n-boyutlu Dijital Görüntülerin Homoloji Grubu

n-boyutlu Dijital Görüntülerin Homoloji Grubu

21. Ulusal Matematik Sempozyumu Bildiriler Kitabı B.1-13
n
∗
†
‡
n
n
M SS18
∗
†
‡
M SS18
Z
Zn n
q = (q1 , ..., qn ) ∈ Z
n
Zn
1 ≤ l ≤ n
p = (p1 , ...., pn ),
|pi − qi | = 1
l
|pj − qj | =
� 1
i
j
p
q
p j = qj
cl
cl
κ
κ
�
�
�
n n−t
κ ∈ 3n (n ≥ 2), 3n − r−2
− 1(2 ≤ r ≤ n − 1, n ≥ 3), 2n(n ≥ 1)
t=0 Ct 2
Z
Z2
Z3
a, b ∈ Z, a ≤ b
[a, b]Z = {z ∈ Z : a ≤ z ≤ b}
c1 := 2
c1 := 4
c2 := 8
κ
κ
(X, κ) ⊂ Z
ε
ε∈N
lκ (x0 , x) x0
n
x0
κ
x
κ
Nκ (x0 , ε) = {x ∈ X | lκ (x0 , x) ≤ ε} ∪ {x0 }
xi
Zn
κ
∀x, y ∈ X x =
� y
xi+1 κ
X
X
f :X→Y
f (x1 )
X
κ
κ0
κ
X ⊂ Zn0
Y ⊂ Zn1 , (X, κ0 )
(Y, κ1 )
U
f (U ) Y
κ1
(κ0 , κ1 )
f : X → Y
X
f (x0 )
f (x1 ) κ1
κ
x = x0 y = xr
X
X ⊂ Zn
i = 0, 1, . . . , r−1
{x0 , x1 , . . . , xr }
(κ0 , κ1 )
{x0 , x1 } κ0
κ
f (x0 ) =
c1 := 6 c2 := 18
(Y, κ1 )
f
c3 := 26
f −1
(κ0 , κ1 )
f : (X, κ0 ) →
(κ1 , κ0 )
(κ0 , κ1 )
R
Z
[1, 3]Z = {1, 2, 3}
[2, 5]Z = {2, 3, 4, 5}
(2, 2)
2
3
f, g : (X, κ0 ) → (Y, κ1 )
H : X × [0, m]Z → Y
f
g
(κ0 , κ1 )
∀x ∈ X
∀x ∈ X
m
f �(κ0 ,κ1 ) g
H(x, 0) = f (x) H(x, m) = g(x)
Hx : [0, m]Z → Y
(2, κ1 )
∀t ∈ [0, m]Z
(κ0 , κ1 )
∀t ∈ [0, m]Z , Hx (t) = H(x, t)
Ht : X → Y ∀x ∈ X, Ht (x) = H(x, t)
(X, κ) ⊂ Zn
f : [0, m]Z −→ X
κ
(2, κ)
f (i)
x
f (0) = x f (m) = y
f (0) = f (m)
f
y
κ
j = i±1
f (j) κ
mod m
f : [0, m − 1]Z −→ X ⊂ Z2
κ
(2, κ)
{f (0), f (1), . . . , f (m − 1)}
M SC8 M SC4 M SC8�
x ∈ X
(X, κ)
y, z ∈ X
X
κ
X
κ
κ
• | X̄ |x x
Dxx
•
κ̄
κ �= 3n − 2n − 1
x
Nκ̄ ∩ C xx �= ∅
X
κ
C xx
κ̄
κ
κ
X̄ = Zn − X
| X |x := N26 (x, 1) − {x}
y ∈ Nκ ∩
X
κ
κ
(κ, κ̄) ∈ {(κ, 2n), (2n, 3n − 1)}
x∈X
x
X
(X, κ) ⊂ Zn , n ≥ 3
X
•
κ
Nκ̄ ∩ Dxx �= ∅
κ
X
(κ, κ̄) = (3n − 2n − 1, 2n)
• X κ
•
x∈X
| X |x
| X |x
κ
X
�
M SS18
1X
f :X→Y
f ◦ g �(κ1 ,κ0 ) 1Y
f
(X, κ0 )
(κ0 , κ0 )
� A ⊂ X
=
∀a ∈ A r ◦ i(a) = a
X κ0
κ
M SS18
M SS6
g ◦ f �(κ0 ,κ1 )
(κ1 , κ0 )
(κ0 , κ1 )
g : Y → X
(κ0 , κ1 )
X
(X, κ0 )
κ0
i : A �→ X κ0
r:X →A
P = {p0 , p1 . . . , pm } ⊂ (Zn , κ)
P
(κ, m)
κ0
ti ∈ Z
{p0 , p1 , . . . , pm }
m
�
ti pi = 0
i=1
ve
m
�
i=1
ti = 0 ise t0 = t1 = · · · = tm = 0 dir.
