zulkaryanto.files.wordpress.com example of the variable-length coding is huffman coding 1. Coding...

1

1

Edge Detection

Yeni Herdiyeni

Departemen Ilmu Komputer FMIPA IPB

2

Outline

Introduction

Image Segmentation

Edge detection method

Differentiation (Gradient Operator)

Hough Transform

Reference

2

3

What are edges?

Edge detection

Local intensity change

Strong edge = the steep areas in a 3D plot (show: blobs-for-edge, surface plot)

4

What are edges?

3

Deteksi Tepi

Deteksi tepi (Edge detection) adalah operasi yang dijalankan untuk mendeteksi garis tepi (edges) atau boundary untuk segmentasi, registrasi, dan identifikasi objek.

Edge adalah beberapa bagian dari citra di mana intensitas kecerahan berubah secara drastis.

5

Deteksi Tepi (edge)


Dalam objek berdimensi 1, perubahan dapat diukur dengan menggunakan fungsi turunan (derivative function).

Perubahan mencapai maksimum pada saat nilai turunannya pertamanya mencapai nilai maksimum atau nilai turunan kedua (2nd derivative) bernilai 0.

6

4

7

Gradient Operators

Motivasi: mendeteksi perubahan changes

Perubahan nilai piksel yang besar Gradient besar

Gradient

operatorimage Thresholding

edge

mapx(m,n)

g(m,n)I(m,n)

otherwise

thnmgnmI

0

|),(|1),(

MATLAB function: > help edge

8

Introduction

Fig.1. Edges are boundaries between different textures.Edge also can be defined as

discontinuities inimage intensity from one pixel to another.. [4]

5

9

EDGE DETECTION METHOD

First-Order Derivative Edge Detection The Roberts operators

The Prewitt operators

The Sobel operators

First-Order of Gausssian (FDOG)

Second-Order Derivative Edge Detection Laplacian

Canny

10

Edge Detectors

Mudah diimplementasikan dan cepat

Berdasarkan grey-level gradient

Dihitung pada setiap piksel => Gradient image: g(x,y)

Menggunakan 16-bit atau 32-bit image untuk merepresentasikan gradient!

Gradient : the first-order derivative:

Bekerja pada arah x dan y:

)1,()1,(),(

),1(),1(),(

yxfyxfyxg

yxfyxfyxg

y

x

),(),( yxgyxf

6

Correlation with this kernel: -1 +1

Normally this kernel to avoid bias:

12

Fist-Order Derivative Edge Detection

x

y

fG xf

fG

x

2 2| | x yf f G G

the direction of the gradient vector

the gradient vector

the magnitude of this vector

Definition:

1 of tany

x

Gangle f

G

7

13

Fist-Order Derivative Edge Detection

Fig.2. Orthogonal gradient generation.[2]

The gradient along the line normal to the edge slope

The spatial gradient amplitude

The gradient amplitude combination

14

The Roberts operators

8

15

The Prewitt operators

16

The Sobel operators

3 6 9 1 4 7( ) ( )yG z z z z z z

9

17

Operators

Fig.3. Impulse response arrays for 3 3 orthogonal

differential gradient edge operators.[2]

18

First-Order of Gausssian

( , ) * ( , )

( , ) * ( , )

x x

y y

M x y G I x y

M x y G I x y

It is hard to find the gradient by using the equation

In order to simplify the computation

M=|Mx|+|My|

10

19

First-order derivative edge detection

Fig.4. Using 1st-order differentiation to detect (a) the sharp edges, (c) the step edges with noise, and (e) the ramp edges. (b)(d)(e) are the results of differentiation of (a)(c)(e).[3]

Second Order Derivation

Laplacian

20

11

21

Second-order derivative edge detection

The Laplacian of a 2-D function f (x, y) is a second-order derivative defined as:

2 22

2 2

f ff

x y

22

Edge Detector: Laplacian

Where exactly is the edge (width)?

Second-order derivative:

The variation of the variation of the gray-level value!

How fast does the gradient change?

Find the zero-crossing

Center of edge

),('' yxf

0Zero-crossing

Prewitt and Sobel0 -1 0

-1 4 -1

0 -1 0

),('' yxf

12

23

24


2 2

22( , )

x y

h x y e

2

22 2 2

2 24

2( , )

rx y

h x y e

The 2-D Gaussian function:

The Laplacian of Gaussian (LOG):

13

25


Fig.7. An 1111 mask approximation to Laplacian of Gaussian (LOG).[3]

26


Sensitive to noise

Gaussian function

Low-pass filter

14

27


(a) Original image.

(b) Laplacian mask of Fig.(a)

(c) Laplacian mask of Fig.(b)

(d) LOG mask of Fig.(b)

(a)

(b)

(c)

(d)

Fig.9. Simulation of second-order of derivative edge detection[3]

28


Fig.10. Using 1st-order differentiation to detect (a) the sharp edges, (c) the step edges with noise, and (e) the ramp edges. (b)(d)(e) are the results of differentiation of (a)(c)(e).[3]

15

29


The Drawbacks of the Differentiation

Method for Edge Detection :

Sensitivity to noise

Not good for ramp edges

Make no difference between the significant edge and the detailed edge

30


Noise reduction

2D Gaussian used for smoothing

Edge enhancement

Second-order derivative (second-order gradient)

Edge localisation

Zero-crossings

16

31


The second-order derivative is very sensitive to noise!

Second Order Derivation

Canny

32

17

33

Canny Edge Detector

Noise reduction

2D Gaussian used for smoothing

Edge enhancement

Magnitude of gradient vector:

Edge localisation

Complicated

yxyx ggggg22

34

Canny Edge Detector

Edge localisation

Wide edges:

Edges give rise to ridges in the gradient image

Thin edges using the principle of non-maximal suppression

Find gradients directions

Each pixel is compared with its two

neighbors in the gradient direction. The

two smallest are suppressed => set to zero

The result is pixel-thin edges

18

35

Canny Edge Detection

Good detection

Good localization

Single response Many edge candidate

The accurate edge

36

Simulation

The original image

The Sobel operatorsFDOG

Fig.5. Comparison with FDOG and the Sobel operators

19

37

Simulation

(a) Original image.

(b) Roberts operator.

(c) Prewitt operator.

(d) Sobel operator.

