��5, 2 = 8 + 21 + 0 − 8 + 22 + 0 = −2 ��5, 2 = 8 + 23 + 8 − 0 + 20 + 0 = 22
�5, 2 = �−2
2
+ 22
2
= 22,09 = 22 ��5, 3 = 8 + 29 + 0 − 2 + 25 + 0 = −13
��5, 3 = 3 + 28 + 8 − 0 + 20 + 0 = 27
�5, 3 = �−13
2
+ 27
2
= 29,9 = 30 ��5, 4 = 5 + 24 + 0 − 8 + 21 + 0 = 3
��5, 4 = 8 + 28 + 8 − 0 + 20 + 0 = 32
�5, 4 = �3
2
+ 32
2
= 32,1 = 32 ��5, 5 = 0 + 20 + 0 − 8 + 29 + 0 = −26
��5,5 = 8 + 25 + 0 − 0 + 20 + 0 = 0
�5, 5 = �−26
2
+ 18
2
= 31,6 = 32
Setelah dilakukan perhitungan seperti diatas, maka didapat lah citra output dari matriks 5x5 yang sebelumnya telah disebutkan.
Citra output dapat dilihat pada matriks berikut ini:
⎣ ⎢
⎢ ⎢
⎡ 3
1 6
2 3
5 15
9 4
6 6
1 10
7 5
8 3
8 8
5 2
5 1
9 4⎦
⎥ ⎥
⎥ ⎤
⎣ ⎢
⎢ ⎢
⎡ 31
45 38
25 18
33 17
17 21
19 21
13 19
17 26
13 6
23 14
25 23
22 30
32 32⎦
⎥ ⎥
⎥ ⎤
3.1.3.5 Analisis Proses Deteksi Tepi dengan Metode Frei-Chen
Berikut ini merupakan contoh penggunaan metode Frei-Chen. Dimisalkan terdapat suatu matriks citra input 5x5 :
⎣ ⎢
⎢ ⎢
⎡ 3
1 6
2 3
5 15
9 4
6 6
1 10
7 5
8 3
8 8
5 2
5 1
9 4⎦
⎥ ⎥
⎥ ⎤
Selanjutnya matriks tersebut dihitung dengan menggunakan kernel 3 x 3, dengan operator Frei-Chen yaitu :
�
� 2
=
�
2
+ ��
3
+ �
4
-
� +
��
7
+ �
6
�
� 2
=
� +
��
1
+ �
2
- �
6
+ ��
5
+ �
4
� = ��
� 2
+ �
� 2
Perhitungan dimulai dari koordinat Fx1,1, Fy1,1 dan dilakukan dengan mengambil matriks 3x3 dari matriks tetangga piksel yang akan direduksi. Untuk dapat
melakukan operasi perhitungan pada koordinat 1, 1 diberikan boundary matrik citra dengan nilai 0. Perhitungan ini dilakukan pada setiap nilai piksel pada matriks citra
dimana c adalah konstanta bernilai √2. Contoh penggunaan operator Frei-Chen dapat
dilihat seperti di bawah ini:
⎣ ⎢
⎢ ⎢
⎡ 3
1 6
2 3
5 15
9 4
6 6
1 10
7 5
8 3
8 8
5 2
5 1
9 4
⎦ ⎥
⎥ ⎥
⎤
⎣ ⎢
⎢ ⎢
⎢ ⎡
3 1
5 15
6 2
3 9
4 6
6 1
8 2
3 5
10 7
5 8
1 8
9 5
4 0⎦
⎥ ⎥
⎥ ⎥
⎤
��1, 1 = �0 + √21 + 15� − �0 + √20 + 0� = 16 ��1, 1 = �0 + √20 + 0� − �0 + √25 + 15� = −22
�1, 1 = �16 + −22
2
= 27,2 = 27 �
�
1
�
2
�
7
�, � �
3
�
6
�
5
�
4
��1, 2 = �0 + √26 + 9� − �0 + √213 + 5� = −6 ��1, 2 = �0 + √20 + 0� − �5 + √215 + 9� = −35
�1, 2 = �−6
2
+ −35
2
= 36 ��1, 3 = �0 + √22 + 4� − �0 + √21 + 15� = −10
��1, 3 = �0 + √20 + 0� − �15 + √29 + 4� = −32
�1, 3 = �−10
2
+ −32
2
= 34 ��1, 4 = �0 + √23 + 6� − �0 + √26 + 9� = −10
��1, 4 = �0 + √20 + 0� − �9 + √24 + 6� = −21
�1, 4 = �−10
2
+ −21
2
= 23,68 = 24 ��1, 5 = �0 + √20 + 0� − �0 + √22 + 4� = −7
��1, 5 = �0 + √20 + 0� − �4 + √26 + 0� = −12
�1, 5 = �−7
2
+ −12
2
= 23,68 = 24 ��2, 1 = �1 + √215 + 1� − �0 + √20 + 0� = 23
��2, 1 = �0 + √23 + 1� − �0 + √26 + 1� = −4
�2, 1 = �23
2
+ −4
2
= 23,68 = 24 ��2, 2 = �6 + √29 + 10� − �3 + √25 + 6� = 13
