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

1. Image (and Video) Coding and ProcessingLecture 2: Basic Filtering Wade Trappe

2. Lecture Overview • Today’s lecture will focus on: • Review of 1-D Signals • Multidimensional signals • Fourier analysis • Multidimensional Z-transforms • Multidimensional Filters

3. 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

4. 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:

5. 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

6. 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:

7. 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

8. 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!)

9. 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

10. 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.

11. 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

12. 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!

13. 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.

14. 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

15. 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

16. 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

17. Properties of Fourier and Z transforms • Linearity • Shift: Hence, the multidimensional Z-transform is analogous to the one-dimensional delay operator • Convolution:

18. 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)

19. 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

20. 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

21. 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

22. 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