Convolution Operators

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

0

0.1

0.2

0.3

0.4

0.5

0.6

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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

-4

-2

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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

5.5

6

6.5

7

7.5

8

8.5

ISNR

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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