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
ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [4]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [5]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [6]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [7]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [9]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [11]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [13]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [14]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [15]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [17]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [18]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [20]
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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ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [21]
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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