introduction to image processing

Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com

© 2002 R. C. Gonzalez & R. E. Woods

Introduction toImage ProcessingIntroduction toImage Processing

Image ProcessingImage Processing



Imaging in the Visible and Infrared BandsImaging in the Visible and Infrared Bands



Other Examples Other Examples

（甲狀腺）



Image Sampling and

Quantization

Image Sampling and

Quantization



Image Sampling and QuantizationImage Sampling and Quantization



Image Sampling and QuantizationImage Sampling and Quantization

• Sampling: digitizing the 2-dimensional spatial coordinate values

• Quantization: digitizing the amplitude values (brightness level)



Representing Digital ImagesRepresenting Digital Images



Representing Digital Images--ExamplesRepresenting Digital Images--Examples

P2320 240255200 215 25 …55 25 25…25 55 255 …

• PGM file • PPM file

P3# Created by Paint Shop Pro 5320 240255200 215 25 200 215 25 …55 25 25 55 25 25 …25 55 255 25 55 255 …



Spatial and Gray-Level ResolutionSpatial and Gray-Level Resolution

Spatial ResolutionSpatial Resolution



Spatial Resolution by Re-samplingSpatial Resolution by Re-sampling



Gray-Level ResolutionGray-Level Resolution

24

816128

3264

256



Some Basic Relationships Between PixelsSome Basic Relationships Between Pixels

• PixelA pixel has a location (the spatial coordinate) and a value.

• Connected component of S: The set of all pixels in S that are connected to a given pixel in S.

• Region of an image• Contour of a region• Edge:

Edge is a path of one or more pixels that separate two regions of significantly different gray levels.



Image Enhancementin the Spatial DomainImage Enhancementin the Spatial Domain



BackgroundBackground

• A mathematical representation of spatial domain enhancement:

where f(x, y): the input image

g(x, y): the processed image

T: an operator on f, defined over some neighborhood of (x, y)

)],([),( yxfTyxg



Gray-level TransformationGray-level Transformation



Piecewise-Linear Transformation FunctionsCase 1: Contrast Stretching

Piecewise-Linear Transformation FunctionsCase 1: Contrast Stretching



Piecewise-Linear Transformation FunctionsCase 2:Gray-level Slicing

Piecewise-Linear Transformation FunctionsCase 2:Gray-level Slicing

An image Result of using the transformation in (a)



Histogram ProcessingHistogram Processing



Histogram EqualizationHistogram Equalization

• Histogram equalization:– To improve the contrast of an image

– To transform an image in such a way that the transformed image has a nearly uniform distribution of pixel values

• Transformation function T(r) (continuous case)




• In discrete version:– The probability of occurrence of gray level rk in an image is

n : the total number of pixels in the image

nk : the number of pixels that have gray level rk

L : the total number of possible gray levels in the image– The transformation function is

– Thus, an output image is obtained by mapping each pixel with level rk in the input image into a corresponding pixel with level sk.

1,...,2,1,0 )()(0 0

Lkn

nrprTs

k

j

k

j

jjrkk

1,...,2,1,0 )( Lkn

nrp k

kr




• Transformation functions (1) through (4) were obtained form the histograms of the images in Fig 3.17(1), using Eq. (3.3-8).



Basics of Spatial FilteringBasics of Spatial Filtering

• In spatial filtering (vs. frequency domain filtering), the output image is computed directly by simple calculations on the pixels of the input image.

• Spatial filtering can be either linear or non-linear.

• For each output pixel, some neighborhood of input pixels is used in the computation.

• In general, linear filtering of an image f of size MXN with a filter mask of size mxn is given by

where a=(m-1)/2 and b=(n-1)/2

• This concept called convolution. Filter masks are sometimes called convolution masks or convolution kernels.

a

as

b

bt

tysxftswyxg ),(),(),(



Basics of Spatial FilteringBasics of Spatial Filtering



Smoothing Spatial FiltersSmoothing Spatial Filters

• Smoothing linear filters– Averaging filters

• Box filter

• Weighted average filter

Box filter Weighted average



Smoothing Spatial FiltersSmoothing Spatial Filters

• The general implementation for filtering an MXN image with a weighted averaging filter of size mxn is given by

where a=(m-1)/2 and b=(n-1)/2

a

as

b

bt

a

as

b

bt

tsw

tysxftswyxg

),(

),(),(),(



Smoothing Spatial FiltersImage smoothing with masks of various sizes

Smoothing Spatial FiltersImage smoothing with masks of various sizes



Smoothing Spatial FiltersAnother Example

Smoothing Spatial FiltersAnother Example



Order-Statistic FiltersOrder-Statistic Filters

• Order-statistic filters– Median filter: to reduce impulse noise (salt-and-

pepper noise)



Image Enhancement in theFrequency Domain

Image Enhancement in theFrequency Domain



BackgroundBackground

• The frequency domain refers to the plane of the two dimensional discrete Fourier transform of an image.

• The purpose of the Fourier transform is to represent a signal as a linear combination of sinusoidal signals of various frequencies.



Introduction to the Fourier Transform and the Frequency Domain


• The one-dimensional Fourier transform and its inverse– Fourier transform (continuous case)

– Inverse Fourier transform:

• The two-dimensional Fourier transform and its inverse– Fourier transform (continuous case)


dueuFxf uxj 2)()(

dydxeyxfvuF vyuxj )(2),(),(

1 where)()( 2

jdxexfuF uxj

dvduevuFyxf vyuxj )(2),(),(

sincos je j





• The one-dimensional Fourier transform and its inverse– Fourier transform (discrete case) DTC


1,...,2,1,0for )(1

)(1

0

/2

MuexfM

uFM

x

Muxj

1,...,2,1,0for )()(1

0

/2

MxeuFxfM

u

Muxj





• Since and the fact

then discrete Fourier transform can be redefined

– Frequency (time) domain: the domain (values of u) over which the values of F(u) range; because u determines the frequency of the components of the transform.

