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Image (and Video) Coding and Processing Lecture 2: Basic FilteringPowerPoint Presentation

Image (and Video) Coding and Processing Lecture 2: Basic Filtering

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### Image (and Video) Coding and ProcessingLecture 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

- 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

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:

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

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:

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

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!)

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

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.

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

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!

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.

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

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

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

Properties of Fourier and Z transforms

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

- Convolution:

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)

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

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

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

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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