IP, José Bioucas Dias, IST, 2015 1
Convolution operators
Spectral Representation
Bandlimited Signals/Systems
Inverse Operator
Null and Range Spaces
Sampling, DFT, and FFT
Tikhonov Regularization/Wiener Filtering
IP, José Bioucas Dias, IST, 2015 2
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
Parseval’s Theorem
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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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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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Example 1: linear motion blur
lens plane
Let a(t)=ct for , then
target velocity
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Example 2: out of focus blur
lens plane
Circle of confusion COC
Geometrical optics
0 5 10 15 20 25 30-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
zeros
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Deconvolution of linear motion blur (TFD)
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
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0.4
0.5
0.6
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0.9
1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-8
-6
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0
2
4
6
8
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 (Total Variation )
Iterative Denoising algorithm
where solves the denoising optimization problem
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