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M. Wu: ENEE631 Digital Image Processing (Spring'09)
Edge Detection and Edge Detection and
Basics on 2-D Random SignalBasics on 2-D Random Signal
Spring ’09 Instructor: Min Wu
Electrical and Computer Engineering Department,
University of Maryland, College Park
bb.eng.umd.edu (select ENEE631 S’09) [email protected]
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ENEE631 Spring’09ENEE631 Spring’09Lecture 6 (2/11/2009)Lecture 6 (2/11/2009)
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [2]
OverviewOverview
Last Time:– Fourier Analysis for 2-D signals– Impulse Response and Frequency Response for 2-D LSI System– Image enhancement via Spatial Filtering
Denoising by averaging filter and median filter
Today– Spatial filtering (cont’d): image sharpening and edge detection– Characterize 2-D random signal (random field)
Assignment 1 Due next Monday– See course website for submission instruction and updated image
zip files
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [4]
Review:Review:2-D Fourier Transforms2-D Fourier Transforms
Separable implementations for 2-D FT, DSFT, DFT due to separable 2-D complex exponentials
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F(x, y) F(x, y)
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2-D FT on continuous-indexed signal
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [5]
Review:Review:Freq. Response & Eigen Function for LSI SystemFreq. Response & Eigen Function for LSI System Relations between signal domain and Fourier domain
– I/O relation for LSI system: Convolve input with impulse response
y[n] = x[n] h[n] Y() = X() H()
– FT of complex exponentials: exp[j 0 n] ( 0)
Eigen function of 1-D LSI system– Output response of complex exponential input x[n]=exp[j 0n]:
Y() = H() ( 0) = H(0) ( 0)
=> y[n] = H(0) exp[j 0n] i.e. output has same signal shape as the input
with a possible change only in amplitude and phase specified by H(0)
– H() is the LSI system’s frequency response Extend to 2-D LSI system
– y[m, n] = x[m, n] h[m, n] Y(u, v) = X(u, v) H(u, v)– Eigen function is 2-D complex exponentials; Freq. response is H(u, v)
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [6]
Spatial Operations with Spatial MaskSpatial Operations with Spatial Mask
Spatial mask is 2-D finite impulse response (FIR) filter
– Usually has small support 2x2, 3x3, 5x5, 7x7
– Convolve this filter with image g(m,n) = f(m-x, n-y) h(x,y)
= f(x,y) h(m-x, n-y) … mirror w.r.t. origin, then shift & sum up
– In frequency domain: multiplying DFT(image) with DFT(filter)
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specified as the already mirrored version of the
equivalent FIR filter.
Image examples are from Gonzalez-Woods 2/e
online slides.
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [7]
Directional Smoothing Directional Smoothing
Simple spatial averaging mask blurs edges
– Improve by avoiding filtering across edges
– Restrict smoothing to along edge direction
Directional smoothing
– Compute spatial average along several directions– Take the result from the direction giving the smallest changes
before and after filtering
Other solutions
– Use more explicit edge detection and adapt filtering accordingly
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independent of directions(aka. circularly symmetric
or rotation invariant)
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [8]
Median FilteringMedian Filtering
Salt-and-Pepper noise– Isolated extreme-valued (white/black) pixels
spread randomly over the image– Spatial averaging filter may lead to blurred
output when averaging with extreme values
Median filtering
– Output the median over a small window Nonlinear operation:
Median{ x(m) + y(m) } Median{x(m)} + Median{y(m)}
– Odd window size is commonly used 3x3, 5x5, 7x7; 5-pixel “+” shaped window
– Even-sized window ~ take the average of two middle values as output
Generalize: apply order statistic operations
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [9]
Image SharpeningImage Sharpening
Use LPF to generate HPF
– Subtract a low pass filtered result from the original signal– HPF identifies the locations of a signal’s transitions
Enhance edges
I0 ILP
IHP = I0 – ILP
I1 = I0 + a*IHP
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [10]
Example of Image SharpeningExample of Image Sharpening
– v(m,n) = u(m,n) + a * g(m,n)– Often use Laplacian operator to obtain g(m,n)– Laplacian operator is a discrete form of 2nd-order derivatives
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [11]
Example of Image SharpeningExample of Image Sharpening
Original moon image is from Matlab Image Toolbox.UMCP ENEE408G Slides (created by M.Wu & R.Liu © 2002)
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [12]
Other Variations of Image SharpeningOther Variations of Image Sharpening High boost filter (Gonzalez-Woods 2/e pp132 & pp188)
I0 ILP IHP = I0 – ILP I1 = (b-1) I0 + IHP
– Equiv. to high pass filtering for b=1– Amplify or suppress original image pixel values when b2
Combine sharpening with histogram equalization
Image example is from Gonzalez-Woods 2/e online slides.
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [13]
Impulse and Frequency Responses of LPF / HPFImpulse and Frequency Responses of LPF / HPF
e.g. Gaussian LPF filter
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Image example is from Gonzalez-Woods 2/e online slides.
