto, se forma la siguiente funci´on de Lagrange:
L(F, λ) = A F 2
− λ( F 2
− 1) (9)
Por consiguiente, se aplica el m´etodo de los multiplicadores de Lagrange:
JL(F, λ){
2AT
AF − λ(2F)
F 2
− 1
, λ ∈ R
+
(10)
Ahora, se procede a resolver la ecuaci´on JL(f , λ) = 0. La cual, es equivalente a hallar los autovalores y
autovectores de la matriz sim´etrica AT
A:
AT
AF = λ.F
F 2
= 1
(11)
Al calcular los autovectores, se habra encontrado la matriz fundamental F.

69. 52
Calculando los autovalores de una matriz (ejemplo)
A =


1 1 0
2 0 1
0 0 3

 A − λI =


1 − λ 1 0
2 −λ 1
0 0 3 − λ

 (12)
det( A - λ I ) = (1 - λ)(- λ )(3 - λ ) - 2( 3 - λ )
det( A - λ I ) = ( - λ + λ2 )(3 - λ )- 6 + 2 λ
det( A - λ I ) = - λ3 + 4 λ2 - λ - 6
det( A - λ I ) = λ3 - 4 λ2 + λ + 6
Resolviendo el polinomio se encuentran las raices (autovalores), los
cuales son: -1, 2 y 3

70. 53
Calculando los autovectores de una matriz (ejemplo)
A =


1 1 0
2 0 1
0 0 3

 A − λI =


1 − λ 1 0
2 −λ 1
0 0 3 − λ

 (13)
I) Para λ = -1
(A - λ I)v =0
(A - (-1) I)v =0
(A + I)v =0


2 1 0
2 1 1
0 0 4


(A+I)


a
b
c

 =


0
0
0

 (14)

71. 54
Calculando los autovectores de una matriz (ejemplo)
Haciendo el m´etodo de Gauss tenemos:


1 1
2 0
0 0 1
0 0 0

 (15)
c=0 (a, b, c) = (−b
2 , b, 0)
a + b
2 = 0 ⇒ a = −b
2 (a, b, c) = b(−1
2, 1, 0)

72. 55
Calculando los autovectores de una matriz (ejemplo)
II) Para λ = 2


−1 1 0
2 −2 1
0 0 1


(A−2×I)


1 −1 0
0 0 1
0 0 0

 (16)
c=0 (a,b,c) = (b,b,0)
a-b=0 ⇒ a = b (a,b,c) = b(1,1,0)

73. 56
Calculando los autovectores de una matriz (ejemplo)
III) Para λ = 3


−2 1 0
2 −3 1
0 0 0


(A−3×I)


1 −1
2 0
0 1 −1
2
0 0 0

 (17)
b − c
2 = 0 a − b
2 = 0 (a,b,c) = (c
4 , c
2 , c)
b = c
2 a = b
2 (a,b,c) = c(1
4, 1
2, 1)
a = c
4
Los autovalores son: {(−1
2, 1, 0), (1, 1, 0), (1
4, 1
2, 1)}

74. 57
Planteamiento inicial de la rectificaci´on
propuesta
slide
La variable π representa a la matriz de mapeamiento
x ∼= π p (18)
Factorizaci´on QR de la matriz π
π = K[R | t] (19)
La matriz π se re-escribe como:
π =


qT
1 |q14
qT
2 |q24
qT
3 |q34

 = Q|q (20)

75. 58
Planteamiento inicial de la rectificaci´on (cont...)
Las coordenadas del centro ´optico c est´a definido como:
c = −Q−1
q (21)
Se hace un despeje de la ecuaci´on 21 en funci´on de q.
π = [Q| − Qc] (22)

76. 59
Desarrollo de la rectificaci´on
Matriz de transformaci´on
xr1 = λ Qr1Q−1
o1
Tl
xo1 λ ∈ R+
(23)
Para lo cual:
πo1 = [Qo1|qo1] πo1 MPP imagen izquierda inicial
πr1 = [Qr1 |qr1] πr1 MPP imagen izquierda rectificada

