ENEE631 Digital Image Processing (Spring'04) Basics on 2-D Random Signal Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland, College Park

ENEE631 Digital Image Processing (Spring'04)

Basics on 2-D Random SignalBasics on 2-D Random Signal

Spring ’04 Instructor: Min Wu

ECE Department, Univ. of Maryland, College Park

www.ajconline.umd.edu (select ENEE631 S’04) [email protected]

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Based on ENEE631 Based on ENEE631 Spring’04Spring’04Section 6Section 6

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [3]

2-D Random Signals2-D Random Signals

Side-by-Side Comparison with 1-D Random Process

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(1) Sequences of random variables & joint distributions(2) First two moment functions and their properties (3) Wide-sense stationarity(4) Unique to 2-D case: separable and isotropic covariance function(5) Power spectral density and properties


Statistical Representation of ImagesStatistical Representation of Images

Each pixel is considered as a random variable (r.v.)

Relations between pixels

– Simplest case: i.i.d.– More realistically, the color value at a pixel may be statistically

related to the colors of its neighbors

A “sample” image

– A specific image we have obtained to study can be considered as a sample from an ensemble of images

– The ensemble represents all possible value combinations of random variable array

Similar ensemble concept for 2-D random noise signals

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Characterize the Ensemble of 2-D SignalsCharacterize the Ensemble of 2-D Signals Specify by a joint probability distribution function

– Difficult to measure and specify the joint distribution for images of practical size=> too many r.v. : e.g. 512 x 512 = 262,144

Specify by the first few moments– Mean (1st moment) and Covariance (2nd moment)

may still be non-trivial to measure for the entire image size

By various stochastic models– Use a few parameters to describe the relations among all pixels

E.g. 2-D extensions from 1-D Autoregressive (AR) model

Important for a variety of image processing tasks– image compression, enhancement, restoration, understanding, …

=> Today: some basics on 2-D random signals

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Discrete Random FieldDiscrete Random Field We call a 2-D sequence discrete random field if each of its

elements is a random variable

– when the random field represents an ensemble of images, we often call it a random image

Mean and Covariance of a complex random field

E[u(m,n)] = (m,n)Cov[u(m,n), u(m’,n’)] = E[(u(m,n) – (m,n)) (u(m’,n’) – (m’,n’))*] = ru( m, n; m’, n’)

For zero-mean random field, autocorrelation function = cov. function

Wide-sense stationary (m,n) = = constant

ru( m, n; m’, n’) = ru( m – m’, n – n’; 0, 0) = r( m – m’, n – n’ ) also called shift invariant, spatial invariant in some literature

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Special Random FieldsSpecial Random Fields

White noise field

– A stationary random field– Any two elements at different locations x(m,n) and x(m’,n’) are

mutually uncorrelated

rx( m – m’, n – n’) = x2

( m, n ) ( m – m’, n – n’ )

Gaussian random field

– Every segment defined on an arbitrary finite grid is Gaussian i.e. every finite segment of u(m,n) when mapped into a vector

have a joint Gaussian p.d.f. ofUM

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Properties of Covariance for Random FieldProperties of Covariance for Random Field

[Similar to the properties of covariance function for 1-D random process]

Symmetry

ru( m, n; m’, n’) = ru*( m’, n’; m, n)

• For stationary random field: r( m, n ) = r*( -m, -n )• For stationary real random field: r( m, n ) = r( -m, -n )

• Note in general ru( m, n; m’, n’) ru( m’, n; m, n’) ru( m’, n; m, n’)

Non-negativityFor x(m,n) 0 at all (m,n): mnm’n’ x(m, n) ru( m, n; m’, n’) x*(m’, n’) 0

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Separable Covariance FunctionsSeparable Covariance Functions

Separable

– If the covariance function of a random field can be expressed as a product of covariance functions of 1-D sequences

r( m, n; m’, n’) = r1( m, m’) r2( n, n’) Nonstationary case

r( m, n ) = r1( m ) r2( n ) Stationary case

Example:

– A separable stationary cov function often used in image proc r(m, n) = 2

1|m|

2|n| , |1|<1 and |2|<1

2 represents the variance of the random field; 1

and 2 are the one-step correlations in the m and n directions

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Isotropic Covariance FunctionsIsotropic Covariance Functions

Isotropic / circularly symmetric

– i.e. the covariance function only changes with respect to the radius (the distance to the origin), and isn’t affected by the angle

Example

– A nonseparable exponential function used as a more realistic cov function for images

– When a1= a2 = a2 , this becomes isotropic: r(m, n) = 2 d

As a function of the Euclidean distance of d = ( m 2 + n 2 ) 1/2

= exp(-|a|)

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Estimating the Mean and Covariance FunctionEstimating the Mean and Covariance Function

Approximate the ensemble average with sample average

Example: for an M x N real-valued image x(m, n)

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Spectral Density FunctionSpectral Density Function The Spectral density function (SDF) is defined as the

Fourier transform of the covariance function rx

– Also known as the power spectral density (p.s.d.)( in some text, p.s.d. is defined as the FT of autocorrelation

function )

Example: SDF of stationary white noise field with r(m,n)= 2

(m,n)

m n

x nmjnmrS )](exp[),(),( 2121

221

221 )](exp[),(),(

m n

nmjnmS

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Properties of Power SpectrumProperties of Power Spectrum

[Recall similar properties in 1-D random process]

SDF is real: S(1, 2) = S*(1, 2) – Follows the conjugate symmetry of the covariance function

r(m, n) = r *(-m, -n)

SDF is nonnegative: S(1, 2) 0 for 1,2

– Follows the non-negativity property of covariance function– Intuition: “power” cannot be negative

SDF of the output from a LSI system w/ freq response H(1, 2)

Sy(1, 2) = | H(1, 2) |2 Sx(1, 2)

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Z-Transform Expression of Power SpectrumZ-Transform Expression of Power Spectrum

The Z transform of ru

– Known as the covariance generating function (CGF) or the ZT expression of the power spectrum

22

11 ,2121

2121

|),(),(

),(),(

jj ezez

m n

nmx

zzSS

zznmrzzS

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2-D Z-Transform2-D Z-Transform

The 2-D Z-transform is defined by

– The space represented by the complex variable pair (z1, z2) is 4-D

Unit surface

– If ROC includeunit surface

Transfer function of 2-D discrete LSI system

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StabilityStability

Recall for 1-D LTI system

– Stability condition in bounded-input bounded-output sense (BIBO) is that the impulse response h[n] is absolutely summable

i.e. ROC of H(z) includes the unit circle

– The ROC of H(z) for a causal and stable system should have all poles inside the unit circle

2-D Stable LSI system

– Requires the 2-D impulse response is absolutely summable

– i.e. ROC of H(z1, z2) must include the unit surface |z1|=1, |z2|=1

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Documents

ENEE631 Digital Image Processing (Spring'04) Basics on 2-D Random Signal Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland, College Park