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

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Lecture Overview

• Todays 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 zejw)
• 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 systems 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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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, Shannons 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 tt0, t1,
  , tD-1T.
• 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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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 nn0,n1,,nD-1T 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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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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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

1. Rotate the impulse response array h( ? , ? )
   around the original by 180 degree
2. Shift by (m, n) and overlay on the input array
   x(m,n)
3. Sum up the element-wise product of the above two
   arrays
4. The result is the output value at location (m, n)

From Jains 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