(a)

(b)

(c)

(d)

Fig.6. Simulation of first-order of derivative edge detection[3]

38


2

5 2 4 6 84 ( )f z z z z z2

5 1 2 3 4 6 7 8 98 ( )f z z z z z z z z z

20

39

Conlusion

Fist-Order Derivative Edge Detection:

the simplest method

Second-Order Derivative Edge Detection:

sensitive to noise

40

Reference

[1] Soo-Chang Pei, Jian-Jiun Ding, Imporved Harris AlgotithmForConer And Edge Detections, vol 1, 2005.

[2] William K. Pratt , Digital Image Processing_William K. Pratt3rd ,chapter 15.

[3] Jiun-De Huang, Image Compression by Segmentation andBoundary Description, chapter 2.

[4] J. Canny, A Computational Approach to Edge Detection, IEEETrans. Pattern Analysis and Machine Intelligence, PAMI-8,6,November 1986, 679698.

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END

Kuliah 10 Deteksi Tepi Canny edit print.pdf

1

Deteksi Tepi Canny

Yeni Herdiyeni

Departemen Ilmu Komputer IPB

Canny Edge Detector 1

Edge Detection Method

The First Derivative - Gradient: The gradient method

detects the edges by looking for the maximum and minimum

in the first derivative of the image.

The Second Derivatives - Laplacian: The Laplacian method

searches for zero crossings in the second derivative of the

image to find edges. An edge has the one-dimensional shape

of a ramp and calculating the derivative of the image can

highlight its location.


Canny Edge Detector 3 Canny Edge Detector 4

Deteksi Tepi (edge)

Perubahan mencapai maksimum pada

saat nilai turunannya pertamanya

mencapai nilai maksimum atau nilai

turunan kedua (2nd derivative) bernilai

0.

5

The Canny edge detector

original image (Lena)

2


Gradient magnitude


thresholding


thinning

(non-maximum suppression)


Steps

1. Doing Convolution :Apply derivative of Gaussian (to

compute Gradient Magnitude)

2. Non-maximum suppression

Thin multi-pixel wide ridges down to single pixel

3. Thresholding

Low, high edge-strength thresholds

Accept all edges over low threshold that are connected to edge

over high threshold

Matlab: edge(I, canny)


Gradient

At each point convolve with

Sobel also can be applied to compute gradient

magnitude and orientation of the Gradient are computed as

Avoid floating point arithmetic for fast computation

11

11xG

11

11yG

22 ],[],[],[ jiQjiPjiM

]),[],,[(tan],[ 1 jiPjiQji

Sobel Edge Detector

Detecting Edges in Image

Image I

101

202

101

121

000

121

*

*

Idx

d

Idy

d

22

Idy

dI

dx

d

ThresholdEdges

3

Teknik Konvolusi


Example:

Non-Maximum Suppression

courtesy of G. Loy

Original image Gradient magnitudeNon-maxima suppressed

Edge Smearing

from Forsyth & Ponce

0

0

0

0

2

2

2

2

20

20

20

20

2

2

2

2

Sobel filter example: Yields2-pixel wide edge band

We want to localize the edge to within 1 pixel

6

8

8

6

6

8

8

6

00

00

00

00

-8

-6

-6

-8

Input

Result


Non-Maxima Suppression

Thin edges by keeping large values of Gradient

not always at the location of an edge

there are many thick edges

12010232

10100232

01001230

00112310

01121200

13121000

31110000


Non-Maxima Suppression (2)

Thin the broad ridges in M[i,j] into ridges that are only one pixel wide

Find local maxima in M[i,j] by suppressing all values along the line of the Gradient that are not peak values of the ridge

12010232

10100232

31001230

00112310

01121200

13121003

31110000

gaps

false

edges

Edge Streaking

Can predict next pixel in edge orthogonal to gradient to make edge chain Can also just use 8-connectedness to define chains

Streaking: Gaps in edge chain due to edge strength dipping below threshold

courtesy of G. Loy

Original image Strong edges

gap

4

Example: Canny Edge Detection

courtesy of G. Loy

gap is gone

Originalimage

Strongedges

only

Strong +connectedweak edges

Weakedges


Suppress the pixels in Gradient Magnitude

Image which are not local maximum

edgean tonormaldirection thealong

in of neighbors theare and Sx,yy,xy,x

otherwise0

,,&

,, if,

, yxSyxS

yxSyxSyxS

yxMyx ,

yx,

yx ,


Gradient Orientation

Reduce angle of Gradient [i,j] to one of the 4 sectors

Check the 3x3 region of each M[i,j]

If the value at the center is not greater than the 2

values along the gradient, then M[i,j] is set to 0


local

maxima

removed

depends

on condition

12010232

10100232

01001230

00112310

01121200

13121000

31110000


The suppressed magnitude image will contain many

false edges caused by noise or fine texture

02010030

10100030

00000230

00000300

00021200

03120000

30000000

false edges


Thresholding

Reduce number of false edges by applying a

threshold T

all values below T are changed to 0

selecting a good values for T is difficult

some false edges will remain if T is too low

some edges will disappear if T is too high

some edges will disappear due to softening of the

edge contrast by shadows

5


Double Thresholding

Apply two thresholds in the suppressed image

T2 = 2T2

two images in the output

the image from T2 contains fewer edges but has gaps in the

contours

the image from T1 has many false edges

combine the results from T1 and T2

link the edges of T2 into contours until we reach a gap

link the edge from T2 with edge pixels from a T1 contour

until a T2 edge is found again


T2=2 T1=1

02000030

00000030

00000230

00000300

00020200

03020000

30000000

02010030

10100030

00000230

00000300

00021200

03120000

30000000

gaps

filled

from

T1

A T2 contour has pixels along the green arrows

Linking: search in a 3x3 of each pixel and connect the

pixel at the center with the one having greater value

Search in the direction of the edge (direction of Gradient)


Line Detection

Model of a line: two edges with opposite polarity in distance less than the size of the smoothing filter

edge detection filters respond to step edges

they do not provide meaningful response to lines

Apply nonmaxima suppression on the smoothed output

a line is the derivative of a step the derivative step of the Canny algorithm is not necessary

Tugas

Buat program deteksi tepi Canny

menggunakan matlab, tidak boleh

menggunakan fungsi dari matlab : edge(I,

canny)

Sertakan penjelasan teori dan kode

programnya pada laporan terpisah dengan

kode program (*baca referensi yang diberikan)

Tugas dikumpulkan tgl 4 Desember 2009


Kuliah 10 Deteksi Tepi Canny.pdf

Euripides G.M. Petrakis 1

Deteksi Tepi Canny

Yeni Herdiyeni




original image (Lena)



