EC 624 Digital Image Processing (3 0 2 8)

Class I Introduction

Instructor: PK. BoraInstructor: PK. Bora

Digital Image ProcessingDigital Image Processing means processing Digital Image Processing means processing

of of digitaldigital imagesimages on digital hardware usually a on digital hardware usually a computercomputer

What is an analog image?Electrical Signal, for example, the output of Electrical Signal, for example, the output of a video camera, that gives the electric a video camera, that gives the electric voltage at locations in an imagevoltage at locations in an image

What is a digital Image2D array of numbers 2D array of numbers representing the sampled representing the sampled version of an imageversion of an imageThe image defined over a The image defined over a grid, each grid location grid, each grid location being called a pixel.being called a pixel.Represented by a finite Represented by a finite grid and each intensity grid and each intensity data is represented a finite data is represented a finite number of bits.number of bits.A binary image is A binary image is represented by one bit represented by one bit graygray--level image is level image is represented by 8 bits.represented by 8 bits.

Pixel and IntensitiesPixel and Intensities

1616 1818 1919 2020

255255 1919 1818 2020

2121 2121 2222 3030

2121 3333 2222 2323

Mathematically

We can think of an We can think of an image image as a function, as a function, ff,,from Rfrom R22 to R:to R:

ff( ( x, y x, y ) gives the ) gives the intensityintensity at position ( at position ( x, y x, y ) ) Realistically, we expect the image only to be Realistically, we expect the image only to be defined over a rectangle, with a finite range:defined over a rectangle, with a finite range:

ff: [: [aa, , bb]]xx[[cc, , dd] ] [0, 1][0, 1]

What is a Colour Image?Three components:Three components:

R,G, B each usually R,G, B each usually represented by 8 bitsrepresented by 8 bits

We call 24We call 24--bit videobit videoThese three primary are These three primary are mixed in different mixed in different proportions to get proportions to get different different colourscoloursFor different processing For different processing applications other formats applications other formats ((YIQ,YCbCrYIQ,YCbCr, HIS etc) are , HIS etc) are usedused

( , )( , ) ( , )

( , )

r x yf x y g x y

b x y

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

A color image is A color image is just a three just a three –– component component function. We can function. We can write this as a write this as a ““vectorvector--valuedvalued”” function:function:

Types of Digital Image

Digital images include Digital images include Digital photosDigital photosImage sequences used for video broadcasting and Image sequences used for video broadcasting and playbackplaybackMultiMulti--sensor data like satellite images in the visible, sensor data like satellite images in the visible, infrared and microwave bandsinfrared and microwave bandsMedical images like ultraMedical images like ultra--sound, Gammasound, Gamma--ray images, ray images, XX--ray images and radioray images and radio--band images like MRI etcband images like MRI etcAstronomical imagesAstronomical imagesElectronElectron--microscope images used to study material microscope images used to study material structurestructure

ExamplesPhotgraphic

Ultrasound Mammogram

Image processing

Digital Image processing deals with manipulation and analysis ofDigital Image processing deals with manipulation and analysis of the the digital image by a digital hardware, usually a computer.digital image by a digital hardware, usually a computer.Emphasizing certain pictorial information for better clarity (huEmphasizing certain pictorial information for better clarity (human man interpretation)interpretation)Automatic machine processing of the scene data.Automatic machine processing of the scene data.Compressing the image data for efficient utilization of storageCompressing the image data for efficient utilization of storage space space and transmission bandwidthand transmission bandwidth

Image ProcessingAn An image processingimage processing operation operation

typically defines a new image typically defines a new image gg in terms in terms of an existing image of an existing image f.f.

We can also transform O the domain of We can also transform O the domain of ff::

Image Processingimage filtering: change image filtering: change rangerange of imageof image

f

x

hf

x

f

x

hf

x

image warping: change image warping: change domaindomain of imageof image

g(xg(x) = ) = h(f(xh(f(x))))

g(xg(x) = ) = f(h(xf(h(x))))

Example Image RestorationImage Restoration

Degraded Image Restored ImageProcessing

Image Processing steps

Acquisition, Sampling/ Quantization/ Acquisition, Sampling/ Quantization/ CompressionCompressionImage enhancement and restorationImage enhancement and restorationFeature ExtractionFeature ExtractionImage SegmentationImage SegmentationObject RecognitionObject RecognitionImage InterpretationImage Interpretation

Image Acquisition

An analog image is An analog image is obtained by scanning the obtained by scanning the sensor output. Some of the sensor output. Some of the modern scanning device modern scanning device such as a CCD camera such as a CCD camera contains an array of photocontains an array of photo--detectors, a set of detectors, a set of electronic switches and electronic switches and control circuitry all in a control circuitry all in a single chipsingle chip

Image Acquisition

Sample and Hold

Sample and Hold

Analog toDigital

Analog toDigital

Digital Image

Takes a measurement and “holds” it for conversion to digital.

Converts a measurement to digital

Image Sensor

Sampling/ Quantization/ Compression

A digital image is obtained by sampling and A digital image is obtained by sampling and quantizing an analog image. quantizing an analog image. The analog image signal is sampled at rate The analog image signal is sampled at rate determined by the application concerneddetermined by the application concernedStill image 512X512, 256X256Still image 512X512, 256X256Video: 720X480, 360X240, 1024x 768 (HDTV)Video: 720X480, 360X240, 1024x 768 (HDTV)The intensity is quantized into a fixed number of The intensity is quantized into a fixed number of levels determined human perceptual limitationlevels determined human perceptual limitation8 bits is sufficient for all but the best applications 8 bits is sufficient for all but the best applications 10 bits 10 bits –– Television production, printingTelevision production, printing1212--16 bits 16 bits –– Medical imageryMedical imagery

Sampling/ Quantization/ Compression (Contd.)

Raw video is very bulky Raw video is very bulky Example:The transmission of highExample:The transmission of high--definition uncompressed digital video at definition uncompressed digital video at 1024x 768, 24 bit/pixel, 25 frames requires 1024x 768, 24 bit/pixel, 25 frames requires 472 Mbps472 MbpsWe have to compress the raw data to store We have to compress the raw data to store and transmitand transmit

Image Enhancement

Improves the qualities of an image byImproves the qualities of an image byenhancing the contrast enhancing the contrast sharpening the edgessharpening the edges

removing noise, etc. removing noise, etc. As an example, let us explain the image As an example, let us explain the image

filtering operation to remove noise. filtering operation to remove noise.

Example: Image Filtering

Original Image Filtered ImageOriginal Image Filtered Image

Histogram Equalization

Enhance the contrast of images by Enhance the contrast of images by transforming the values in an intensity transforming the values in an intensity image to its normalized histogramimage to its normalized histogramthe histogram of the output image is the histogram of the output image is

uniformly distributed.uniformly distributed.Contrast is betterContrast is better

Feature Extraction

Extracting Features like edgesExtracting Features like edgesVery important to detect the boundaries of the objectVery important to detect the boundaries of the objectDone through digital differentiation operationDone through digital differentiation operation

Example: Edge Detection

Original Saturn Image Edge Image

Segmentation

Partitioning of an image into connected Partitioning of an image into connected homogenous regions.homogenous regions.Homogeneity may be defined in terms of:Homogeneity may be defined in terms of:Gray valueGray valueColourColourTextureTextureShapeShapeMotionMotion

Segmented Image

Object Recognition

An object recognition system finds An object recognition system finds objects in the real world from an image objects in the real world from an image of the world, using object models which of the world, using object models which are known a prioriare known a priorilabelling problem based on models of labelling problem based on models of known objectsknown objects

Object Recognition (Contd.)

••

Object or model representationObject or model representationFeature extractionFeature extractionFeatureFeature--model matchingmodel matchingHypotheses formationHypotheses formationObject verificationObject verification

Image Understanding

Inferring about the scene on the basis of the Inferring about the scene on the basis of the recognized objectsrecognized objectsSupervision is requiredSupervision is requiredNormally considered as part of artificial Normally considered as part of artificial intelligenceintelligence

Books

1.1. R. C. Gonzalez and R. E. Woods, R. C. Gonzalez and R. E. Woods, Digital Digital Image ProcessingImage Processing, Pearson Education, , Pearson Education, 2001 (Main Text2001 (Main Text))

2.2. A. K. Jain, A. K. Jain, Fundamentals of Digital Image Fundamentals of Digital Image processingprocessing, Pearson Education, 1989., Pearson Education, 1989.

3. R. C. Gonzalez , R. E. Woods and S. L. 3. R. C. Gonzalez , R. E. Woods and S. L. EddinsEddins, , Digital Image Processing using MATLABDigital Image Processing using MATLAB, , Pearson Education, 2004. (Lab Ref)Pearson Education, 2004. (Lab Ref)

Evaluation Scheme

End Sem 50End Sem 50Mid Sem 25Mid Sem 25Quiz 5Quiz 5Matlab Assignment 10Matlab Assignment 10Mini Project 10Mini Project 10

Total 100 Total 100

1. MINI PROJECT

Matlab Implementation and preparing Report and DemonstratiMatlab Implementation and preparing Report and Demonstration of any on of any advanced topic like:advanced topic like:

Video compressionVideo compressionVideo Video mosaicingmosaicingVideoVideo--based trackingbased trackingMedical Image CompressionMedical Image CompressionVideo WatermarkingVideo WatermarkingMedical Image SegmentationMedical Image SegmentationImage and Video RestorationImage and Video RestorationBiometric recognitionBiometric recognition

2D Discrete Time (Space) Fourier Transform

• Recall DTFT of a 1D sequence, Given is

and

Note that

exists if and only if is absolutely summable, i.e.,

[ ]{ }, .....x n n = − ∝ ∝

( ) [ ]n=-

= x n j nX e ωω∝

−

∝∑

[ ] ( )1x n = X2

j ne dπ

ω

πω ω

π −∫

( ) ( )2 = XX ω π ω+

( )X ω [ ]x n

[ ]n

x n∝

=−∝< ∝∑

Relationship between CTFT and DTFT

• Consider a discrete sequence obtained by sampling an analog signal at a uniform sampling rate , where T is the sampling period.

• We can represent the sampling process by means of the Dirac delta function with the relation

• Now the sampled signal can be represented in continuous domain as,

• Thus, the analog and discrete frequencies are related as

[ ]{ }, ....x n n = − ∝ ∝

( )ax t 1TsF =

[ ] ( ) , n=0, 1......ax n x nT= ±

( ) ( ) ( )

( ) ( )

an=-

an=-

= x

= x

sx t t t nT

nT t nT

δ

δ

∝

∝∝

∝

−

−

∑

∑

.w T= Ω

2D DSFT• Consider the signal defined over the

two-dimensional space.

• Also assume

• Then the two-dimensional discrete-space Fourier transform

(2D DSFT) and its inverse are defined by the following relations:

and

{ [ , ], ,....., , ,....., }f m n m n= −∞ ∞ = −∞ ∞

[ , ] .m n

f m n∞ ∞

=−∞ =−∞

< ∞∑ ∑

)(],[ ),( vnumj

n m

enmfvuF +−∞

−∞=

∞

−∞=∑ ∑=

( )2

1[ , ] ( , )4

j um vnf m n F u v e du dvπ

∞ ∞+ +

−∞ −∞

= ∫ ∫

• Note that is doubly periodic in u and v. • Following properties of are easily verified:

– Linearity– Separability– Shifting theorem:

– Convolution theorem:

– Eigen function– Modulation– Correlation– Inner product– Parseval’s theorem

( , )F u v( , )F u v

( )0 0

2 DSFT

2 DSFT0 0

If [ , ] ( , ), then

[ , ] ( , ),

D

j um vnD

f m n F u v

f m m n n e F u v− +

←⎯⎯⎯→

− − ←⎯⎯⎯→

2 DSFT 2 DSFT1 1 2 2

2 DSFT1 2 1 2

If [ , ] ( , ) and [ , ] ( , ), then

[ , ]) * [ , ] ( , ) ( , )

D D

D

f m n F u v f m n F u v

f m n f m n F u v F u v

←⎯⎯⎯→ ←⎯⎯⎯→

←⎯⎯⎯→

2D DFT

Motivation

• Consider the 1D DTFT which is uniquely defined for

each

• Numerical evaluation of involves very large (infinite) data and is to be done for each

• An easier way is the Discrete Fourier transform (DFT) which is obtained by

sampling at a regular interval .

• Sampling periodically in the frequency domain at a rate means that the data sequence will be periodic with a period

• The relation between the Fourier transform of an analog signal and the DFT of the sampled version is illustrated in the Figure below.

( ) [ ] jwn

nX x n eω

∞−

=−∞

= ∑. [0, 2 ].ω π∈

( )X ω.ω

( )X ω1N

.N(t)ax

1N

2D DFT

1 2 and k k

The 2D DFT of a 2D sequence is defined as

and the inverse 2D DFT is given by

[ ] [ ] 1 22 2-1 -1

1 2 1 20 0

, , , 0,1,..., 1, =0,1,...,N-1 N M j mk nk

M N

n m

F k k f m n e k M kπ π⎛ ⎞− +⎜ ⎟

⎝ ⎠

= =

= = −∑∑

[ ] [ ] 1 2

2 1

2 2N-1 M-1

1 2k =0 k =0

1, k , , 0,1,..., 1, =0,1,...,N-1 j mk nk

M Nf m n F k e m M nMN

π π⎛ ⎞+⎜ ⎟⎝ ⎠= = −∑ ∑

2D DFT is periodic in both

Thus [ ] [ ]1 2 1 2, ,F k k F k M k N= + +

Properties of 2D DFT

0 1 0 2

2 DFT1 2

2 22 DFT

0 0 1 2

If [ , ] [ , ], then

[ , ] [ , ]

D

j m k n kD M N

f m n F k k

f m m n n e F k kπ π⎛ ⎞− +⎜ ⎟

⎝ ⎠

←⎯⎯⎯→

− − ←⎯⎯⎯→

•Shifting property

•Separability propery

Since 1 2 2

2 2 22j mk nk j nkj mkM N NMe e eπ π ππ⎛ ⎞− + −−⎜ ⎟

⎝ ⎠ =

Properties of 2D DFT

1 2 22 2 22j mk nk j nkj mkM N NMe e eπ π ππ⎛ ⎞− + −−⎜ ⎟

⎝ ⎠ =

•Separability propery

Since

We can write

[ ] [ ]

[ ]

[ ]

1 2

21

2

1

2 2-1 -1

1 20 0

22-1 -1

0 0

2-1

1 10

2-1

1 10

, ,

,

[ , ]

w h e r e [ , ] ,

N M j m k n kM N

n m

N M j n kj m kNM

n m

N j n kN

n

M j m kM

m

F k k f m n e

f m n e e

F k n e

F k n f m n e

π π

ππ

π

π

⎛ ⎞− +⎜ ⎟⎝ ⎠

= =

−−

= =

−

=

−

=

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

=

=

∑ ∑

∑ ∑

∑

∑

Thus the 2D DFT can be computed from a 1D FFT routine

2D Fourier Transform

• Frequency – domain representation of 2D signal ::• Consider a two-dimensional signal • The signal and its two-dimensional Fourier transform

are related by ::

• u and v represent the spatial frequency in radian/length.• F(u,v) represents the component of f(x,y) with frequencies u and v.• A sufficient condition for the existence of F(u,v) is that f(x,y) is absolutely

integrable.

( ), .f x y( ),f x y ( ),F u v

( ) 2DFT

( )

, ( , )

( , ) ( , ) j xu yv

f x y F u v

F u v f x y e dx dy∞ ∞

− +

−∞ −∞

←⎯⎯→

= ∫ ∫( )

2

1( , ) ( , ) 4

j x u y vf x y F u v e d u d vπ

∞ ∞+

−∞ −∞

= ∫ ∫

( , ) f x y dxdy∝ ∝

−∝ −∝< ∝∫ ∫

2D Fourier Transform

• u and v represent the spatial frequency in horizontal and vertical directions inradian/length.

• F(u,v) represents the component of f(x,y) with frequencies u and v.

Illustration of 2D Fourier transform

2D Fourier Transform

• A sufficient condition for the existence of F(u,v) is that f(x,y) is absolutely integrable.

( , ) f x y dxdy∝ ∝

−∝ −∝< ∝∫ ∫

Properties of 2D Fourier Transform1. The 2D Fourier transform is in general a complex function of the

real variables and As such, it can be expressed in terms of the magnitude and the phase .

2. Linearity Property:

3. Shifting Property:

Phase information changes, no change in amplitude.

4. Modulation Property:

.vu( , )F u v ( , )F u v∠

2D FT1 1

2D FT2 2

2D FT1 2 1 2

( , ) ( , )

( , ) ( , )

( , ) ( , ) ( , ) ( , )

f x y F u v

f x y F u v

a f x y bf x y a F u v bF u v

←⎯⎯→

←⎯⎯→

+ ←⎯⎯→ +

),(),( )( 2 vuFeyyxxf vyuxjFTDoo

oo +−⎯⎯ →←−−

),(),( )(oo

yvxuj vvuuFeyxf oo −−⇔++

5. Complex exponentials are the eigen functions of linear shift invariant systems.

The Fourier bases are the Eigen functions of linear systems

For an imaging system, h(x, y) is called the point spread function and H (u, v) is called the optical transfer function

6. Separability property:

Particularly if then .

