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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 • 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:
Properties • A is bounded: Let • A is linear and bounded is continuous Adjoint of a convolution operator
Adjoint of convolution operator (cont.) since or has isolated zeros Inverse of a convolution operator as is not bounded is defined only if
Bandlimited convolution operators/systems is bandlimited with band B, i.e., are orthogonal
Convolution of Bandlimited 2D Signals Approximate using periodic sequences
From Continuous to Discrete Representation Let Assume that is N-periodic sequences such that Let Discrete Fourier Transform (DFT)
Fast Fourier Transform (FFT) Efficient algorithm to compute When N is a power of 2
Vector Space Perspective Let vectors defined in Euclidian vector space with inner product Parseval generalized equality Basis
2D Periodic Convolution 2D N-periodic signals (images) Periodic convolution DFT of a convolution Hadamard product
Spectral Representation of 2D Periodic Signals Can be represented as a block cyclic matrix Spectral Representation of A eingenvalues of A
Adjoint operator Operator
Inverse operator Let
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)
Example 1: Linear Motion Blur lens plane , then Let a(t)=ct for
Example 2: Out of Focus Blur lens plane Circle of confusion COC Geometrical optics zeros
Deconvolution of Linear Motion Blur (Tikhonov regularization) Wiener filter Assuming that D is cyclic convolution operator Regularization filter
Deconvolution of Linear Motion Blur (Tikhonov regularization) Regularization filter Effect of the regularization filter is a frequency selective threshold
Deconvolution of Linear Motion Blur (Total Variation ) Iterative Denoising algorithm solves the denoising optimization problem where
Deconvolution of Linear Motion Blur Tikhonov (D=I) TFD TV