Gradient magnitude


thresholding



thinning

(non-maximum suppression)

Sobel Edge Detector

Detecting Edges in Image

Image I

101

202

101

121

000

121

*

*

Idx

d

Idy

d

22

Idy

dI

dx

d

ThresholdEdges


Canny Edge Detector

xS

yS

I

Canny Edge Detector

I

22

yx SSS

25ThresholdS



Steps

1. Apply derivative of Gaussian : Gradient Magnitude

2. Non-maximum suppression

Thin multi-pixel wide ridges down to single pixel

3. Thresholding

Low, high edge-strength thresholds

Accept all edges over low threshold that are connected to edge

over high threshold

Matlab: edge(I, canny)


Canny Edge Detector

1) Smooth image with a Gaussian

optimizes the trade-off between noise filtering and edge localization

2) Compute the Gradient magnitude using approximations of partial derivatives

2x2 filters

3) Thin edges by applying non-maxima suppression to the gradient magnitude

4) Detect edges by double thresholding



Gradient

At each point convolve with

magnitude and orientation of the Gradient are computed as

Avoid floating point arithmetic for fast computation

11

11xG

11

11yG

22 ],[],[],[ jiQjiPjiM

]),[],,[(tan],[ 1 jiPjiQji

Example:


courtesy of G. Loy

Original image Gradient magnitudeNon-maxima suppressed


Edge Smearing

from Forsyth & Ponce

0

0

0

0

2

2

2

2

20

20

20

20

2

2

2

2

Sobel filter example: Yields2-pixel wide edge band

We want to localize the edge to within 1 pixel

6

8

8

6

6

8

8

6

00

00

00

00

-8

-6

-6

-8

Input

Result


Non-Maxima Suppression

Thin edges by keeping large values of Gradient

not always at the location of an edge

there are many thick edges

12010232

10100232

01001230

00112310

01121200

13121000

31110000



Non-Maxima Suppression (2)

Thin the broad ridges in M[i,j] into ridges that are only one pixel wide

Find local maxima in M[i,j] by suppressing all values along the line of the Gradient that are not peak values of the ridge

12010232

10100232

31001230

00112310

01121200

13121003

31110000

gaps

false

edges

Edge Streaking

Can predict next pixel in edge orthogonal to gradient to make edge chain Can also just use 8-connectedness to define chains

Streaking: Gaps in edge chain due to edge strength dipping below threshold

courtesy of G. Loy

Original image Strong edges

gap


Example: Canny Edge Detection

courtesy of G. Loy

gap is gone

Originalimage

Strongedges

only

Strong +connectedweak edges

Weakedges


Suppress the pixels in Gradient Magnitude

Image which are not local maximum

edgean tonormaldirection thealong

in of neighbors theare and Sx,yy,xy,x

otherwise0

,,&

,, if,

, yxSyxS

yxSyxSyxS

yxMyx ,

yx,

yx ,



Gradient Orientation

Reduce angle of Gradient [i,j] to one of the 4 sectors

Check the 3x3 region of each M[i,j]

If the value at the center is not greater than the 2

values along the gradient, then M[i,j] is set to 0


local

maxima

removed

depends

on condition

12010232

10100232

01001230

00112310

01121200

13121000

31110000



The suppressed magnitude image will contain many

false edges caused by noise or fine texture

02010030

10100030

00000230

00000300

00021200

03120000

30000000

false edges


Thresholding

Reduce number of false edges by applying a

threshold T

all values below T are changed to 0

selecting a good values for T is difficult

some false edges will remain if T is too low

some edges will disappear if T is too high

some edges will disappear due to softening of the

edge contrast by shadows



Double Thresholding

Apply two thresholds in the suppressed image

T2 = 2T2

two images in the output

the image from T2 contains fewer edges but has gaps in the

contours

the image from T1 has many false edges

combine the results from T1 and T2

link the edges of T2 into contours until we reach a gap

link the edge from T2 with edge pixels from a T1 contour

until a T2 edge is found again


T2=2 T1=1

02000030

00000030

00000230

00000300

00020200

03020000

30000000

02010030

10100030

00000230

00000300

00021200

03120000

30000000

gaps

filled

from

T1

A T2 contour has pixels along the green arrows

Linking: search in a 3x3 of each pixel and connect the

pixel at the center with the one having greater value

Search in the direction of the edge (direction of Gradient)



Line Detection

Model of a line: two edges with opposite polarity in distance less than the size of the smoothing filter

edge detection filters respond to step edges

they do not provide meaningful response to lines

Apply nonmaxima suppression on the smoothed output

a line is the derivative of a step the derivative step of the Canny algorithm is not necessary

Tugas

Buat program deteksi tepi Canny

menggunakan matlab, tidak bolehmenggunakan fungsi dari matlab : edge(I,

canny)

Sertakan penjelasan programnya pada laporan

terpisah dengan kode program

Tugas dikumpulkan tgl 18 Desember 2008


Kuliah 13 Kompresi Citra 2 yhy.pptx

KOMPRESI CITRA

Yeni Herdiyeni

Departmen Ilmu Komputer IPB

Kompresi Citra

Kompresi Citra adalah aplikasi kompresi data yang dilakukan terhadap citra digital dengan tujuan untuk mengurangi redundansi dari data-data yang terdapat dalam citra sehingga dapat disimpan atau ditransmisikan secara efisien.

Teknik Kompresi Citra (4)

Loseless Compression :

Teknik kompresi citra dimana tidak ada satupun informasi citra yang dihilangkan.

Biasa digunakan pada citra medis.

Metode loseless : Run Length Encoding, Entropy Encoding (Huffman, Aritmatik), dan Adaptive Dictionary Based (LZW)

Contoh Citra Medik

Run Length Encoding (RLE)

Contoh 1

Citra biner:

0000000000000000000000010

0000000000010000000000000

Total = 50 bit

Kode string 0 dan 1 dikodekan menjadi

(format : kode(jumlah frekuensi)):

0(23) 1(1) 0(12) 1(1) 0(13)

Jika untuk menyimpan nilai jumlah frekuensi membutuhkan 5 bit untuk penyimpanan dan kode string membutuhkan 1 bit, maka diperlukan kapasitas sebesar : (1*5) + (5*5) = 30 bit.

Run Length Encoding (RLE)

Exercise


Lossy Compression :

Ukuran file citra menjadi lebih kecil dengan menghilangkan beberapa informasi dalam citra asli.

