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Convolution Operators

Convolution Operators. Spectral Representation Bandlimited Signals/Systems Inverse Operator Null and Range Spaces Sampling, DFT and FFT Tikhonov Regularization/Wiener Filtering. Convolution Operators. Definition :. FT. Spectral representation of a convolution operator :.

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Convolution Operators

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  1. Convolution Operators • Spectral Representation • Bandlimited Signals/Systems • Inverse Operator • Null and Range Spaces • Sampling, DFT and FFT • Tikhonov Regularization/Wiener Filtering

  2. Convolution Operators Definition: FT Spectral representation of a convolution operator:

  3. Properties • A is bounded: Let • A is linear and bounded is continuous Adjoint of a convolution operator

  4. Adjoint of convolution operator (cont.) since or has isolated zeros Inverse of a convolution operator as is not bounded is defined only if

  5. Bandlimited convolution operators/systems is bandlimited with band B, i.e., are orthogonal

  6. Convolution of Bandlimited 2D Signals Approximate using periodic sequences

  7. From Continuous to Discrete Representation Let Assume that is N-periodic sequences such that Let Discrete Fourier Transform (DFT)

  8. Fast Fourier Transform (FFT) Efficient algorithm to compute When N is a power of 2

  9. Vector Space Perspective Let vectors defined in Euclidian vector space with inner product Parseval generalized equality Basis

  10. 2D Periodic Convolution 2D N-periodic signals (images) Periodic convolution DFT of a convolution Hadamard product

  11. Spectral Representation of 2D Periodic Signals Can be represented as a block cyclic matrix Spectral Representation of A eingenvalues of A

  12. Adjoint operator Operator

  13. Inverse operator Let

  14. Deconvolution Examples Imaging Systems Linear Imaging System System noise + Poisson noise Impulsive Response function or Point spread function (PSF) Invariant systems Is the transfer function (TF)

  15. Example 1: Linear Motion Blur lens plane , then Let a(t)=ct for

  16. Example 1: Linear Motion Blur

  17. Example 1: Linear Motion Blur

  18. Example 2: Out of Focus Blur lens plane Circle of confusion COC Geometrical optics zeros

  19. Deconvolution of Linear Motion Blur and Let

  20. Deconvolution of Linear Motion Blur

  21. Deconvolution of Linear Motion Blur (TFD) ISNR

  22. Deconvolution of Linear Motion Blur (Tikhonov regularization) Wiener filter Assuming that D is cyclic convolution operator Regularization filter

  23. Deconvolution of Linear Motion Blur (Tikhonov regularization) Regularization filter Effect of the regularization filter is a frequency selective threshold

  24. Deconvolution of Linear Motion Blur ISNR

  25. Deconvolution of Linear Motion Blur (Total Variation ) Iterative Denoising algorithm solves the denoising optimization problem where

  26. Deconvolution of Linear Motion Blur Tikhonov (D=I) TFD TV

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