i, j ∈ {0, 1, . . . , m} i �= j
P = �p0 , p1 . . . , pm �
pi
pj κ
m
2, 2, 8, 26
P = {p0 , p1 . . . , pm } ⊂ (Zn , κ)
K
(κ, m)
s∈K
s
{p0 , p1 . . . , pm }
s, t ∈ K
s
s∩t
s
t
(K, κ)
(K, κ)
(K, κ)
|K|
|K| =
�
s∈K
(κ, 0)
s.
(X, κ0 )
h : |K| −→ X (κ0 , κ1 )
(K, κ1 )
X
P ⊂ (Zn0 , κ0 ) Q ⊂ (Zn1 , κ1 )
P
Q
(κ0 , κ1 )
P = {p0 , p1 , . . . , pm }
(κ0 , m)
(κ1 , m)
Q = {q0 , q1 , . . . , qm }
h : (P, κ0 ) −→ (Q, κ1 ) pi �→ h(pi ) = qi
pi ∈ P
Cqκ (K)
p̂i pi
(K, κ)
(K, κ) ⊂ Zn
�
(κ0 , κ1 )
(κ, q)
m
κ
∂q : Cqκ (K) −→ Cq−1
(K)
∂q (< p0 , p1 , . . . , pq >) =
m≥q
��
q
i
i=0 (−1)
0,
∂q−1 ◦ ∂q = 0
< p0 , p1 , . . . , p̂i , . . . , pq >, m ≥ q
m<q
�p0 , p1 , . . . , pq � ∈ Cqκ (K)
q
�
∂q−1 ◦ ∂q (�p0 , p1 , . . . , pq �) = ∂q−1 ( (−1)i �p0 , . . . , p�i , . . . , pq �)
i=0
=
q−1
�
j=0
=
�
q
�
(−1) ( (−1)i �p0 , . . . , p�j , . . . , p�i , . . . , pq �)
j
i=0
i+j
(−1)
i<j
+
�
j�i
=
(−1)i+j �p0 , . . . , p�j , . . . , p�i , . . . , pq �
�
i<j
= 0.
�p0 , . . . , p�i , . . . , p�j , . . . , pq �
[(−1)i+j + (−1)i+j+1 ]�p0 , . . . , p�i , . . . , p�j , . . . , pq �
(K, κ) m
C∗κ (K)
∂m+1
∂m−1
∂
∂
∂
m
1
0
κ
κ
0 −→ Cm
(K) −→
Cm−1
(K) −→ · · · −→
C0κ (K) −→
0
(K, κ) m
Zqκ (K) = Ker ∂q
Bqκ (K) = Im ∂q+1
κ
κ
q.
q.
Hqκ (K) = Zqκ (K)/Bqκ (K)
f : X → Y
Hqκ0 (X) ∼
= Hqκ1 (Y )
κ
(κ0 , κ1 )
q.
∀m ≥ q
f : X → Y
(κ0 , κ1 )
f
x 1 , x2 ∈ X x 1
⇔ f (x1 )
f (x2 ) κ1
f (x1 ) = f (x2 )
κ0
m≥q�0
�p0 , p1 , . . . , pq � ∈ Cq (X)
φ : Cqκ0 (X) −→ Cqκ1 (Y ),
x2 κ0
φ(�p0 , p1 , . . . , pq �) = �f (p0 ), f (p1 ), . . . , f (pq )�
φ
f
κ0
κ1
∼
Cq (X) = Cq (Y )
Hqκ0 (X) ∼
= Hqκ1 (Y ).
(X, κ)
�
Z, q = 0
Hq (X) =
0, q �= 0.
X = {x0 }
m≥q >0
Cqκ (X) = 0
C0κ (X)
(κ, 0)
C0κ (X) ∼
∂1 : 0 −→ C0κ (X) ∼
=Z
=Z
∼
Im ∂1 = 0, ve Ker ∂0 = Z
κ
Hqκ (X) ∼
=
X
X
Hqκ (X)
(κ, q)
q=0
=0
∂0 : C0κ (X) ∼
= Z −→ 0
κ
∼
H0 (X) = Z
{Xλ : λ�Λ}
�
X
Hqκ (Xλ ).
λ
κ
�
Z, q = 0, 1
Hq (X) =
0, q �= 0, 1.