Suppose , then

1

1

( , ) ( , ) .

( , )

where ( , ) is 1-D Fourier Transform

jux jvy

jvy

F u v f x y e dx e dy

F u v e dy

F u v

∞ ∞− −

−∞ −∞

∞−

−∞

=

=

∫ ∫

∫

)()(),( 21 yfxfyxf =1 2( , ) ( ) ( )F u v F u F v=

( ) x, .a

yf x y rect recta

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) ( )2

, = asincau sin

= a sincau sincav

F u v a cav

7. 2D Convolution:

Thus the convolution of two functions is equivalent to product of the corresponding Fourier transforms.

2

If ( , ) ( , )* ( , ) ( , ) ( , ) ( , )Similarly if ( , ) ( , ) ( , )

1 ( , ) ( , )* ( , )4

g x y f x y h x yG u v F u v H u v

g x y f x y h x y

G u v F u v H u vπ

==

=

=

8. Preservation of inner product:Recall that the inner product of two functions is defined by

The inner product is preserved through Fourier transform. Thus,

Particularly,

Hence Norm is preserved through 2D Fourier transform.

2

1( , ), ( , ) ( , ), ( , )4

( , ), ( , ) ( , ) ( , )

f x y h x y F u v H u v

where F u v H u v F u v H u v du dv

π∞ ∞

−∞ −∞

=

= ∫ ∫

2

2 22

1( , ), ( , ) ( , ), ( , )4

1( , ) ( , ) 4

f x y f x y F u v F u v

f x y dx dy F u v du dv

π

π

∞ ∞ ∞ ∞

−∞ −∞ −∞ −∞

=

∴ =∫ ∫ ∫ ∫

( ) ( ), and h x,yf x y

( , ), ( , ) ( , ) ( , ) f x y h x y f x y h x y dx dy∞ ∞

−∞ −∞

= ∫ ∫

2D Discrete Time (Space) Fourier Transform

• Recall DTFT of a 1D sequence, Given is

and

Note that

exists if and only if is absolutely summable, i.e.,

[ ]{ }, .....x n n = − ∝ ∝

( ) [ ]n=-

= x n j nX e ωω∝

−

∝∑

[ ] ( )1x n = X2

j ne dπ

ω

πω ω

π −∫

( ) ( )2 = XX ω π ω+

( )X ω [ ]x n

[ ]n

x n∝

=−∝< ∝∑

Relationship between CTFT and DTFT

• Consider a discrete sequence obtained by sampling an analog signal at a uniform sampling rate , where T is the sampling period.

• We can represent the sampling process by means of the Dirac delta function with the relation

• Now the sampled signal can be represented in continuous domain as,

• Thus, the analog and discrete frequencies are related as

[ ]{ }, ....x n n = − ∝ ∝

( )ax t 1TsF =

[ ] ( ) , n=0, 1......ax n x nT= ±

( ) ( ) ( )

( ) ( )

an=-

an=-

= x

= x

sx t t t nT

nT t nT

δ

δ

∝

∝∝

∝

−

−

∑

∑

.w T= Ω

2D DSFT• Consider the signal defined over the

two-dimensional space.

• Also assume

• Then the two-dimensional discrete-space Fourier transform

(2D DSFT) and its inverse are defined by the following relations:

and

{ [ , ], ,....., , ,....., }f m n m n= −∞ ∞ = −∞ ∞

[ , ] .m n

f m n∞ ∞

=−∞ =−∞

< ∞∑ ∑

)(],[ ),( vnumj

n m

enmfvuF +−∞

−∞=

∞

−∞=∑ ∑=

( )2

1[ , ] ( , )4

j um vnf m n F u v e du dvπ

∞ ∞+ +

−∞ −∞

= ∫ ∫

• Note that is doubly periodic in u and v. • Following properties of are easily verified:

– Linearity– Separability– Shifting theorem:

– Convolution theorem:

– Eigen function– Modulation– Correlation– Inner product– Parseval’s theorem

( , )F u v( , )F u v

( )0 0

2 DSFT

2 DSFT0 0

If [ , ] ( , ), then

[ , ] ( , ),

D

j um vnD

f m n F u v

f m m n n e F u v− +

←⎯⎯⎯→

− − ←⎯⎯⎯→

2 DSFT 2 DSFT1 1 2 2

2 DSFT1 2 1 2

If [ , ] ( , ) and [ , ] ( , ), then

[ , ]) * [ , ] ( , ) ( , )

D D

D

f m n F u v f m n F u v

f m n f m n F u v F u v

←⎯⎯⎯→ ←⎯⎯⎯→

←⎯⎯⎯→

• Colour plays an important role in image processing• Colour image processing can be divided into two

major areas• Full-colour processing: Colour sensors such as colour

cameras and colour scanners are used to capture coloured image. Processing involves enhancement and other image processing tasks

• Pseudo-colour processing : Assigning a colour to a particular monochrome intensity range of intensities to enhance visual discrimination.

Colour Image Processing

• Visible spectrum: approx. 400 ~ 700 nm

• The frequency or mix of frequencies of the light determines the colour

• Visible colours: VIBGYOR with UV and IR at the two extremes (excluding)

Colour Fundamentals

• Cones are the sensors in the eye responsible for colour vision • Humans perceive colour using three types of cones• Primary colours: RGB – because the cones of our eyes can

basically absorb these three colours. • The sensation of a certain colour is produced due to the mixed response of these three types of cones in a certain proportion

• Experiments show that 6-7 million cones in the human eye can be divided into red, green and blue vision.

• 65% cones are sensitive to red vision, 33% are for green and only 2% are for blue vision (blue cones are the most sensitive)

HVS review

Experimental curves for colour Sensitivity

Absorption of light by red, green and blue cones in the human eye as a function of wavelength

Colour representations: Primary colours

• According to the CIE (Commission Internationale de l’Eclairage, The International Commission on Illumination) the wavelength of

each primary colour is set as follows: blue=435.8nm, green=546.1nm, and red=700nm.

• However this standard is just an approximate; it has been found experimentally that no single colour may be called red, green, or blue There is no pure red, green or blue colour.

• The primary colours can be added in certain proportions to produce different colours of light.

• The colour produced by mixing RGB is not a natural colour. • A natural colour will have a single wavelength, say λ. • On the other hand, the same colour is artificially produced by combining weighted R, G and B each having different wavelength.

• The idea is that these three colours together will produce the same amount of response as that would have been produced by wavelength λ

alone (proportion of RGB is

taken accordingly), thereby giving the sensation of the colour with wavelength λ

to some extent.

Natural and Artificial Colour

• Mixing two primary colours in equal proportion produces a secondary colour of light: magenta (R+B), cyan (G+B) and yellow (R+G).

• Mixing RGB in equal proportion produces white light.

• The second figure shows primary/secondary colours of pigments.

Colour representations: Secondary colours

There is a difference between the primary colours of light and primary colours of pigments.

Primary colour of a pigment is defined as one that subtracts or absorbs a primary colour of light and reflects or transmits the other two.• Hence, the primary colours of

pigments are magenta, cyan, and yellow.

• Corresponding secondary colours are red, green, and blue.

Colour representations: Secondary colours

• Brightness perceived (subjective brightness) is a logarithmic function of light intensity. In other words it embodies the chromatic notion of intensity.

• Hue is an attribute associated with the dominant wavelength in a mixture of light waves. It represents the dominant colour as perceived by an observer. Thus, when we call an object red, orange, or yellow, we are specifying its hue.

• Saturation refers to the relative purity or the amount of white light mixed with hue. The pure spectrum colours are fully saturated. colour such as pink (red and white) is less saturated. The degree of saturation is inversely proportional to the amount of white light added.

Brightness, Hue, and Saturation

• Red, Green, Blue, Yellow, Orange, etc. are different hues. Red and Pink have the same hue, but different saturation. A faint red and a piercing intense red have different brightness.

• Hue and saturation taken together are called chromaticity. • So, brightness + chromaticity defines any colour.

Brightness, Hue, and Saturation (contd..)

XYZ Colour System

• CIE (Commision Internationale de L’Eclairage), 1931. Spectral RGB primaries (scaled, such that X=Y=Z matches spectrally flat white).

• The entire colour gamut can be produced by the three primaries used in CIE 3-colour system. A particular colour (of wavelength λ) be represented by three components X, Y, and Z. These are called tri-stimulus values.

0.490 0.310 0.2100.177 0.813 0.0110.000 0.010 0.990

denotes corresponding spectral component

RXY GZ B

λ

λ

λ

λ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

• A colour is then specified by its tri-chromatic coefficients, defined asx= X/(X+Y+Z) y= Y/(X+Y+Z)z = Z/(X+Y+Z)so thatx + y +z=1

• For any wavelength of light in the visible spectrum, these values can be obtained directly from curves or tables compiled from experimental results.

Colour composition using XYZ

XYZ Colour System

• Shows colour composition as a function of x and y (only two of x, y and z are independent z = 1 – (x + y) and so not independent of them)

• The triangle in the diagram below shows the colour gamut for a typical RGB system plotted as the XYZ system.

The axes extend from 0 to 1. The origin corresponds to BLUE. The extreme points on the axes correspond to RED and GREEN.The point corresponding to x= y= 1/3 (marked by the white spot) corresponds to WHITE.

x

y

01

1

Chromaticity Diagram

• The positions of various spectrum colours from violet (380nm) to red (700 nm) are indicated around the boundary (100% saturation). These are pure colours.

• Any inside point represents mixture of spectrum colours.

• A straight line joining a spectrum colour point to the equal energy point shows all the different shades of the spectrum colour.

Actual Chromaticity Diagram

• Any color in the interior of the “horse shoe" can be achieved

through the linear combination of two pure spectral colors

• A straight line joining any two points shows all the different

colours that may be produced by mixing the two colours

corresponding to the two points

• The straight line connecting red and blue is referred to as the

line of purples

RGB primaries form a triangular color gamut. The white colour falls in the center of the diagram

W

• Colour models are normally invented for practical reasons, and so a wide variety exist.

• The RGB colour space (model) is a linear colour space that formally uses single wavelength primaries.

• Informally, RGB uses whatever phosphors a monitor has as primaries

• Available colours are usually represented as a unit cube — usually called the RGB cube — whose edges represent the R, G, and B weights.

RGB 24-bit colour cube

Schematic of the RGB colour cube

Colour vision model: RGB colour Model

111

C RM GY B

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Cyan, Magenta, Yellow Primary pigment colour

Subtractive color space

Related to RGB by

111

CMY

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Should produce black

Practical printing devices additional black pigment is needed. This gives the CMYK colour space

CMY and CMYK colour models

Decoupling the intensity from colour components has several advantages:• Human eyes are more sensitive to the intensity than to the hue• We can distribute the bits for encoding in a more effective way. • We can drop the colour part altogether if we want gray-scale images. • In this way, black-and-white TVs can pick up the same signal as color ones.

• We can do image processing on the intensity and color parts separately.

Example:Histogram equalization on the intensity part to contrast enhance the image while leaving the relative colors the same

Decoupling the colour components from intensity

SIH

I

• Hue is the colour corresponding to the dominant wavelength measured in angle with reference to the red axis

• Saturation measures the purity of the colour. In this sense impurity means how much white is present. Saturation is 1 for a pure colour and less than 1 for an impure colour.• Intensity is the chromatic equivalent of brightness also means the grey level component.

HSI Colour system

• HSI model can be obtained from the RGB model.

• The diagonal of the joining Black and White in the

RGB cube is the intensity axis

HSI Colour Model

HSI Model

• HSI colour model based on a

triangle and a circle are shown

• The circle and the triangle are

perpendicular to the intensity axis.

HSI Colour Model

[ ]

[ ]

( ) [ ]

)(31

),,min(31

))(()(

)()(21

cos

if360 if

212

1

BGRI

BGRBGR

S

BGBRGR

BRGR

GBGB

H o

++=

++−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−+−

−+−=

⎩⎨⎧

>−≤

=

−θ

θθ

Conversion from RGB space to HSI

The following formulae show how to convert from RGB space to HSI:

oo H 1200 ≤≤oo H 240120 ≤≤

oo H 360240 ≤≤

( )( )BRIG

HHSIR

SIB

+−=

⎥⎦

⎤⎢⎣

⎡−°

+=

−=

360coscos1

)1(

( )( )

120(1 )

cos1cos 60

3

H HR I S

S HG IH

B I R G

= − °= −

⎡ ⎤= +⎢ ⎥°−⎣ ⎦= − +

( )( )BGIR

HHSIB

SIGHH

+−=

⎥⎦

⎤⎢⎣

⎡−°

+=

−=°−=

360coscos1

)1(240

Conversion from RGB space to HSITo convert from HSI to RGB, the process depends on which colour sector H lies in. For the RG sector:For the GB sector:For the BR sector:

• YIQ colour model is the NTSC standard for analog video transmission

Y stands for intensityI is th e in phase component, orange-cyan axisQ is the quadrature component, magenta-green axis

• Y component is decoupled because the signal has to be made compatible for both monochrome and colour television.

• The relationship between the YIQ and RGB model is

0.299 0.581 0.1140.596 0.274 0.3220.211 0.523 0.312

Y RI GQ B

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

YIQ model

• International standard for studio-quality video• This colour model is chosen in such a way that it achieves

maximum amount of decorrelation. • This colour model is obtained by extensive experiments on human observers

0.299 0.587 0.114

b

r

Y R G BC B YC R Y

= + += −

= −

Y-Cb-Cr colour model

Colour balancingRefers to the adjustment of the relative amounts of red, green, and blue primary colors in an image such that neutral colors are reproduced correctly Colour imbalance is a serious problem in Colour Image Processing

• Select a gray level, say white, where RGB components are equal• Examine the RGB values. Keep one component fixed and

match the other components to it, there by defining a

transformation for each of the variable components

• Apply the transformation to all the images

Example: Colour Balanced Image

Histograms of a colour image

1 2 3 4 5 6 7 8S1

S4

S7

0

1

2

3

4

5

6

7

8

Hue

Saturation

-Histogram of Luminance and chrominance components separately

-Colour histograms ( H-S components or normalized R-G components

-Useful way to segment objects like skin non-skin

-Colour based indexing of images

Contrast enhancement by histogram equalisationHistogram equalisation cannot be applied separately for each channel• Convert to HIS space

• Apply histogram equalisation to the I component

• Correct the saturation if needed

• Convert back to RGB valuesDigital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.

www.imageprocessingbook.com

© 2002 R. C. Gonzalez & R. E. Woods

Chapter 6Color Image Processing

Chapter 6Color Image Processing

Colour image smoothing• Vector processing is used

• Averaging in vector is equivalent to averaging separately in each channel

Example- Averaging low pass filter: Averaging a vector is equivalent to averaging all the components

Colour image sharpening

Vector median filter

•We cannot apply the median filtering to the component images separately because that will result in colour distortion

•If each channel is separately median filtered then the net median will be completely different from the values of the pixel in the window

•Vector median filter will minimize the sum of the distances of a vector pixel from the other vector pixels in the window

•The pixel with the minimum distance will give the vector median.

•The set of all vector pixels inside the window is given by

W 1 2{ }N=X x x x, , ....,

Computation of vector median filter

(1) Find the sum of the distances of the vector pixel from all other neighbouring vector pixels in the window given by

th (1 )i i N≤ ≤iδ

1( , )

N

i i jj

dδ=

= ∑ x x

where represents an appropriate distance measure between the ith and jth neighbouring vector pixels

( , ) d i jx x

(2) Arrange s in the ascending order. Assign the vector pixel a rank

equal to that of

Thus, an ordering implies the same ordering of

the corresponding vectors given as

iδ ix

.iδ

(1) (2) ( ).... Nδ δ δ≤ ≤ ≤

(1) (2) ( ).... N≤ ≤ ≤x x x

where are the rank-ordered vector pixels with the number inside the parentheses denoting the corresponding rank.

(1) (2) ( ).... N≤ ≤ ≤x x x

(1) Find the sum of the distances of the vector pixel from all other neighbouring vector pixels in the window given by

th (1 )i i N≤ ≤iδ

1( , )

N

i i jj

dδ=

= ∑ x x

where represents an appropriate distance measure between the ith and jth neighbouring vector pixels

( , ) d i jx x

(2) Arrange s in the ascending order. Assign the vector pixel a rank

equal to that of

Thus, an ordering implies the same ordering of

the corresponding vectors given as

iδ ix

.iδ

(1) (2) ( ).... Nδ δ δ≤ ≤ ≤

(1) (2) ( ).... N≤ ≤ ≤x x x

where are the rank-ordered vector pixels with the number inside the parentheses denoting the corresponding rank.