– Frequency (time) component: each of the M terms of F(u).

1,...,2,1,0for Mu

1

0

]/2sin/2)[cos(1

)(M

x

MuxjMuxxfM

uF

sincos je j cos)cos(



The One-Dimensional Fourier Transform Example

The One-Dimensional Fourier Transform Example





• The two-dimensional Fourier transform and its inverse– Fourier transform (discrete case) DTC


• u, v : the transform or frequency variables

• x, y : the spatial or image variables

1,...,2,1,0,1,...,2,1,0for

),(1

),(1

0

1

0

)//(2

NvMu

eyxfMN

vuFM

x

N

y

NvyMuxj

1,...,2,1,0,1,...,2,1,0for

),(),(1

0

1

0

)//(2

NyMx

evuFyxfM

u

N

v

NvyMuxj





• Some properties of Fourier transform:

)(symmetric ),(),(

(average) ),(1

)0,0(

(shift) )2

,2

()1)(,(

1

0

1

0

vuFvuF

yxfMN

F

Nv

MuFyxf

M

x

N

y

yx



The Two-Dimensional DFT and Its InverseThe Two-Dimensional DFT and Its Inverse

(a) f(x,y) (b) F(u,y) (c) F(u,v)

The 2D DFT F(u,v) can be obtained by 1. taking the 1D DFT of every row of image f(x,y), F(u,y), 2. taking the 1D DFT of every column of F(u,y)




shift



The Property of Two-Dimensional DFT Rotation

The Property of Two-Dimensional DFT Rotation

DFT

DFT



The Property of Two-Dimensional DFT Linear Combination

The Property of Two-Dimensional DFT Linear Combination

DFT

DFT

DFT

A

B

0.25 * A + 0.75 * B



The Property of Two-Dimensional DFT Expansion

The Property of Two-Dimensional DFT Expansion

DFT

DFT

A

Expanding the original image by a factor of n (n=2), filling the empty new values with zeros, results in the same DFT.

B



Two-Dimensional DFT with Different FunctionsTwo-Dimensional DFT with Different Functions

Sine wave

Rectangle

Its DFT

Its DFT



Two-Dimensional DFT with Different FunctionsTwo-Dimensional DFT with Different Functions

2D Gaussianfunction

Impulses

Its DFT

Its DFT



Filtering in the Frequency DomainFiltering in the Frequency Domain



Basics of Filtering in the Frequency DomainBasics of Filtering in the Frequency Domain



Some Basic Filters and Their FunctionsSome Basic Filters and Their Functions

• Multiply all values of F(u,v) by the filter function (notch filter):

– All this filter would do is set F(0,0) to zero (force the average value of an image to zero) and leave all frequency components of the Fourier transform untouched.

otherwise. 1

)2/,2/(),( if 0),(

NMvuvuH




Lowpass filter

Highpass filter



Ideal Lowpass Filters (ILPFs)Ideal Lowpass Filters (ILPFs)



Ideal Lowpass FiltersIdeal Lowpass Filters



Additional Examples of Lowpass FilteringAdditional Examples of Lowpass Filtering



Ideal Highpass FiltersIdeal Highpass Filters

),( if 1

),( if 0),(

0

0

DvuD

DvuDvuH



Color Image ProcessingColor Image Processing



Color FundamentalsColor Fundamentals



Color FundamentalsColor Fundamentals

• Secondary colors: magenta (red + blue), cyan (green + blue), and yellow (red + green)



Color Models -- RGB ModelColor Models -- RGB Model




• For most graphics images used for Internet applications, a set of 216 colors has been selected to represent “safe colors” which should be reliably displayed on computer

monitors.



Color Models -- CMY and CMYK ModelsColor Models -- CMY and CMYK Models

• An RGB to CMY conversion

B

G

R

Y

M

C

1

1

1



Color Models -- HSI ModelColor Models -- HSI Model




• Converting colors from RGB to HSI

21

))(()(

)()(cos

2

21

1

BGBRGR

BRGR

)(3

1BGRI

GB

GBH

if 360

if

),,min()(

31 BGR

BGRS




• Converting colors from HSI to RGB– RG sector ( )

– GB sector ( )

– BR sector ( )

oo H 1200

)60cos(

cos1

H

HSIR

o

)(3 BRIG

)1( SIB

oo H 360240

oo H 240120



Full Color Image ProcessingFull Color Image Processing

• Two processing methods: – (1) process each channel (or color component)

separately, as if the color image were three gray scale images;

– (2) process all channels with each pixel represented as a vector.



Color TransformationsRGB<->HSI<->CMYKColor TransformationsRGB<->HSI<->CMYK



Color TransformationsExample

Color TransformationsExample



Color Complement TransformationsColor Complement Transformations

• To the human visual system, colors have complements.

• Complements are basically given by subtracting one color from white, or by changing a hue by 180 degrees.



Color Complement TransformationsExample

Color Complement TransformationsExample



Tone and Color CorrectionsTone Corrections

Tone and Color CorrectionsTone Corrections



Tone and Color CorrectionsColor Corrections

Tone and Color CorrectionsColor Corrections



Smoothing and SharpeningColor Image Smoothing

Smoothing and SharpeningColor Image Smoothing



Color Edge DetectionColor Edge Detection



Noise in Color ImagesNoise in Color Images



Noise in Color ImagesNoise in Color Images

Fig. 6.48(d)

introduction to image processing

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