ENEE631 Digital Image Processing (Spring'04) Basics on 2-D Random Signal Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland, College Park   (select ENEE631 S’04)   UMCP ENEE631 Slides (created by M.Wu © 2004) Based on ENEE631 Spring’04 Section 6

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [2] Overview Last Time: –Fourier Analysis for 2-D signals –Image enhancement via spatial filtering Today –Spatial filtering (cont’d): image sharpening and edge detection –Characterize 2-D random signal (random field) UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [3] 2-D Random Signals 2-D Random Signals  Side-by-Side Comparison with 1-D Random Process UMCP ENEE631 Slides (created by M.Wu © 2004) (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 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 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [5] Characterize 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 (1 st moment) and Covariance (2 nd moment) u 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 u 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 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [6] Discrete 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’)) * ] = r u ( m, n; m’, n’) u For zero-mean random field, autocorrelation function = cov. function Wide-sense stationary  (m,n) =  = constant r u ( m, n; m’, n’) = r u ( m – m’, n – n’; 0, 0) = r ( m – m’, n – n’ ) u also called shift invariant, spatial invariant in some literature UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [7] 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 r x ( m – m’, n – n’) =  x 2 ( 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 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [8] 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 r x ( m – m’, n – n’) =  x 2 ( 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 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [9] Properties of Covariance for Random Field [Similar to the properties of covariance function for 1-D random process] Symmetry r u ( m, n; m’, n’) = r u * ( 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 r u ( m, n; m’, n’)  r u ( m’, n; m, n’)  r u ( m’, n; m, n’) Non-negativity For x(m,n)  0 at all (m,n):  m  n  m’  n’ x(m, n) r u ( m, n; m’, n’) x * (m’, n’)  0 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [10] Properties of Covariance for Random Field [Recall similar properties of covariance function for 1-D random process] Symmetry r u ( m, n; m’, n’) = r u * ( 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 r u ( m, n; m’, n’)  r u ( m’, n; m, n’)  r u ( m’, n; m, n’) Non-negativity For x(m,n)  0 at all (m,n):  m  n  m’  n’ x(m, n) r u ( m, n; m’, n’) x * (m’, n’)  0 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [11] Separable 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’) = r 1 ( m, m’) r 2 ( n, n’) Nonstationary case r( m, n ) = r 1 ( m ) r 2 ( 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 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [12] Separable 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’) = r 1 ( m, m’) r 2 ( n, n’) Nonstationary case r( m, n ) = r 1 ( m ) r 2 ( 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 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [13] Isotropic 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 a 1 = a 2 = a 2, this becomes isotropic: r(m, n) =  2  d u As a function of the Euclidean distance of d = ( m 2 + n 2 ) 1/2 u  = exp(-|a|) UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [14] 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) UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [15] Spectral Density Function The Spectral density function (SDF) is defined as the Fourier transform of the covariance function r x –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) UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [16] Spectral Density Function The Spectral density function (SDF) is defined as the Fourier transform of the covariance function r x –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) UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [17] Properties 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 ) S y (  1,  2 ) = | H(  1,  2 ) | 2 S x (  1,  2 ) UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [18] Z-Transform Expression of Power Spectrum The Z transform of r u –Known as the covariance generating function (CGF) or the ZT expression of the power spectrum UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [19] Rational Spectrum Rational Spectrum is the SDF that can be expressed as a ratio of finite polynomials in z 1 and z 2 Realize by Linear Shift-Invariant systems –LSI system represented by finite-order difference equations between the 2-D input and output UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [20] 2-D Z-Transform The 2-D Z-transform is defined by –The space represented by the complex variable pair (z 1, z 2 ) is 4-D Unit surface –If ROC include unit surface Transfer function of 2-D discrete LSI system UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [21] Stability 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 u 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(z 1, z 2 ) must include the unit surface |z 1 |=1, |z 2 |=1 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [22] Stability 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 u 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(z 1, z 2 ) must include the unit surface |z 1 |=1, |z 2 |=1 UMCP ENEE631 Slides (created by M.Wu © 2004)

ENEE631 Digital Image Processing (Spring'04) Lec6 – 2-D Random Signal [23] Summary of Today’s Lecture Spatial filter: LPF, HPF, BPF –Image sharpening –Edge detection Basics on 2-D random signals Next time –Image restoration Readings –Jain’s book 7.4; 9.4; Gonzalez’s book UMCP ENEE631 Slides (created by M.Wu © 2004)