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Image (and Video) Coding and Processing Lecture 2: Basic Filtering. Wade Trappe. Lecture Overview. Today’s lecture will focus on: Review of 1-D Signals Multidimensional signals Fourier analysis Multidimensional Z-transforms Multidimensional Filters. 1-D Discrete Time Signals.

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Image and video coding and processing lecture 2 basic filtering l.jpg

Image (and Video) Coding and ProcessingLecture 2: Basic Filtering

Wade Trappe


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Lecture Overview

  • Today’s lecture will focus on:

    • Review of 1-D Signals

    • Multidimensional signals

    • Fourier analysis

    • Multidimensional Z-transforms

    • Multidimensional Filters


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1-D Discrete Time Signals

  • A one-dimensional discrete time signal is a function x(n)

  • The Z-transform of x(n) is given by

  • The Z-transform is not guaranteed to exist because the summation may not converge for arbitrary values of z.

  • The region where the summation converges is the Region of Convergence

  • Example: If U(n) is the unit step sequence, an x(n)=anU(n), then


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1-D Discrete Time Signals, pg. 2

  • If the ROC includes the unit circle, then there is a discrete Fourier transform (found by evaluating at z=ejw):

  • The inverse transform is given by:

  • Observe: The DFT is defined in terms of radians! It is therefore periodic with period 2p!

  • Parseval/Plancherel Relationship:


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1-D Discrete Time Signals, pg. 3

  • Discrete time linear, time-invariant systems are characterized by the impulse response h(n), which define the relationship between input x(n) and output y(n)

  • This is convolution, and is expressed in the transform domain as:

  • Causality: A discrete-time system is causal if the output at time n does not depend on any future values of the input sequence. This requires that


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1-D Filters

  • The impulse response for a system is also called the system’s transfer function.

  • In general, transfer functions are of the form

  • A system is a finite impulse response (FIR) system if H(z)=A(z), i.e. we can remove the denominator B(z)

    • That is, the impulse response has a finite amount of terms.

  • An infinite impulse response system is one where H(z) has an infinite amount of non-zero terms.

  • Example of an IIR system:


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1-D Filters, pg. 2

  • A discrete-time system is said to be bounded input bounded output (BIBO stable), if every input sequence that is bounded produces an output sequence that is bounded.

  • For LTI systems, BIBO stability is equivalent to

  • Stability in terms of the poles of H(z):

    • If H(z) is rational, and h(n) is causal, then stability is equivalent to all of the poles of H(z) lying inside of the unit circle


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Sampling: From Continuity to Discrete

  • The real world is a world of continuous (analog) signals, whether it is sound or light.

  • To process signals we will need “sampled” discrete-time signals

  • Analog signals xa(t) have Fourier transform pairs

  • Let us define the sampled function x(n)=xa(nT). The Fourier transforms are related as:

  • (Note: This is a good, little homework problem… will be assigned!)


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0

2p/T

-4p/T

-2p/T

4p/T

Sampling: From Continuity to Discrete

  • The effect of the sampling in the frequency domain is essentially

    • Duplication of Xa(W) at intervals of 2p/T

    • Addition of these “copies”

  • Pictorially, we have something like the following:

  • Note: If the shifted copies overlap, then its “impossible” to recover the original signal from X(w).

1/T Xa(W)

Shifted Copies

Aliasing


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Sampling: From Continuity to Discrete, pg. 2

  • Aliasing occurs when there is overlap between the shifted copies

  • To prevent aliasing, and ensure recoverability, we can apply an “anti-aliasing” filter to ensure there is no overlap.

  • The overlap-free condition amounts to ensuring that

  • If , then we say that xa(t) is W-bandlimited.

  • As a consequence of the overlap-free condition, if we sample at a rate at least W, then we can avoid aliasing.

  • This is, essentially, Shannon’s sampling theorem.


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Multidimensional signals

  • A D-dimensional signal xa(t0,t1,…,tD-1) is a function of D real variables.

  • We will often denote this as xa(t), where the bold-faced t denotes the column vector t=[t0, t1, …, tD-1]T.

  • The subscript “a” is just used to denote the analog signal. Later, we shall use the subscript “s” to denote the sampled signal, or no subscript at all.

  • The Fourier transform of xa(t) is defined by


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Multidimensional signals, pg. 2

  • The Fourier transform is thus a scalar function of D variables.

  • The Fourier transform is (in general) complex!

  • The Inverse Fourier transform of Xa(W) is defined by

  • Define the column vector of frequencies

  • We get these relationships

Note the difference that D-dimensions introduces compared to 1-d Fourier Transform!


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Example 2D Fourier Transform

Note: Ringing artifacts, just like 1-D case when we Fourier Transform a square wave

Image example from Gonzalez-Woods 2/e online slides.


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W1

W1

W0

W0

Bandlimited Signals

  • The notion of a bandlimited multidimensional signal is a straight-forward extension of the one-dimensional case:

    • xa(t) is bandlimited if Xa(W) is zero everywhere except over a region with finite area.

Bandlimited

Not Bandlimited


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Multidimensional Sampled Signals

  • We will use n=[n0,n1,…,nD-1]T to denote an arbitrary D-dimensional vector of integer values

  • A signal x(n) is just a function of D integer values

  • The Fourier transform of x(n) and the inverse transform are given by

  • Key point: X(w) is periodic in each variable wi with period 2p


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Multidimensional Z transform

  • The Z transform of x(n) is

  • Plugging in gives X(w).

  • We will often use the notation

  • This notation will be useful later as it allows us to represent things in a way similar to the 1-dimensional Z-transform


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Properties of Fourier and Z transforms

  • Linearity

  • Shift:

    Hence, the multidimensional Z-transform is analogous to the one-dimensional delay operator

  • Convolution:


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Multidimensional Filters

  • The basic scenario for multidimensional digital filters is:

  • Convolution:

  • Here, the transfer function is

  • If x(n) has finite support, then y(n) will generally have larger support than x(n)

y(n)

x(n)

H(z)


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w1

w1

w1

w0

w0

w0

Multidimensional Filter Response

  • Just as in 1-D, the filter H can be characterized in terms of its frequency response.

  • In this case, the frequency response is

Rectangular Lowpass

Diamond Lowpass

Circular Lowpass


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w1

w1

w0

w0

Multidimensional Filters

  • Multidimensional filters can be built by applying 1-D filters to each dimension separately

  • These types of filters are separable.

  • A separable filter is one for which the frequency response can be represented as:

Rectangular Lowpass

Not Separable


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2-D Convolution, by hand…

  • Rotate the impulse response array h(  ,  ) around the original by 180 degree

  • Shift by (m, n) and overlay on the input array x(m’,n’)

  • Sum up the element-wise product of the above two arrays

  • The result is the output value at location (m, n)

From Jain’s book Example 2.1


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For Next Time…

  • Next time we will focus on multidimensional sampling.

    • This lecture will be a blackboard/whiteboard style lecture.

  • To prepare, read paper provided on website, and the discussion on lattices in the textbook


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