Teknik ini mengubah detail dan warna pada file citra menjadi lebih sederhana tanpa terlihat perbedaan yang mencolok dalam pandangan manusia, sehingga ukurannya menjadi lebih kecil.

Biasanya digunakan pada citra foto atau image lain yang tidak terlalu memerlukan detail citra, dimana kehilangan bit rate foto tidak berpengaruh pada citra.


Beberapa teknik lossy

Color reduction : untuk warna-warna tertentu yang mayoritas dimana informasi warna disimpan dalam color palette.

Chroma subsampling : teknik yang memanfaatkan fakta bahwa mata manusia merasa brightness (luminance) lebih berpengaruh daripada warna (chrominance) itu sendiri, maka dilakukan pengurangan resolusi warna dengan disampling ulang. Biasanya digunakan pada sinyal YUV.

Chorma Subsampling terdiri dari 3 komponen :

Y (luminance) : U (CBlue) : V (CRed)

11

Downsampling

The human eye can see more detail in the Y component (brightness) than in Cb (blue) and Cr (red). Using this knowledge, encoders can be designed to compress images more efficiently. (http://en.wikipedia.org)

Transformasi warna dari RGB ke ruang warna (color space) dinamakan YCbCr.

Pengurangan komponen Cb dan Cr dinamakan "downsampling" atau "chroma subsampling

Downsampling the chroma components menghemat 33% atau 50% space yang digunakan image.

Contoh Downsampling

Rasio downsampling pada JPEG adalah 4:4:4 (no downsampling), 4:2:2 (reduce by factor of 2 in horizontal direction), dan 4:2:0 (reduce by factor of 2 in horizontal and vertical directions).


Transform coding : menggunakan Fourier Transform seperti DCT (Discrete Cosine Transform).

Fractal Compression : adalah suatu metode lossy untuk mengkompresi citra dengan menggunakan kurva fractal. Sangat cocok untuk citra natural seperti pepohonan, pakis, pegunungan, dan awan.

Fractal Compression bersandar pada fakta bahwa dalam sebuah image, terdapat bagian-bagian image yang menyerupai bagian image yang lain.

Proses kompresi Fractal lebih lambat daripada JPEG sedangkan proses dekompresinya sama.

Hal-Hal Penting Dalam Kompresi Citra

Scalability/Progressive Coding/Embedded Bitstream

Adalah kualitas dari hasil proses pengkompresian citra karena manipulasi bitstream tanpa adanya dekompresi atau rekompresi.

Biasanya dikenal pada loseless codec.

Contohnya pada saat preview image sementara image tersebut didownload. Semakin baik scalability, makin bagus preview image.


Tipe scalability :

Quality progressive : dimana image dikompres secara perlahan-lahan dengan penurunan kualitasnya.

Resolution progressive : dimana image dikompresi dengan mengenkode resolusi image yang lebih rendah terlebih dahulu baru kemudian ke resolusi yang lebih tinggi.

Component progressive : dimana image dikompresi berdasarkan komponennya, pertama mengenkode komponen grey baru kemudian komponen warnanya.


Region of Interest Coding : daerah-daerah tertentu dienkode dengan kualitas yang lebih tinggi daripada yang lain.

Meta Information : image yang dikompres juga dapat memiliki meta information seperti statistik warna, tekstur, small preview image, dan author atau copyright information

Algoritma Kompresi/Dekompresi Citra (1)

Algoritma umum untuk kompresi image adalah :

1. Menentukan bitrate dan toleransi distorsi image dari inputan user.

2. Pembagian data image ke dalam bagian-bagian tertentu sesuai dengan tingkat kepentingan yang ada (classifying).

Menggunakan salah satu teknik : DWT (Discreate Wavelet Transform) : mencari frekuensi nilai pixel masing-masing, menggabungkannya menjadi satu dan mengelompokkannya sebagai berikut:


Dimana

LL : Low Low Frequency (most importance)

HL : High Low Frequency (lesser importance)

LH : Low High Frequency (more lesser importance)

HH : High High Frequency (most less importance)


Hasil dekomposisi 3 level decomposition


3. Pembagian bit-bit di dalam masing-masing bagian yang ada (bit allocation).

4. Lakukan kuantisasi (quantization).

Kuantisasi Scalar : data-data dikuantisasi sendiri-sendiri

Kuantisasi Vector : data-data dikuantisasi sebagai suatu himpunan nilai-nilai vektor yang diperlakukan sebagai suatu kesatuan.

5. Lakukan pengenkodingan untuk masing-masing bagian yang sudah dikuantisasi tadi dengan menggunakan teknik entropy coding (huffman dan aritmatik) dan menuliskannya ke dalam file hasil.

Algoritma Kompresi/Dekompresi Citra

Sedangkan algoritma umum dekompresi image adalah :

1. Baca data hasil kompresi menggunakan entropy dekoder.

2. Dekuantisasi data.

3. Rebuild image.

Beberapa Metode Kompresi Citra

Teknik Kompresi GIF

GIF (Graphic Interchange Format) dibuat oleh Compuserve pada tahun 1987 untuk menyimpan berbagai file bitmap menjadi file lain yang mudah diubah dan ditransmisikan pada jaringan komputer.

GIF merupakan format citra web yang tertua yang mendukung kedalaman warna sampai 8 bit (256 warna), menggunakan 4 langkah interlacing, mendukung transparency, dan mampu menyimpan banyak image dalam 1 file.

Byte ordering : LSB MSB

Interlacing

Interlacing separates the odd and even lines and transmits them separately. It allows the overall frame rate to be half what it would need to be if the whole display were delivered progressively. Thus, it reduces the bandwidth required to 50% and is therefore a form of compression.

Struktur file GIF (1)

Header : menyimpan informasi identitas file GIF (3 bytes, harus string GIF) dan versinya (3 bytes, harus string 87a or 89b)

Global Screen Descriptor (GSD) : mendefinisikan logical screen area dimana masing-masing file GIF ditampilkan.


Global Color Table : masing-masing image dalam GIF dapat menggunakan global color table atau tabel warnanya sendiri-sendiri. Penggunaan GCT akan memperkecil ukuran file GIF.

Image1, Image2, Image3, ... Image-n : dimana masing-masing image memiliki struktur blok sendiri-sendiri dan terminator antar file.

Trailer : Akhir dari sebuah file GIF



Kompresi GIF menggunakan teknik LZW : gambar GIF yang berpola horizontal dan memiliki perubahan warna yang sedikit, serta tidak bernoise akan menghasilkan hasil kompresan yang baik.

