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IPIM, IST, José Bioucas, 2007 1 Convolution Operators Spectral Representation Bandlimited Signals/Systems Inverse Operator Null and Range Spaces Sampling,

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Presentation on theme: "IPIM, IST, José Bioucas, 2007 1 Convolution Operators Spectral Representation Bandlimited Signals/Systems Inverse Operator Null and Range Spaces Sampling,"— Presentation transcript:

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

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

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

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

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

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

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

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

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

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

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

12 IPIM, IST, José Bioucas, Adjoint operator Operator

13 IPIM, IST, José Bioucas, Inverse operator Let

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

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

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

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

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

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

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

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

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

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

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

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


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