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Linear operations on signals and images

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

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  1. Linear operations on signals and images 236327 | Signal and Image Processing by Computer Written by : MiLa & RouvenStrauss Edited by : Yael Erez

  2. Signals, images and videos ubiquitous • Need for convenient manipulation Motivation 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss

  3. Definition (Linearity): Linear shift-invariant operations DEMO Definition (Shift-invariance): 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss

  4. Definition (Linearity): Linear shift-invariant operations DEMO Definition (Shift-invariance): 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss

  5. 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

  6. Convolution Some important properties: • Commutativity • Associativity • Distributivity • Convolution & Fourier transform 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss

  7. Convolution DEMO

  8. Equivalence of LSI operations and convolution 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss

  9. 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

  10. 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

  11. Desired: Motivation for Fourier transform Let . That is, is eigenfunction of LSI operator . 236327 | Signal and Image Processing by Computer | MiLa & Rouven Strauss

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