Kompresi LZW (Lempel, Ziv dan Welch) pada GIF

Proses encoding yang mencari rangkaian pixel yang sama pada gambar. Pola yang lebih sering muncul mendapatkan sebuah kode yang mewakili rangkaian tersebut dalam file terkompresi.

Format file GIF

GIF87a : mendukung interlacing dan mampu manyimpan beberapa image dalam 1 file, ditemukan tahun 1987 dan menjadi standar.

GIF89a : kelanjutan dari 87a dan ditambahkan dengan dukungan transparency, mendukung text, dan animasi.

Animated GIF

Animated GIF : tidak ada standar bagaimana harus ditampilkan sehingga umumnya image viewer hanya akan menampilkan image pertama dari file GIF. Animated GIF memiliki informasi berapa kali harus diloop.

Tidak semua bagian dalam animated GIF ditampilkan kembali, hanya bagian yang berubah saja yang ditampilkan kembali.

JPEG

JPEG (Joint Photograpic Experts Group) menggunakan teknik kompresi lossy sehingga sulit untuk proses pengeditan.

JPEG cocok untuk citra pemandangan (natural generated image), tidak cocok untuk citra yang mengandung banyak garis, ketajaman warna, dan computer generated image

JPEG Compression Models (1)

Sequential : kompresi dilakukan secara top-down, left-right menggunakan proses single-scan dan algoritma Huffman Encoding 8 bit secara sekuensial.


Progressive : kompresi dilakukan dengan multiple-scan secara progresif, sehingga kita dapat mengira-ngira gambar yang akan kita download.


Hierarchical : super-progressive mode, dimana image akan dipecah-pecah menjadi sub image yang disebut frame. Frame pertama akan membentuk image dalam resolusi rendah hingga berangsur-angsur ke resolusi tinggi.

Loseless (JPEG-LS) : exact image

Teknik Kompresi JPEG

JPEG merupakan nama teknik kompresi, sedangkan nama format filenya adalah JFIF (JPEG File Interchange Format)

Tingkat kompresi yang baik untuk JPEG adalah 10:1-20:1 untuk citra foto, 30:1-50:1 untuk citra web, dan 60:1-100:1 untuk kualitas rendah seperti citra untuk ponsel.

Byte order : MSB-LSB

JPEG

Tahapan Kompresi JPEG (1)

Sampling : adalah proses pengkonversian data pixel dari RGB ke YUV/YIQ dan dilakukan down sampling. Biasanya sampling dilakukan per 8x8 blok, semakin banyak blok yang dipakai makin bagus kualitas sampling yang dihasilkan.

DCT (Discreate Cosine Transform) : hasil dari proses sampling akan digunakan sebagai inputan proses DCT, dimana blok 8x8 pixels akan diubah menjadi fungsi matriks cosinus


Quantization : proses membersihkan koefisien DCT yang tidak penting untuk pembentukan image baru. Hal ini yang menyebabkan JPEG bersifat lossy.

Entropy Coding : proses penggunaan algoritma entropy, misalnya Huffman atau Aritmatik untuk mengenkodekan koefisien hasil proses DCT yang akan mengeliminasi nilai-nilai matriks yang bernilai nol secara zig-zag order.


Baseline sequential JPEG encoding and decoding processes

Step 1: Picture Preparation (a)

Color space transformation

First, the image is converted from RGB into a different color space called YCbCr. This is the same as the color space used by PAL, MAC and Digital color television transmission.

The Y component represents the brightness of a pixel

The Cb and Cr components together represent the chrominance

475 x 330 x 3 = 157 KB

luminance

RGB Components

Y U V (Y Cb Cr) Components

Assign more bits to Y, less bits to Cb and Cr

Y U V (Y Cb Cr) Components

Assign more bits to Y, less bits to Cb and Cr

Step 1: Picture Preparation (b)

Downsampling

The human eye can see more detail in the Y component than in Cb and Cr. Using this knowledge, encoders can be designed to compress images more efficiently.

The above transformation enables the next step, which is to reduce the Cb and Cr components (called "downsampling" or "chroma subsampling").

Typically each group of 4 pixels would be sampled with 4 values for Y and only 1 each for U & V (sampling ratio of 4:1:1).

For the rest of the compression process, Y, Cb and Cr are processed separately and in a very similar manner.

Step 1: Picture Preparation (b)

Downsampling

Original: 12 blocks.

Sampled: 4 + 1 + 1 = 6 blocks.

Compression ratio = 50%.

There is 75% lost in chrominance components

For the rest of the compression process, Y, Cb and Cr are processed separately and in a very similar manner.

Step 2: Picture Processing

Discrete Cosine Transform

Next, each component (Y, Cb, Cr) of the image is "tiled" into sections of eight by eight pixels each, then each tile is converted to frequency space using a two-dimensional forward discrete cosine transform.

Within the MCU, each component is processed separately one 8x8 block at a time.

The luminance block is shifted from numbers 0..255 to numbers 128..127 by subtracting 128.

Chrominance values are already in the range 128..127.

The 64 (8 X 8) DCT Basis Functions

Each 8x8 block can be looked at as a weighted sum of these basis functions.

The process of 2D DCT is also the process of finding those weights.

50

This is another explanation of 2D DCT.


Each block is transformed by the forward DCT into low- and high-frequency information.

This step is lossless, except that there may be little lost due to rounding of DCT coefficients

If one such 88 8-bit subimage is:

JPEG Image Compression II

DCT: similar to FT; only real (cosine) part is kept.

Identifies average value (dc coefficient) and remaining components (ac coefficients).


which, if we now subtract 128 from each element, results in

and then taking the DCT and rounding to the nearest integer results in

Note the rather large value of the top-left corner. This is the DC coefficient. The remaining 63 coefficients are called the AC coefficients.

Step 3: Quantization

The human eye is good at seeing small differences in brightness over a relatively large area, but not so good at distinguishing the exact strength of a high frequency brightness variation.

This fact allows one to get away with greatly reducing the amount of information in the high frequency components.

This is done by simply dividing each component in the frequency domain by a constant for that component, and then rounding to the nearest integer.

This is the main lossy operation in the whole process.


As a result of this, it is typically the case that many of the higher frequency components are rounded to zero, and many of the rest become small positive or negative numbers.

A common quantization matrix is:


Using this quantization matrix with the DCT coefficient matrix from above results in:

For example, using 415 (the DC coefficient) and rounding to the nearest integer

The quantization matrix may be scaled by a quality factor

Step 4: Entropy Encoding

Entropy coding is a special form of lossless data compression.

