5.7.1 Lowpass Filters

5.7 Filtering

주파수 영역에서 특정 주파수 부분을 조작(수정, 삭제, 추가)

그러나, 공간 영역에서의 고주파, 저주파 분리의 어려움이 있다.

Lowpass filter

removing high frequency information Î blurring an image

Highpass filter

removing low frequency information Î sharpening an image

Bandpass filter

removing specific part of frequency information Î removing unwanted noise

(page 231)

5.7.1 Lowpass Filters

frequency

gain

f0

cutoff frequency

passband

stopband

(a) 1-D lowpass ideal filter (b) 2-D lowpass ideal filter

u v

gain

frequency

gain

f0

cutoff frequency

Transition

(c) 1-D lowpass non-ideal filter (d) 2-D lowpass non-ideal filter

(e) 2-D lowpass ideal filter for Walsh-Hadamard and Cosine

transform

[ ^{( , ) ( , )} ]

) ,

( r c T

T u v H u v I

=

filter function filtered image transform

point-by-point method

) 0 , 0 ( ) 0 , 0

( H

T T(0,1)H(0,1)

(a)Original image (b)Filtered image (non-ideal lowpass filter) Blurring that softens the image

(b)Filtered image (ideal lowpass filter) Ripple artifacts at boundaries

Frequency cutoff=32

Butterworth filter

A commonly used non-ideal filter

With the Butterworth filter we can specify the order of the filter, which determines how steep the slope is in the transition of the filter function

A higher order to the filter creates a steeper slope, and the closer we get to an ideal filter

n

f v u

v u

H

0 2 2

1 ) 1 , (

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ +

+

=

= 1

n

c) Fourier spectrum, filter order = 3

a) Fourier spectrum, filter order = 1

b) Resultant image with order =1

d) Resultant image with order =3

e) Fourier spectrum, filter order = 5

g) Fourier spectrum, filter order = 8

f) Resultant image with order =5

h) Resultant image with order =8

Lowpass Butterworth Filtering Various Filter Orders

n

f v u v

u

H ₂

0 2 2

1 ) 1 , (

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +

+

=

Butterworth Lowpass Filtering

(Filter Order = 3, Various Cutoff Frequencies)

a) Fourier spectrum, cutoff frequency = 64

b) Resultant image with cutoff frequency = 64

c) Fourier spectrum, cutoff frequency = 32

d) Resultant image with cutoff frequency = 32

e) Fourier spectrum, cutoff frequency = 16

g) Fourier spectrum, cutoff frequency = 8 f) Resultant image with

cutoff frequency = 16

h) Resultant image with cutoff frequency = 8

n

f v u v

u

H ₂

0 2 2

1 ) 1 , (

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +

+

=

5.7.2 Highpass Filters

frequency

gain

(a) 1-D highpass ideal filter

frequency

gain

(c) 1-D highpass non-ideal filter

(b) 2-D highpass ideal filter

(d) 2-D highpass non-ideal filter

A highpass filter will keep high frequency information, which corresponds to areas of rapid change in brightness, such as edges or fine textures and attenuate low

frequencies

Used for edge enhancement, since it passes only high frequency information

n

v u

f v

u

H

2 2

1

) 1 , (

⎥ ⎦

⎢ ⎤

⎣

⎡ + +

=

(e) 2-D highpass ideal filter for Walsh-Hadamard and Cosine

transform

a) Original image b) Butterworth filter;

order = 2; cutoff = 32

c) Ideal filter; cutoff = 32

High-frequency emphasis(고주파 강조) filter

This filter function boosts the high frequencies and retains some of the low frequency information and by adding an offset value to the function

Therefore we do not lose the overall image information

frequency

gain

(a) Ideal highpass emphasis filter for 1-D

frequency

gain

(c) non-ideal highpass emphasis filter for 1-D

(b) Ideal highpass emphasis filter for 2-D

(d) non-ideal highpass emphasis filter for 2-D Offset

Offset

a) Original image b) Butterworth filter;

order = 2; cutoff = 32

c) Ideal filter; cutoff = 32

d) High-frequency emphasis filter;

offset = 0.5, order = 2, cutoff = 32

e) High frequency emphasis filter;

offset = 1.5, order = 2, cutoff = 32

5.7.3 Bandpass and Bandreject Filters

Bandpass filter

It retains specific parts of the spectrum (특정 부분 남김)

Bandreject filter

It removes specific parts of the spectrum (특정 부분 제거)