(1) (2) ( ).... N≤ ≤ ≤x x x

Computation of vector median filter (contd..)

The set of rank ordered vector pixels is given by

R (1) (2) ( ){ , ,..., }N=X x x x

(3) Take the vector median as VMF (1)=x x

The vector median is defined as the vector that corresponds to the minimum SOD to all other vector pixels

Edge Detection and Colour image segmentation

Considering the vector pixels as feature vectors we can apply clustering technique to segment the colour image

EDGE DETECTION•

Edge detection is one of the important and difficult operations in image processing.

•

It is important step in image segmentation the process of partitioning image into constituent objects.

•

Edge indicates a boundary between object(s) and background.

Edge

•

When pixel intensity is plotted along a particular spatial dimension the existence of edge should mean sudden jump or step.

Magnitude of first derivative is maximum

Second derivative crosses zero at the edge point

X0

dfdx

X0

2

2

d fdx

•

All edge detection methods are based on the above two principles.

•

In two dimensional spatial coordinates the intensity function is a two dimensional surface. We have to consider the maximum of the magnitude of the gradient.

•

The gradient magnitude gives the edge location.

•

For simplicity of implementation, the gradient magnitude is approximated by

•

The direction of the normal to the edge is obtained from

•

Second derivative is implemented as a Laplacian given by

•

differentiation is highly prone to high frequency noise. •

An ideal differentiation corresponds to the function being changed in the frequency domain by the addition of a zero at origin. Thus there is an increase of 20dB per decade. This will lead to high frequency noise being amplified.

•

To circumvent this problem, low pass filtering has to be performed.

•

Differentiation is implemented as finite difference operation.

•

Three types of differences generally done are:

•

forward difference = f(x+1) –

f(x)•

backward difference = f(x) –

f(x+1)

•

centre difference = { f(x+1) –

f(x-1) } / 2

•

The most common kernels used for the gradient edge detector are the Roberts, Sobel and Prewitt edge operators.

Roberts Edge Operator

Disadvantage: High sensitivity to noise

Prewitt Edge Operator

•

Does some averaging operation to reduce the effect of noise.

•

May be considered as the forward difference operations in all 2-pixel blocks in a 3 x 3 window.

Sobel Edge Operator

•

Does some averaging operation to reduce the effect of noise, like the Prewitt operator.

•

May be considered as the forward difference operations in all 2 x 2 blocks in a 3 x 3 window.

Gradient Based Edge detection

•

Find fx and fy using a suitable operator.

•

Compute gradient

•

Edge pixels are those for which where T is a suitable threshold

Example

Second derivative Based

•

For the two-dimensional image, we can consider the orientation-free Laplacian operator as the second derivative. The Laplacian of the image f is given by

Laplacian Operator

•

Advantages:No thresholding symmetric operation

•

Disadvantages:Noise is more amplifiedIt does not give information about edge orientation

Model based edge detection

•

Marr studied the literature on mammalian visual systems and summarized these in five major points:–

In natural images, features of interest occur at a variety of scales. No single operator can function at all of these scales, so the result of operators at each of many scales should be combined.

–

A natural scene does not appear to consist of diffraction patterns or other wave-like effects, and so some form of local averaging (smoothing) must take place.

–

The optimal smoothing filter that matches the observed requirements of biological vision (smooth and localized in the spatial domain and smooth and band-limited in the frequency domain) is the Gaussian.

–

When a change in intensity (an edge) occurs there is an extreme value in the first derivative or intensity. This corresponds to a zero crossing in the second derivative.

–

The orientation independent differential operator of lowest order is the Laplacian.

•

Based on the five observations an edge detection algorithm is proposed as follows:

–

Convolve the image with a two dimensional Gaussian function. –

Compute the Laplacian of the convolved image. –

Edge pixels are those for which there is a zero crossing in the

second derivative.

LOG Operation

Convolving the image with Gaussian and Laplacian operator can be combined into convolution with Laplacian of Gaussian (LoG) operator ( Inverted Maxican Hat)

� Continuous function and discrete approximation

Canny Edge detector

• Canny’s criterion:

–

Minimizing the error of detection.–

Localization of edge i.e. edge should be detected where it is present in the image.

–

single response corresponding to one edge.

Canny Algorithm1.

Smooth the image with Gaussian filter •

If one wants more detail of the edge, then the variance of the filter is made large .If less detail is required, then the variance is made small.

•

Noise is smoothed out

2.

Gradient Operation3.

Non Maximal Suppression:

•

Consider the pixels in the neighborhood of the current pixel. If

the gradient magnitude in either of the pixels is greater than the current pixel, mark the current pixel as non edge.

4.

Thresholding with hysterisis: •

mark all pixels with Δf > TH

as the edges. •

mark all pixels with Δf < TL

as non edges. •

a pixel with TL

> Δf < TH

marked as an edge only if it is connected to a strong edge.

Example

Canny

Edge linking

•

After labeling the edges, we have to link the similar edges to get the object boundary.

•

Two neighboring points (x1

,y1

)

and (x2

,y2

) are linked if

Line Detection and Hough transform

• Many edges can be approximated by straight lines.

• For edge pixels, there are possible lines. n 1 2

n(n )−

• To find whether a point is closer to a line we have to perform

comparisons. Thus, a total of comparisons.1 2

n(n )− 30( )n

y mx c= +( , )m c

x

y

m

c

• Hough transform uses parametric representation of a

straight line for line detection.

( , )x y

c mx y= − +

x

y

m

c

P

1l2l

1 1( , )x y

• The points and are mapped to lines and

respectively in space

),( yx 1 1( , )x y1l 2l

m c−• and will intersect at a point P representing the

values of the line joining and2l1l ( , )m c

),( yx 1 1( , ).x y

• The straight line map of another point collinear with these two

points will also intersect at P The intersection of multiple lines in the m- c plane will give the (m,c) values of lines in the edge image plane.

The transformation is implemented by an accumulator array A, each

accumulator corresponding to a quantized value of ( , ).m c

Corresponding to each edge point for each in the range

find

( , ),x yim

1by ),( jiAIncrement

The array A is initialized to zero.

[ ]min max, ,m m

j ic m x y= − +

m

c

maxM m→

maxN c→

Hough transform algorithm

Initialize a 2-D array A of accumulators to zero.

For each edge point find ( , ),x y c mx y= − +

Increment 1by ),( jiA

Threshold the accumulators: the indices of accumulators with

entry greater than a threshold give values of the lines.( , )m c

Group the edges that belong to each line by traversing each line.

Hough transform variation

• and are, in principle, unbounded: cannot handle

all situations.

m c

• Rewrite the line equation as

cos( ) sin( ) x yθ θ ρ+ =

• Instead of we can consider as the parameters

with varying between -90o and 90o and varying from

0 to for an image.

( , ),m c ( , )ρ θα p

2 2M N+ M N×

θρ

Example

Circle detection• Other parametric curves like circle

ellipse etc. can be detected by Hough transform technique.

2 2 2( ) ( ) constanto ox x y y r− + − = =For circles of undetermined radius, use 3-d Houghtransform for parameters

Example

• Today’s world is dependent upon a lot of data either stored in a computer or transmitted through a communication system

• Compression involves reducing the number of bits to represent the data for storing and transmission.

• Particularly, Image compression is the application of compression

on digital

images.

Compression Basics

Storage Requirement

• Example:One second of digital video without compression requires 720X480X24X25~24.8 MB

• Example: One 4-minute song: 44100samples per second X16 bits per sampleX4X60~20 MB

•How to store these data efficiently?

• The large data rate also means a larger bandwidth requirement for transmission

• For an available bandwidth of the maximum allowable bit-rate is 2B.

2B bits/s can be resolved without ambiguity

How to send large amount of data in real-time data through a limited bandwidth channel, say a telephone channel?

We have to compress the raw data to store and transmit.

Band-width requirement

,B

Lossless Vs Lossy Compression

Lossless Lossy• A compressed image can be

restored without any loss of information

• Perfect reconstruction is not possible but visually useful information is retained

• Provides large compressionApplications

Medical imagesDocument imagesGIS

ExamplesVideo broadcastingVideo conferencing

Progressive transmission of imagesDigital libraries and image databases

Encoder and Decoder

A digital compression system requires two algorithms:

• Compression of data at the source (encoding) • Decompression at the destination (decoding)

What are the principles behind compression?

• Compression is possible if the signal data contains redundancy.

• Statistically speaking the data contain highly correlated sample values.

Example : Speech data, Image dataTemperature and Rainfall Data

Types of Redundancy

• Coding Redundancy• Spatial Redundancy• Temporal Redundancy • Perceptual Redundancy

Coding Redundancy

• Some symbols may be used more often than othersIn English text, the letter E is far more common than the letter Z.

• More common symbols are given shorter code-lengths• Less common symbols are given bigger code-lengths

Coding redundancy is exploited in loss-less coding like Huffman coding

( ) ( ) ( ) ( ). , . , . - - -e a q and j− − −

• Morse Code

• Morse noticed that certain letters occurred more frequently than others. In order to reduce the average time required to send a telegraph message, frequent letters were given shortersymbols.

• Example:

Example of Coding Redundancy

Spatial Redundancy

( ) ( )1 , 2 ,...,x n x n− −

• Neighboring data samples are correlated

• Given a sample a part of

can be predicted if the data are correlated

•

( )x n

Spatial Correlation: Example

• In video, same objects may be present in consecutive

frames so that objects may be predicted

Temporal Redundancy

Frame k Frame k+1

Perceptual Redundancy

• Humans are sensible to only limited changes in the amplitude of the signal

• While choosing the levels of quantization, this fact may be considered

• ‘Visually lossless’ means that the degradation is not visible to the human eye

Example: Humans are less sensitive to variation of colour

32 levels64 levels

Principle Behind lossless Compression

Lossless compression methods work by identifying some frequently occurring symbols in the data, and by representing these symbols in an efficient way.

Examples:

– Run-Length Encoding (RLE).

– Huffman Coding.

– Arithmetic coding.

Elements of Information Theory

• Information is a measure of uncertainty• A more uncertain (probable) symbol has more

information

• Information of a symbol is related to probability

Information Theory (Contd.)

1 2

1 2

2

Source is a random variable that takes the symbols , ,...,

with probabilities , ,...,

The self information or information ( ) is defined as

1( ) log

n

n

i

ii

Xx x x

p p pI x

I xp

⎛ ⎞= ⎜ ⎟

⎝ ⎠

• Suppose a symbol x always occurs. Thenp(x) = 1 => I(x) = 0 ( no information)If the base of the logarithm is 2, then the unit of information is called a bit.If p(x) = 1/2 , I(x) = -log2(1/2) = 1 bit.

• Example: Tossing of a fair coin: outcome of this experiment requires one bit to convey the information.

Information Theory (Contd.)

•

Average Information content of the source.

•

Measures the uncertainty associated with a source

and is called entropy

21

1( ) log /n

ii i

H X p bits symbolp=

⎛ ⎞=∑ ⎜ ⎟

⎝ ⎠

Information Theory (Contd.)

Entropy

• Introduced by Ludwig Boltzmann• His only epitaph

lnS k W=

Properties of Entropy

2

2

2

1. 0 ( ) log ( )2. ( ) log ( ) when all symbols are equally likelyIf is a binary soure with symbols 0 and 1 emitted with probability and (1- ) respectively, then

1( ) log ( ) (1 ) log

H X nH X n n

Xp p

H X p pp

≤ ≤=

= + − 21( )

1 p−

Properties of a Code· Codes should be uniquely decodable.· Should be instantaneous ( we can decode the code by

reading from left to right as soon as the code word is received.

· Instantaneous codes satisfy the prefix property (no code word is a prefix to any other code) .

• The average codeword length is given by

1

nL l pi ia v g i

∑==

avgL

Kraft’s Inequality

1 2

i=1

2

There is an instantaneous binary code withcodewords having lengths , ,.... if and only if

2 1

For example, there is an instantaneous binary codewith lengths 1, 2, 3;,3, since

1 12 2

i

In l

l l l

−

⎛ ⎞⎜ ⎟⎝ ⎠

≤

+ +

∑

2 2 3

3 3

1 1 1 1 1.125 12 2 2 2

1 1 12 2

An example of such a code is 0; 10; 110; 111. There is no instantaneous binary codewith lengths 1, 2, 2, 3, since

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + = >⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+ =

Shannon’s Noiseless Coding theorem

1 2

Given a discrete memoryless source X with symbols , ,...,

the average code word length for any instantaneous code is given by ( )

More over there exists at least one code such that

n

avg

avg

x x x

L H X

L

≥

≤ ( ) 1 H X +

Shannon’s Noiseless Coding theorem (Contd..)

1 2

Given a discrete memory less source X with symbols , ,..., if we code strings of symbols at a time,

the average code word length for any instantaneous code is given by ( )

n

avg

x x x

L H X→

Example

1

2

3

4

2 2

2

Symbol Probability 0.125 0.125 0.250 0.500( ) 0.125log (1/ 0.125) 0.125log (1/ 0.125)

0.125log (1/ 0.25) 0.

xxxxH X = +

+ + 2125log (1/ 0.5) 1.125 bit/symbol=

Example (Contd..)

1

2

3

4

Symbol Probability code 0.125 000 0.125 001 0.250 01 0.500 1

0.125 3 0.125 3 0.25av

xxxxL X X X= + + 2 0.5 1 1.125 bit/symbol

X+

=

• A prefix code can be represented by a binary tree, each branch being denoted by 0 or 1, emanating from a root node and having n leaf nodes

• A prefix code is obtained by tracing out branches from the root node to each leaf node.

Prefix code and binary tree

• Based on a loss-less statistical method of the 1950s.

0.125 0.125 0.25 0.5

0

0

0

11

1

Huffman coding

• Creates a probability tree and combines the two lowest probabilities to obtain the code

Huffman Coding ( Contd..)

• Most common data value (with the highest frequency)

has the shortest code• Huffman table of data value versus code must be sent • Time of coding and decoding can be long• Typical compression ratios 2:1 – 3:1

ip

Steps in Huffman Coding

• Arrange symbol probabilities in decreasing order while there is more than one node

• Merge the two nodes with the smallest probabilities to form a new node with probabilities equal to their sum

• Arbitrarily assign 1 and 0 to each pair of branches merging in to a node

• Read sequentially from the root node to the leaf node where the symbol is located

40 40 10 10 10 10 10 10 10 10 10 0 0 0 0

40 2 10 9 0 4

Run-length coding

• Looks for sequential pixel valuesExample: 1 row of an image with the new code

below

• Has reduced the size from 18 bytes to 6• Higher compression ratios when predominantly low frequency information

•Typical compression ratios of 4:1 to 10:1•Used in Fax machine•Used for coding the quantized transform coefficients in a lossy coder

Arithmetic coding

• Codes a sequence of symbols rather than a single symbol at a time

• Now, the sequence has to be coded

• A single number lying between 0 and 1 is generated,

corresponding to all the symbols in the sequence, this

number is called tag1 2 3( ) 0.7, ( ) 0.8, ( ) 1F a F a F a= = =

1 2 2 3a a a a

1 2 3 1 2 3{ , , }; ( ) 0.7, ( ) 0.1, ( ) 0.2A a a a p a p a p a= = = =

• Choose the interval corresponding to the first symbol; the tag willlie in the interval

•Go on subdividing the subintervals according to the symbol probability •The code is the AM of the final subinterval

•The tag is sent to decoder which has to know the symbol probabilities

•The decoder will repeat the same procedure to decode the symbols

0 0 0.49 0.539 0.5446

0.5460.5460.560.7

a1a1

a2

a1

a3

a1a1

a3 a3 a3a3

a3a2a2

a2a2

a2 Tag

Decoding algorithm for Arithmetic coding

( )

0 0

1*

1 1

*1

k

In itia lise 0 , 0 , 1R epeat 1

such that ( )

U pdate u ,U ntil k= size o f the sequence

k

k k

k X k x k

k

k l uk k

T A G ltu l

x F x t F x

l

−

− −

−

= = == +

−=

−≤ <

Disadvantage• Assumes data to be stationary, does not consider dynamics of data

•Arithmetic code is used to code the symbols in JPEG2000

• Similar to run-length coding but with some statistical methods similar to Huffman

• Dynamically builds up a dictionary by both encoders and decoders

• Examples:Unix command compressionImage Compression- Graphics Interchange Format (GIF)Portable document format (PDF)

LZW (Lemple, Ziv and Welch) coding

•

Initialize table with single character strings

STRING = first input character WHILE not end of input stream CHARACTER = next input character IF STRING + CHARACTER is in the string table

STRING = STRING + CHARACTER ELSE

output the code for STRING add STRING + CHARACTER

to the string table STRING = CHARACTER END WHILE

output code for STRING

LZW Coding (contd..)