DC Coefficient Coding the difference between the quantized DC coefficient of the current block and that of the previous block of the same component is taken.

Step 4: Entropy Encoding

AC Coefficient Coding

the 63 quantized coefficients are converted into a one-dimensional zig-zag sequence to increase the run length of zeros.

- zero run-length encoding is then applied.

- Huffman coding or arithmetic encoding is then further applied.

Zig-zag Scan DCT Blocks

Why? -- To group low frequency coefficients in top of vector.

Maps 8 x 8 to a 1 x 64 vector.

Horizontal frequency

Vertical frequency

59

The zig-zag sequence for the above quantized coefficients would be:

26,

3, 0,

3, 2, 6,

2, 4, 1, 4,

1, 1, 5, 1, 2,

1, 1, 1, 2, 0, 0,

0, 0, 0, 1, 1, 0, 0,

0, 0, 0, 0, 0, 0, 0, 0,

0, 0, 0, 0, 0, 0, 0,

0, 0, 0, 0, 0, 0,

0, 0, 0, 0, 0,

0, 0, 0, 0,

0, 0, 0,

0, 0,

0

DC coefficient is -26

Zero run-length encoding is

(0, -3),

(1, -3), (0, -2), (0, -6), (0, 2), (0, -4),

(0, 1), (0, -4), (0, 1), (0, 1), (0, 5), (0, 1),

(0, 2), (0, -1), (0, 1), (0, -1), (0, 2),

(5, -1), (0, -1),

(0, 0)

Decoding

Decoding to display the image consists of doing all the above in reverse.

Taking the DCT coefficient matrix (after adding the difference of the DC coefficient back in)

Decoding

and multiplying it by the quantization matrix from above results in

which closely resembles the original DCT coefficient matrix for the top-left portion.

Original

63

Decoding

Taking the inverse DCT (type-III DCT) results in an image with values (still shifted down by 128)

and adding 128 to each entry

Original

64

Latihan

Given the following matrices

Matrix A : 8 x 8 luminance block.

Matrix B : 8 x 8 Quantization matrix (luminance)

For this luminance block, obtain a Matrix C by subtracting each entry from 128.

Obtained the DCT coefficient matrix as a result of apply DCT to Matrix C.

Compute the quantized coefficients using truncation to the nearest integers and then write them out in a zig-zag order.

Compute the quantized coefficients by rounding to the nearest integers and then write them out in a zig-zag order.

Compare the merits of using each of the techniques in parts (d) and (e) above.

JPEG 2000

JPEG 2000 merupakan pengembangan kompresi JPEG.

Didesain untuk internet, scanning, foto digital, remote sensing, medical imegrey, perpustakaan digital dan e-commerce

Kelebihan :

Dapat digunakan pada bit-rate rendah sehingga dapat digunakan untuk network image dan remote sensing

Menggunakan Lossy dan loseless tergantung kebutuhan bandwidth. Loseless digunakan untuk medical image

Transmisi progresif dan akurasi & resolusi pixel tinggi

Menggunakan Region of Interest (ROI)

JPEG 2000

Kelebihan (lanjutan) :

Robustness to bit error yang digunakan untuk komunikasi jaringan dan wireless

Open architecture : single compression/decompression

Mendukung protective image security : watermarking, labeling, stamping, dan encryption

Mendukung image ukuran besar 64k x 64k, size up to 232 - 1

Mendukung meta data dan baik untuk computer-generated imagenary. Dulu JPEG standar baik untuk natural imagenary.

2-D Wavelet Transform via Separable Filters

From Matlab Wavelet Toolbox Documentation

1/18/2010

69

2-D Example

From Usevitch (IEEE Sig.Proc. Mag. 9/01)

015614152728

2471316262942

38121725304143

911182431404453

1019233239455254

2022333846515560

2134374750565961

3536494857586263

Sheet1

0.2

-0.6

0.8

-1

1

-0.8

0.6

-0.2

1

Sheet1

U=7

Amplitude

Sheet2

015614152728

2471316262942

38121725304143

911182431404453

1019233239455254

2022333846515560

2134374750565961

3536494857586263

-10-10-101010-10-10-10

-10101010101010-10

1010101010101010

1010101010101010

1010101010101010

1010101010101010

-10101010101010-10

-10-10-101010-10-10-10

400-26000-110

00000000

-450-24080-100

00000000

-2000020000

00000000

-3010018040

00000000

Sheet3

Kuliah09-Deteksi%20Tepi%20(Edge%20Detection)_01_print_version.pdf

11/22/2009

1

Deteksi Tepi (Edge Detection)Yeni Herdiyeni

Departemen Ilmu Komputer FMIPA IPB

www.ilkom.fmipa.ipb.ac.id/yeni

Deteksi Tepi

Deteksi tepi (Edge detection) adalah operasi yang dijalankan untuk mendeteksi garis tepi (edges) yang membatasi dua wilayah citra homogen yang memiliki tingkat kecerahan yang berbeda (Pitas 1993).

Tujuannya adalah untuk mengubah citra 2D menjadi bentuk kurva.


http://www.ilkom.fmipa.ipb.ac.id/yeni

11/22/2009

2

Edge

Deteksi Edge


Dalam objek berdimensi 1, perubahan dapatdiukur dengan menggunakan fungsi turunan(derivative function).

Perubahan mencapai maksimum pada saat nilaiturunannya pertamanya mencapai nilaimaksimum atau nilai turunan kedua (2nd

derivative) bernilai 0.

11/22/2009

3

Edge Detection Using the Gradient

Properties of the gradient:

The magnitude of gradient

provides information about

the strength of the edge

The direction of gradient is

always perpendicular to the

direction of the edge

Main idea:

Compute derivatives in x and y directions

Find gradient magnitude

Threshold gradient magnitude


Estimating the gradient with finite differences

Approximation by finite differences:

11/22/2009

4


Using pixel-coordinate notation (remember: j corresponds

to the x direction and i to the negative y direction):


Example:

Suppose we want to

approximate the gradient

magnitude at z5

We can implement I/ x and I/ y using the following masks:

Note: Mx is the approximation at (i, j + 1/2)

and My is the approximation at (i + 1/2, j)

11/22/2009

5


The Roberts edge detector

This approximation can be implemented by the following masks:

Note: Mx and My are approximations at (i + 1/2, j + 1/2)


The Prewitt edge detector

The partial derivatives can be computed by:

Note: Mx and My are approximations at (i, j))

Consider the arrangement of

pixels about the pixel (i, j):

The constant c implies the emphasis given to pixels closer to the

center of the mask.

