Aplikasi Differensial Numerik Dalam Pengolahan Citra Digital

Abstrak

Artikel ini bertujuan untuk mengkaji aplikasi differensial numerik dalam pengolahan citra digital, khususnya dalam mendeteksi tepi objek yang terdapat dalam citra digital. Pendeteksian tepi dapat dilakukan dengan cara yang mirip dengan differensial pertamanya secara parsial dalam ruang diskrit. Aplikasi differensial numeric turunan kedua dalam mendeteksi tepi menggunakan persamaan Laplace, perumusan teori tersebut dalam system digital menggunakan metode hampiran selisih mundur (backward difference approximation). Hasil deteksi tepi dengan menggunakan differensial numeric turunan kedua pada bahasa Matlab memberikan hasil tepi yang jelas dan tajam.

1. Pendahuluan

Citra merupakan kumpulan elemen-elemen gambar (pixel) yang secara keseluruhan merekam suatu adegan (scene) melalui pengindera visual (kamera). Untuk kebutuhan pengolahan dengan berbantuan computer,  citra disajikan dalam bentuk diskrit yang disebut citra digital. Salah satu proses yang terdapat dalam pengolahan citra digital adalah pendeteksian tepi yang menggunakan pendekatan kemiringan differensial.

Penggunaan teori differensial untuk pengolahan citra digital harus dirumuskan dalam bentuk fungsi diskrit, dan secara matematis teori tersebut merupakan persoalan differensial numerik. Differensial numerik adalah metode untuk menentukan hampiran nilai differensial fungsi f dengan menggunakan metode hasil bagi selisih yaitu pengurangan dua buah nilai yang besar (f(x+h) – f(x)) dan membaginya dengan bilangan yang kecil (h). Berdasarkan pada teori tersebut artikel ini ingin mengkaji konsep differensial numerik dan aplikasinya dalam pengolahan citra digital khususnya untuk pendeteksian tepi citra.

2. Metode Penelitian

Bahan yang digunakan dalam penelitian ini berupa citra digital gedung rektorat UNIB dalam format JPEG. Alat yang digunakan pada penelitian ini adalah sebagai berikut:

  • Perangkat keras berupa: 1 set komputer Intel Pentium 4 yang dilengkapi 1 buah monitor warna, 1 buah printer dan 1 buah scanner.
  • Perangkat lunak Matlab ver. 6,5

a. Perumusan Teori Derivatif Dalam Bentuk Digital

Perumusan teori differensial diawali dengan memahami konsep-konsep aplikasi kalkulus khususnya teori differensial dalam rumusan digital (differensial numerik). Metode yang digunakan dalam perumusan teori ini adalah mencari, menghimpun dan mempelajari beberapa teori yang terkait dengan differensial dan aplikasinya.
b. Implementasi Teori Differensial Numerik

Implementasi teori differensial numerik dalam mengolah citra digital untuk mendeteksi tepi citra menggunakan bahasa Matlab. Citra uji (test image) yang digunakan adalah citra (foto) gedung rektorat UNIB yang sudah dikonversi dalam format digital. Hasil implementasi berupa citra tepi akan dianalisis secara visual untuk melihat dampak dari teori differensial numerik dalam mendeteksi tepi citra digital.

3. Hasil Dan Pembahasan
Pendeteksian tepi merupakan langkah pertama untuk melingkupi informasi didalam citra. Tepi mencirikan
batas-batas objek dan karena itu tepi berguna untuk proses segmentasi dan identifikasi objek di dalam citra. Tujuan operasi pendeteksian tepi adalah untuk meningkatkan penampakan garis batas suatu daerah atau objek di dalam citra. Karena tepi termasuk ke dalam komponen berfrekuensi tinggi, maka pendeteksian tepi dapat dilakukan dengan filter lolos tinggi. Tepi (edge) adalah perubahan nilai intensitas derajat keabuan yang mendadak (besar) dalam jarak yang singkat. Perbedaan intensias inilah yang menampakkan rincian pada gambar. Tepi biasanya terdapat pada batas antara dua daerah berbeda pada suatu citra. Tepi dapat diorientasikan dengan suatu arah, dan arah ini berbeda-beda pada bergantung pada perubahan intensitas.

 

jurnal1

jurnal2

Karena menghitung akar adalah persoalan rumit dan menghasilkan nilai riil, maka dalam praktek kekuatan tepi biasanya disederhanakan perhitungannya dengan menggunakan pendekatan rumus berikut (Dulimarta dalam Munir, 2004):

jurnal3

Hasil pendeteksian tepi adalah citra tepi (edge image) g(x,y), yang nilai setiap pixel-nya menyatakan kekuatan tepi:

jurnal4

Keputusan apakah suatu pixel merupakan tepi atau bukan tepi dinyatakan dengan operasi treshold berikut:

jurnal5

Yang dalam hal ini T adalah nilai ambang, pixel tepi dinyatakan putih sedangkan pixel bukan tepi dinyatakan hitam.

a. Differensial Numerik Turunan kedua
Operator turunan kedua disebut juga operator laplace. Operator Laplace mendeteksi lokasi tepi lebih akurat khususnya pada tepi yang curam. Pada tepi yang curam, turunan keduanya mempunyai persilangan nol (zero-crossing), yaitu titik di mana terdapat pergantian tanda nilai turunan kedua, sedangkan pada tepi yang landai tidak terdapat persilangan nol. Persilangan nol merupakan lokasi tepi yang akurat.

Turunan kedua fungsi dengan dua peubah adalah:

jurnal6

Dengan menggunakan definisi hampiran selisih mundur (backward difference approximation):

jurnal7

jurnal8

jurnal9

b. Implementasi Differensial Numerik pada Citra Digital

Implementasi teori differensial numerik (turunan kedua) pada citra digital didahului dengan menerapkan algoritma-algoritma dasar dalam sistem pengolahan citra digital pada software Matlab. Membaca dan menampilkan citra digital

Perintah yang digunakan adalah:

jurnal10

Contoh Membaca dan menampilkan citra UNIB Perintah yang digunakan

jurnal11

Algoritma pengolahan citra digital dalam mendeteksi tepi menggunakan turunan kedua yang berhasil dirancang adalah operator Laplace. Secara umum algoritma tersebut dapat disajikan sebagai berikut:

c. Algoritma Differensial Numerik

jurnal12

Berdasarkan pada hasil rancangan algoritma differensial numeric turunan kedua langkah selanjutnya adalah menyusun program sederhana dengan memanfaatkan fasilitas m-file yang terdapat pada bahasa Matlab. Program yang dirancang adalah program yang disadur dan dikembangkan dari program m-file > edge.m. Hasil implementasi dari algoritma differensial numeric turunan kedua berupa citra gedung rektorat UNIB dengan kenampakan tepi-tepi dari bangunan gedung rektorat UNIB, output citra dari algoritma ini disajikan dalam gambar berikut:

jurnal134. Kesimpulan
Perumusan Operator Laplace menggunakan teori differensial numerik turunan kedua dengan menggunakan metode hampiran selisih-mundur (backward difference approximation) dan dalam aplikasinya dapat digunakan untuk mendeteksi tepi dari objek yang terekam dalam citra digital. Citra yang dihasilkan dari operator Laplace mampu memberikan kenampakan tepi yang cukup baik yang terdapat dalam dari citra asli (citra gedung rektorat UNIB). Tepi pada citra ditandai dengan warna yang putih dan berupa kenampakan linier (garis) yang sangat tegas.

Sumber : gradienfmipaunib.files.wordpress.com/2008/07/yulian-fauzi1.pdf

 

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