Bandreject filters are often used for noise removal

Bandpass and bandreject filters

They require high and low frequency cutoff values

They are typically used in image restoration(복원), en hancement(개선), and compression(압축)

frequency

gain

High cutoff Low

cutoff

frequency

gain

High cutoff Low

cutoff

a) bandpass filter

d) bandreject filter

frequency

gain

High cutoff Low

cutoff

frequency

gain

High cutoff Low

cutoff

a) 1-D ideal bandpass filter

d) 1-D ideal bandreject filter

b) 2-D ideal bandpass filter

e) 2-D ideal bandreject filter

c) 2-D ideal notch filter for rejecting specific frequencies

c) 2-D ideal notch filter for passing specific frequencies

목차

5.1 Introduction

5.2 Fourier Transform

5.2.1 The One-Dimensional Discrete Fourier Transform 5.2.2 The Two-Dimensional Discrete Fourier Transform 5.2.3 Fourier Transform Properties

5.2.4 Displaying the Fourier Spectrum 5.3 Cosine Transform

5.4 Walsh-Hadamard Transform 5.7 Filtering

5.7.1 Lowpass Filters 5.7.2 Highpass Filters

5.7.3 Bandpass and Bandreject Filters 5.6 Principal Components Transform 5.5 Haar Transform

5.8 Wavelet Transform

5.6 Principal Components Transform

Mathematically, it involves finding eigenvectors of covariance matrix It decomposes the image into its principal components

Î이미지 공분산의 특징벡터로부터 이미지의 기본요소(시각적)만을 찾아냄

It differs from the transforms considered so far, as it is not related to extracting frequency or sequency information from images

Used to find optimal basis images for a specific image, but this use of it is not very practical due to the extensive processing required (광범위한 연산-실용적이지 못함)

a) Red band of a color image b) Green band c) Blue band

d) Principal component band 1 e) Principal component band 2 f) Principal component band 3

The three step procedure for finding the PCT for a color, RGB, image is as follows:

1) Find the covariance matrix in RGB space, given by:

where

Similar equations are used for C_GG and C_BB (the auto-covariance terms) 10

20 20 10

(10 20 10 20) 15 4

1 + + + =

R = m

-5 5

5 -5

( 5 5 5 5) 0

4

1 − + − + =

RR = C

R

i m

R −

Red channel ⎥⎥⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

−

−

−

−

−

−

−

= 0

] [COV _RGB

The elements of the covariance matrix that involve more than one of the RGB variables, C_GR, C_BR, C_RG, C_BG, C_RB, and C_GB, are called cross-covariance terms and are found as follows:

2. Find the eigenvalues of the covariance matrix, e₁, e₂ and e₃, and their corresponding eigenvectors(고유벡터):

Order them such that e₁ is the largest eigenvalue, and e₃ is the smallest. (e₁>e₂>e₃)

[ ]

[ ]

[

]

3

23 22 21 2

13 12 11 1

E E E e

E E E e

E E E e

, ,

, ,

, ,

⇒

⇒

⇒

(e₁>e₂>e₃)

3. Perform the linear transform on the RGB data by using the eigenvectors as follows:

The PCT data is P₁, P₂ and P₃ where the P₁ data is the principal component and contains the most variance (see Fig. 4.3.9)

correct the third equation on page 231

compression

In pattern recognition theory the measure of variance(변이치) is a measure of

information(정보치), in this sense most information is in the principal component band

Used in image compression, since this transform is optimal in the least-square-error(평균 제곱 오차) sense (most of the information, assumed to be directly correlated with variance, is in a reduced dimensionality)

It has been experimentally determined (application involving a database of medical images) that the dimension with the largest variance after the PCT was performed contained approximately 91% of the variance

This would allow at least a 3:1 compression and still retain 91% of the information

91% information of original image 1/3 volume of original image

PCT

P1 P2

P3

R

G B

목차

5.1 Introduction

5.2 Fourier Transform

5.2.1 The One-Dimensional Discrete Fourier Transform 5.2.2 The Two-Dimensional Discrete Fourier Transform 5.2.3 Fourier Transform Properties

5.2.4 Displaying the Fourier Spectrum 5.3 Cosine Transform

5.4 Walsh-Hadamard Transform 5.7 Filtering

5.7.1 Lowpass Filters 5.7.2 Highpass Filters

5.7.3 Bandpass and Bandreject Filters 5.6 Principal Components Transform

5.5 Haar Transform 5.8 Wavelet Transform

영상 피라미드

N N×

2 / 2

/ N

N ×

1×1 2 2× 4 4×

피라미드 영상구조

9 고해상도 영상을 표현

Î 개개의 물체 특징을 분석하는데 유용

9 저해상도 영상을 표현

Î 큰 구조 또는 전체적인 배경을 분석하는데 유용

키보드, 마우스 분석 (1024X1024)

모니터 분석 (512X512)

책상 분석 (265X265)

부 대역 부호화(Subband Coding)

frequency

gain Low band High band

5.5 Harr Transform

The Haar transform has rectangular waves as basis functions Basis vectors contain not just +1 and -1, but also contain zeros The Haar transform is derived from the Haar matrices

Basis vectors for a Haar transform of two basis vectors (N = 2), four basis vectors (N

= 4) and eight basis vectors (N = 8) are:

??