77. 60
Pasos para hallar la matriz de transformaci´on
Se hace una factorizaci´on QR de las matrices iniciales
π1 = K[R | − R c1] π1 MPP de la imagen izquierda
π2 = K[R | − R c2] π2 MPP de la imagen derecha
(24)
Los centros ´opticos se hallan con la ecuaci´on 21
La matriz K es la matriz de par´ametros intr´ınsecos
La matriz de rotaci´on R es la misma para ambas matrices de
mapeamiento

78. 61
Pasos para hallar la matriz de transformaci´on (cont...)
Hallando la matriz de rotaci´on R
R =


rT
1
rT
2
rT
3

 (25)
El nuevo eje X es paralelo a la l´ınea base: r1 = ( c1−c2
c1−c2
)
El nuevo eje Y es ortogonal a X, k : r2 = k ∧r1
El nuevo eje Z es ortogonal a XY r3 = r1 ∧ r2

79. 62
Vector unitario k
Demostraci´on de la ortogonalidad del vector Y a trav´es del
vector unitario k.
Plano R3
Z × X =
i j k
0 0 1
1 0 0
Z × X = i(0) - j(-1) + k(0)
Z × X = 0i + 1j + 0k
Z × X
Y
= (0,1,0)
return

80. 63
Intersecci´on de rectas (triangulaci´on)
slide
p = c1 + tQ−1
r1 x1 t ∈ R
p = c2 + sQ−1
r2 x2 s ∈ R
(26)

81. 64
Intersecci´on de rectas (ejemplo)
return
L1 : (X, Y , Z)
p
= (1, 2, 1)
c1
+t (2, 0, 3)
Q−1
r1 x1
L2 : (X, Y , Z)
p
= (5, 4, 1)
c2
+s (−2, −2, 3)
Q−1
r2 x2
L1 : (X, Y , Z) = (1 + 2t, 2, 1 + 3t)
L2 : (X, Y , Z) = (5 − 2s, 4 − 2s, 1 + 3s)
t = 1 , s = 1
(X, Y , Z) = (3, 2, 4)
(27)

82. 65
Representaci´on del punto medio
Calculando el punto medio
c1Q1 + λ = R1 tx1 +
Q2−Q1
2
= R1 tx1 +
(c2+sx2)−(c1+tx1)
2
= R1 tx1 +
(c2−c1)+(sx2−tx1)
2
= R1
c2Q2 − λ = R2 sx2 −
Q1−Q2
2
= R2 sx2 −
(c1+tx1)−(c2+sx2)
2
= R2 sx2 −
(c1−c2)+(tx1−sx2)
2
= R2
L1 = c1 + mR1 M = (
Q1+Q2
2
)
L2 = c2 + nR2 M = (
c1+tx1+c2+sx2
2
)
L1 ∩ L2 = M M = (
c1+c2
2
+
tx1+sx2
2
)

83. 66
Factorizaci´on QR
La factorizaci´on de la matriz de mapeamiento π consta de la
siguientes dos matrices:
π = Q × R
donde:
La matriz Q se obtiene a trav´es del proceso de Gram-Schmidt
La matriz R se consigue a trav´es de la siguiente multiplicaci´on
R = QT × π
return

84. 67
Proceso Gram-Schmidt
u1 = v1,
uk = vk - k−1
j=1
vk ,uj
uj
2 , ; j = 2,. . .,k
return

85. 68
Proceso Gram-Schmidt (ejemplo)
A = {(1, 0, 1)
v1
, (0, 0, 1)
v2
, (1, 1, −1)
v3
}
u1 = v1 = (1, 0, 1)
u2 = (0, 0, 1) − 1
2(1, 0, 1)
u2 = (- 1
2 ,0, 1
2 )
u3 = (1,1,-1) - 0
2 (1,0,1) - (−1)
1
2
(- 1
2 ,0, 1
2 )
u3 = (1,1,-1) + 2(- 1
2 ,0, 1
2 )
u3 = (0,1,0)
La base ortogonal de A es {(1, 0, 1), (−1
2, 0, 1
2), (0, 1, 0)}