ExampleLet ‘aabbbaa’ be the sequence to be encoded; the dictionary will be

The output for the given sequence is ‘11253, which is ‘aabbbaa’ according to dictionary

• Throws away both non-relevant information and a part of

relevant information to achieve required compression

• Usually, involves a series of algorithm-specific

transformations to the data, possibly from one domain

to another (e.g to frequency domain in Fourier Transform)

without storing all the resulting transformation terms and

thus, loosing some of the information contained

Lossy Compression

• Perceptually unimportant information is discarded.

• The remaining information is represented efficiently to achieve compression

• The reconstructed data contains degradations with respect to the original data

Lossy Compression (contd..)

Example

• Differential Encoding: Stores the difference between consecutive data samples using a limited number of bits.

• Discrete Cosine Transform (DCT): Applied to image data.

• Vector Quantization

• JPEG (Joint Photographic Experts Group)

Fig. Original Lena image, and Reconstructed image from

lossy Compression

Rate distortion theory

•

Rate distortion theory deals with the problem of representing information allowing a distortion

-Less exact representation requires less bits

X Y

Lossy Coder

Average distortion ( ) ( )2 2 ( ) ( / )x y

E X Y x y p x p y x∑∑− = −

Rate Distortion theory

X Ysource Lossy

Coder

I( X,Y ) = H(X) – H(X| Y)

Hence: minimize I( X,Y ) under the constraint D

Minimize the bit rate

Constraint: Average Distortion between X and Y ≤

D

Rate Distortion function for a Gaussian source

If the source X is a Gaussian random variable, the rate distortionfunction is given by

( ) 2 21 log , D<2

0 , otherwise

R D Dσ σ⎛ ⎞= ⎜ ⎟⎝ ⎠

2σ is the variance of Gaussian random variable

σ2

D

R(D)

Rate Distortion

•

Gaussian case presents the worst case of coding

•

For a non Gaussian case, achievable bit error rate is lower than that of Gaussian.

• If we do not know about anything about

the distribution of X, then Gaussian case gives us the pessimistic bound.

•

An increase of 1 bit improves the SNR by about 6 dB.

Lossy Encoder

Fig. A Typical Lossy Signal/Image Encoder

Compressed Data

Entropy CodingQuantization

Quantization tableEntropy-coding table

Input Data Prediction/ Transformation

Differential Encoding• Given a sample a part of

can be predicted if the data are correlated.

• A simple prediction scheme expresses the predicted value as a linear combination of past p samples:

[ ] [ ] [ ]1 , 2 ,..., ,x n x n x n p− − −

[ ] [ ]1

ˆp

ii

x n a x n i=

= −∑

[ ]x n

• The prediction parametersare estimated using correlation among data

• The prediction parameters and the prediction error are transmitted

1 2, ..... pa a a

1 2, ..... pa a aˆ( ) ( )x n x n−

Linear Prediction Coding (LPC)

LPC (contd..)

• Variants of LPC (10) are used for coding speech for mobile communication

• Speech is sampled at 8000 samples per second• Frames of 240 samples ( 30 msec of data) are

considered for LPC• Corresponding to each frame, quantized versions of

10 prediction parameters and approximate prediction errors are transmitted

Transform coding• Transform coding applies an invertible linear coordinate

transformation to the image.

• Most of the energy will be stored in a few transform coefficients

• Example: Discrete Cosine transform (DCT), Discrete wavelet transform (DWT)

TransformCorrelated data

Less correlated data

Transform Merits Demerits

KLT Theoretically optimal •Data dependent•Not fast

DFT Very fast • Assumes periodicity of data• High frequency distortion is more because of Gibb’s phenomenon

DCT • Less high frequencydistortion

•High energy compactionHigh Energy CompactionScalabilty

Blocking Artifacts

Computationally complex

•Also, DCT is theoretically closer to KLT and implementation wise closer to DFT

Transform selection

DWT

• Reversible transform, like Fourier Transform• N samples of signal

[ , ], 0,1, ..., 1, 0,1, ..., 1f m n m N n N= − = −

Discrete Cosine Transform (DCT)

( )( )1 1

0 0

(2 1) 2 1( , ) ( ) ( ) [ , ]cos , 0,1,.. 1, 0,1,.., 1

2

1 1 0 0 ( ) ( )

2 2 1, 2,.., -1 1,2,.., -1

N N

cm n

m u n vF u v u v f m n u N v N

N

u vN N

with u and vu N v N

N N

πα α

α α

− −

= =

⎛ ⎞+ + += = − = −⎜ ⎟⎜ ⎟

⎝ ⎠⎧ ⎧= =⎪ ⎪⎪ ⎪= =⎨ ⎨⎪ ⎪= =⎪ ⎪⎩ ⎩

∑ ∑

DCT is given by

x 50 54 49 55 52 53 54 49DCT 147.07 -0.32 -2.54 1.54 -1.41 1.13 -4.30 -3.0366

Round 147 0 -3 2 -1 1 -4 -3Threshold 147 0 0 0 0 0 -4 0

IDCT 51 54 50 53 53 50 54 51

• We see that only two DCT coefficients contain most information about the original signal

• DCT can be easily extended to 2D

DCT (contd..)

Block DCT• DCT can be efficiently implemented in blocks using FFT and other

fast methods.• FFT based transform is more computationally efficient if applied

in blocks rather than on the entire data.

• For a data length and point FFT, computational

complexity is of order 2logN N

N N

• If the data is divided into sub-blocks of length then

the number of sub-blocks is and the

nNn

computational complexity 2 2log logN n n N nn

= =

How to choose block size?• Smaller block-size gives more computational efficiency• Neighboring blocks will be correlated causing inter block redundancy.• If blocks are coded independently, blocking artifact will appear

Block size

Reconstruction error

2 2× 4 4× 8 8×

• Beyond block size, reduction in error is not significant. 8 8×

4 DCT co-efficients per 8X8 block

16 DCT co-efficients per 8X8 block

8 DCT co-efficients per 8X8 block

• Replaces the transform coefficients with lower-precision approximations which can be coded in a more compact form

• A many-to-one function.

• Precision is limited by the number of bits available.

• X= 147.07 -0.32 -2.54 1.54 -1.41

Quant(X)= 147 0 -3 2 -1

Quantization

Quantization (contd..)

• Information theoretic significance

• More the variance of the co-efficients, more is the information

• Estimate the variance of each transform coefficient from

given image or determine the variance from the assumed model

• In the DCT,

DC co-efficients: Raleigh distribution

AC co-efficients: Generalized Gaussian distribution model

•Two methods for quantization are zonal coding and threshold coding

Zonal coding

• The co-efficients with more information content (more variance)

are retained

Threshold coding• The co-efficients with higher energy are retained, the rest are

assigned zero

• More adaptive

• Computationally exhaustive

Zonal Coding mask and the number of bits allotted for each coefficient

Original image and its DCT

Reconstructed image from truncated DCT

• Joint Photographic Expert Group• A generally used lossy image coding format• Allows tradeoff between compression ratio and image

quality• Can achieve high compression ratio(20+) with almost

invisible difference

JPEG

8x8DCT

HuffmanCoding

quantizationImage

HuffmanTable

QuantizationTable

Coded Image

JPEG (contd..)

Baseline JPEG

•

Divide image into blocks of size 8X8.•

Level shift all 64 pixels values in each

block by subtracting, (where is themaximum number of gray levels).•

Compute 2D DCT of a block.

•

Quantize DCT coefficients using a quantization table.•

Zig-zag

scan the quantized DCT coefficients to

form 1-D sequence.•

Code 1-D sequence (AC and DC) usingJPEG

Huffman variable length codes.

12n− 2n

An image block and DCT

An 8X8 intensity block

QuantizationQuantization table

Zig-zag

scanning

Zigzag scanning

AC Coefficients18 –9 –3 –8 1 –3 1 –2 4 –2 4 0 3 –1 0 1 1 0 –1 –1 0 0 0 –1 (39 zeros)

Wavelet Based Compression

•

Recall that DWT is implemented through Row-wise and column-wise filtering and the down-sampling by 2 after each filtering. The approximate image is further decomposed.

•

First and second stages of decomposition are illustrated in the figure below.

LL1

LH1

HL1

HH 1

First stage

LH1

HL1

HH 1

Second stage

LL2

LH2

HL2

HH2

Embedded Tree Image Coding

Embedded bit stream: A bit stream at a lower rate is contained in a higher rate bit stream (good for progressive transmission)

• Embedded Zero-tree Wavelet (EZW) codingalgorithm, Shapiro [1993]

•Set Partitioning In Hierarchical Trees (SPIHT)- based algorithm, Said and Pearlman [1996]

•EBCOT (Embedded Block Coding with Optimized Truncation) proposed by Taubman

in 2000.

Tree representation of Wavelet Decomposition

EZW

EZW scans wavelet coefficients subband

by subband. Parents are scanned before any of their children, but only after all neighboring parents have been scanned.

EZW coding

Each coefficient is compared against the current threshold T. A coefficient is significant if its amplitude is greater then T;such a coefficient is then encoded as

Positive significant (PS) Negative significant (NS)

Zerotree root (ZTR)

is used to signify a coefficient below T, with all its children also below TIsolated zero (IZ)

signify a coefficient below T, but with

at least one child not below T2 bits are needed to code this information

Successive Approximation quantization

•Sequentially applies a sequence of thresholds T0,···,TN-1 to determine significance•Three-level mid-tread quantizer•Refined using 2-level quantizer

Example

l g l g 522 max 22 2 32o C oThreshold T ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦= = =

Quantization

+8

-8

2nd

Pass

• Not only better efficiency, but also more functionality

• Superior low bit-rate performance• Lossless and lossy compression• Multiple resolution• Region of interest(ROI)

JPEG 2000

Transform Quantization EntropyCoding

JPEG

J2K

DCT DiscreteCosineTransform

DWT DiscreteWaveletTransform

8x8Quantization

Table

Quantizationfor each

sub-band

HuffmanCoding

ArithmeticCoding

JPEG2000 v.s. JPEG

low bit-rate performance

JPEG2000 v.s. JPEG

• A video sequence consists of a number of pictures, containing a lot of time domain redundancy.

• This is often exploited to reduce data rates of a video sequence leading to video compression.

• Motion-compensated frame differencing can be used very effectively to reduce redundant information in sequences

• Finding corresponding points between frames (i.e., motion estimation) can be difficult because of occlusion, noise, illumination changes, etc

• Motion vectors (x,y-displacements) are sent

Video Compression

Reference frame Current frame

Predicted frame Error frame

Motion-compensated Prediction

Search procedure

Current frame

Reference frame

Current block

Best match Search region

Search Algorithms

• Exhaustive Search• Three-step search• Hierarchical Block Matching

Three step search algorithm

Search window of ±(2N –1) pixels is selected (N=3)Search at location (0,0)Set S= 2N-1 (the step size)Search at eight locations ±S pixels around location (0,0)From the nine locations searched so far, pick the location with smallest Mean Absolute Difference (MAD) and make this the new search origin.Set S=S/2.Repeat stages 4-6 until S=1

Minimum atfirst iteration

Minimum atseconditeration

Minimum atthird iteration

First iteration

Seconditeration

Thirditeration

Three step search algorithm

Video Compression Standards

Two Formal OrganizationsInternational Standardization Organization/ International Electro- technical Commission (ISO/IEC) International Telecommunications Organization (ITU-T)

ITU-T Standard •H.261 (1990) •H.263 (1995)

ISO/IEC standards•MPEG-1 (1993)•MPEG-4 (1998)

ITU-T and ISO/IEC MPEG-2 (1995)H.264/AVC (2003)

Applications And The Bit-rates Supported

Standard Application Target bit-rateH.261 Video conferencing and video

telephonyPx64 kbps

1≤P≤30MPEG-1 CD-ROM video applications 1.5Mbps

MPEG-2/ H.262 HDTV and DVD 15-30 MbpsH.263 Transmissions over PSTN

networksUp to 64kbps

MPEG-4 Multimedia applications 5kbps-50Mbps

H.264 Broad cast over cable, Video on demand, Multimedia streaming services

64kbps to 240Mbps

• Motion Pictures Expert Group• Standards for coding video and the associated

audio• Compression ratio above 100• MPEG 1, MPEG 2, MPEG 4, MPEG 7, MPEG

21….

MPEG Video Standards

MPEG 2 Coder Decoder

• The MPEG system specifies three types of frames within a sequence:

• Intra-coded picture (I-frame): Coded independently from all other frames

• Predictive-coded picture (P-frame): Coded based on a prediction from a past I- or P-frame

• Bidirectionally predictive coded picture (B- frame): Coded based on a prediction from a past and/or future I- or P-frame(s). Uses the least

MPEG 2 Frame Types

MPEG 2 GOP structure

Includes both spatial- and frequency-domain techniques:• Basic gray level transformations• Histogram Modification • Average and Median Filtering • Frequency domain operations • Edge enhancement

Image Enhancement

Aimed at improving the quality of an image for

•Better human perception or

•Better interpretation by machines.

Enhancement technique

Better imageInput image

Image enhancement

•Application specific•No general theory•Can be done in

- Spatial domain: Manipulate pixel intensity directly- Frequency domain: Modify the Fourier transform

Spatial Domain technique

X-Simplest case• depends only on the value of

does not depend on the position of the pixel in the image. • called brightness transform or point processing

[.]g at [ , ];f x y

[ , ] ( [ , ])

( )

g x y T f x yors T r

=

=

Contrast stretching

( )s T r=

Some useful transformations

r

255s r= −

( )s T r=

r Enhanced in the range 100-150 and 150-255

Thresholding [ , ]

[ , ] 255 else[ , ] 0

If I m n ThI m n

I m n

>=

=

Th= 120

log( 1)s c r= +

Log transformation

• Compresses the dynamic range

where c is the scaling factor.

Example : Used to display the 2D Fourier Spectrum

Log transformation (Contd..)

s crγ=

γ c

• Expands dynamic range

Example : Image scaling, same effect as adjusting camera exposure time.

Power law transformation

where and are positive constants

• Often referred to as gamma-correction1,γ =

Example: Image Display in the monitor

Sample Input to Monitor Monitor output

Sample Input

Gamma Corrected Input

Monitor Output

Gamma Correction

Gamma corrected image

Original Image Corrected by 1.5γ =

Gamma correction (Contd..)

r

ss

r

Gray level slicing

Results of slicing in the black and white regions

• Highlights the contribution of specific bits to image intensity

• Analyses relative importance of each bit; aids determining the

number of quantization levels for each pixel

Bit-plane slicing

MSB plane obtained by

thresholding at 128

MSB plane

Original

Original Image and Eight bit-planes

Includes• Histogram• Histogram Equalization• Histogram specification

Histogram Processing

{0,1,.., 1}

( )

Number of pixels with gray level

Total number of pixels

k

kk

k

k

r Lnp rn

nr

n

∈ −

=

→

→

Histogram

histogram:For B-bit image, initialize bins with 0For each pixel x,yIf f(x,y)=i, increment bin with index Iendifendfor

2B

Histogram

Low-contrast image Histogram

Improved-contrast image Histogram

Suppose r represents continuous gray levels0 1r≤ ≤ . Consider a transformation of the form ( )s T r= that satisfies the following conditions (1) ( )s T r= is single valued, monotonically increasing in r . (2) ( )0 1T r≤ ≤ for 0 r 1≤ ≤ [ ] [ ]0,1 0,1T⎯⎯→ Inverse transformation is ( )1T s r− = , 0 1s≤ ≤

Histogram Equalisation

Histogram Equalisation (contd.)

r can be treated as a random variable in [ ]0, 1 with pdf ( )rp r .The pdf of ( )s T r= is

( ) ( )

( )1

rs

r T s

p rp s

dsdr −=

=

( ) ( )

( )0

s=T r ,0 r 1

ds then, dr

r

r

r

Suppose p u du

p r

= ≤ ≤

=

∫

( ) ( )( )

1, 0 1rs

r

p rp s s

p r∴ = = ≤ ≤

Histogram Equalisation

0

{0,1,.., 1}

( )

( )

k

kk

kk i

i

r Ln

p rn

g p r=∑

∈ −

=

=

The resulting needs to be scaled and rounded.

kg

Histogram-equalized image image Histogram

Histogram-equalized Image

Example

The following table shows the process of histogram equalization for a 128X128 pixel 3-bit (8-level) image.

Gray level ( )kr kn kn

n

07

ki

ki

ns round Xn=

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑

0 1116 0.0681 0 1 4513 0.2755 2 2 5420 0.3308 5 3 2149 0.1312 6 4 1389 0.0848 6 5 917 0.0560 7 6 654 0.0399 7 7 226 0.0138 7

Histogram specification Given an image with a particular histogram, another image which has a specified histogram can be generated and this process is called histogram specification or histogram matching.

( )( )

r

z

p r original histogram

p z desired histogram

→

→

( )0

r

rs p u du= ∫

( )0

z

zr p w dw= ∫

( ) ( )1 1z G s G T r− −= =

Image filtering

Image filtering involves • Neighbourhood operation • taking a filter mask from point to point in an image and perform operations

on pixels inside the mask.