Setting c = 1, we get the Prewitt operator:

11/22/2009

6

Algoritme Deteksi Tepi

Robert Operator

Sobel Operator

Prewitt Operator

Canny Operator

Laplacian operator

dan lain-lain.


The Sobel edge detector

Note: Mx and My are approximations at (i, j))

Setting c = 2, we get the Sobel operator:

11/22/2009

7


(an example using the Prewitt edge detector - dont divide by 2)

Operator Robert

Robert Operator menggunakan operator gradient berukuran 2 x 2 :

Gradient magnitude

Karena operator Robert hanya menggunakanconvolution mask berukuran 2 x 2, maka operator Robert sangat sensitive terhadap noise.

11/22/2009

8

Operator Sobel #1

Operator Gradient 3 x 3

Operator Sobel melakukan deteksi tepidengan memperhatikan tepi vertical danhorizontal.

Operator Sobel #2

Gradient Magnitude

11/22/2009

9

Operator Prewitt

Operator Prewitt menggunakan 8 (delapan) buah kernel operator gradient

Implementasi Matlab Deteksi tepi dengan operator Prewitt

citra=imread('cameraman.tif');ic = citra (:,:,1);

px=[-1 0 1;-1 0 1;-1 0 1]; %% Deteksi Vertikalicx=filter2(px,ic); % convolutionfigure,imshow(icx/255);

py=px'; %% Deteksi Horizontalicy=filter2(py,ic);figure,imshow(icy/255);

edge_p=sqrt(icx.^2+icy.^2);figure,imshow(edge_p/255);

edge_t=im2bw(edge_p/255,0.3);figure, imshow(edge_t);

11/22/2009

10


Example:

Idx

d

Idy

d


Example cont.:

22

Idy

dI

dx

d

11/22/2009

11


Example cont.:

100Threshold


11/22/2009

12


Image Gradient

Perubahan intensitas kecerahan dapatdihitung dengan menggunakan gradient citra(image gradient).

11/22/2009

13


Definition of the gradient:

To save computations, the magnitude of gradient is usually

approximated by:

Gradient Magnitude

gradient magnitude. Gradient Magnitude dapat dihitung dengan cara:

11/22/2009

14

Effects of noise Consider a single row or column of the image

Plotting intensity as a function of position gives a signal

Where is the edge?

Where is the edge?

Solution: smooth first

Look for peaks in

11/22/2009

15

Derivative theorem of convolution

This saves us one operation:

Tugas #3

Buat program deteksi Tepi dengan menggunakan operator

Robert

Prewitt

Sobel

Program tidak boleh menggunakan fungsi MATLAB

Kuliah10 Hough Transform.pdf

12/11/2008

1

Hough Transform

Yeni Herdiyeni


http:www.ilkom.fmipa.ipb.ac.id/yeni

2

Feature extraction

Citra dijital

Ekstraksi ciri : bentuk

1. Segmentasi

2. Ekstraksi Bentuk

- Edge detection

- Hough transform

Ciri bentuk untuk

pengindeksan

12/11/2008

2

3

Intensitas gradien

yang berbeda (Pitas 1993)

Edge Detection

Good Detection

Good Localization

Clear Response

Canny Edge Detector:

Dikembangkan oleh

John F. Canny in

1986.

Citra dijital

4

12/11/2008

3

Hough Transform

5

Hough Transform

6

12/11/2008

4

Hough Transform

7

8

Diperkenalkan oleh Paul Hough, 1962

The Hough Transform (HT) is a technique that locates shapes in images.

.there are an infinite number of potential lines that pass

through any point, each at a different orientation.

y = mx + c

Hough Transform

c = (-x)m + y

12/11/2008

5

Hough Transform

sebuah titik (xi,yi) dalam ruang koordiant-xy dapat

ditransformasi menjadi sebuah persamaan garis pada

ruang koordinat-mc.

9

10

Titik perpotongantiga garis (-1,4)

A

12/11/2008

6

Hough Transform

11

12

Hough Transform Line (r, ):

Local maxima

Parameter space (r, )

r

x

y

Dikembangkan oleh Richard Duda dan

Peter Hart, 1972

Standard Hough Transform (SHT)

r = xi cos + yi sin

Hough Transform

xi

yi

r

12/11/2008

7

Hough Transform

13

Hough Transform

14

12/11/2008

8

Hough Transform

15

Algoritme

1. Tentukan nilai

2. Untuk setiap piksel (x,y) hitung nilai r = x

cos + y sin untuk semua nilai

3. Buat array nilai dan r

16

12/11/2008

9

17

Hough Transform Circle (x0, y0, r):

Local maxima

x = x0 + r cos

y = y0 + r sin

x0

y0

x0

y0

Hough Transform

r (x, y)

x0

y0

18

Hough Transform

Hough transform circle

12/11/2008

10

19

Hough Transform Ellipse (x0, y0, rx, ry):

rx

ry

y0

x0

(x, y)x = x0 + rx cos

y = y0 + ry sin

x0

y0

x0

y0

Local maxima

Hough Transform

20

Hough Transform

Hough transform ellipse

12/11/2008

11

21

1. DaffodilCitra asli Citra biner Hasil Hough Transform

(a) Circle

(b) Ellipse

22

2. LilyCitra asli Citra biner Hasil Hough Transform

(a) Circle

(b) Ellipse

12/11/2008

12

23

3. Morning GloryCitra asli Citra biner Hasil Hough Transform

(a) Circle

(b) Ellipse

24

Citra asli Citra biner Hasil Hough Transform

(a) Circle

(b) Ellipse

4. Rose

12/11/2008

13

25

Shape Indexing

nnnn

n

n

ppp

ppp

ppp

P

21

22221

11211

1

nn

nn

nn

p

p

p

p

p

P

1

2

12

11

1

nn

nn

nn

p

p

p

p

p

Pm

1

2

12

11

HT Matrik 50x50 Matrik 2500x1

nn

nn

nn

p

p

p

p

p

P

1

2

12

11

1

nn

nn

nn

p

p

p

p

p

P

1

2

12

11

2

P = [ P1 P2 Pm ]

nn

nn

nn

nn

nn

nn

nn

nn

nn

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

P

1

2

12

11

1

2

12

11

1

2

12

11

26

Similarty Measure

n

n

n

q

q

q

q

q

Q

1

2

2

1

Kueri

n

n

n

n

n

n

n

n

n

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

P

1

2

2

1

1

2

2

1

1

2

2

1

Euclidean distance (P, Q):

n

i

ii

nn

)q(pd

)q(p...)q(p)q(pd

1

2

22

22

2

11

(Gonzalez 2004)

12/11/2008

14

27

Terima kasih.

