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

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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

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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Sampling 2D signals

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Sampling 2D signals

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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 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 1: Linear motion blur

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Example 1: Linear Motion Blur

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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

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

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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 Blur

TFD Tikhonov (D=I)

TV