Linear Filtering

he case of linear filtering, the mask is placed over the pixel; the gray valueof the image are multiplied with the corresponding mask weights and thenadded up to give the new value of the pixel.

Thus the filtered image g[m,n] is given by ', '

' '[ , ] [ ', ']m n

m ng m n w f m m n n= − −∑∑

Where summations are performed over the window. The filtering window is usually symmetric about the origin so that we can write

', '' '

[ , ] [ ', ']m nm n

g m n w f m m n n= + +∑∑

Linear Filtering Illustrated

Averaging Low-pass filter

An example of a linear filter is the averaging low-pass

filter. The output avgI of an averaging filter at any pixel is the average of the neighbouring pixels inside the filter mask. It can be given as ,[ , ] [ , ]avg i j

i jf m n w f m i n j= + +∑∑ ,

where the filter mask is of size m n× and , ,,i j i jf w are the image pixel values and filter weights respectively. Averaging filter can be used for blurring and noise reduction.

• Show that averaging low-pass filter reduces noise. • Large filtering window means more blurring

1/9 1/9 1/9

1/9 1/9 1/9

1/9 1/9 1/9

Original Image Noisy Image

Filtered Image

Averaging filter

Low-pass filter example

• Filtered with 7X7 averaging mask

High-pass filter

0 0 0

0 1 0

0 0 0

1/9 1/9 1/9

1/9 1/9 1/9

1/9 1/9 1/9

A highpass

filtered image can be computed as the difference betweenthe original and a lowpass

filtered version.Highpass

= Original −

Lowpass

-

-1/9 -1/9 -1/9

-1/9 8/9 -1/9

-1/9 -1/9 -1/9

=

−1

-1/9 -1/9 -1/9

-1/9 8/9 -1/9

-1/9 -1/9 -1/9

High-pass filtering

-1/9 -1/9 -1/9

-1/9 8/9 -1/9

-1/9 -1/9 -1/9

[ , ] [ , ] [ , ] 1 ( 1) [ , ] [ , ] [ , ] ( 1) [ , ] [ , ]

s av

av

high

f m n Af m n f m n AA f m n f m n f m n

A f m n f m n

= − >

= − + −

= − +

Unsharp

Masking

Median filtering

• The median filter is a nonlinear filter that outputs

the median of the data inside a moving window of

pre-determined length. This filter is easily

implemented and has some attractive properties

• Useful in eliminating intensity spikes( salt & pepper noise)

• Better at preserving edges

• Works up to 50% of noise corruption

•Verify that median filter is a nonlinear filter.

18 20 20

15 255 17

20 20 20

Sorted data

15 17 18 20 20 20 20 20 255

Median= 20, and 255 will be replaced by 20

Median

Median Filtering

IMAGE TRANSFORMS

Image transform

Signal data are represented as vectors. The transform changes the basis of the signal space. The transform is usually linear but not shift-invariant.

Useful for compact representation of dataseparation noise and salient image featuresefficient compression. .

A transfom may beorthonormal/unitary or non-orthonormalcomplete, overcomplete, or undercomplete.applied to image blocks or the whole image

. .

1D TRANSFORMDATA

UNITARY TRANSFORM

Unitary transform and basis

0,0 0 ,1 0 , 1

1,0 1,1 1, 1

1,0 1,1 1, 1

* * *

0,0 1,0 1,0

* * *

0,1 1,1 1, 1

1

* * *

1,0 1,1 1, 1

......

......:

.....

Then............

:.....

N

N

N N N N

N

N

N N N N

t t tt t t

t t t

t t tt t t

t t t

−

−

− − − −

−

−

−

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

T

T

Unitary transform and basis (Contd..)

* * *

0,0 1,0 1,0

* * *

0,1 1,1 1, 1

* * *

0, 1 1, 1 1, 1

*

0 ,0

*

0 ,1

*

1,0

Therefore,......[0] [0]......[1] [1]

: : :[ 1] ..... [ 1]

[0] :

N

N

N N N N

N

t t tf Ft t tf F

f N t t t F N

tt

Ft

−

−

− − − −

−

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢⎢⎢⎣ ⎦

* *

1,0 1,0

* *

1,1 1,1

* *

1, 1 1, 1

[1] : ... [1] :

N

N

N N N

t tt t

F Ft t

−

−

− − −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+ +

⎥ ⎢ ⎥ ⎢ ⎥⎥ ⎢ ⎥ ⎢ ⎥⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

Since the columns of are independent, they form a basis for the N-

dimensional space

1T −

Examples of Unitary Transform

2D DFT

Other Examples : DCT (Discrete Cosine Transform)

DST

(Discrete Sine Transform)

DHT

(Discrete Hardmard

Transform)

KLT

(Karhunen

Loeve

Transform)

1.

The rows of T form an orthogonal basis

for the N-

dimensional complex space.

2.

All the eigen vectors of T have unit magnitude

Tf

= λf

Properties of Unitary Transform

3.

Parseval’s theorem: T is energy preserving transformation, because it is an unitary transform which preserves energy

F*' F = energy in transform domainf*'

f = energy in data domain

F*' F =

[Tf]*'

Tf=

f*'

T*' Tf

=

f*' I f= f*'

f

4.

Unitary transform is a length and distance preserving transform.

5. Energy is conserved, but often will be unevenly distributedamong

coefficients.

Decorrelating propertymakes the data sequence uncorrelated. useful for compression

Let f = [ f[0], … ,f[N-1] ] T be data vector,

= Covariance matrix

= Covariance matrix of transformation

Diagonal elements of CFt– variance, off-diagonal elements of CFt

–

Covariance

Perfect Decorrelation – off-diagonal elements are zero.

2D CASE

Separable Property

Matrix representation

If we consider separable, unitary and symmetric transform.

Thus we can write

2 1

1 1

1 2 1 1 2 20 0[ , ] [ , ] [ , ]N N

n nf n n t k n t n k− −

= =∑ ∑=

2D Separable, unitary and symmetric transforms have following properties:

i.

Energy preservingii.

Distance preservingiii.

Energy compactioniv.

Other properties specific to particular transforms.

.

Karhunen

Loeve

(KL) transform

Let f = [ f[0], … ,f[N-1] ] T be data vector,

= Covariance matrix

Cf can be diagonalised by

where is diagonal matrix formed by eigen valuesU is matrix formed by eigen vectors as columns

Transform matrix of KL transform is

fUCU′ =Λ

Λ

T U ′=

KL transform is

E(FKLT) = 0

KLT KLT fF =T (f -μ )

KL Transform

KL Taransform

Covariance Matrix isKLTFC

For KLT eigen values are arranged in descending order and the transformation matrix is formed by considering the eigen vectorsin order of the eigen values.

Reconstructed value is

Suppose we want to retain only k- transform coefficients we will retain the transformation matrix formed by the k largest eigen vectors.

Principal Component Analysis (PCA) – Linear combination of largest principal eigen vectors.

KLT Illustrated

F1

F2

KL transformMean square error of approximation is

Mean square error is minimum over all affine transforms.

Transform matrix is data dependent so computationally extensive.

1 1

0 0

1

ˆ ˆ( ) ( )N J

i ii i

N

ii j

E Energyof Energyofλ λ

λ

− −

= =

−

=

∑ ∑

∑

′ ′= −

= −

=

f - f f - f f f

1D-DCTLet f = [ f[0], … ,f[N-1] ] T be data vector

1D-DCT of data vector f and its IDCT are

where

The transformation can be written asF = TC f

where TC

is transformation matrix with elements

DCT is a real transform.

TC is orthogonal transformTC

‘

TC = I so TC

-1

= TC‘

Relation with DFTDCT :

Let

DCT of is given by

21 2 2( ) ( )Re ( )0

C

j j knkN N NF k k al f n e en

π πα

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

− −−= ∑

=

( ) ( ) , 1f n f n n N′ = ≤ −0 , 1 2 1N n N= − < ≤ −

( )f n 22 1 2 2( ) ( )Re ( )0

C

j j knkN N NF k k al f n e en

π πα

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

− −−′= ∑

=

[ ], for 0,1,..., 10, otherwise

( ) f n n Nf n = −⎧⎨⎩

′ =

Another interpretationExtend the data symmetrically about n=N-1 so that

DFT of is given by2 2 (2 1 )

' 2 2( ) ( ) (2 1 )10

j jnk k N nN Nk f n e f N n e

NFn

π π− − − −⎧ ⎫⎪ ⎪+ − −⎨ ⎬

⎪ ⎪⎩ ⎭

−= ∑=

(21),fNN N=− −<≤−

(2 1)2

22

2 21 2 2 2 2( )0

1 ( )cos( )0

k nN

j nkNe

j nkNe

j j j jnk nk nk nkN N N N Nf n e e e en

N f nn

π

π

π

π π π π⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

= +

− − − −−= +∑

=

−∑=

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

( ) f n n Nf N n n N N N

f n = −⎧⎨ − − = + −⎩

′ =

( )f n′

( )

( )

( ) '( )

( ) (2 1)2

(2 1)2

1 ( )cos( )0

1 ( )cos( )0

kc

k

k F k

kk nN

k nN

j jnk nkN Ne e

j nkNe

NF f nn

N f nn

α

α

α

π

ππ π

π=

+

−= +

−

−∴ = ∑=

−∑=

DCT

Interpretation of DCT with DFT

2D-DCT

The 2D –DCT of is

2D – DCT can be implemented from 1D – DCT by

Performing 1D – DCT column wiseThen performing 1D – DCT row wise

1 2( , )f nn

DCT is close to KLTFirstly, DCT of basis vectors will be eigen vectors of triangular matrix

Secondly, a first order markov process with correlation coefficient ρ has a covariance matrix

If ρ is close to 1 then

Therefore for a Markov first order process with ρ close to 1 DCT is close to KLT.

Because of closeness to KLT, energy compaction, data decorrelation and ease of computation DCT is used in many applications.

Many image processing operations are efficiently implemented in terms of matrices

Particularly, many linear transforms are used

Simple example: colur transformation

0.299 0.587 0.114 0.596 -0.275 -0.321 0.212 -0.523 0.311

Y RI GQ B

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Matrices

Matrix representation of linear operations• Let be data samples.

• These data points can be represented by a vector

• Consider a transformation

• Denoting we get where

[ ] [ ] [ ]0 , 1 ,......, 1x x x N − N

0

1

-1

x

x

x N

⎡ ⎤⎡ ⎤⎣ ⎦⎢ ⎥⎡ ⎤⎢ ⎥⎣ ⎦

⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎣ ⎦⎣ ⎦

x =

[ ] [ ]1

0

, 0,1,......, 1N

njj

y n a x j n N−

=

= = −∑[ ][ ]

[ ]

0

1...

-1

y

y

y N

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

y = y = Ax

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

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

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

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

. . . . . . . . .

.

. . . .

N

N

N

N N N N N

a a a aa a a aa a a a

a a a a

−

−

−

− − − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

Example: Discrete Fourier transform

[ ] [ ]21

0 k=0,1,.........N-1

N i nkN

nx k x n e

π− −

=

=∑

[ ][ ]

( )

2 2 ( 1)

2 21 (

0 1 1 . . . 1

1 1 . . . . .. ..

1 . . . [ 1]

j j NN N

j N j NN N

X

X e e

e eX N

π π

π π

− − −

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

( )

[ ][ ]

1) 1

0

1...

[ 1]N

x

x

x N−

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦

Example: Rotation operation

x

y

r

r

θ0θ

( ),x y′ ′

( ),x y cos sinx x yθ θ′ = −

sin cosy x yθ θ′ = +

cos sinsin cos

x xy y

θ θθ θ

′ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Matrices - Basic Definitions• Transpose of a matrix:

• Symmetric Matrix: =

• Hermitian Matrix: =

Example:

• Inverse of a Matrix: for a non singular matrix

TA A

TA

*TA A

T1 i 1 i, =

-i 1 -i 1∗⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

A A

1− =A A I A

• A matrix of complex elements is called unitary if

Example =

A 1 T− ∗=A A

1 1 1 3 3 3

1 1 1 3 1 1 3 2 2 2 23 3 3

1 1 1 3 1 1 3 2 2 2 23 3 3

j j

j j

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥+ − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢⎢ ⎛ ⎞ ⎛ ⎞

− − − +⎢ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎝ ⎠ ⎝ ⎠⎣ ⎦

⎥⎥⎥⎥

A

Unitary Matrix

•

Orthogonal Matrix

1 T− =A A

cos sinx x yθ θ′ = −

sin cosy x yθ θ′ = +

-sinsin coscosθ θ

θ θ⎡ ⎤

= ⎢ ⎥⎣ ⎦

A1 cos sin

-sin cosTθ θ

θ θ− ⎡ ⎤= =⎢ ⎥⎣ ⎦

A A

Example: Rotation operation

For an orthogonal matrix A

Real-valued unitary matrices are orthogonal

Example

• Is the following matrix orthogonal?

1 1 2 2

1 1 2 2

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

• A matrix is called Toeplitz if the diagonal contains the same element, each of the sub-diagonals contains the same element and each of the super-diagonals contains the same element.

• The system matrix corresponding to linear convolution of two sequences is a Toeplitz matrix.

• The autocorrelation matrix of a wide-sense stationary process is Toeplitz.

Example

0,0 0,1 0, 1

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

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

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

. . . . . . . . .

.

. . .

N

N

N

N N

a a aa a a aa a a a

a a a a

−

−

−

− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

Toeplitz Matrix

Circulant Matrix

• A matrix is called a Circulant Matrix if each row is obtained by the circular shift of the previous row.

Example

• The system matrix corresponding to circular convolution of two sequences is a circulant matrix.

0,0 0,1 0, 1

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

0, 2 0, 1 0,0 0, 3

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

. . . . . . . . .

.

. . . ...

N

N N

N N N

N

a a aa a a aa a a a

a a a a

−

− −

− − −

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A

Eigen Values and Eigen Vectors• If is a square matrix and is a scalar such that

then is the eigen value and is the corresponding eigen vector.

• Thus the eigen vectors are invariant in direction under an operation by a matrix.

Example: Rotation operation with then

then No real eigen values and eigen vectors exist.

Now consider Rotation operation with then

so that each vector is invariant under rotation by

A λ λ=Ax xλ x

90θ =0 -11 0

x xy y′⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0 -11 0⎡ ⎤

= ⎢ ⎥⎣ ⎦

A

180θ = -1 00 -1⎡ ⎤

= ⎢ ⎥⎣ ⎦

A

0180

Basically, a filtering operation. •Set-theoretic operations are used for image processing

applications. Such operations are simple and fast.• In normal image filtering, we have an image and

filtering is done through the convolution mask.• In morphological image filtering, the same is done

through the structuring element

Morphological Image Processing

{ }2( , ) | ( , )x y x y ∈ Z

{ }( , ) | ( , ) 1A x y I x y= =

{ }2( , , ) | ( , ) , ( , )A x y z x y I x y z= ∈ =Z

• Image plane is represented by a set

in Binary Image Morphology

The value of ‘z’ gives the gray value and (x, y) gives the gray point.

• A binary object is represented by

A

Binary Morphology Basics

• In Gray-scale Morphology,{ }( , ) | ( , ) 0cA x y I x y= =Background

Structuring elementA

Entire image

A

Structuring element• It is similar to mask in convolution

• It is used to operate on the object image A

B

{ }ˆ ( , ) | ( , )A x y x y A= − − ∈

(1) Reflection Operation: Reflection of is given by

B̂

B

Reflection and Translation operations

A

Translation of by an amount is given by

( ) { }|x

A a x a A= + ∈

(2) Translation Operation:

xA

x

Binary image morphology operations

a. Dilation operation

b. Erosion operation

c. Closing operation

d. Opening operation

e. Hit or Miss transform

A

( ){ }ˆ|x

A B x A B⊕ = ∩ ≠ ∅

B

Dilation operation

Given a set A and the structuring element B, we define dilation of A with B as

A

B

A

Why dilation?• If there is a very small object, say (hole) inside the object ‘A’, then

this unfilled hole inside the object is filled up.• Small disconnected regions outside the boundary may be

connected by dilation.• Irregular boundary may be smoothened out.

Properties of Dilation

A B B A⊕ = ⊕

( ) ( )A B C A B C⊕ ⊕ = ⊕ ⊕

( )x xA B A B⊕ = ⊕

1.

2.

3. Dilation is translation invariant

Dilation is associative

Dilation is commutative

Erosion operationA

BErosion of A with B is given by,

{ }| xA B x B AΘ = ⊆

• Here, structuring element ‘B’ should be completely inside the

object ‘A’.

Why Erosion?