As the order increases the number of zeros in the basis vectors increase

This has the unique effect of allowing a multiresolution decomposition (다해상도 분 해) of an input image

The Haar transform retains both spatial and sequency information

a) Original image b) Haar transform image ( Log Remapped )

Note: Haar provides edge information at increasing levels of resolution

??

5.8 Wavelet Transform

The wavelet transform contains not just frequency information, but also spatial information

One of the most common models for a wavelet transform uses the Fourier transform and highpass and lowpass filters

The wavelet transform breaks an image down into four subsampled, or decimated, images by keeping every other pixel

4 7 10

The wavelet results consist of

One subsampled image that has been highpass filtered in both the horizontal and vertical directions

One that has been highpass filtered in the vertical and lowpassed in the horizontal One that has been highpassed in the horizontal and lowpassed in the vertical

One that has been lowpass filtered in both directions

1단계 Wavelet Transform 2단계 Wavelet Transform

LL HL

HH LH

Uses the convolution property of the Fourier transform to perform the wavelet transform in the spatial domain

주파수 영역에서의 곱 Î 공간 영역에서의 convolution

Uses the separable property to perform a 2-D wavelet with two 1-D filters

A special type of convolution called circular convolution must be used in order to perform the wavelet transform with convolution filters

Circular convolution is performed by taking the underlying image array and extending it in a periodic manner to match the symmetry implied by the discrete Fourier transform

Circular convolution allows us to retain the outer rows and columns

This is important since we may want to perform the wavelet transform on small blocks, and eliminating the outer row(s) and column(s) is not practical

Haar basis vectors

Many different convolution filters are available for use with the wavelet transform An example of Haar basis vectors are:

An example of Daubechies basis vectors are:

As these are separable, so they can be used to implement a wavelet transform by first convolving them with the rows and then the columns

The wavelet transform is performed by doing the following:

1. Convolve the lowpass filter with the rows (this is done by sliding, multiplying coincident terms and summing the results) and save the results

(Note: For the basis vectors as given, they do not need to be reversed for convolution) 2. Convolve the lowpass filter with the columns (of the results from step 1), and subsample this result by taking every other value; this gives us the lowpass-lowpass version of the image

3. Convolve the result from step 1, the lowpass filtered rows, with the highpass filter on the columns. Subsample by taking every other value to produce the lowpass-highpass image

4. Convolve the original image with the highpass filter on the rows, and save the result 5. Convolve the result from step 4 with the lowpass filter on the columns; subsample to yield the highpass-lowpass version of the image

6. To obtain the highpass-highpass version, convolve the columns of the result from step 4 with the highpass filter

Wavelet Transform

Wavelet Transform Display

Location of frequency bands in a four-band wavelet

transformed image. Designation is row/column.

LL HL

HH LH

Fourier, Walsh-Hadamard, Cosine Spectra

Contain No Obvious Spatial Information (공간 정보 없음)

a) Original image b) Fourier spectrum

c) Walsh-Hadamard spectrum d) Cosine spectrum

Fourier, Walsh-Hadamard, Cosine Spectra

Performed on Small Blocks Resembles Image(공간 정보 표현)

a) Original image b) Fourier spectrum 8x8 blocks

c) Fourier spectrum 4x4 blocks d) Walsh-Hadamard spectrum 8x8 blocks

256x256 image

8x8 subimage

Fourier, Walsh-Hadamard, Cosine Spectra

Performed on Small Blocks Resembles Image (공간 정보 표현)

e) Walsh-Hadamard spectrum, 4x4 blocks

f) Cosine spectrum, 8x8 blocks

g) Cosine spectrum, 4x4 blocks

Note: All spectra are log remapped

The inverse wavelet transform is performed in the following way:

Enlarging the wavelet transform data to its original size Inserting zeros between each value

Convolving the corresponding (lowpass and highpass) inverse filters to each of the four subimages

Summing the results to obtain the original image

∑

Enlarging inverse

For the Haar filter, the inverse wavelet filters are identical to the forward filters

For the Daubechies, the inverse wavelet filters are

Daubechies wavelet filters

Daubechies inverse wavelet filters

The wavelet transform is used in image compression, for example in JPEG2000 The multiresolution decomposition property of the wavelet transform, makes it useful in applications where it is desirable to have coarse information available fast such as perusing an image database or progressively transmitting images on the Internet

Reference

Umbaugh, Computer Imaging : Digital Image Analysis and Processing, CRC Press, 2005.

Gonzalez Woods, Digital Image Processing, Prentice Hall, 2002.

천인국, 이강승, 영상처리 기초편, 기한재, 2004.

정성환, 이문호, JAVA를 이용한 디지털 영상처리, 정익사, 2005.

이강웅, 백중환, Signal & System, 복두출판사, 2006.

이승훈, 윤동한, 알기쉬운 웨이블렛 변환, 진한도서, 2002.

Vinay K. Ingle, John G. Proakis, Digital Signal Processing Using MATHLAB, 시그마 프레스, 1998.