Linear Shift Invariant Systems (LSI)

Transcription

1 Linear Shift Invariant Systems (LSI) Baba C. Professor, Department of Computer & Information Science and Engineering University of Florida

2 Linear Shift Invariant Systems 1 Image Processing (Chapter 6 from BKP Horn)

3 Continuous Images Key task in Image Processing: Transform an image to a form more amenable to further manipulation. Imaging systems or image formation systems can be approximated by Linear Shift Invariant Systems (LSI), a powerful analytic tool. Consider a 2D system with inputs f 1 (x, y) and f 2 (x, y) and outputs g 1 (x, y) and g 2 (x, y).

4 System is linear if Input = αf 1 + βf 2 and Output = αg 1 + βg 2. Which of the following systems are linear? (a) g(x) = e π f (x), (b) g(x) = f (x) + 1, (c) g(x) = xf (x), (d) g(x) = (f (x)) 2 Shift Invariant: Input = f (x a, y b), Output = g(x a, y b), for arbitrary a, b. In practice, images are limited in area and so, shift invariance holds only for limited areas.

5 Why are LSIs important? Can model image formation systems using LSIs System shortcomings can be discussed in terms of the LSI model. Study of LSI leads to useful algorithms for processing images digitally or even optically.

6 Convolution and Point Spread Functions Consider a system with f (x, y) as input that produces, g(x, y) = f (x ξ, y η)h(ξ, η)dξdη (1) i.e. g out = f in h in (convolution integral) (2) This system is an LSI system: Input : αf 1 + βf 2 ; Output : αg 1 + βg 2 Input : f (x a, y b); Output : g(x a, y b) Thus, a system whose response is described by a convolution, is an LSI and conversely, any LSI system performs a convolution.

7 Convolution Example Figure: Convolution Example

8 Impulse Functions Question: Given an arbitrary function h(x, y), can we find a function f (x, y) that causes the output to be h(x, y) i.e., h(x, y) = f (x ξ, y η)h(ξ, η)dξdη? (3) Ans: Yes; called an impulse function. δ(x, y) = 0 if (x, y) 0 δ(x, y) = 1

9 Impulse Function Definition & Properties Figure: Impulse Function Further, δ(x, y) f (x, y) = f (x, y) δ(x, y) = f (x, y). δ(ax, by) = (1/ ab )δ(x, y)

10 Sampling and Replicating Property Consider an infinite sequence of impulses denoted by the comb or shah symbol, Properties: 1/2 1/2 Ш(x, y) = δ(x n, y m) (4) Ш(ax, by) = 1 ab δ(x n/a, y m/b) (5) Ш( x, y) = Ш(x, y) (6) Ш(x, y)dxdy = 1 because its periodic with unit period (7)

11 Figure: Sampling a function Multiplication of a function by Ш(x, y) effectively samples it at unit intervals. Ш(x, y)f (x, y) = f (n, m)δ(x n, y m) (8)

12 Ш(x, y) f (x, y) = f (x n, y m) Seaparability: δ(x, y) = δ(x)δ(y), similarly, Ш(x, y) = Ш(x)Ш(y). Derivative of the Impulse function: We use the limit definition of the impulse function and take the derivative of the function whose limit defines the impulse function. Then take the limit. Derivative Sifting Property: δ (x, y) f (x, y) = δ (x ξ, y η)f (ξ, η)dξdη = f (x, y) Similarly, δ (x, y) f (x, y) = f (x, y)

13 LSI Performs Convolution Proof: Using the sifting property of the impulse function, f (x, y) = f (ξ, η)δ(x ξ, y η)dξdη (9) Decompose f (x, y) into elementary functions and then we can determine the overall output g(x, y) by summing the shifted scaled impulses. Why? (because we can use the fact that the system is linear) Response of the LSI system to αδ(x ξ, y η) is αh(x ξ, y η) as the system is shift invariant. But α = f (ξ, η). Hence, g(x, y) = f (ξ, η)h(x ξ, y η)dξdη (10) Which is the same as g(x, y) = f (x, y) h(x, y). QED.

14 Properties of Convolution Figure: Convolution Properties Convolution is commutative: f g = g f. Convolution is associative: (f g) h = f (g h). Can cascade systems etc. Reading Assignment: read section 6.3 on MTFs.