86. 69
Pseudo-c´odigo del algoritmo de c´alculo de disparidad
dispComp(imDerecha,imIzquierda,maxDisp)
1 thNorm ← escalar ∗ (2 ∗ r + 1)
2 for i = 1 + r to col − r do
3 for j = 1 + r to fil − maxDisp − r do
5 pBase ← imDerecha(i − r : i + r, j − r : j + r)
6 pBase ← pBase − promedio(pBase)
7 nBase ← norma(pBase)
8 if nBase <= thNorm then
9 continue
10 end if
11 pBase ← pBase/nBase
12 for sh = 1 to maxDisp do
13 pShift ← imIzquierda(i − r : i + r, j + sh − r : j + sh + r)
14 pShift ← pShift − promedio(pShift)
15 nShift ← norma(pShift)
16 if nShift <= thNorm then
17 corr[sh] ← 0
18 continue
19 end if
20 corr[sh] ← sum((pShift/nShift). ∗ (pBase))
21 end for
22 [valor indice] ← max(corr)
23 if valor == 0 then
24 imDisp[i, j] ← 0
25 else
26 imDisp[i, j] ← indice
27 end if
28 end for
29 end for
30 return(imDisp)

87. 70
Mapa de disparidad
propuesta
slide
Correlaci´on Normalizada Cruzada (CNC)
u,v (I1(u,v)−I1)(I2(u+d,v)−I2)
u,v (I1(u,v)−I1)2
u,v (I2(u+d,v)−I2)2

88. 70
Mapa de disparidad
propuesta
slide
Correlaci´on Normalizada Cruzada (CNC)
u,v (I1(u,v)−I1)(I2(u+d,v)−I2)
u,v (I1(u,v)−I1)2
u,v (I2(u+d,v)−I2)2
d(x, y) = arg supr∈RC(x, y, r) (28)

89. 71
Filtro Gaussiano
slide
Valores de la m´ascara
1 2 1
2 3 2
1 2 1
La cantidad de “suavizamiento” que
realiza el filtro gaussiano se puede
controlar variando la desviaci´on est´andar y
el tama˜no de la m´ascara Matriz deslizante de filtrado en el dominio espacial

90. 72
Filtro de la mediana
slide
Valores de ejemplo
6 2 0
3 97 4
19 3 10
En orden ascendente los
n´umeros ser´ıan : 0, 2, 3,
3, 4, 6, 10, 15, 97

91. 72
Filtro de la mediana
slide
Valores de ejemplo
6 2 0
3 97 4
19 3 10
En orden ascendente los
n´umeros ser´ıan : 0, 2, 3,
3, 4, 6, 10, 15, 97
Valor actualizado
* * *
* 4 *
* * *
El valor inicial fue 97 y
luego de utilizar el filtro
de la mediana fue
reemplazado por 4

92. 73
Suavizaci´on del Laplaciano
slide
Calcula la posici´on de un
v´ertice q a partir del
promedio de los v´ertices
adyacentes.
Ejemplo:
3,6
5,4
9,2
14,3
16,10
7,6
———
54, 31
p(9,5) = 54
6 , 31
6
Representaci´on de la suavizaci´on del Laplaciano [Vollmer et al., 1999]

93. 74
Propiedades de la Triangulaci´on de Delaunay
slide
Figura: Ilustraci´on de la primera propiedad de la Triangulaci´on de
Delaunay.

94. 75
Propiedades de la Triangulaci´on de Delaunay (cont...)
Figura: Ilustraci´on de la segunda propiedad de la Triangulaci´on de
Delaunay.

95. 76
Propiedades de la Triangulaci´on de Delaunay (cont...)
(a) Arista ilegal (b) Correcci´on de la
arista ilegal

96. 77
Consideraciones del patr´on de calibraci´on
Detecci´on de las esquinas del patr´on Detecci´on de los puntos internos del patr´on

97. 78
Algoritmo de reducci´on de pol´ıgonos
Reducci´on de pol´ıgonos