UMCP ENEE631 Slides (created by M.Wu © 2004)
Based on ENEE631 Spring’04
Section 6
Basics 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]
ENEE631 Digital Image Processing (Spring'04)
UMCP ENEE631 Slides (created by M.Wu © 2004)
2-D Random Signals
 Side-by-Side Comparison with 1-D Random Process
(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
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [3]
UMCP ENEE631 Slides (created by M.Wu © 2004)
Statistical 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
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [4]
Characterize the Ensemble of 2-D Signals
UMCP ENEE631 Slides (created by M.Wu © 2004)

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
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [5]
Discrete Random Field
UMCP ENEE631 Slides (created by M.Wu © 2004)

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
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [6]
UMCP ENEE631 Slides (created by M.Wu © 2004)
Special 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. of
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [7]
Properties of Covariance for Random Field
UMCP ENEE631 Slides (created by M.Wu © 2004)
[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-negativity
For x(m,n)  0 at all (m,n):
mnm’n’ x(m, n) ru( m, n; m’, n’) x*(m’, n’)  0
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [9]
Separable Covariance Functions
UMCP ENEE631 Slides (created by M.Wu © 2004)

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
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [11]
Isotropic Covariance Functions
UMCP ENEE631 Slides (created by M.Wu © 2004)

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|)
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [13]
UMCP ENEE631 Slides (created by M.Wu © 2004)
Estimating the Mean and Covariance Function

Approximate the ensemble average with sample average

Example: for an M x N real-valued image x(m, n)
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [14]
Spectral Density Function
UMCP ENEE631 Slides (created by M.Wu © 2004)

The Spectral density function (SDF) is defined as the
Fourier transform of the covariance function rx

S (1 , 2 ) 


  r (m, n)  exp[  j ( m   n)]
m   n  
x
1
2
– 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)
S (1 , 2 ) 



2
2


(
m
,
n
)

exp[

j
(

m


n
)]



1
2
m   n  
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [15]
Properties of Power Spectrum
UMCP ENEE631 Slides (created by M.Wu © 2004)
[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)
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [17]
UMCP ENEE631 Slides (created by M.Wu © 2004)
Z-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

S ( z1 , z 2 ) 


  r (m, n) z
m   n  
m n
1
2
x
z

S (1 , 2 )  S ( z1 , z2 ) | z e j1 , z
1
ENEE631 Digital Image Processing (Spring'04)
j2

e
2
Lec6 – 2-D Random Signal [18]
2-D Z-Transform
UMCP ENEE631 Slides (created by M.Wu © 2004)

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 include
unit surface

Transfer function of 2-D discrete LSI system
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [20]
Stability
UMCP ENEE631 Slides (created by M.Wu © 2004)

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
ENEE631 Digital Image Processing (Spring'04)
Lec6 – 2-D Random Signal [21]