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Fast Algorithms for the Removal of Non-Uniform Motion Blurs James - - PowerPoint PPT Presentation

Jul 14, 2023 946 likes •1.42k views

Fast Algorithms for the Removal of Non-Uniform Motion Blurs James G. Nagy Emory University Atlanta, GA Joint work with Julianne Chung, Eldad Haber, John Oshinski Motivating Example Motion in MRI cardiac image Motivating Example Motion in

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SLIDE 1

Fast Algorithms

for the

Removal of Non-Uniform Motion Blurs

James G. Nagy Emory University Atlanta, GA Joint work with Julianne Chung, Eldad Haber, John Oshinski

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SLIDE 2

Motivating Example Motion in MRI cardiac image

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SLIDE 3

Motivating Example Motion in MRI cardiac image Restored MRI cardiac image

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SLIDE 4

Outline

  • Uniform motion blurs, matrix models
  • Non-uniform motion blurs, matrix models
  • Algorithms
  • Difficulties
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SLIDE 5

Uniform Motion Blur

True Image Horizontal Blur Vertical Blur Angled Blur

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SLIDE 6

Uniform Motion Blur

True Image Horizontal Blur Vertical Blur Angled Blur

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SLIDE 7

Uniform Motion Blur Linear model:

b = Ax + n

The matrix A is structured: Horizontal blur ⇒ A = T ⊗ I Vertical blur ⇒ A = I ⊗ T

where T =

     

1 d

. . .

1 d

... ...

1 d

· · ·

1 d

     

Angled blur ⇒ A = block Toeplitz, Toeplitz blocks

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SLIDE 8

Non-Uniform Motion Blur

True Image Non−Uniform Blur

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SLIDE 9

Non-Uniform Motion Blur

True Image Non−Uniform Blur

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SLIDE 10

Non-Uniform Motion Blur

True Image Non−Uniform Blur

We still have linear model

b = Ax + n

However, A has no obvious structure.

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SLIDE 11

Non-Uniform Motion Blur To explicitly construct A:

  • Estimate direction and magnitude of

motion at each pixel

  • Construct corresponding column of A
  • Use sparse data structure for A
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SLIDE 12

Sparsity Pattern of A

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SLIDE 13

Sparsity Pattern of A

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 nz = 223734

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SLIDE 14

Sparsity Pattern of A

100 200 300 400 500 600 700 800 900 1000 100 200 300 400 500 600 700 800 900 1000 nz = 13017

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SLIDE 15

Non-Uniform Motion Blur Problem: It may be difficult to estimate motion at every pixel

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SLIDE 16

Non-Uniform Motion Blur To approximate A:

  • Partition image into regions
  • Assume motion is uniform in each region
  • Estimate direction and magnitude of ”uniform” mo-

tion in each region

  • Use interpolation:

A ≈

  • IkAk

where Ak is defined by uniform motion in kth region and Ik is diagonal, with Ik = I

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SLIDE 17

Region Partitioning

One region

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SLIDE 18

Region Partitioning

4 regions

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SLIDE 19

Region Partitioning

9 regions

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SLIDE 20

Region Partitioning

16 regions

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SLIDE 21

Region Partitioning

25 regions

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SLIDE 22

Region Partitioning

64 regions

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SLIDE 23

Deblurring Algorithms Given A (or its approximation), need to solve

b = Ax + n

where

  • A is ill-conditioned

⇒ need regularization

  • A is large

⇒ usually need iterative method

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SLIDE 24

Example Singular Value Distribution

10 20 30 40 50 60 70 80 90 100 110 10

−18

10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

singular values Horizontal blur

10 20 30 40 50 60 70 80 90 100 110 10

−18

10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

singular values Angle blur

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SLIDE 25

Deblurring Algorithms Possible regularization methods

  • Truncated singular value decomposition (TSVD)

A = UΣV T ⇒

xtsvd =

k

  • i=1

uT

i b

σi

vi

(can use efficiently for horizontal or vertical blurs)

  • Tikhonov:

min

  • ||b − Ax||2 + µ2||Lx||2
  • Iterative: Terminate iterations early

(e.g., conjugate gradients)

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SLIDE 26

Deblurring Algorithms Example of iterative regularization ...

5 10 15 20 25 30 35 40 45 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

iteration relative error

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SLIDE 27

Numerical Examples First consider text data:

True Image Non−Uniform Blur

  • Use CGLS, iterative regularization.
  • For A,

– Approximate motion on regions – Use motion information at every pixel

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SLIDE 28

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

Blurred image

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SLIDE 29

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

1 region, 0 iterations

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SLIDE 30

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

4 region, 0 iterations

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SLIDE 31

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

16 region, 1 iterations

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SLIDE 32

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

64 region, 3 iterations

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SLIDE 33

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

256 region, 9 iterations

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SLIDE 34

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

1024 region, 15 iterations

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SLIDE 35

Numerical Examples

10 20 30 40 50 60 70 80 90 100 0.2 0.3 0.4 0.5 0.6 0.7

iteration relative error

Every pixel, 79 iterations

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Numerical Examples

2 4 6 8 10 12 14 16 18 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2

iteration relative error

Blurred image

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SLIDE 37

Numerical Examples

2 4 6 8 10 12 14 16 18 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2

iteration relative error

1 region, 1 iterations

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SLIDE 38

Numerical Examples

2 4 6 8 10 12 14 16 18 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2

iteration relative error

4 regions, 1 iteration

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SLIDE 39

Numerical Examples

2 4 6 8 10 12 14 16 18 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2

iteration relative error

16 regions, 3 iterations

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SLIDE 40

Numerical Examples

2 4 6 8 10 12 14 16 18 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2

iteration relative error

64 regions, 7 iterations

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SLIDE 41

Numerical Examples

2 4 6 8 10 12 14 16 18 20 0.08 0.1 0.12 0.14 0.16 0.18 0.2

iteration relative error

Every pixel, 14 iterations

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SLIDE 42

Computational Issues

  • Choosing a regularization parameter
  • Fast matrix-vector multiplication

should scale well with number of regions

  • Preconditioning

especially when using a lot of regions, or all pixel information

  • How to get motion information?
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SLIDE 43

Preconditioning

  • A is ill-conditioned ⇒ preconditioner is ill-conditioned
  • How to regularize preconditioner?
  • Relatively easy for circulant,

and other transform based preconditioners.

  • But, A is not well approximated by circulant matrix.
  • Alternative approach – try:

A ≈ C ⊗ D

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SLIDE 44

Preconditioning using Kronecker Products Using the model: A =

p

  • k=1

IkAk

  • Each Ak can be decomposed as:

Ak =

r

  • j=1

C(k)

j

⊗ D(k)

j

  • Therefore, our model for A is:

A =

p

  • k=1

Ik

 

r

  • j=1

C(k)

j

⊗ D(k)

j

 

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SLIDE 45

The End!

  • It is possible to efficiently implement iterative meth-
  • ds for non-uniform motion blurs.
  • More

information provides substantially better restorations.

  • However, more information results in slower conver-

gence of iterative methods. www.mathcs.emory.edu/∼nagy