1. Two nearly connected regions will be separated by erosion operation.

1. Shrinks the size of the objects3. Removes peninsulas and small objects .4. Boundary may be smoothened

Properties of Erosion

1. Erosion is translation invariant ( ) xxA B A BΘ = Θ

( )A B B AΘ ≠ Θ

( ) ( )A B C A B CΘ Θ = Θ ⊕

Dilation and erosion are dual operations in the sense that,

( ) ˆC CA B A BΘ = ⊕

( )ˆ C CA B A B⊕ = Θ

2. Erosion is not commutative

3. Erosion is not associative

Opening and Closing operation

Dilation, and Erosion changes size

By opening and closing operations, irregular boundaries may be smoothened without changing the overall size of the object

Original Dilated

Closing operation

Dilation, followed by Erosion( )A B A B B• = ⊕ Θ

After Dilation operation, the size of the object is increased and it is brought back to the original size by erosion operation

Object AStructuring element B

⇒Closing

By closing operation, irregular boundaries may be smoothened depending upon the structural element

Example

Original Dilated Closed

Opening operation

Erosion, followed by Dilation

( )A B A B B= Θ ⊕o

• Opens the weak links between the nearby objects

• Smoothens the irregular boundary

In all operations, performance depends upon the structuring element and is applied to binary images.

Example: Edge detection

Example

• A={(1,0), (1,1), (1,2),(1,3), (0,3)}

• B={(0.0), (1.0)}

={(1,0), (1,1), (1,2),(1,3), (0,3),(2,0), (2,1), (2,2),(2,3)}A B⊕ =

{(1,3), (0,3)}A BΘ =

We can similarly do other morphological operations

Hit or Miss transform

The main aim of this transform is pattern matching

( ) ( )1 2CA B A B A B∗ = Θ ∩ Θ

2 1B W B= −

( ) ( )1 2ˆ C

A B A B A B∗ = Θ ∩ ⊕

The transform is given by

where

W is the window around the structuring element

Procedure for Hit-or-Miss Transform

Object AStructuring element B1

W

B2=W-B1

After B2 is obtained, find

( ) ( )1 2CA B A B A B∗ = Θ ∩ Θ

CA

ExampleIn a text, how many F’s are present?

Structuring element

or

The structuring element will match both E & F .In such cases, background is also considered for pattern matching

Hit or miss transform will match F only

Example

Applications of binary morphology operations

1. Boundary extraction( ) ( )A A A Bβ = − Θ

Applications of binary morphology operations

2. Region filling

(a)Starts with a pixel inside the unfilled region

(b)Select a proper structuring element B and perform

( )1C

n nX X B A−= ⊕ ∩

Repeat this operation 1n nX X −=

After getting findnA X∪nX to get the filled region

0X

3. Thinning operation

• This gives the image as single pixel width line

Thinningobject

objectskeletonizing

3. Thinning operation

• Thinning and skeletonizing are impact operations

( )A B A A B⊗ = − ∗Thinning operator

Go on matching until exact matching is achievedOriginal image A

1’s 0’s

1B

Now, consider another structuring element 2B

( ) ( )1 21A B A A B A A B B⊗ = − ∗ − − ∗ ∗⎡ ⎤⎣ ⎦

Then, the thinning operation is

In this manner, we can do hit or miss with structuring elements to get ultimately thinned object

The skeleton is given by the operation

0

( ) ( )

( ) ( ) ( )

K

kk

k

S A S A

whereS A A kB A kB B

=

=

= Θ − Θ o

U

Skeletonizing

•.Generalization of binary morphology to gray-level images• Max and Min operations are used in place of OR and AND • Nonlinear operation• The generalization only applies to flat structuring elements. .

Gray-scale morphology

Gray scale morphological operations

( ),f x y

( ),b x y

Object (intensity surface)

Structuring element (will also have an

intensity surface)

Domain of ( ),f x y fD

Domain of ( ),b x y bDDomain of structuring element

Domain of object

Dilation at a point (s,t) of f with b

( )( ) ( ) ( ) ( ){ }, max , , , , , ,f bf b s t f s x t y b x y s x t y D x y D⊕ = − − + − − ∈ ∈

• It can also be written like in convolution ( ) ( ), ,f x y b s x t y+ − −

• Here, mask is rotated by 180 degree and placed over the object;

from overlapping pixels, maximum value is considered.

• Since it is a maxima operation, darker regions become bright

Applications1. Pepper noise can be removed

2. Size of the image is also changed.

Dilation Illustrated

Dilation Result

Erosion operation

( )( ) ( ) ( ) ( ){ }, min , , ,( , ) , ,f bf b s t f s x t y b x y s x t y D x y DΘ = + + − + + ∈ ∈

• It is a minimum operation and hence, bright details

will be reduced.Application: Salt noise is removed

• We can erose and dilate with a flat structuring element

Erosion Result

Closing operation

( )f b f b b• = ⊕ Θ

• Removes pepper noise

• Keeps intensity approximately constant

• Keeps brightness features

Opening operation

( )f b f b b= Θ ⊕o

• Removes salt noise

• Brightness level is maintained

• Dark features are preserved

Opening and Closing illustration

Closing and Opening results

closing opening

Original image

• Gray-scale dilation and erosion are duals with respect to

function complementation and reflection

( ) ( ) ( )( )ˆ, ,c cf b s t f b s tΘ = ⊕

• Gray-scale opening and closing are duals with respect to

function complementation and reflection

( ) ˆc cf b f b• = o

Duality

• Opening followed by closing

• Removes bright and dark artifacts, noise

Smoothing

g f b f b= ⊕ − Θ• Subtract the Eroded image from the dilated image

• Similar to boundary detection in the case of binary image

• Direction Independent

Morphological gradient

Probability and Random processes• Probability and random processes are widely used in image processing. For example

inImage enhancementImage coding Texture Image processingPattern matching

• Two ways of applications:• (a)

The intensity levels can be considered as the values of a discrete random variable with a probability mass function.

• (b) Image intensity as two-dimensional random process

• The following figure shows an image and its histogram.

• We can use this distribution of grey levels to extract meaningful information about the image.

Example: Coding application and segmentation application

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

We may also model the analog image as a continuous random variable

Probability concepts

• Random Experiment: An experiment is a random experiment if its outcome cannot be predicted precisely. One out of a number of outcomes is possible in a random experiment. A single performance of the random experiment is called a trial.

• 2. Sample Space: The sample space is the collection of all possible outcomes of a random experiment. The elements of are called sample points.

• 3. Event: An event A is a subset of the sample space such that probability can be assigned to it.

Probability Definitions

Classical definition of probability)Consider a random experiment with a finite number of outcomes. If all the

outcomes of the experiment are equally likely, the probability of an event is defined by

Where is the number of outcomes favourable to A.Example• A fair die is rolled once. What is the probability of getting a‘6’?• Here and

AN

( ) ANP AN

=

{'1', '2 ', '3', '4 ', '5 ', '6 '}S = { '6 '}.A =

1( ) .6

P A∴ =

Relative Frequency Definition

( ) A

n

nP A Lim

n→∞=

An

Face 1 2 3 4 5 6

Frequency 82 81 88 81 90 78

If an experiment is repeated n times under similar conditions and the event A occurs in times, then

Example: Suppose a die is rolled 500 times. The following table shows the frequency each face.

Then 78 1( )500 6

P A =

Image Intensity as a Random Variable

0p1p 1Lp −

x 0 1 . L-1

.( )XP x

•The probability is estimated from the histogram of the image.

For each gray level ,

[ ] i

ifh iN

=

where is the number of pixels with intensity i.

if

Some Important Properties

( ) ( ) ( ) ( )P AUB P A P B P A B= + − ∩

( ) 0P A B∩ =

Probability that either A or B or both occurs is given by

If A and B are mutually exclusive or disjoint, then

( ) 1P S =

Conditional Probability

• the conditional probability of B given A denoted by is defined byP(B/A)

( )( )

Similarly( )( / )

( )

A B

A

A B

A

NN

NNNN

P A BP A

P A BP A BP B

∩

∩

∩

∩=

P(B/A) =

=

=

Independent events

• Two events are called independent if the probability of occurrence of one event does not affect the probability of occurrence of the other. Thus the events A and B are independent if and only if

• or ( / ) ( )P B A P B=

( / ) ( )and hence

( ) ( ) ( )

P A B P A

P A B P A P B

=

∩ =

A random variable associates the points in the sample space with real numbers. Consider the probability space and function mapping the sample space into the real line.

Random variable

S

Domain of XRange of X

Figure Random variable X

Real line

Probability Distribution FunctionThe probability ({ }) ({ | ( ) , })P X x P s X s x s S≤ = ≤ ∈ is called the probability distribution function (also called the cumulative distribution function abbreviated as CDF) of X and denoted by ( ).XF x Thus ( , ],x−∞ ( ) ({ })XF x P X x= ≤

Properties of Distribution Function • 0 ( ) 1XF x≤ ≤ • )(xFX is a non-decreasing function of .X Thus, 1 2 1 2, then ( ) ( )X Xx x F x F x< < • 0)( =−∞XF

• 1)( =∞XF • 1 2 2 1({ }) ( ) ( )X XP x X x F x F x< ≤ = −

Value of the random variable

Random variable

( )XF x

Example

Suppose { , }S H T= and :X S → is defined by ( ) 1X H = and ( ) 1.X T =−

Therefore, X is a random variable that can take values 1 with

probability 12

and -1 with probability 12

.

SH

T

1

-1

X

o x

( )XF x

x

If )(xFX is differentiable, the probability density function (pdf) of ,X denoted by ( ),Xf x is defined as

)()( xFdxdxf XX =

Properties of the Probability Density Function

• ( ) 0.Xf x ≥

This follows from the fact that ( )XF x is a non-decreasing function

• ( ) ( )x

X XF x f u du−∞

= ∫

• ∫∞

∞−

= 1)( dxxf X

• ∫−

=≤<2

1

)()( 21

x

xX dxxfxXxP

0x

( )Xf x

0 0x x+ Δ x

- 0 0 0 0 0({ }) ( )XP x X x x f x x< ≤ + Δ Δ

Probability Density Function

Example Uniform Random Variable

1

( ) -0

X

a x bf x b a

otherwise

⎧ ≤ ≤⎪= ⎨⎪⎩

Gaussian or Normal Random Variable

2121( )

2

X

X

x

XX

f x eμ

σ

πσ

⎛ ⎞−− ⎜ ⎟

⎝ ⎠=

, x−∞ < < ∞

Functions of a Random variable

Let X be a random variable and (.)g be a function of X . Then ( )Y g X= is a random variable. We are interested to find the pdf of .Y For example, suppose X represents the random voltageinput to a full-wave rectifier. Then the rectifier output Y is given by .Y X= We have to find the probability description of the random variable .Y We consider the casewhen ( )g X is a monotonically increasing or monotonically decreasing function . Here

1 ( )

( ) ( )( )( )

X XY

x g y

f x f xf ydy g xdx

−=

⎤= = ⎥′ ⎦

Example: Probability density function of a linear function of a random variable Suppose , 0.Y aX b a= + >

Then and

( )( )( )X

XY

y b dyx aa dx

y bff x af y dy adx

−= =

−

∴ = =

The expectation operation extracts a few parameters of a random variable and provides asummary description of the random variable in terms of these parameters.

The expected value or mean of a continuous random variable X is defined by

1

( ) for a continuousRV

( ) for a discrete RV

X

X N

i X ii

xf x dxEX

x p xμ

∞

−∞

=

⎧∫⎪⎪= = ⎨

⎪∑⎪⎩

Generally for any function ( )g X of a RV X

( ) ( ) ( )XEY Eg X g x f x dx∞

−∞= = ∫

Particularly,

Mean-square value ∞

−∞= ∫2 2

XE X x f (x )d x

Variance ∞

−∞∫2 2 2

X X X Xσ = E(X - μ ) = (x - μ ) f (x)dx

EXPECTATION AND MOMENTS OF A RANDOM VARIABLE

Multiple Random variablesIn many applications we have to deal with many random variables. For example, the noise affecting the RB channels of colour video may be represented by three random variables. In such situations, it is convento define the vector-valued random variables where each component of the vector is a random variable. Joint CDF of n random variables Consider nrandom variables 1 2 , ,.., nX X X defined on the same sample space. We define the random vectas,

1

2

.

.

n

XX

X

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X

A particular value of the random vector X is denoted by

1

2= .

n

xx

x

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

x

The CDF of the random vector X is defined as follows

( , ,.. ), ,.., 1 21 2 ({ , ,.. })1 1 2 2

F x x x FX X X nnP X x X x X xn n

=

= ≤ ≤ ≤

(x)X

Multiple random variablesIf X is a continuous random vector, that is, ( , ,.. ), ,.. 1 21 2

F x x xX X X nnis continuous in each of

its arguments, then X can be specified by the joint probability density function

( , ,.. ) ( , ,.. ), ,.. 1 2 , ,.. 1 2...1 2 1 21 2

nf f x x x F x x xX X X n X X X nx x xn nn

∂= =

∂ ∂ ∂(x)X

We also define the following important parameters.

The mean vector of ,X denoted by ,Xμ is defined as

1

2

1

2

( )( )

( )

( )

n

n

X

X

X

E XE X

E

E Xμ

μ

μ

⎡ ⎤⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

XμX

Multiple Random Variables

Similarly for each ( , ) 1,2,.., , 1,2,..,i j i n j n= = we can define the covariance

,

( , ) ( )( )

and the ( , )

i j i j

i j

i j

i j i X j X i j X X

i jX X

X X

Cov X X E X X EX X

correlation coefficientCov X X

μ μ μ μ

ρσ σ

= − − = −

=

All the possible covariances and variances can be represented in terms of amatrix called the covariance matrix XC defined by

1 1 2 1

2 1 2 2

1 2

( )( )var( ) cov( , ) cov( , )cov( , ) var( ) . cov( , )

cov( , ) cov( , ) var( )

n

n

n n n

EX X X X XX X X X X

X X X X X

μ μ ′= − −

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

X X XC X X

Multi-dimensional Gaussian

Suppose for any positive integer ,n 1 2, ,....., nX X X represent n jointly random variables.

These random variables are called jointly Gaussian if the random variables 1 2, ,.....,

nX X X

have joint probability density function given by

( )1

1 2

1( ) ' ( )2

, ,....., 1 2( , ,... )2 det( )

X

nnX X X n

ef x x xπ

−− − −

=X XX μ C X μ

XC

where ( )( ) 'E= − −X X X

C X μ X μ is the covariance matrix and

[ ]1 2( ) ( ), ( )...... ( ) 'nE E X E X E X= =Xμ X is the vector formed by the means of the random

variables.

Example 2D Gaussian

Two random variables YX and are called jointly Gaussian if their joint density function is 2 2( ) ( )( ) ( )1

2 2 22 (1 ),2

,

21

, 2 1( , )

x x y yX X Y YXY

X YX Y X Y

X Y X YX Yf x y e

μ μ μ μσ σρ σ σ

ρ

πσ σ ρ

− − − −

−

⎡ ⎤− − +⎢ ⎥

⎣ ⎦

−=

- , - x y∞ < < ∞ ∞ < < ∞

The joint pdf is determined by 5 parameters

• means and X Yμ μ

• variances 2 2

and X Yσ σ

• correlation coefficient , .X Yρ

Karhunen-Loe’ve transform (KLT)

• Let a random vector be given as and characterized by mean and

auto co-variance matrix . • is a positive definite matrix. This matrix can be digitalized as

Where are the eigen values of the matrix and is the matrix formed with the eigen vectors as its columns.

• Consider the transformation . Then, will be a diagonal matrix and the transformation is called the Karhunen

Loe’ve

transform

(KLT).

X

[ ][ ]

[ ]

1

2..

X

X

X N

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X xμ

( )( )'x xE μ μ= − −XC X X

XC

1

2

N

0 . .0 . .

= . . . .. 0

T

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Xφ C φ

1 2, ..... Nλ λ λ XC φ

( )T= − xY φ X μ YC

Recall that a random variable maps each sample point in the sample space to a point in the

real line. A random process maps each sample point to a waveform. A random process is

thus a function of t and s. The random process X(t,s) is usually denoted by X(t).

Discrete time random process X[n].

2( , )X t s 3s

2s 1s

S

1( , )X t s

3( , )X t s

Random Process

The discrete random process is a function of one- dimensional variable n.

• Some important parameters of a random process are:• Mean:

• Variance:

• Auto correlation:

• Auto covariance:

[ ]{ }X n

[ ] [ ] [ ]( ) [ ] [ ]( ),XC n m E X n n X m mμ μ= − −

[ ] [ ] [ ]( ),XR n m E X n X m=

[ ] [ ]2 2( [ ] )n E X n nσ μ= −

[ ] [ ]n EX nμ =

Wide sense stationary (WSS) random process

• For WSS process the Mean

and the autocorrelation function

is a function of the lag only.• We denote the autocorrelation function of a WSS process at lag k by • Autocorrelation function is even symmetric with a maximum at k=0.

[ ] constantXEX n μ= =

[ ], [ ] [ ]XR m n EX m X n=n m−

[ ],X n

[ ]XR k[ ],X n

• Mean vector:

• Autocorrelation matrix:

• Auto covariance matrix:

[ ][ ]

[ ]

1

2..