��2, 2 = �3 + √21 + 6� − �6 + √21 + 10� = −7
�2, 2 = �13
2
+ −7
2
= 14,76 = 15 ��2, 3 = �2 + √24 + 7� − �1 + √22 + 1� = 10
��2, 3 = �1 + √26 + 2� − �1 + √210 + 7� = −11
�2, 3 = �10
2
+ 11
2
= 14,8 = 15 ��2, 4 = �3 + √26 + 5� − �6 + √29 + 10� = −12
��2, 4 = �6 + √22 + 3� − �10 + √27 + 5� = 13
�2, 4 = �−12
2
+ 13
2
= 17,6 = 18 ��2, 5 = �0 + √20 + 0� − �2 + √24 + 7� = −15
��2, 5 = �2 + √23 + 0� − �7 + √25 + 0� = −8
�2, 5 = �−15
2
+ −8
2
= 17 ��3, 1 = �15 + √21 + 3� − �0 + √20 + 0� = 19
��3, 1 = �0 + √25 + 15� − �0 + √28 + 3� = 8
�3, 1 = �19
2
+ 8
2
= 20,6 = 21 ��3, 2 = �9 + √210 + 8� − �5 + √26 + 8� = 10
��3, 2 = �5 + √22 + 9� − �8 + √23 + 8� = 3
�3, 2 = �10
2
+ 3
2
= 10,3 = 10 ��3, 3 = �4 + √27 + 8� − �2 + √21 + 3� = 15
��3, 3 = �2 + √29 + 4� − �3 + √28 + 8� = 4
�3, 3 = �15
2
+ 4
2
= 15,52 = 16 ��3, 4 = �6 + √25 + 5� − �9 + √210 + 8� = 13
��3, 4 = �9 + √24 + 6� − �8 + √28 + 5� = 4
�3, 4 = �13
2
+ 4
2
= 13,6 = 14 ��3, 5 = �0 + √20 + 0� − �4 + √27 + 8� = −22
��3, 5 = �4 + √26 + 0� − �8 + √25 + 0� = 3
�3, 5 = �−22
2
+ 3
2
= 26,07 = 22 ��4, 1 = �3 + √25 + 0� − �0 + √20 + 0� = 10
��4, 1 = �0 + √26 + 1� − �0 + √22 + 5� = 2
�4, 1 = �10
2
+ 2
2
= 10 ��4, 2 = �10 + √28 + 1� − �6 + √28 + 2� = 3
��4, 2 = �6 + √21 + 10� − �2 + √25 + 1� = 7,3
�4, 2 = �3
2
+ 7,3
2
= 7,6 = 8 ��4, 3 = �7 + √28 + 9� − �1 + √23 + 5� = 17
��4, 3 = �1 + √210 + 7� − �5 + √21 + 9� = 11
�4, 3 = �17
2
+ 11
2
= 20,32 = 20 ��4, 4 = �5 + √25 + 4� − �10 + √28 + 1� = −6
��4, 4 = �10 + √27 + 5� − �1 + √29 + 4� = 7 �4, 4 = �−6
2
+ 7
2
= 9,4 = 9 ��4, 5 = �0 + √20 + 0� − �0 + √28 + 9� = −20
��4, 5 = �7 + √25 + 0� − �9 + √24 + 0� = −1
�4, 5 = �−20
2
+ −1
2
= 20 ��5, 1 = �3 + √25 + 0� − �0 + √20 + 0� = 10
��5, 1 = �0 + √28 + 1� − �0 + √20 + 0� = 8
�5, 1 = �10
2
+ 8
2
= 13 ��5, 2 = �8 + √21 + 0� − �8 + √22 + 0� = −1
��5, 2 = �8 + √23 + 8� − �0 + √20 + 0� = 20
�5, 2 = �−1
2
+ 20
2
= 20 ��5, 3 = �8 + √29 + 0� − �2 + √25 + 0� = 12
��5, 3 = �3 + √28 + 8� − �0 + √20 + 0� = 22
�5, 3 = �12
2
+ 22
2
= 25 ��5, 4 = �5 + √24 + 0� − �8 + √21 + 0� = 2
��5, 4 = �8 + √28 + 8� − �0 + √20 + 0� = 27
�5, 4 = �2
2
+ 27
2
= 27 ��5, 5 = �0 + √20 + 0� − �8 + √29 + 0� = −21
��5, 5 = �8 + √25 + 0� − �0 + √20 + 0� = 15
�5, 5 = �−21
2
+ 15
2
= 26
Setelah dilakukan perhitungan seperti diatas, maka didapat lah citra output dari matriks 5x5 yang sebelumnya telah disebutkan.
Citra output dapat dilihat pada matriks berikut ini:
⎣ ⎢
⎢ ⎢
⎡ 3
1 6
2 3
5 15
9 4
6 6
1 10
7 5
8 3
8 8
5 2
5 1
9 4⎦
⎥ ⎥
⎥ ⎤
⎣ ⎢
⎢ ⎢
⎡ 27
36 34
24 24
24 15
15 18
17 21
10 16
14 22
10 8
20 9
20 13
20 25
27 26⎦
⎥ ⎥
⎥ ⎤
3.1.3.6 Analisis Proses Deteksi Tepi dengan Metode Morphologi
