ipim, ist, josé bioucas, 2007 1 convolution operators spectral representation bandlimited...
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1IPIM, IST, José Bioucas, 2007
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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Convolution Operators
Definition:
Spectral representation of a convolution operator:
FT
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• A is linear and bounded
• A is bounded:
Let
is continuous
Adjoint of a convolution operator
Properties
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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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Bandlimited convolution operators/systems
is bandlimited with band B, i.e.,
are orthogonal
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Convolution of Bandlimited 2D Signals
Approximate using periodic sequences
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From Continuous to Discrete Representation
Assume that
Let
Let is N-periodic sequences such that
Discrete Fourier Transform (DFT)
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Fast Fourier Transform (FFT)
Efficient algorithm to compute
When N is a power of 2
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Vector Space Perspective
Let vectors defined in Euclidian vector space with inner product
Parseval generalized equality
Basis
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2D Periodic Convolution
2D N-periodic signals (images)
Periodic convolution
DFT of a convolution
Hadamard product
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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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Adjoint operator
Operator
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Inverse operator
Let
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Deconvolution Examples
Imaging Systems
Linear ImagingSystem
System noise + Poisson noise
Impulsive Response functionorPoint spread function (PSF)
Invariant systems
Is the transfer function (TF)
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Example 1: Linear Motion Blur
lens plane
Let a(t)=ct for , then
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Example 1: Linear Motion Blur
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Example 1: Linear Motion Blur
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Example 2: Out of Focus Blurlens plane
Circle of confusion COC
Geometrical optics
0 5 10 15 20 25 30-0.1
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zeros
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Deconvolution of Linear Motion Blur
Let and
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Deconvolution of Linear Motion Blur
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Deconvolution of Linear Motion Blur (TFD)
-4 -3 -2 -1 0 1 2 3 40
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ISNR
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Deconvolution of Linear Motion Blur (Tikhonov regularization)
Assuming that D is cyclic convolution operator
Wiener filter
Regularization filter
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Deconvolution of Linear Motion Blur (Tikhonov regularization)
Regularization filter
Effect of the regularization filter
is a frequency selective threshold
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Deconvolution of Linear Motion Blur
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
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Deconvolution of Linear Motion Blur (Total Variation )
Iterative Denoising algorithm
where solves the denoising optimization problem
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Deconvolution of Linear Motion BlurTFD Tikhonov (D=I)
TV