X

X

X N

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

X

[ ][ ]

[ ]

[ ][ ]

[ ]

1 1

2 2. .. .

x

x

x

EX

EXE

EX N N

μ

μ

μ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Xμ X

X E ′=R XX

( )( )′x X XC = E X -μ X - μ

We can represent N samples by an N-dimensional random vector

Check that are symmetric Toeplitz matrixand X XR C

Matrix Representation

Frequency-domain Representation

• A host of tools are available to study a WSS process. • Particularly, we may have the frequency domain representation of a WSS process in

terms of the power spectral density (PSD) given by

• The autocorrelation function is obtained by the inverse transform of as given by

( ) [ ] j kX X

k

S R k e ωω∞

−

=−∞

= ∑( )XS ω

[ ] 1 ( )2

j kX XR k S e d

πω

π

ω ωπ −

= ∫

Gaussian random process

• A random process is called a Gaussian process if for any N, the joint density function is given by,

( )( ) ( )

1( ) ' ( )2

[1], [2],..., [ ] 1 2 31, , ,...,

2 det

where the vecors and the matrices are as interpreted earlier.

X X X N N Nf x x x x eπ

−− −

=-1

X X Xx μ C x μ

xC

[ ]{ }X n

Markov process

• is a random process with discrete state .i.e. can take the one of discrete values with certain probabilities as shown below.State:Probabilities:

• is called first-order Markov if

• Thus for a first-order Markov process, the current state depends on the immediate past.

• Similarly, is called p th-order Markov if

• A first-order Markov process is generally called a Markov process.

{ [ ]}X n [ ]X n L0 , 1 2 1, , . . . . Lx x x x −

0 1 2 1, , ,.... Lx x x x −

0 1 2 1, , ,..., Lp p p p −

{ [ ]}X n[ ] [ ] [ ]{ } [ ] [ ]{ }1 2 1( | 1 , 2 ,... ) ( | 1 )n n n n nP X n x X n x X n x P X n x X n x− − −= − = − = = = − =

{ [ ]}X n

[ ] [ ] [ ]{ }( )[ ] [ ] [ ] [ ]{ }( )

1 2

1 2

| 1 , 2 ,...

| 1 , 2 ,...,

n n n

n n n n p

P X n x X n x X n x

P X n x X n x X n x X n p x

− −

− − −

= − = − =

= = − = − = − =

Random field

• A two dimensional random sequence is called a random field. For a random field we can define the mean and the autocorrelation functions as follows:

• Mean:

• Autocorrelation:

• A random field is called a wide-sense stationary (WSS) or homogeneous random field if

• Thus, for a WSS random field the autocorrelation function can be defined by

{ [ , ]}X m n{ [ , ]},X m n

[ ], [ , ]EX m n m nμ=

[ ], , , [ , ] [ , ]XR m n m n EX m n X m n′ ′ ′ ′=

{ [ , ]}X m n[ ], , , is a function of the lags [ , ].XR m n m n m m n n′ ′ ′ ′− −

{ [ , ]},X m n [ ],XR k l

[ ]

[ ]

, [ , ] [ , ) [ , ] [ , ] ,

X

X

R k l EX m n X m k n lEX m k n l X m nR k l

= + +

= + +

= − −

• A random field is called an isotropic random field if

• A random field is called a separable random field if can be separated as

Two-dimensional power spectral density• We have the frequency domain representation of a WSS random field in terms of the

two-dimensional power spectral density given by

• The autocorrelation function is given by

• A random field is called a Markov random field if the current state at a location depends only on the states of the neighboring locations.

{ [ , ]}X m n

[ ] 2 2, , , is a function of the distance ( ) ( )XR m n m n m m n n′ ′ ′ ′− + −

{ [ , ]}X m n [ ],XR m n[ ] [ ] [ ]

1 2,X X XR m n R m R n=

( )( , ) [ , ] j uk vlX X

l kS u v R k l e

∞ ∞− +

=−∞ =−∞

= ∑ ∑

[ ],fR k l[ ] ( )

2

1, ( , )4

j uk vlX XR k l S u v e dudv

π π

π ππ+

− −

= ∫ ∫

{ [ , ]}f m n

Divide the image into homogenous segments.

Homogeneity may be in terms of(1) Gray values (within a region the gray values don’t vary much )

Ex: gray level of characters is < gray level of background colour (2) Texture : some type of repetitive statistical uniformity(3) Shape(4) Motion (used in video segmentation)

Segmentation

Example

Segmentation based on Colour

Segmentation based on texture

Optical character recognitionIndustrial inspectionRoboticsDetermining the microstructure of biological ,metallurgical specimensRemote sensing Astronomical applicationsMedical image segmentation Object based compression techniques (MPEG 4)Related area in object representation

Application

• Histogram-based segmentation• Region-based segmentation

–Edge detection–Region growing–Region splitting and merging.

• Clustering–K-means–Mean shift

• Motion segmentation

Main approaches

•Problem: we are given an image of a paper, and we would like to extract the text from the image.•Thresholding: define a threshold Th such that

If I(x,y) < ThThen object(character)

Else background (paper)•How do we determine the threshold ?

– Just choose 128 as a threshold (problematic for dark images)

– Use the median/mean (both are not good, as most of the paper is white)

Example: Text and Background

Example

• Assumption– Regions are distinct in terms of gray level range

Gray levelFreq

uen

cy

Histogram-based Threshold

•Compute the gray level histogram of the image.•Find two “clusters”: black and white.•Minimizing the L2 error:

–Select initial estimate T–Segment the image using T.–Compute the average gray level of each segment mb ,mw

–Compute a new threshold value: T = ½ (mb +mw )–Continue until convergence.

•We are already familiar with this algorithm !

Histogram-based threshold

• Noise• Many holes and discontinuities in the segmentation.• Changes in the illumination • We do not use spatial information.Some of the problems can be solved using image processing techniques. For example, we can enhance the result using morphological operations.

Yet – How can we overcome the changes in the illumination ?

Problems with this approach

• Divide the image into sub-images. • Assume that the illumination in each sub-images is constant.• Use a different threshold for each sub-image.• Alternatively – use a running window (and use the threshold of the

window only for the central pixel )

Problems:• Rapid illumination changes.• Regions without text: we can try to recognize that these regions is

unimodal.

Adaptive Thresholding

• We may use probability based approach.• Intensity histogram is a mixture of Gaussian distribution.• Normalized histogram = sum of two Gaussian with different mean and variance.• Identify the Gaussian• Decide the threshold

Optimal Thresholding

We would like to use spatial information.We assume that neighboring pixels tend to belong to the same segment (not always true)

• Edge detection: Looking for the boundaries of the segments.

• Problem: Edges usually do not determine close contours. We can try to do it with edge linking

Region-based segmentation

Basic FormulationLet R represent the entire image region.

Segmentation: Partitioning R into n subgroups Ri s.t:a) b) is a connected regionc)d)e)

P is the partition predicate

Ui

i RR =

iRI φ=ji RR

TrueRP i =)(

FalseRRP ji =)( U

Partition should be such that each region is homogenous as well as connected.

Region-based segmentation

A predicate P has values TRUE or False• The intensity variation within the region is not much. Not

a valid predicate• The intensity difference between two pixels is less than

5.Valid Predicate• The distance between two R, G, B vectors is less than

10.Valid Predicate

Example of a Predicate

• Choose a group of points as initial regions.• Expand the regions to neighboring pixels using a predicate:

– Color distance from the neighbors.– The total error in the region (till a certain threshold):

•Variance•Sum of the differences between neighbors.•Maximal difference from a central pixel.

– In some cases, we can also use structural information: the region size and shape.

• In this way we can handle regions with a smoothly varying gray level or color.

• Question: How do we choose the starting points ? It is less important if we also can merge regions.

Region growing

• One solution of selecting the seed point is to choose the modes of histogram.

• Select pixels corresponding to modes as seed points.

IntensityS1 S2 S3

Frequency

Selection of seed points

FalseRP i =)(TrueRRP ji =)( U

Region merging and splitting

• In region merging, we start with small regions (it can be pixels), and iteratively merge regions which are similar.

• In region splitting, we start with the whole image, and split regions which are not uniform.

• These methods can be combined. Formally:1. Choose a predicate P.2. Split into disjoint regions any region Ri for which3. Merge any adjacent regions Ri and Rj for which 4. Stop when no further merging and splitting is possible.

R1

R3 R4

R21 R22

R23 R24 R1 R3 R4

R21 R22 R24

R2

R23

R

With quadtree, one can use a variation of the split & merge scheme: •Start with splitting regions.•Only at the final stage: merge regions.

QuadTree

R4

• Address the image as a set of points in the n-dimensional space:– Gray level images: p=(x,y,I(x,y)) in R3

– Color images: p =(x,y,R(x,y),G(x,y),B(x,y)) in R5

– Texture: p= (x,y,vector_of_fetures)– Color Histograms: p=(R(x,y),G(x,y),B(x,y)) in R3. we ignore

the spatial information.

• From this stage, we forget the meaning of each coordinate. We deal with arbitrary set of points.

• Therefore, we first need to “normalize” the features (For example - convert a color image to the appropriate linear space representation)

Segmentation as clustering

Given two vectors , we can use measures like• Euclidean distance• Weighted Euclidean distance• Normalized correlation

and i jX X

Similarity Measure

Here, we merge each time the closest neighbors.

Again, we can use splitting & merging

Idea:• Determine the number of clusters• Find the cluster centers and point-cluster correspondences to minimize error

Problem: Exhaustive search is too expensive.Solution: We will use instead an iterative search [Recall the ideal

quantization procedure.]

Algorithm– Fix cluster centers – Allocate points to closest cluster– Fix allocation; compute best cluster centers

K-means

Error function =

1, 2 ,..., kμ μ μ

• Data set (72,180) (65,120) (59,119) (64,150) (65,162) (57,88) (72,175) (44,41) (62,114) (60,110) (56,91) (70,72)

• Intitial Cluster centres (45,50) (75,117) (45,117) (80,180)

• Iteration 1

• (45,50) (44,41) New mean (44,41)

• (75,117) (62,114), (65,120) New mean (63,117)

• (45,117) (57,88),(59,119),(56,91),(60,110)New mean ( 58,102)

• (80,180) (72,180), (64,150),(65,162), (72,175),(70,172)New mean ( 39,170)

Illustration of K-means

Example – clustering with K-means using gray-level and color histograms

• K-means is a powerful and popular method for clustering. However:– It assumes a pre-determined number of clusters– It “likes” compact clusters. Sometimes, we are looking for

long but continues clusters.

• Mean Shift:– Determine a window size (usually small).– For each point p:

• Compute a weighted mean of the shift in the window:

• Set p := p + m• Continue until convergence.

– At the end, use a more standard clustering method.

∑∈

−=windowi

ii ppwm )( rrr ),( ii ppdw =

Mean Shift

This method is based on the assumption that points are more and more dense as we are getting near the

cluster “central mass”.

Mean Shift (contd..)

• Background subtraction:– Assumes the existence of a dominant background.

• Optical flow (use the motion vectors as features)• Multi model motion:

– Divide the image to layers such that in each layer, there exist a parametric motion model.

Motion segmentation

Texture

• Texture may be informally defined as a structure composed of a large number of more or less ordered similar patterns or structures

• Textures provide the idea about the perceived smoothness, coarseness or regularity of the surface.

• Texture has played increasingly important role in diverse application of image processing

– Computer vision– Pattern recognition– Remote sensing– Industrial inspection and – Medical diagnosis.

Texture Image Processing

Texture analysis: how to represent and model texture• Texture synthesis: construct large regions of texture from small example images• Shape from texture: recovering surface orientation orsurface shape from image texture.

• In image processing texture analysis is aimed at two main issues:

– Segmentation of the scene in an image into different homogeneously textured regions without a priori knowing the textures.

– Classification of the textures present in an image into a finite number of known texture classes. A closely related field is the image data retrieval on the basis of texture. Thus a speedy classification can help in browsing images in a database.

• Texture classification methods can be broadly grouped into one of the two approaches: – Non-filtering approach and – Filtering approach.

Co-occurrence Matrix

( , )C i jd

Pd

Cd

Objective: Capture spatial relationsA co-occurrence matrix is a 2D array in which• Both the rows and columns represent a set of possibleimage values• indicates how many times gray value i co-occurs withvalue j in a particular spatial relationship d.• The spatial relationship is specified by a vector d = (dr,dc).From we can compute , the normalized gray-level co-occurrence matrix, where each value is divided by the sum of all the values.

Cd

Example

1 1 0 0 1 1 0 0

1 1 0 0 1 1 0 0

0 0 1 1 0 0 1 1

0 0 1 1 0 0 1 1

1 1 0 0 1 1 0 0

1 1 0 0 1 1 0 0

0 0 1 1 0 0 1 1

0 0 1 1 0 0 1 1

d

d

1 pixel right16 12

C =12 16

16 1256 56P =12 1656 56

=

⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

d

Measures Extracted from GLCC Matrix

,( , )( )k

i jC i j i j−∑∑ d

,( ( , ))

i jMax C i jd

,

( , ) /( )k

i jC i j i j−∑∑ d

From Co-occurrence matrices extract some quantitative features:

1. the maximum element of C

2. the element difference moment of order k

3. the inverse element difference moment of order k

4. entropy

5. uniformity

,

( , ) log ( , )i j

C i j C i j−∑∑ d d

2 ( , )i j

C i j∑∑ d

Disadvantages

• Computationally expensive• Sensitive to gray scale distortion (co-occurrence matricesdepend on gray values)• May be useful for fine-grain texture. Not suitable forspatially large textures.

Non- Filtering approach

• Structural texture analysis methods that consider texture as a composition of primitive elements arranged according to some placement rule. These primitives are called texels. Extracting the texels from the natural image is a difficult task. Therefore these methods have limited applications.

• Statistical methods that are based on the various joint probabilities of gray values. Co-occurrence matrices estimate the second order statistics by counting the frequencies for all the pairs of gray values and all displacements in the input image. Several texture features can be extracted from the co-occurrence matrices such as uniformity of energy, entropy, maximum probability, contrast, inverse difference moments, and correlation and probability run-lengths.

• Model based methods that include fitting of model like Markov random field, autoregressive, fractal and others. The estimated model parameters are used to segment and classify textures.

Filtering approach

• In the filtering approach, the input image is passed through a linear filter followed by some energy measure.

• Feature vectors are extracted based on these energy outputs. • Texture classification is based on these feature vectors. • The following figure shows the basic filtering approach for texture classification.

• Filtering approach includes Laws mask, ring/wedge filters, dyadic Gabor filter banks, wavelet transforms, quadrature mirror filters, DCT, Eigen filters etc.

Gabor filters

•

Gabor

Filters•

Fourier coefficients depend on the entire image (Global): We losespatial information.•

Objective: Local Spatial Frequency Analysis•

Gabor

kernels: Fourier basis multiplied by a Gaussian–

( )2

00 22

( )1( ) exp ( ) exp22

x xg x i x x iω θσπσ

⎛ ⎞−= − − −⎜ ⎟

⎝ ⎠

The product of a symmetric Gaussian with an oriented sinusoid–

Gabor filter

Gabor filters come in pairs: symmetric and antisymmetric–

Each pair recover symmetric and antisymmetric

components in aparticular direction.–:the spatial frequency to which the filter responds strongly–

σ: the scale of the filter. When σ

= infinity, similar to FT•

We need to apply a number of Gabor filters at different scales,orientations, and spatial frequencies.

Two Dimensional Signals and Systems

• An analog image is modeled as a two-dimensional (2D) signal. • Consider an image plane where a point is denoted by the coordinate

• The intensity at is a two-dimensional function and is denoted by

• The video is modeled as a three-dimensional function

• The digital image is defined over a grid, each grid location being called a pixel.