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [14]
Edges and Gradient VectorEdges and Gradient Vector Edge: pixel locations of abrupt luminance change
For binary image– Take black pixels with immediate white neighbors as edge pixel
Detectable by XOR operations
For continuous-tone image– How to represent edge?
by intensity + direction => Edge map ~ edge intensity + directions
– Detection Method-1: prepare edge examples (templates) of different intensities and directions, then find the best match
– Spatial luminance gradient vector of an edge pixel: edge gradient gives the direction with highest rate of luminance
changes is a vector of partial derivatives along two orthogonal directions
– Detection Method-2: measure transitions along 2 orthogonal directions
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [15]
Edge DetectionEdge Detection
Measure gradient vector– Along two orthogonal directions ~ usually horizontal and vertical
gx = L / x gy= L / y
– Magnitude of gradient vector g(m,n) 2 = gx(m,n) 2 + gy(m,n) 2
g(m,n) = |gx(m,n) | + |gy(m,n)| (preferred in hardware implement.)
– Direction of gradient vector tan –1 [ gy(m,n) / gx(m,n) ]
Characterizing edges in an image– (binary) Edge map: specify “edge point” locations with g(m,n) > thresh.– Edge intensity map: specify gradient magnitude at each pixel– Edge direction map: specify directions
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [16]
Common Gradient Operators for Edge DetectionCommon Gradient Operators for Edge Detection
– Move the operators across the image and take the inner products Magnitude of gradient vector g(m,n) 2 = gx(m,n) 2 + gy(m,n) 2
Direction of gradient vector tan –1 [ gy(m,n) / gx(m,n) ]
– Gradient operator is HPF in nature ~ could amplify noise Prewitt and Sobel operators compute horizontal and vertical
differences of local sum to reduce the effect of noise
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [17]
Derivative Operators: A Closer LookDerivative Operators: A Closer Look
Spatial averaging filter perpendicular to direction of discrete derivative
discrete derivative filter in horizontal direction
Combination of the two masks gives single “averaged gradient mask” in horizontal direction.
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [18]
Examples of Examples of
Edge DetectorsEdge Detectors– Quantize edge
intensity to 0/1: set a threshold white pixel
denotes strong edge
Roberts Prewitt Sobel
UMCP ENEE408G Slides (created by M.Wu & R.Liu © 2002)
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [20]
Robust Edge DetectorRobust Edge Detector Apply LPF to suppress noise, then apply edge detector or
derivative operations
E.g. Laplacian of Gaussian: in shape of Mexican hat
Figures from Gonzalez-Woods 2/e online slides.
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [22]
Summary: Spatial LPF, HPF, & BPFSummary: Spatial LPF, HPF, & BPF
HPF and BPF can be constructed from LPF
Low-pass filter
– Useful in noise smoothing and downsampling/upsampling
High-pass filter
– hHP(m,n) = (m,n) – hLP(m,n)
– Useful in edge extraction and image sharpening
Band-pass filter
– hBP(m,n) = hL2(m,n) – hL1(m,n)
– Useful in edge enhancement– Also good for high-pass tasks in the presence of noise
avoid amplifying high-frequency noise
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [23]
2-D Random Signals (aka Random Field)2-D Random Signals (aka Random Field)
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
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [24]
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– Each 2-D location can take a real value
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [25]
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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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [26]
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 (or wide-sense homogeneity)
(m,n) = = constant
ru( m, n; m’, n’) = ru( m – m’, n – n’; 0, 0) = r( m – m’, n – n’ ) also called shift invariant or spatial invariant in some literature
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [27]
Special Random Fields of InterestsSpecial Random Fields of Interests
White noise field
– A stationary random field– Any two elements at different locations x(m1,n1) and x(m2,n2) are
mutually uncorrelated
rx( m, n ) = x2
( m, 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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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [29]
Properties of Covariance for Random FieldProperties of Covariance for Random Field
[Recall similar 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
mnm’n’ x(m, n) ru( m, n; m’, n’) x*(m’, n’) 0
~ Recall for 1-D case, correlation matrix is non-negative definite: xH R x 0 for all x .
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Recall: ru ( m, n; m’, n’) = E[ (u(m,n) – (m,n)) (u(m’,n’) – (m’,n’))* ]
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [31]
Separable Covariance FunctionsSeparable Covariance Functions
Separability
– 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 covariance function often used in image proc for its simplicity
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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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [33]
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 often used as a more realistic model of the covariance 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|), i.e. correlation is decaying as distance increases
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [34]
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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Note: similar to the 1-D case, the cov estimates here are biased, in order to achieve smaller variance in estimation and to avoid the possible negative definiteness by unbiased estimate
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [36]
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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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [38]
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)
DSFT of cross correlation: Syx(1, 2) = H(1, 2) Sx(1, 2)
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [39]
Summary of Today’s LectureSummary of Today’s Lecture
Spatial filter: LPF, HPF, BPF
– Image sharpening and edge detection
Basics on 2-D random signals
Next time
– Continue on 2-D random field; – Image restoration
Readings
– Gonzalez’s book 3.6-3.7; 10.2; 5.2; Wood’s book 7.1 (on random field)
– For further readings: Woods’ book 6.6; 3.1, 3.2, 3.5.0;Jain’s book 7.4; 9.4; 2.9-2.11
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [40]
For Next LectureFor Next Lecture
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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
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [41]
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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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [43]
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 Region of Convergence(ROC) include unit surface
Transfer function of 2-D discrete LSI system
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M. Wu: ENEE631 Digital Image Processing (Spring'09)Lec6 – Edge Detection & 2-D
Random Signal [44]
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
– 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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