X = {x0 , x1 . . . , xq } ⊂ Z2
κ
xi
xj κ
κ
⇔ i = j ± 1 mod q
C0κ (X) = {�x0 �, �x1 �, �x2 �, . . . , �xq �} ∼
= Zq+1
C1κ (X) = {�x0 , x1 �, �x1 , x2 �, . . . , �xq , x0 �} ∼
= Zq+1
m>q>1
Cqκ (X) = 0
Hqκ (X) = 0
∂
∂
∂
2
1
0
0 −→
C1κ (X) −→
C0κ (X) −→
0
Im ∂2 = 0
Ker ∂0 = Zq+1
∂1 (a0 �x0 , x1 � + a1 �x1 , x2 � + . . . + aq �xq , x0 �) = a0 (x1 − x0 ) + a1 (x2 − x1 )
+ . . . + aq (x0 − xq )
Im ∂1 = Zq
∂1 (a0 �x0 , x1 � + a1 �x1 , x2 � + . . . + aq �xq , x0 �) = 0
a0 (x1 − x0 ) + a1 (x2 − x1 ) + . . . + aq (x0 − xq ) = 0
(aq − a0 )x0 + (a0 − a1 )x1 + . . . + (aq−1 − aq )xq = 0
Ker ∂1 = Z
a0 = a1 = . . . = aq
H1κ (X) = Z = H0κ (X).
X
κ0
κ1
Hq (X)
Hq (Y )
Y
3.11
M SC8� = {(1, 0), (0, 1), (−1, 0), (0, −1)}
�
M SC8 �(8,8) {∗}
p 1 < p2
H18 (M SC8 ) = Z
H18 (X)
H18 ({∗})
H18 ({∗}) = 0
X = {po = (0, 0), p1 = (1, 0), p2 = (1, 1)}
8
8
C0 (X)
(X, 8)
{�p0 �, �p1 �, �p2 �} ,
C18 (X)
C28 (X)
�
{�p0 p1 �, �p1 p2 �, �p0 p2 �}
{�p0 p1 p2 �}
C08 (X) ∼
= Z3 , C18 (X) ∼
= Z3 , ve C82 (X) ∼
= Z.
p0 <
∂
∂
∂
∂
3
2
1
0
0 −→
C28 (X) −→
C18 (X) −→
C08 (X) −→
0.
Ker ∂2 = 0, Ker ∂1 = Z2 , Ker ∂0 = Z3 ,
Im ∂2 = Z, Im ∂1 = Z2
�
Z, q = 0, 1
Hq8 (X) =
0, q �= 0, 1.
Im ∂3 = 0,
M SS18 ∼
= {c0 = (0, 0, 1), c1 = (−1, 1, 1), c2 = (−2, 0, 1), c3 = (−2, −1, 1)
c4 = (−1, −2, 1), c5 = (0, −1, 1), c6 = (−1, 0, 0), c7 = (−1, −1, 0)
c8 = (−1, 0, 2), c9 = (−1, −1, 2)}
C018 (M SS18 )
{�c0 �, �c1 �, . . . , �c9 �}
C118 (M SS18 )
(18, 0)
{�c0 c1 �, �c1 c2 �, �c2 c3 �, �c3 c4 �, �c4 c5 �, �c5 c0 �, �c6 c7 �, �c8 c9 �, �c9 c0 �, �c0 c6 �,
�c1 c9 �, �c6 c1 �, �c2 c8 �, �c7 c2 �, �c3 c8 �, �c7 c3 �, �c4 c8 �, �c7 c4 �, �c5 c9 �, �c6 c5 �,
�c2 c6 �, �c5 c8 �}
C218 (M SS18 )
(18, 1)
{�c0 c1 c9 �, �c0 c6 c1 �, �c0 c6 c5 �, �c9 c0 c5 �, �c3 c4 c8 �, �c7 c3 c4 �, �c2 c3 c8 �, �c7 c2 c3 �,
�c1 c2 c6 �, �c2 c6 c7 �, �c4 c5 c8 �, �c5 c8 c9 �}
(18, 2)
∂
∂
∂
∂
3
2
1
0
0 −→
C28 (M SS18 ) −→
C18 (M SS18 ) −→
C08 (M SS18 ) −→
0.
Im ∂2 ∼
= Z12
Ker ∂2 = 0
Im ∂1 ∼
= Z9
Ker ∂1 ∼
= Z13 , Ker ∂0 ∼
= Z10 , Im ∂3 = 0
�
Z, q = 0, 1
Hq18 (M SS18 ) =
0, q =
� 0, 1.
15th