• We will denote this 2D discrete space signal as

( , ).x y

( , )x y( , ).f x y

( , , ).f x y t

[ , ].f m n

Some of the useful one-dimensional(1 D) functions

1. Dirac Delta or Impulse function ::

• Sifting property

• Scaling property

( ) ( )x u xδ ′=

( ) 0 0

( ) 1

x x

x dx

δ

δ∞

−∞

= ≠

=∫( ) ( ) ( )f x x x dx f xδ

∞

−∞

′ ′ ′− =∫( )( ) xaxa

δδ =

•Dirac Delta function is the derivative of the unit step function

a x -a

12a

2. Kronecker Delta or discrete-time impulse function ::

• Sifting property :

3. Rectangle function ::

0 0[ ]

1 1n

nn

δ≠⎧

= ⎨ =⎩

[ ] [ - ] = [ ]m

mf m n m f nδ

=∞

=−∞∑

11, ( ) 2

0 otherwise

xrect x

⎧ ≤⎪= ⎨⎪⎩

4. Sinc function ::

5. Complex exponential function ::

• These functions are defined in two or multiple dimensions by the seperability property ::

• For example, the complex exponential function is separable.

sinsin ( ) xc xxπ

π=

j xe ω

1 2( , ) ( ) ( )f x y f x f y=( , ) is symmetric if ( , ) ( ) ( )

f x y f x y f x f y=

1 2 1 2( )j x y j x j ye e eω ω ω ω+ =

The two-dimensional delta functions• 2D Dirac delta function ::

• 2D Kronecker Delta function ::

Linear Systems and Shift Invariance

• A system is called linear if :• And the input and output are :

( , ) ( ) ( )x y x yδ δ δ=[ , ] [ ] [ ]m n m nδ δ δ=

1 2 1 2[ [ , ] [ , ]] [ , ] [ , ]T af m n bf m n aTf m n bTf m n+ = +

' '

' '

[ , ] [ ', '] [ ', ']

[ , ] [ , ] [ ', '] [ ', ']m n

m n

f m n f m n m m n n

g m n Tf m n f m n T m m n n

δ

δ

= − −

= = − −

∑∑

∑∑

• Shift invariance ::

•

For a 2-D linear shift invariant system with input and impulse response ,the output is given by

0 0

' '

' '

[ ] [ ][ ] [ ][ , ] [ , ] [ ', '] [ , , ', ']

[ ', '] [ ', ']m n

m n

f n g nf n n g n ng m n Tf m n f m n h m n m n

f m n h m m n n

∑ ∑

∑ ∑

→− → −

= =

= − −

],[ nmf

],[ nmh ],[ nmg

[ , ] [ , ]* [ , ]g m n h m n f m n=

• 2D convolution involves:1. Rotate by to get

2. Shift the origin of to

3. Multiply/overlap elements and sum up.

• 2D convolution can be similarly defined in the continuous domain through

1 1

2 2

Suppose [ , ] is defined for 0,1, . . . . 1 and 0,1, . . . . , 1and [ , ] is defined for 0,1, . . . . , 1 and 0,1, . . . . , 1Then [ , ] is defined for 0,1,

h m n m M n Nf m n m M n N

g m n m

= − = −= − = −= 1 2 1 2. . . . , ( 2) and 0,1, . . . . ,( 2) M M n N N+ − = + −

[ ],h m n′ ′ 180 [ ],h m n′ ′− −

[ ],h m n′ ′− − [ , ]m n

( , )* ( , )

( , ) ( , ) ,

f x y h x y

f x y h x x y y dx dy∞ ∞

−∞ −∞

′ ′ ′ ′ ′ ′= − −∫ ∫

Illustration of convolution of x[m,n] and h[m,n]

Illustration of convolution of x[m,n] and h[m,n] (contd)

[m,n]

Example

[0 0] 0[1 0] 2[2 1] 12

g ,g ,g ,

=== and so on…

Causality

The concept of causality is also extended. Particularly important is the non-symmetrical half-plane (NSHP) model.

For a causal system, present output depends on present and past inputs. Other wise, the system is non-causal.

[m,n]

Lectures on

WAVELET TRANSFORM

OUTLINE

FT, STFT, WS & DWT

Multi-Resolution Analysis (MRA)

Perfect Reconstruction Filter Banks

Filter Bank Implementation of DWT

Extension to 2D Case (Image)

Applications in Denoising, Compression etc.,

Fourier Transform

Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies. a serious drawback In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place.

dtetfF tj∫∞

∞−

−= ωω )()(

FT – Sine wave –two frequencies stationary

Peaks corresponding to 5 Hz and 10 Hz

1( ) 0.25sin100 sin 200f t t tπ π= +

FT – Sine wave – two frequencies- nonstationary

Peaks corresponding to 50 Hz and 100 Hz

2sin 200 50

( )0.25sin100 sin 200 otherwise

t tf t

t tπ

π π<⎧

= ⎨ +⎩

The Short Time Fourier Transform Gabor

( , ) ( ) ( ) j tSTFT

tF f t w t e dtωτ ω τ∫

−= −

Time parameter Frequency parameter

Window function centered at τ

FT – Sine wave – two frequencies- nonstationary

Short Time Fourier Transform (STFT)

Take FT of segmented consecutive pieces of a signal.Each FT then provides the spectral content of that time segment onDifficulty is in selecting time window.

NOTELow frequency signal better resolved in frequency domainHigh frequency signal better resolved in time domain

Uncertainty Theorem

Uncertainty Theorem - We cannot calculate frequency and time of a signal with absolute certainty (Similar to Heisenberg’s uncertainty principle involving momentum and velocity of a particle).

In FT we use the basis which has infinite support and infinite energy.

In wavelet transform we have to localize both in time domain (through translation of basis function) and in frequency domain (through scaling).

Example

The Wavelet Transform

Analysis windows of different lengths are used for different frequencies:Analysis of high frequencies Use narrower windows for better time resolutionAnalysis of low frequencies Use wider windows for better frequency resolutionHeisenberg principle still holds!!!The function used to window the signal is called the wavelet

Mother Wavelet

Shannon Wavelet

sin(2 ) sin( )( ) x xxx

π πψπ−

=

In wavelet we have a mother wavelet as the basic unit.

Daubechies Haar

Translation and Scaling

( )f x ( )f x b−

b

Translation( )f ax

1( )

| | | |f ax F

a a

ω↔

⎛ ⎞⎜ ⎟⎝ ⎠

ScalingMother Wavelet

1( ) translated to b and scaled by ,

x bx aa b a a

−Ψ = Ψ →

⎛ ⎞⎜ ⎟⎝ ⎠

Continuous Wavelet Transform

∫∞

∞−Ψ=Ψ= )(),( )( )( ,,, xxfdxxxfW bababa

( ) ∫ ∫∞

∞−

∞

∞−Ψ

Ψ= 2,,.)( 1

adbdaxW

Cxf baba

2( )FC dω

ωω

∞

Ψ − ∞

Ψ= ∫

Where energy is

and ( ) of ( )F FT xωΨ = Ψ

Admissibility Criterion

Requirement is =>

1) term of the mother wavelet must be zero.

2) Ψ(x) should be of finite energy => Ψ(x) should have finite support (asymptotically decaying signal).

3) spectrum should be concentrated in a narrow band.

∞<ΨC

(0) 0F DCΨ = ⇒

2

FΨ ( )dω ω∫ < ∞ ⇒

Wavelet series expansion

,1( )s

ttssττ−⎛ ⎞Ψ = Ψ⎜ ⎟

⎝ ⎠

m mo o owhere scale s s and translation n sτ τ− −= =

If scale and translation take place in discrete steps

/ 2, ( ) ( )m m

m n o o ot s s t nτΨ = Ψ −

Discrete wavelet transform

Dyadic wavelet

/ 2,

If 2 & 1

( ) 2 (2 )o o

m mm n

s

t t n

τ= =

Ψ = Ψ −

can be represented as a wavelet series iff( )f t

22 2,

,( ), ( )m n

m nA f f t w t B f∑≤ ≤

where A and B are positive constants independent of ( )f t

Such a family of discrete wavelet functions is called a frame

When the wavelets form a family of orthogonalbasis functions

,A B=

can be represented as a series combination of wavelets( )f t

, ,( ) ( )m n m nm n

f t w t∑∑= Ψ

, ,

/20

( ), ( )

( ) ( )

m n m n

m mo o

t

W f t w t

s f t s t n dtτ∫

=

= Ψ −

/ 2, ( ) 2 (2 )m m

m n t t nΨ = Ψ −

where

and

Discrete Wavelet transform

Multi resolution analysis (MRA)A scaling function φ(t) is introduced

Eg: Haar scaling function is given by

( ) 1 0 1 0 elsewhere

t tφ = ≤ ≤=

( )tφ

0 1

1

This scaling function is also to be scaled and translated

to generate a family of scaling functions/ 2

, ( ) 2 (2 )j jj k t t kφ φ= −

A function can be generated using the basis set of

translated scaling functions.

( )f t

,( ) ( )k j kk

f t a t kφ∑= −

In the case of Haar basis, comprises of all piecewise

continuous functions.

( )f t

This set of functions is called span of { }, ( ),j k t k Zφ ∈

Let this space be denoted by Vj

Requirements of MRA

Requirement 1

The scaling functions should be orthogonal with respect to their integral translates

, ,

, ,

( ), ( ) 0

( ), ( ) 0 j k j l

j k j lt

t t l k

t t dt l k

φ φ

φ φ∫

= ≠

= ≠

• Haar basis function is orthogonal

Requirement 2

1 1 2... ....

OV V V V−⊂ ⊂ ⊂ ⊂

21

1,0 ( )tφ2

The Haar scaling function at resolution 0

0,0 1,0 1,01 1 1( ) ( ) ( )

22 21 1( ) 2 (2 ) 2 (2 1)2 2

t x t

t t t

φ φ φ

φ φ φ

= + −

∴ = + −

scaling functions at low scale is nested with in the subspace spanned by the higher scale. In general, we can write

basis function at lower scale

basis function at the higher scale

( ) 2 (2 )kk

t h t kφ φ∑= −

Similarly,

( ) 2 (2 )kk

t b t kψ φ∑= −

Requirement 2(contd..)

11 1, 2 2

0 1

o

k

h h

h k

= =

= >

2

1

For example

Haar

wavelet

otherwise

khk

0

...2,1,0 ,22

1

=

==

Triangular wavelet

Requirement 30 01 V ... V V V ... C−∞ ∞⊂ ⊂ ⊂ ⊂ ⊂ ⊂

{ } function zero 0=∞V

Requirement 4All square integrable functions can be represented with arbitrary precision.In particular

V0

V1

V2

001 WVV ⊕= 0W

1W2( )V L R∞ =

2 1 1 0 0 1V V W V W W= ⊕ = ⊕ ⊕and so on. Here denotes the direct sum.⊕

Thus at a scale, the function can be represented by a scale part and a number of wavelet parts.

MRA equation or Dilation equation

( ) 2 (2 )kk

t h t kφ φ∑= −

( ) 2 (2 )kk

t b t kψ φ∑= −

For scaling function,

For the wavelet bases,

MRA and DWT

1

1

0

φ(t)φ1,0 (t)

12

2

( )1,0

1 0, 0, 1 1

1( ) ( ) ( )2

( ) ( ) ( ) ( )k k k kk k

t t t

f t C t d t f t V

φ φ ψ

φ ψ∑ ∑

= +

= + ∈

Scaling part + wavelet part

Low-pass part + high-pass part

Ψ(t)

0

12 1

1

MRA and DWT ( Contd.)

1W

0W

V0

V1

V2

1 12 1, 1, 2 2

0 0 11, 1, 1,

( ) ( ) ( ), ( )

( ) ( ) ( )

k k k kk k

k k k k k kk k k

f t C t d t f t V

C t d t d t

φ ψ

φ ψ ψ

∑ ∑

∑ ∑ ∑

= + ∈

= + +

Similarly,

and so on

How to find those c and d coefficients?

We have to learn a bit of filterbank theory to have an answer.

Perfect Reconstruction bank

[ ]f n

[ ]0

LPFh n

[ ]1

HPFh n

2↓

2↓

[ ]0g n2↑

2↑ [ ]1g n [ ]f̂ n

Analysis filter bank Synthesis filter bank

Note that

2↓

2↑( )X z

1( ) ( ( ) ( ))2

Y z X z X z= + −

On the synthesis side

To avoid aliasing, can be selected by a simple

relationship with

0 1[ ]and [ ]g n g n

[ ] [ ]0 1and .h n h n

0 1 01

0 1 0

( ) ( ), ( ) ( )

[ ] [ ], [ ] ( 1) [ ]o

no

G z H z G z H z

g n h n g n h n+

= = − −

= = −

Orthonormal filters

A class of perfect reconstruction filters needed for the filter

bank implementation of discrete wavelet transform (DWT)

1 0 [ ] ( 1) [ 1 ]nh n h N n≡ − − −

These filters satisfy the relation

where is the tap length required to be evenN

The synthesis filters are given by

[ ] [ ] {0,1}i ig n h n i= − ∈

Orthonormal filter banks

[ ]f n

0[ ]h n

1[ ]h n

2↓

2↓

[ ]0h n−2↑

2↑ [ ]1h n− [ ]f̂ n

Analysis filter bank Synthesis filter bank

is the original signal

is the reconstructed signal

[ ]f n

[ ]f̂ n

Filter bank implementation of DWT

The MRA equation (or) dilation equation for the scaling function is

( ) [ ] 2 (2 )k

t h k t kφφ φ∑= −

By slightly modifying the notations, we have

, , , ,

( ) [ ] 2 (2 )

( ) [ ] 2 (2 )

( ) ( ) ( )

k

k

j k j k j k j kk k

t h k t k

t h k t k

f t c t d t

φφ φ

φ

Ψ

= −

Ψ = Ψ −

= + Ψ

∑

∑

∑ ∑

, ,( ), ( )j k j kt tφ Ψ are orthogonal individually and to each

otherFurther, orthonormality is assumed

,j kcThe co-efficient can be given as the convolution of

1, and [ ]j kc h kφ+ − at alternate points

, 1,[ 2 ] j k j ll

c h l k cφ += −∑

, 1,[ 2 ]j k j kl

d h l k CΨ += −∑Similarly

is an anti-causal filter, but it is not a serious issue

and can be addressed by proper translation of the function.

[ ]h kφ −

Contd…Using the orthogonality

of the scaling and the wavelet bases

[ ]h kφ − 2↓ 1,j kC +

,j kC

How to get the filter co-efficients?Integrating the dilation equation on both sides

( ) 2 [ ] (2 ) 2 -

1 [ ] ( )2

[ ] 2 ------ (1)

k

k

k

t dt h k t k dt t k u

h k u du

h k

φ

φ

φ

φ φ

φ

∞ ∞

−∞ −∞

∞

−∞

= − =

=

⇒ =

∑∫ ∫

∑ ∫

∑

2

2

( ) 1

[ ] 1 (2)k

x dx

h kφ

φ∞

−∞

=

⇒ = − − − − −

∫

∑

Similarly

Due to the orthogonality of the scaling function and its integer

translates, we have

( ) ( ) [ ]x x m dx mφ φ δ∞

−∞

− =∫

[ ] [ 2 ] [ ]k

h k h k m mφ φ δ− =∑ ------- (3)

If dilation equation is applied, we get

Considering the orthogonality of scaling function and wavelet

functions at a particular scale

[ ] ( 1) [ ]kh k h N kφΨ = − −

Hence, form perfect reconstruction

orthonormal

filter banks

[ ] and [ ]h k h kφ Ψ

Filter bank representation for 2-tap filter

2 2

[0] [1] 2

& [0] [1] 1

h h

h hφ φ

φ φ

+ =

+ =

The unique solution for these equations s

0

1

1[0] [1] 2

1[0] ( 1) [1]21[1] ( 1) [0]2

h h

h h

h h

φ φ

φ

φ

Ψ

Ψ

= =

= − =

= − = −

There will be no filter with odd tap length

The equations representing the filter co-efficients are

4-tap wavelet (Daubachies wavelet )3

0

32

0

[ ] 2

[ ] 1

[0] [2] [1] [3] 0

k

k

h k

h k

h h h h

φ

φ

φ φ φ φ

=

=

=

=

+ =

∑

∑

For a 4-tap filter,

With these 3 equations, Daubachies

wavelets can be generated

1 3 3 3[0] [1]4 2 4 2

3 3 1 3[2] [3]4 2 4 2

h h

h h

φ φ

φ φ

+ += =

+ −= =

Where to start and where to stop?

The raw data is considered as a scaled version of data

at infinite resolution

V0

V1

V2

0W

1W

The process of approximation from highest resolution can be

explained as shown by the following figure

Lowest resolution

Detail 2

Highest resolution

Detail 1[ ]f k

[ ]h kφ −

[ ]h kΨ −

2↓

2↓

[ ]h kφ −

[ ]h kΨ −

2↓

2↓

Approximate

Reconstruction

Synthesis filter banks can be applied to get back the original signal

Approximation

Reconstructed

signal

+

+Detail - 1

Detail - 2

2

2

2

Processing

Processing

Processing

[ ]g kφ

[ ]g kψ

[ ]g kψ

[ ]g kφ2

2D Case

For 2 dimensional case, separability property enables the use

of 1D filters

1 2 1 1 2 2( , ) ( ) ( )t t t tΨ = Ψ Ψ

The corresponding filters can be first applied in first dimension

and then in other dimension.

First, the LPF and HPF operations are done row wise and then

column wise. This can be explained with the following figure

[ ],f m n

LP 2↓row

HP 2↓row

LP

HP

2↓

2↓

column

column

LL-scaling co-eff

LH-scaling co-eff

LP

HP

2↓

2↓

column

column

HL-scaling co-eff

HH-scaling co-eff

LL LH

HL HH

Original image

Co-efficient of diagonal detail at level 1

Co-efficient of vertical detail at level 1

Co-efficient of horizontal detail at level 1

Co-efficient of approximation at level 1

Decomposition at level 1 Decomposition at level 2

Original image