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IPIM, IST, José Bioucas, 2007 1 Convolution Operators Spectral Representation Bandlimited Signals/Systems Inverse Operator Null and Range Spaces Sampling, DFT and FFT Tikhonov Regularization/Wiener Filtering

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IPIM, IST, José Bioucas, 2007 2 Convolution Operators Definition: Spectral representation of a convolution operator: FT

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IPIM, IST, José Bioucas, 2007 3 A is linear and bounded A is bounded: Let is continuous Adjoint of a convolution operator Properties

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IPIM, IST, José Bioucas, 2007 4 Adjoint of convolution operator (cont.) since Inverse of a convolution operator or has isolated zeros as is not bounded is defined only if

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IPIM, IST, José Bioucas, 2007 5 Bandlimited convolution operators/systems is bandlimited with band B, i.e., are orthogonal

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IPIM, IST, José Bioucas, 2007 6 Convolution of Bandlimited 2D Signals Approximate using periodic sequences

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IPIM, IST, José Bioucas, 2007 7 From Continuous to Discrete Representation Assume that Let is N-periodic sequences such that Discrete Fourier Transform (DFT)

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IPIM, IST, José Bioucas, 2007 8 Fast Fourier Transform (FFT) Efficient algorithm to compute When N is a power of 2

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IPIM, IST, José Bioucas, 2007 9 Vector Space Perspective Let vectors defined in Euclidian vector space with inner product Parseval generalized equality Basis

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IPIM, IST, José Bioucas, 2007 10 2D Periodic Convolution 2D N-periodic signals (images) Periodic convolution DFT of a convolution Hadamard product

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IPIM, IST, José Bioucas, 2007 11 Spectral Representation of 2D Periodic Signals Can be represented as a block cyclic matrix Spectral Representation of A eingenvalues of A

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IPIM, IST, José Bioucas, 2007 12 Adjoint operator Operator

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IPIM, IST, José Bioucas, 2007 13 Inverse operator Let

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IPIM, IST, José Bioucas, 2007 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)

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IPIM, IST, José Bioucas, 2007 15 Example 1: Linear Motion Blur lens plane Let a(t)=ct for, then

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IPIM, IST, José Bioucas, 2007 16 Example 1: Linear Motion Blur

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IPIM, IST, José Bioucas, 2007 17 Example 1: Linear Motion Blur

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IPIM, IST, José Bioucas, 2007 18 Example 2: Out of Focus Blur lens plane Circle of confusion COC Geometrical optics zeros

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IPIM, IST, José Bioucas, 2007 19 Deconvolution of Linear Motion Blur Let and

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IPIM, IST, José Bioucas, 2007 20 Deconvolution of Linear Motion Blur

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IPIM, IST, José Bioucas, 2007 21 Deconvolution of Linear Motion Blur (TFD) ISNR

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IPIM, IST, José Bioucas, 2007 22 Deconvolution of Linear Motion Blur (Tikhonov regularization) Assuming that D is cyclic convolution operator Wiener filter Regularization filter

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IPIM, IST, José Bioucas, 2007 23 Deconvolution of Linear Motion Blur (Tikhonov regularization) Regularization filter Effect of the regularization filter is a frequency selective threshold

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IPIM, IST, José Bioucas, 2007 24 Deconvolution of Linear Motion Blur ISNR

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IPIM, IST, José Bioucas, 2007 25 Deconvolution of Linear Motion Blur (Total Variation ) Iterative Denoising algorithm where solves the denoising optimization problem

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IPIM, IST, José Bioucas, 2007 26 Deconvolution of Linear Motion Blur TFD Tikhonov (D=I) TV

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