Transformasi fourier 1 dimensi.ppt

Transformasi fourier

1 dimensi

Kuliah PPCD

aziz kustiyo

departemen ilmu komputer fmipa ipb

Transformasi Fourier

Mengapa perlu transformasi ?

Setiap orang pada suatu saat pernah menggunakan suatu teknik analisis dengan transformasi untuk menyederhanakan penyelesaian suatu masalah [Brigham,1974]

Contoh: penyelesaian fungsi y = x/z

Analisa konvensional : pembagian secara manual

Analisa transformasi : melakukan transformasi

log(y) = log(x) log(z)

look-up table pengurangan look-up table

Transformasi juga diperlukan bila kita ingin mengetahui suatu informasi tertentu yang tidak tersedia sebelumnya

Contoh :

jika ingin mengetahui informasi frekuensi kita memerlukan transformasi Fourier

Jika ingin mengetahui informasi tentang kombinasi skala dan frekuensi kita memerlukan transformasi wavelet

Transformasi citra, sesuai namanya, merupakan proses perubahan bentuk citra untuk mendapatkan suatu informasi tertentu

Transformasi bisa dibagi menjadi 2 :

Transformasi piksel/transformasi geometris:

Transformasi ruang/domain/space

Transformasi Citra

Transformasi Pixel

Transformasi piksel masih bermain di ruang/domain yang sama (domain spasial), hanya posisi piksel yang kadang diubah

Contoh: rotasi, translasi, scaling, invers, shear, dll.

Transformasi jenis ini relatif mudah diimplementasikan dan banyak aplikasi yang dapat melakukannya (Paint, ACDSee, dll)

Transformasi Ruang

Transformasi ruang merupakan proses perubahan citra dari suatu ruang/domain ke ruang/domain lainnya, contoh: dari ruang spasial ke ruang frekuensi

Masih ingat istilah ruang ? Ingat-ingat kembali pelajaran Aljabar Linier tentang Basis dan Ruang

Contoh : Ruang vektor. Salah satu basis yang merentang ruang vektor 2 dimensi adalah [1 0] dan [0 1]. Artinya, semua vektor yang mungkin ada di ruang vektor 2 dimensi selalu dapat direpresentasikan sebagai kombinasi linier dari basis tersebut.

Ada beberapa transformasi ruang yang akan kita pelajari, yaitu :

Transformasi Fourier (basis: cos-sin)

Transformasi Hadamard/Walsh (basis: kolom dan baris yang ortogonal)

Transformasi DCT (basis: cos)

Transformasi Wavelet (basis: scaling function dan mother wavelet)

Pada tahun 1822, Joseph Fourier, ahli matematika dari Prancis menemukan bahwa: setiap fungsi periodik (sinyal) dapat dibentuk dari penjumlahan gelombang-gelombang sinus/cosinus.

Contoh : Sinyal kotal merupakan penjumlahan dari fungsi-fungsi sinus berikut (lihat gambar pada halaman berikut)

Transformasi Fourier

f(x) = sin(x) + sin(3x)/3 + sin(5x)/5 +

sin(7x)/7 + sin(9x)/9

Hasil dalam transformasi fourier

Cobakan juga program matlab berikut untuk melihat sampai batas n berapa fungsi yang dihasilkan sudah berbentuk fungsi kotak.

function kotak(n)

t = 0:pi/200:8*pi;

kot = sin(t);

for i = 3 : 2: n

kot = kot + (sin(i*t))/i;

end

plot(kot)

Fungsi kotak sebagai penjumlahan fungsi-fungsi sinus

Gambar a) n = 1, b) n =3, c) n = 7, d) n = 99

(a)

(c)

(d)

(b)

Contoh lain

Contoh lain

Contoh lain

FT - Motivasi

Jika semua sinyal periodik dapat dinyatakan dalam penjumlahan fungsi-fungsi sinus-cosinus, pertanyaan berikutnya yang muncul adalah:

Jika saya memiliki sebuah sinyal sembarang, bagaimana saya tahu fungsi-fungsi cos sin apa yang membentuknya ?

Atau dengan kata lain

Berapakah frekuensi yang dominan di sinyal tersebut ?

Pertanyaan di atas dapat dijawab dengan menghitung nilai F(u) dari sinyal tersebut. Dari nilai F(u) kemudian dapat diperoleh kembali sinyal awal dengan menghitung f(x), menggunakan rumus:

Rumus FT 1 dimensi

Rumus FT kontinu 1 dimensi

Rumus FT diskret 1 dimensi

Contoh FT 1 dimensi

Contoh berikut diambil dari Polikar

(http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial.html)

Misalkan kita memiliki sinyal x(t) dengan rumus sbb:

x(t) = cos(2*pi*5*t) + cos(2*pi*10*t) +

cos(2*pi*20*t) + cos(2*pi*50*t)

Sinyal ini memiliki empat komponen frekuensi yaitu 5,10,20,50

Contoh sinyal 1 Dimensi x(t)

Gambar sinyal satu dimensi dengan rumus

x(t)= cos(2*pi*5*t) +

cos(2*pi*10*t) +

cos(2*pi*20*t) +

cos(2*pi*50*t)

(Sumber: Polikar)

FT dari sinyal tersebut

FT dari sinyal tersebut.

Terlihat bahwa FT dapat menangkap frekuensi-frekuensi yang dominan dalam sinyal tersebut, yaitu 5,10, 20, 50

(nilai maksimum F(u) berada pada angka 5,10, 20, 50)

Contoh Penghitungan FT 1 dimensi

Contoh Penghitungan FT 1 dimensi

Hasil penghitungan FT biasanya mengandung bilangan real dan imajiner

Fourier Spectrum didapatkan dari magnitude kedua bilangan tersebut shg|F(u)| = [R 2(u) + I 2(u)]1/2

Untuk contoh di halaman sebelumnya, Fourier Spectrumnya adalah sebagai berikut:

|F(0)| = 3.25|F(1)| = [(-0.5)2+(0.25)2]1/2 = 0.5590

|F(2)| = 0.25|F(3)| = [(0.5)2+(0.25)2]1/2 = 0.5590

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