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Linear operations on signals and images. 236327 | Signal and Image Processing by Computer Written by : Mi La & Rouven Strauss Edited by : Yael Erez. Signals, images and videos ubiquitous Need for convenient manipulation. Motivation.
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Linear operations on signals and images 236327 | Signal and Image Processing by Computer Written by : MiLa & RouvenStrauss Edited by : Yael Erez
Signals, images and videos ubiquitous • Need for convenient manipulation Motivation 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Definition (Linearity): Linear shift-invariant operations DEMO Definition (Shift-invariance): 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Definition (Linearity): Linear shift-invariant operations DEMO Definition (Shift-invariance): 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Definition (Convolution): Convolution DEMO • Reverse signal x and shift result by t • Multiply element-wise with signal y • Integrate 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Convolution Some important properties: • Commutativity • Associativity • Distributivity • Convolution & Fourier transform 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Convolution DEMO
Equivalence of LSI operations and convolution 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Signal approximated by Linear operations and orthonormal functions Applying LSI operator yields 1. Question: Can be represented as ? 2. Question: Is still a set of orthogonal functions? 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
1. Question: Can be represented as ? 2. Question: Is still a set of orthogonal functions? Linear operations and orthonormal functions Answer: Not in general • Basis functions: • Impulse response of LSI system: 1 • 0 1 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss
Desired: Motivation for Fourier transform Let . That is, is eigenfunction of LSI operator . 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss