ENEE631 Digital Image Processing (Spring'04) Signal Processing: From 1-D to 2-D (m-D) Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland,

ENEE631 Digital Image Processing (Spring'04)

Signal Processing: From 1-D to 2-D (m-D)Signal Processing: From 1-D to 2-D (m-D)

Spring ’04 Instructor: Min Wu

ECE Department, Univ. of Maryland, College Park

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

Based on ENEE631 Based on ENEE631 Spring’04Spring’04Section 4 Section 4

ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [3]

Signals and Systems: 1-D to 2-DSignals and Systems: 1-D to 2-D

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1-D and 2-D Sig. Proc: Similarity and Differences1-D and 2-D Sig. Proc: Similarity and Differences

Many signal processing concepts can be extended from 1-D to 2-D to multi-dimension

Major differences– The amount of data involved becomes several magnitude higher

Audio: CD quality 44.1K samples/second Video: DVD quality 720*480 at 30 frames/sec => 10.4 M

samples/sec

– Less complete mathematic foundations for multi-dimension SP E.g. A 1-D polynomial can be factored as a product of first-order

polynomials (as we see and use in ZT, filter design, etc) A general 2-D polynomial cannot always be factored as a product

of lower-order polynomials

– Notion of causality: Causal processing a 2-D signal: from top to bottom & left to right Causality often matters more for temporal signal than spatial

signal


2-D Signals2-D Signals

Continuously indexed vs. discretely indexed (sampled)

2-D Impulse (unit sample function)

Any 2-D discrete function can be represented as linear combination of impulses


2-D Signals (cont’d)2-D Signals (cont’d) 2-D step function

Extensions: line impulse and 1-sided step function


PeriodicityPeriodicity

x[n1,n2] is periodic with a period T1 -by- T2 if

x[n1,n2] = x[n1+T1, n2] = x[n1, n2+T2] for (n1, n2)

Example:

cos[ n1/2 + n2] is periodic with a period 4-by-2


SeparabilitySeparability

x[n1,n2] is called a separable signal if it can be expressed as

x[n1,n2] = f[n1] g[n2]

– E.g. the impulse signal is separable: [n1,n2] = [n1] [n2]

Separable signals form a special class of multi-dimensional signals

– Consider indices range: 0 n1 N11, 0 n2 N21

– A general 2-D signal x[n1,n2] has N1 N2 degrees of freedom

– A separable signal has only N1+ N2 1 degrees of freedom


2-D System2-D System A 2-D system often refers to a system that maps a 2-D input

signals to a 2-D output signal

– Such as system can be represented by y[n1,n2] = H ( x[n1,n2] ) Linearity of a system H( ): for all a, b, x1[ ], x2[ ]

H (ax1[n1,n2] + bx2[n1,n2]) = a H (x1[n1,n2]) + b H (x2[n1,n2])

Impulse response h(m,n; m’,n’) = H ( [m-m’, n-n’] ) is the output at location (m,n) in response to a unit impulse at (m’,n’)

=> Point Spread Function (PSF): impulse response for system with positive inputs & outputs (such as intensity of light in imaging system)

A linear sys can be characterized by its impulse response


Shift InvarianceShift Invariance Shift invariance: If H (x[m, n]) = y[m,n], then

H ( x[m - m0, n - n0] ) = y[m - m0, n - n0 ]

Impulse response for Linear Shift-Invariant (LSI) System

– A function of the two displacement index variable only: h(m,n; m’,n’) = h[ m-m’, n-n’]– i.e. the shape of the impluse response does not change as the input

impulse move in the (m,n) plane

I/O relation for a LSI system:

– Equal to the convolution of the input with the impulse response


2-D Convolution2-D Convolution1. Rotate the impulse response array h( , ) around the original by 180 degree2. Shift by (m, n) and overlay on the input array x(m’,n’)3. Sum up the element-wise product of the above two arrays4. The result is the output value at location (m, n)

From Jain’s book Example 2.1

Documents

ENEE631 Digital Image Processing (Spring'04) Signal Processing: From 1-D to 2-D (m-D) Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland,