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Tomographic Image Reconstruction in Noisy and Limited Data Settings. Syed Tabish Abbas International Institute of Information Technology, Hyderabad [email protected] July 1, 2016 Tabish (IIIT-H) July 1, 2016 1 / 39
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Page 1: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Tomographic Image Reconstruction in Noisy and LimitedData Settings.

Syed Tabish Abbas

International Institute of Information Technology, Hyderabad

[email protected]

July 1, 2016

Tabish (IIIT-H) July 1, 2016 1 / 39

Page 2: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 3: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 4: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 5: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 6: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 7: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 8: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Problem Statement and Contributions

We investigate the tomographic reconstruction under 2 scenarios

- Noisy Data case.- Limited Data case.

and consider the following questions.

◦ In linear Radon transform, does reconstruction lattice play a role inquality of reconstructed image?

◦ How to reconstruct an image under limited view circular RadonTransform: the Circular arc Radon transform?

◦ How to remove the artifacts which arise in the Circular arc Radontransform due to the limited view?

Tabish (IIIT-H) July 1, 2016 2 / 39

Page 9: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction onto Hexagonal Lattices.

In linear Radon transform, does reconstruction lattice play a role in quality of reconstructed image?

Tabish (IIIT-H) July 1, 2016 3 / 39

Page 10: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Positron Emission Tomography(PET)

→ Positron Emission Tomography(PET) is aninvasive, nuclear imaging technique involvesinjecting the patient with a radioactivematerial(tracer)

→ PET imaging allows collecting metabolicinformation about different tissues.

→ Due to physics of imaging process, PETscans are very noisy.

Figure: ForwardProjection.

Tabish (IIIT-H) July 1, 2016 4 / 39

Page 11: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Positron Emission Tomography(PET)

→ Positron Emission Tomography(PET) is aninvasive, nuclear imaging technique involvesinjecting the patient with a radioactivematerial(tracer)

→ PET imaging allows collecting metabolicinformation about different tissues.

→ Due to physics of imaging process, PETscans are very noisy.

Figure: ForwardProjection.

Tabish (IIIT-H) July 1, 2016 4 / 39

Page 12: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Positron Emission Tomography(PET)

→ Positron Emission Tomography(PET) is aninvasive, nuclear imaging technique involvesinjecting the patient with a radioactivematerial(tracer)

→ PET imaging allows collecting metabolicinformation about different tissues.

→ Due to physics of imaging process, PETscans are very noisy.

Figure: ForwardProjection.

Tabish (IIIT-H) July 1, 2016 4 / 39

Page 13: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Positron Emission Tomography(PET)

→ Positron Emission Tomography(PET) is aninvasive, nuclear imaging technique involvesinjecting the patient with a radioactivematerial(tracer)

→ PET imaging allows collecting metabolicinformation about different tissues.

→ Due to physics of imaging process, PETscans are very noisy.

Figure: ForwardProjection.

Tabish (IIIT-H) July 1, 2016 4 / 39

Page 14: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

PET Image Reconstruction

PET Images are reconstructed from noisy sinogram data by essentiallyinverting the forward emission process.

An approximate inversion is achieved by high pass filtered backprojection.

Tabish (IIIT-H) July 1, 2016 5 / 39

Page 15: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

PET Image Reconstruction

PET Images are reconstructed from noisy sinogram data by essentiallyinverting the forward emission process.

An approximate inversion is achieved by high pass filtered backprojection.

Tabish (IIIT-H) July 1, 2016 5 / 39

Page 16: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

PET Image Reconstruction

PET Images are reconstructed from noisy sinogram data by essentiallyinverting the forward emission process.

An approximate inversion is achieved by high pass filtered backprojection.

Tabish (IIIT-H) July 1, 2016 5 / 39

Page 17: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Handling Noisy Data

More sophisticated methods, like algebraicinversion1, Statistical inversion2, etc. have alsobeen proposed

Other methods, follow a two step process ofreconstruction followed by denoising3.

Reconstruction onto a different lattice hasreceived very little attention.

ReconstructedImage

1

1Herman,’80, 2Fessler, ’00 3 Valiollahzadeh, ’13Tabish (IIIT-H) July 1, 2016 6 / 39

Page 18: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Handling Noisy Data

More sophisticated methods, like algebraicinversion1, Statistical inversion2, etc. have alsobeen proposed

Other methods, follow a two step process ofreconstruction followed by denoising3.

Reconstruction onto a different lattice hasreceived very little attention.

ReconstructedImage

1

1Herman,’80, 2Fessler, ’00 3 Valiollahzadeh, ’13Tabish (IIIT-H) July 1, 2016 6 / 39

Page 19: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Handling Noisy Data

More sophisticated methods, like algebraicinversion1, Statistical inversion2, etc. have alsobeen proposed

Other methods, follow a two step process ofreconstruction followed by denoising3.

Reconstruction onto a different lattice hasreceived very little attention.

ReconstructedImage

1

1Herman,’80, 2Fessler, ’00 3 Valiollahzadeh, ’13Tabish (IIIT-H) July 1, 2016 6 / 39

Page 20: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Handling Noisy Data

More sophisticated methods, like algebraicinversion1, Statistical inversion2, etc. have alsobeen proposed

Other methods, follow a two step process ofreconstruction followed by denoising3.

Reconstruction onto a different lattice hasreceived very little attention.

ReconstructedImage

1

1Herman,’80, 2Fessler, ’00 3 Valiollahzadeh, ’13Tabish (IIIT-H) July 1, 2016 6 / 39

Page 21: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Our Pipeline

We propose a 2 step reconstruction process onto Hexagonal lattice:

→ Step 1: Noisy Reconstruction using Filtered Back Projection.

→ Step 2: Denoising using a sparse dictionary learned for the noisyimage.

Tabish (IIIT-H) July 1, 2016 7 / 39

Page 22: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Our Pipeline

We propose a 2 step reconstruction process onto Hexagonal lattice:

→ Step 1: Noisy Reconstruction using Filtered Back Projection.

→ Step 2: Denoising using a sparse dictionary learned for the noisyimage.

Tabish (IIIT-H) July 1, 2016 7 / 39

Page 23: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Our Pipeline

We propose a 2 step reconstruction process onto Hexagonal lattice:

→ Step 1: Noisy Reconstruction using Filtered Back Projection.

→ Step 2: Denoising using a sparse dictionary learned for the noisyimage.

Tabish (IIIT-H) July 1, 2016 7 / 39

Page 24: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Tiling of Euclidean Plane

Figure: Square Tiling of EuclideanPlane

Figure: Hexagonal Tiling ofEuclidean Plane

3 Packing density.

3 Larger, symmetricneighbourhood

7 Irrational Coordinates

Tabish (IIIT-H) July 1, 2016 8 / 39

Page 25: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Tiling of Euclidean Plane

Figure: Square Tiling of EuclideanPlane

Figure: Hexagonal Tiling ofEuclidean Plane

3 Packing density.

3 Larger, symmetricneighbourhood

7 Irrational Coordinates

Tabish (IIIT-H) July 1, 2016 8 / 39

Page 26: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Tiling of Euclidean Plane

Figure: Square Tiling of EuclideanPlane

Figure: Hexagonal Tiling ofEuclidean Plane

3 Packing density.

3 Larger, symmetricneighbourhood

7 Irrational Coordinates

Tabish (IIIT-H) July 1, 2016 8 / 39

Page 27: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Tiling of Euclidean Plane

Figure: Square Tiling of EuclideanPlane

Figure: Hexagonal Tiling ofEuclidean Plane

3 Packing density.

3 Larger, symmetricneighbourhood

7 Irrational Coordinates

Tabish (IIIT-H) July 1, 2016 8 / 39

Page 28: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Addressing Hexagonal Lattices

→ Use base 7 indices

→ Start numbering from centerand move out spirally.

Figure: Addressing HexagonalLattice

2

2L Middleton and J Sivaswamy, 2006.

Tabish (IIIT-H) July 1, 2016 9 / 39

Page 29: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Addressing Hexagonal Lattices

→ Use base 7 indices

→ Start numbering from centerand move out spirally.

Figure: Addressing HexagonalLattice

2

2L Middleton and J Sivaswamy, 2006.

Tabish (IIIT-H) July 1, 2016 9 / 39

Page 30: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Addressing Hexagonal Lattices

→ Use base 7 indices

→ Start numbering from centerand move out spirally.

Figure: Addressing HexagonalLattice

2

2L Middleton and J Sivaswamy, 2006.

Tabish (IIIT-H) July 1, 2016 9 / 39

Page 31: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Addressing Hexagonal Lattices

→ Use base 7 indices

→ Start numbering from centerand move out spirally.

Figure: Addressing HexagonalLattice

2

2L Middleton and J Sivaswamy, 2006.

Tabish (IIIT-H) July 1, 2016 9 / 39

Page 32: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Hexagonal Patch and vectorization

Figure: Hexagonal Patch of order 2

Tabish (IIIT-H) July 1, 2016 10 / 39

Page 33: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Hexagonal Patch and vectorization

Figure: Hexagonal Patch of order 2

Tabish (IIIT-H) July 1, 2016 10 / 39

Page 34: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Our Pipeline

We propose a 2 step reconstruction process onto Hexagonal lattice:

→ Step 1: Noisy Reconstruction using Filtered Back Projection.

→ Step 2: Denoising using a sparse dictionary learned for the noisyimage.

Tabish (IIIT-H) July 1, 2016 11 / 39

Page 35: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Filtered Back-Projection (FBP)

7 Image reconstruction (especially in nuclear modalities) is very noisy.

3 Back-projection (and also other reconstruction methods) allows achoice of reconstruction lattice.

Tabish (IIIT-H) July 1, 2016 12 / 39

Page 36: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Filtered Back-Projection (FBP)

7 Image reconstruction (especially in nuclear modalities) is very noisy.

3 Back-projection (and also other reconstruction methods) allows achoice of reconstruction lattice.

Tabish (IIIT-H) July 1, 2016 12 / 39

Page 37: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Filtered Back-Projection (FBP)

7 Image reconstruction (especially in nuclear modalities) is very noisy.

3 Back-projection (and also other reconstruction methods) allows achoice of reconstruction lattice.

Tabish (IIIT-H) July 1, 2016 12 / 39

Page 38: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Our Pipeline

We propose a 2 step reconstruction process onto Hexagonal lattice:

3 Step 1: Noisy Reconstruction using Filtered Back Projection.

→ Step 2: Denoising using a sparse dictionary learned for the noisyimage.

Tabish (IIIT-H) July 1, 2016 13 / 39

Page 39: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Dictionary based denoising

Figure: Sample Dictionary atoms

→ Learn a dictionary of patchesof size 49 (a level 2 patch).

→ Use the learned dictionaryfor Denoising.

→ Dictionary is learned bysolving the followingoptimization problem.

minD∈C,α∈Rk×n

1

2‖ X−Dα ‖2

F +λ ‖ α ‖1,1

C = {D ∈ Rm×ks.t∀j = 1, ...k , ||dTj ||2 ≤ 1}

3

3M Elad and M Aharon, 2006Tabish (IIIT-H) July 1, 2016 14 / 39

Page 40: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Dictionary based denoising

Figure: Sample Dictionary atoms

→ Learn a dictionary of patchesof size 49 (a level 2 patch).

→ Use the learned dictionaryfor Denoising.

→ Dictionary is learned bysolving the followingoptimization problem.

minD∈C,α∈Rk×n

1

2‖ X−Dα ‖2

F +λ ‖ α ‖1,1

C = {D ∈ Rm×ks.t∀j = 1, ...k , ||dTj ||2 ≤ 1}

3

3M Elad and M Aharon, 2006Tabish (IIIT-H) July 1, 2016 14 / 39

Page 41: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Dictionary based denoising

Figure: Sample Dictionary atoms

→ Learn a dictionary of patchesof size 49 (a level 2 patch).

→ Use the learned dictionaryfor Denoising.

→ Dictionary is learned bysolving the followingoptimization problem.

minD∈C,α∈Rk×n

1

2‖ X−Dα ‖2

F +λ ‖ α ‖1,1

C = {D ∈ Rm×ks.t∀j = 1, ...k , ||dTj ||2 ≤ 1}

3

3M Elad and M Aharon, 2006Tabish (IIIT-H) July 1, 2016 14 / 39

Page 42: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Dictionary based denoising

Figure: Sample Dictionary atoms

→ Learn a dictionary of patchesof size 49 (a level 2 patch).

→ Use the learned dictionaryfor Denoising.

→ Dictionary is learned bysolving the followingoptimization problem.

minD∈C,α∈Rk×n

1

2‖ X−Dα ‖2

F +λ ‖ α ‖1,1

C = {D ∈ Rm×ks.t∀j = 1, ...k , ||dTj ||2 ≤ 1}

3

3M Elad and M Aharon, 2006Tabish (IIIT-H) July 1, 2016 14 / 39

Page 43: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice

Noisy Image

Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 15 / 39

Page 44: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice Noisy Image

Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 15 / 39

Page 45: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice Noisy Image Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 15 / 39

Page 46: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice Noisy Image Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 15 / 39

Page 47: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice

Noisy Image

Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 15 / 39

Page 48: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice

Noisy Image

Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 16 / 39

Page 49: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice Noisy Image

Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 16 / 39

Page 50: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice Noisy Image Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 16 / 39

Page 51: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice Noisy Image Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 16 / 39

Page 52: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Qualitative Results

Square lattice

Noisy Image

Hexagonal lattice

Tabish (IIIT-H) July 1, 2016 16 / 39

Page 53: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Quantitative Results

Figure: PSNR Comparison

Figure: Line Profile

Tabish (IIIT-H) July 1, 2016 17 / 39

Page 54: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Quantitative Results

Figure: PSNR Comparison Figure: Line Profile

Tabish (IIIT-H) July 1, 2016 17 / 39

Page 55: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work

3 We Proposed that the change of lattice can improve thereconstruction quality of PET images.

3 The change in lattice improves both the quality and fidelity of thefinal denoised image.

3 Include the noise model in the denoising step.

3 Provide an analytical explanation for the improvement inreconstruction.

Tabish (IIIT-H) July 1, 2016 18 / 39

Page 56: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work

3 We Proposed that the change of lattice can improve thereconstruction quality of PET images.

3 The change in lattice improves both the quality and fidelity of thefinal denoised image.

3 Include the noise model in the denoising step.

3 Provide an analytical explanation for the improvement inreconstruction.

Tabish (IIIT-H) July 1, 2016 18 / 39

Page 57: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work

3 We Proposed that the change of lattice can improve thereconstruction quality of PET images.

3 The change in lattice improves both the quality and fidelity of thefinal denoised image.

3 Include the noise model in the denoising step.

3 Provide an analytical explanation for the improvement inreconstruction.

Tabish (IIIT-H) July 1, 2016 18 / 39

Page 58: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work

3 We Proposed that the change of lattice can improve thereconstruction quality of PET images.

3 The change in lattice improves both the quality and fidelity of thefinal denoised image.

3 Include the noise model in the denoising step.

3 Provide an analytical explanation for the improvement inreconstruction.

Tabish (IIIT-H) July 1, 2016 18 / 39

Page 59: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction In Limited View Scenario

How to reconstruct an image under limited view circular Radon Transform: the Circular arc Radon transform?

Tabish (IIIT-H) July 1, 2016 19 / 39

Page 60: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Imaging setup

Type: Photoacoustic type sensorswhere source of excitement is EMwaves and measurement is acousticwaves.

Geometry: The sensors are assumedto be along a circle at points Pφ.

Sensor Structure: Each sensor isassumed to have a limited conicalview equal to α

Object

x axis

y axis

O

φ

α

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 20 / 39

Page 61: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Imaging setup

Type: Photoacoustic type sensorswhere source of excitement is EMwaves and measurement is acousticwaves.

Geometry: The sensors are assumedto be along a circle at points Pφ.

Sensor Structure: Each sensor isassumed to have a limited conicalview equal to α

Object

x axis

y axis

O

φ

α

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 20 / 39

Page 62: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Imaging setup

Type: Photoacoustic type sensorswhere source of excitement is EMwaves and measurement is acousticwaves.

Geometry: The sensors are assumedto be along a circle at points Pφ.

Sensor Structure: Each sensor isassumed to have a limited conicalview equal to α

Object

x axis

y axis

O

φ

α

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 20 / 39

Page 63: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Imaging setup

Type: Photoacoustic type sensorswhere source of excitement is EMwaves and measurement is acousticwaves.

Geometry: The sensors are assumedto be along a circle at points Pφ.

Sensor Structure: Each sensor isassumed to have a limited conicalview equal to α

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 20 / 39

Page 64: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Mathematial Model

We define Circular arc Radon(CAR)Transform gα of a function f as follows

gα(ρ, φ)

︸ ︷︷ ︸Measured Data

=

∫Aα(ρ,φ)

Object︷ ︸︸ ︷

f (r , θ) ds (1)

where is α is the view angle and s is thearc length measure.

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 21 / 39

Page 65: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Mathematial Model

We define Circular arc Radon(CAR)Transform gα of a function f as follows

gα(ρ, φ)

︸ ︷︷ ︸Measured Data

=

∫Aα(ρ,φ)

Object︷ ︸︸ ︷f (r , θ) ds (1)

where is α is the view angle and s is thearc length measure.

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 21 / 39

Page 66: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Mathematial Model

We define Circular arc Radon(CAR)Transform gα of a function f as follows

gα(ρ, φ)

︸ ︷︷ ︸Measured Data

=

∫Aα(ρ,φ)

Object︷ ︸︸ ︷f (r , θ) ds (1)

where is α is the view angle and s is thearc length measure.

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 21 / 39

Page 67: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Mathematial Model

We define Circular arc Radon(CAR)Transform gα of a function f as follows

gα(ρ, φ)︸ ︷︷ ︸Measured Data

=

∫Aα(ρ,φ)

Object︷ ︸︸ ︷f (r , θ) ds (1)

where is α is the view angle and s is thearc length measure.

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

Figure: Measurement Setup.

Tabish (IIIT-H) July 1, 2016 21 / 39

Page 68: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Back projection based inversion

An approximate inversion of the transform may be done using an algorithmbased on Backprojection, such that

(a) OriginalPhantom

(b) α = 5 (c) α = 17 (d) α = 21

Examples of image reconstructions using a naive Backprojection Algorithm

Tabish (IIIT-H) July 1, 2016 22 / 39

Page 69: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Back projection based inversion

An approximate inversion of the transform may be done using an algorithmbased on Backprojection, such that

f (x , y) =

2π∫0

g(ρ,√

(x − cosφ)2 + (y − sinφ)2)dφ

(a) OriginalPhantom

(b) α = 5 (c) α = 17 (d) α = 21

Examples of image reconstructions using a naive Backprojection Algorithm

Tabish (IIIT-H) July 1, 2016 22 / 39

Page 70: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Back projection based inversion

An approximate inversion of the transform may be done using an algorithmbased on Backprojection, such that

f (x , y) =

2π∫0

g(ρ,√

(x − cosφ)2 + (y − sinφ)2)dφ

(a) OriginalPhantom

(b) α = 5 (c) α = 17 (d) α = 21

Examples of image reconstructions using a naive Backprojection Algorithm

Tabish (IIIT-H) July 1, 2016 22 / 39

Page 71: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Back projection based inversion

(a) OriginalPhantom

(b) α = 5 (c) α = 17 (d) α = 21

Examples of image reconstructions using a naive Backprojection Algorithm

The BP based algotithm is an approximate inversion and leads to lot of artifacts as well asblurring.

Due the form of transform, it is non-trivial to derive the exact form of the filter.

To improve the quality of reconstruction, we adopt a Fourier series based solution.

Tabish (IIIT-H) July 1, 2016 22 / 39

Page 72: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Back projection based inversion

(a) OriginalPhantom

(b) α = 5 (c) α = 17 (d) α = 21

Examples of image reconstructions using a naive Backprojection Algorithm

The BP based algotithm is an approximate inversion and leads to lot of artifacts as well asblurring.

Due the form of transform, it is non-trivial to derive the exact form of the filter.

To improve the quality of reconstruction, we adopt a Fourier series based solution.

Tabish (IIIT-H) July 1, 2016 22 / 39

Page 73: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Back projection based inversion

(a) OriginalPhantom

(b) α = 5 (c) α = 17 (d) α = 21

Examples of image reconstructions using a naive Backprojection Algorithm

The BP based algotithm is an approximate inversion and leads to lot of artifacts as well asblurring.

Due the form of transform, it is non-trivial to derive the exact form of the filter.

To improve the quality of reconstruction, we adopt a Fourier series based solution.

Tabish (IIIT-H) July 1, 2016 22 / 39

Page 74: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Fourier Series based analysis

Since both f , g are 2π periodic in angular variable, we may expand theminto their Fourier series such that,

then,∞∑

n=−∞gαn (ρ) e inφ =

∞∑n=−∞

∫Aα(ρ,φ)

fn(r)e inθdθ.

Tabish (IIIT-H) July 1, 2016 23 / 39

gα(ρ, φ) =

∫Aα(ρ,φ)

f (r , θ) ds

Page 75: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Fourier Series based analysis

Since both f , g are 2π periodic in angular variable, we may expand theminto their Fourier series such that,then,

∞∑n=−∞

gαn (ρ) e inφ =∞∑

n=−∞

∫Aα(ρ,φ)

fn(r)e inθdθ.

Tabish (IIIT-H) July 1, 2016 23 / 39

gα(ρ, φ) =

∫Aα(ρ,φ)

f (r , θ) ds

Page 76: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Fourier Series based analysis

On Simplifying and equating the Fourier coefficients, the equation reducesto

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Kn(ρ, u)√ρ− u

Fn(u)du

whereFn(u) = fn(R − u)

and

Kn(ρ, u) =2ρ(R − u)Tn

[(R−u)2+R2−ρ2

2R(R−u)

]√

(u + ρ)(2R + ρ− u)(2R − ρ− u). (2)

where, Tn(x) = cos(n cos−1(x))

Tabish (IIIT-H) July 1, 2016 24 / 39

Page 77: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Fourier Series based analysis

On Simplifying and equating the Fourier coefficients, the equation reducesto

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Kn(ρ, u)√ρ− u

Fn(u)du

whereFn(u) = fn(R − u)

and

Kn(ρ, u) =2ρ(R − u)Tn

[(R−u)2+R2−ρ2

2R(R−u)

]√

(u + ρ)(2R + ρ− u)(2R − ρ− u). (2)

where, Tn(x) = cos(n cos−1(x))

Tabish (IIIT-H) July 1, 2016 24 / 39

Page 78: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Fourier Series based analysis

On Simplifying and equating the Fourier coefficients, the equation reducesto

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Kn(ρ, u)√ρ− u

Fn(u)du

whereFn(u) = fn(R − u)

and

Kn(ρ, u) =2ρ(R − u)Tn

[(R−u)2+R2−ρ2

2R(R−u)

]√

(u + ρ)(2R + ρ− u)(2R − ρ− u). (2)

where, Tn(x) = cos(n cos−1(x))

Tabish (IIIT-H) July 1, 2016 24 / 39

Page 79: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Integral equation

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Singural Kernel︷ ︸︸ ︷

Kn(ρ, u)√ρ− u

Fn(u)du (3)

Functions

The equation is a non-standard Volterra integral equation of first kindwith a weakly singular kernel.

/ The exact (closed form) solution of such an equation is not known.

, A direct numerical solution of the equation does not require closedform solution.

Tabish (IIIT-H) July 1, 2016 25 / 39

Page 80: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Integral equation

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Singural Kernel︷ ︸︸ ︷Kn(ρ, u)√ρ− u

Fn(u)du (3)

Functions

The equation is a non-standard Volterra integral equation of first kindwith a weakly singular kernel.

/ The exact (closed form) solution of such an equation is not known.

, A direct numerical solution of the equation does not require closedform solution.

Tabish (IIIT-H) July 1, 2016 25 / 39

Page 81: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Integral equation

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Singural Kernel︷ ︸︸ ︷Kn(ρ, u)√ρ− u

Fn(u)du (3)

Functions

The equation is a non-standard Volterra integral equation of first kindwith a weakly singular kernel.

/ The exact (closed form) solution of such an equation is not known.

, A direct numerical solution of the equation does not require closedform solution.

Tabish (IIIT-H) July 1, 2016 25 / 39

Page 82: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Integral equation

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Singural Kernel︷ ︸︸ ︷Kn(ρ, u)√ρ− u

Fn(u)du (3)

Functions

The equation is a non-standard Volterra integral equation of first kindwith a weakly singular kernel.

/ The exact (closed form) solution of such an equation is not known.

, A direct numerical solution of the equation does not require closedform solution.

Tabish (IIIT-H) July 1, 2016 25 / 39

Page 83: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

CAR Transform: Integral equation

gαn (ρ) =

ρ∫R−√

R2+ρ2−2ρR cosα

Singural Kernel︷ ︸︸ ︷Kn(ρ, u)√ρ− u

Fn(u)du (3)

Functions

The equation is a non-standard Volterra integral equation of first kindwith a weakly singular kernel.

/ The exact (closed form) solution of such an equation is not known.

, A direct numerical solution of the equation does not require closedform solution.

Tabish (IIIT-H) July 1, 2016 25 / 39

Page 84: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

gαn (ρk) =k∑

q=1

ρq∫ρq−1

Fn(u)Kn(ρ, u)√ρ− u

du.

where

bkq =

43{(k − q + 1)

32 + 4

3(k − q)

32 + 2(k − q)

12 q = l

43

((k − q + 1)

32 − 2(k − q)

32 + (k − q − 1)

32

)q = l + 1, ...k − 1.

43

q = k.

and l = max(

0,⌊R −

√R2 + ρ2

k − 2ρkR cosα⌋)

where bxc is the greatest integer less than

equal to x .

Tabish (IIIT-H) July 1, 2016 26 / 39

gαn (ρ) =

ρ∫R−

√R2+ρ2−2ρR cosα

Kn(ρ, u)√ρ− u

Fn(u)du.

Page 85: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

gαn (ρk) =k∑

q=1

ρq∫ρq−1

Fn(u)Kn(ρ, u)√ρ− u

du.

Approximating the integrand as a linear function over each interval [ρq−1, ρq ], and integrating

we get

gn(ρk) =√h

k∑

q=l

bkqKn(ρk , ρq)Fn(ρq)

where

bkq =

43{(k − q + 1)

32 + 4

3(k − q)

32 + 2(k − q)

12 q = l

43

((k − q + 1)

32 − 2(k − q)

32 + (k − q − 1)

32

)q = l + 1, ...k − 1.

43

q = k.

and l = max(

0,⌊R −

√R2 + ρ2

k − 2ρkR cosα⌋)

where bxc is the greatest integer less than

equal to x .

Tabish (IIIT-H) July 1, 2016 26 / 39

Page 86: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

gαn (ρk) =k∑

q=1

ρq∫ρq−1

Fn(u)Kn(ρ, u)√ρ− u

du.

Approximating the integrand as a linear function over each interval [ρq−1, ρq ], and integrating

we get

gn(ρk) =√h

k∑

q=l

bkqKn(ρk , ρq)Fn(ρq)

where

bkq =

43{(k − q + 1)

32 + 4

3(k − q)

32 + 2(k − q)

12 q = l

43

((k − q + 1)

32 − 2(k − q)

32 + (k − q − 1)

32

)q = l + 1, ...k − 1.

43

q = k.

and l = max(

0,⌊R −

√R2 + ρ2

k − 2ρkR cosα⌋)

where bxc is the greatest integer less than

equal to x .

Tabish (IIIT-H) July 1, 2016 26 / 39

Page 87: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

gn(ρk ) =√h

k∑

q=l

bkqKn(ρk , ρq)Fn(ρq)

The previous equation can be written in the matrix from as

gαn = BnFn (4)

Matrix Bn lower triangular matrix which is a piecewise linear, discreteapproximation of the integral in Equation (3).

Diagonal entries of Bn, bii = 43

√h 6= 0, hence the matrix is invertible.

Tabish (IIIT-H) July 1, 2016 27 / 39

Page 88: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

gn(ρk ) =√h

k∑

q=l

bkqKn(ρk , ρq)Fn(ρq)

The previous equation can be written in the matrix from as

gαn = BnFn (4)

where

gαn =

gαn (ρ0)

.

.

.gαn (ρM−1)

Fn =

Fn(ρ0)...

Fn(ρM−1)

.

Matrix Bn lower triangular matrix which is a piecewise linear, discreteapproximation of the integral in Equation (3).

Diagonal entries of Bn, bii = 43

√h 6= 0, hence the matrix is invertible.

Tabish (IIIT-H) July 1, 2016 27 / 39

Page 89: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

gn(ρk ) =√h

k∑

q=l

bkqKn(ρk , ρq)Fn(ρq)

The previous equation can be written in the matrix from as

gαn = BnFn (4)

where

gαn =

gαn (ρ0)

.

.

.gαn (ρM−1)

Fn =

Fn(ρ0)...

Fn(ρM−1)

.

Matrix Bn lower triangular matrix which is a piecewise linear, discreteapproximation of the integral in Equation (3).

Diagonal entries of Bn, bii = 43

√h 6= 0, hence the matrix is invertible.

Tabish (IIIT-H) July 1, 2016 27 / 39

Page 90: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Discrete CAR Transform

The previous equation can be written in the matrix from as

gαn = BnFn (4)

where

gαn =

gαn (ρ0)

.

.

.gαn (ρM−1)

Fn =

Fn(ρ0)...

Fn(ρM−1)

.

Matrix Bn lower triangular matrix which is a piecewise linear, discreteapproximation of the integral in Equation (3).

Diagonal entries of Bn, bii = 43

√h 6= 0, hence the matrix is invertible.

Tabish (IIIT-H) July 1, 2016 27 / 39

Page 91: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Numerical inversion of CAR Transform.

gαn = BnFn

, Matrix Bn lower triangular matrix which is a piecewise linear,discrete approximation of the integral in Equation (3).

, Diagonal entries of Bn, bii = 43

√h 6= 0, hence the matrix is

invertible.

/ The matrix is Bn has a high condition number (O(1015)), hencedirect inversion is unstable.

Tabish (IIIT-H) July 1, 2016 28 / 39

Page 92: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Numerical inversion of CAR Transform.

gαn = BnFn

, Matrix Bn lower triangular matrix which is a piecewise linear,discrete approximation of the integral in Equation (3).

, Diagonal entries of Bn, bii = 43

√h 6= 0, hence the matrix is

invertible.

/ The matrix is Bn has a high condition number (O(1015)), hencedirect inversion is unstable.

We use a Truncated SVD based r -rank inverse (r < M) such that,

Fn ≈ B−1n,r gαn

Tabish (IIIT-H) July 1, 2016 28 / 39

Page 93: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Effect of Rank

Full rank inversion isexpected to be unstable.

If the rank r is set to be toolow reconstructed image isexpected to have ringingartifacts.

original phantom (f ) used inexperiments.

Tabish (IIIT-H) July 1, 2016 29 / 39

Page 94: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Effect of Rank

Full rank inversion isexpected to be unstable.

If the rank r is set to be toolow reconstructed image isexpected to have ringingartifacts.

(a) r = n/6 (b) r = n/2

(c) r = 9n/10 (d) r = n

Effect of rank r of matrix Bn,r on the reconstruction

quality. n = 300

Tabish (IIIT-H) July 1, 2016 29 / 39

Page 95: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Effect of Rank

Full rank inversion isexpected to be unstable.

If the rank r is set to be toolow reconstructed image isexpected to have ringingartifacts.

(a) r = n/6 (b) r = n/2

(c) r = 9n/10 (d) r = n

Effect of rank r of matrix Bn,r on the reconstruction

quality. n = 300

Tabish (IIIT-H) July 1, 2016 29 / 39

Page 96: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Choosing Rank

original phantom (f ) used inexperiments.

Tabish (IIIT-H) July 1, 2016 30 / 39

Page 97: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Choosing Rank

(a) r = n/3 (b) r = n/2

(c) r = 9n/10 (d) r = n

Effect of rank r of matrix Bn,r on the reconstruction

quality. n = 300

original phantom (f ) used inexperiments.

Tabish (IIIT-H) July 1, 2016 30 / 39

Page 98: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Choosing Rank

(a) r = n/3 (b) r = n/2

(c) r = 9n/10 (d) r = n

Effect of rank r of matrix Bn,r on the reconstruction

quality. n = 300

Plot of Mean Square Error as afunction of rank r .

Tabish (IIIT-H) July 1, 2016 30 / 39

Page 99: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Choosing Rank

(a) r = n/3 (b) r = n/2

(c) r = 9n/10 (d) r = n

Effect of rank r of matrix Bn,r on the reconstruction

quality. n = 300

original phantom (f ) used in experiments where region to

be zoomed is shown in red.

Tabish (IIIT-H) July 1, 2016 30 / 39

Page 100: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Choosing Rank

(a) r = n/3 (b) r = n/2

(c) r = 9n/10 (d) r = n

Effect of rank r of matrix Bn,r on the reconstruction

quality. n = 300

Tabish (IIIT-H) July 1, 2016 30 / 39

Based on our experiments,as a rule of thumb, droppinghighest 10% of singular val-ues gives a fairly stable re-construction.

Page 101: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction In Limited View Scenario

How to remove the artifacts which arise in the Circular arc Radon transform due to the limited view?

Tabish (IIIT-H) July 1, 2016 31 / 39

Page 102: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction of Singularities under Radon Transforms.

Figure: Image with visualization of

projection value along direction shown.

Tabish (IIIT-H) July 1, 2016 32 / 39

Page 103: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction of Singularities under Radon Transforms.

Figure: Image with visualization of

projection value along direction shown.

Tabish (IIIT-H) July 1, 2016 32 / 39

Page 104: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction of Singularities under Radon Transforms.

Figure: Image with visualization of

projection value along direction shown.

Tabish (IIIT-H) July 1, 2016 32 / 39

Page 105: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction of Singularities under Radon Transforms.

Figure: Image with visualization of

projection value along direction shown.

Tabish (IIIT-H) July 1, 2016 32 / 39

Page 106: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reconstruction of Singularities under Radon Transforms.

Figure: Image with visualization of

projection value along direction shown.

Tabish (IIIT-H) July 1, 2016 32 / 39

Let C be the set of curves,along which we measure pro-jections. Then for an edge tobe visible there must be atleast one element in the in-terior of set C, tangential tothe edge

Page 107: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Effect of limited view

Due to limited view not all edges are visible,in the sense of meeting tangency criterionw.r.t set C.

The end points of the arc lie inside the object,which leads to curves C having discontinuitiesat end points.

The presence of these sharp discontinuities indata set C and limited view will lead to streakand circular artifacts.

Object

x axis

y axis

Figure: A sharp circular artifact is

observed due to discontinuity in angular,

as well as radial direction

Tabish (IIIT-H) July 1, 2016 33 / 39

Page 108: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Effect of limited view

Due to limited view not all edges are visible,in the sense of meeting tangency criterionw.r.t set C.

The end points of the arc lie inside the object,which leads to curves C having discontinuitiesat end points.

The presence of these sharp discontinuities indata set C and limited view will lead to streakand circular artifacts.

Object

x axis

y axis

Figure: A sharp circular artifact is

observed due to discontinuity in angular,

as well as radial direction

Tabish (IIIT-H) July 1, 2016 33 / 39

Page 109: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Effect of limited view

Due to limited view not all edges are visible,in the sense of meeting tangency criterionw.r.t set C.

The end points of the arc lie inside the object,which leads to curves C having discontinuitiesat end points.

The presence of these sharp discontinuities indata set C and limited view will lead to streakand circular artifacts.

Object

x axis

y axis

Figure: A sharp circular artifact is

observed due to discontinuity in angular,

as well as radial direction

Tabish (IIIT-H) July 1, 2016 33 / 39

Page 110: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Effect of limited view

Due to limited view not all edges are visible,in the sense of meeting tangency criterionw.r.t set C.

The end points of the arc lie inside the object,which leads to curves C having discontinuitiesat end points.

The presence of these sharp discontinuities indata set C and limited view will lead to streakand circular artifacts.

Object

x axis

y axis

Figure: A sharp circular artifact is

observed due to discontinuity in angular,

as well as radial direction

Tabish (IIIT-H) July 1, 2016 33 / 39

Page 111: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reducing artifacts in reconstructed images

To reduce the artifacts in the reconstructed images we smooth outthe discontinuities of the elements of C

This is achieved by gracefully decaying arcs to zero at the edges.

Algorithmically, this achieved by weighing rows of Bn by a factor of

the form e(i−h)2

σ2 ; visualized below.

Visualization of structure of unmodifiedoriginal matrix Bn.

Visualization of structure of modifiedmatrix Bn for artifact suppression.

Tabish (IIIT-H) July 1, 2016 34 / 39

Page 112: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reducing artifacts in reconstructed images

To reduce the artifacts in the reconstructed images we smooth outthe discontinuities of the elements of CThis is achieved by gracefully decaying arcs to zero at the edges.

Algorithmically, this achieved by weighing rows of Bn by a factor of

the form e(i−h)2

σ2 ; visualized below.

Visualization of structure of unmodifiedoriginal matrix Bn.

Visualization of structure of modifiedmatrix Bn for artifact suppression.

Tabish (IIIT-H) July 1, 2016 34 / 39

Page 113: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reducing artifacts in reconstructed images

To reduce the artifacts in the reconstructed images we smooth outthe discontinuities of the elements of CThis is achieved by gracefully decaying arcs to zero at the edges.

Algorithmically, this achieved by weighing rows of Bn by a factor of

the form e(i−h)2

σ2 ; visualized below.

Visualization of structure of unmodifiedoriginal matrix Bn.

Visualization of structure of modifiedmatrix Bn for artifact suppression.

Tabish (IIIT-H) July 1, 2016 34 / 39

Page 114: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Reducing artifacts in reconstructed images

To reduce the artifacts in the reconstructed images we smooth outthe discontinuities of the elements of CThis is achieved by gracefully decaying arcs to zero at the edges.

Algorithmically, this achieved by weighing rows of Bn by a factor of

the form e(i−h)2

σ2 ; visualized below.

Visualization of structure of unmodifiedoriginal matrix Bn.

Visualization of structure of modifiedmatrix Bn for artifact suppression.

Tabish (IIIT-H) July 1, 2016 34 / 39

Page 115: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction.

(a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76

(e) α = 21 (f) α = 31 (g) α = 46 (h) α = 76

Reconstructed images corresponding to different α before (row 1), and after artifact suppression(row 2).

Tabish (IIIT-H) July 1, 2016 35 / 39

Page 116: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction.

(a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76

(e) α = 21 (f) α = 31 (g) α = 46 (h) α = 76

Reconstructed images corresponding to different α before (row 1), and after artifact suppression(row 2).

Tabish (IIIT-H) July 1, 2016 35 / 39

Page 117: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction

Object

α

ρ

R

φ

x axis

y axis

Figure: Setup with support outside the acquisition circle

Tabish (IIIT-H) July 1, 2016 36 / 39

Page 118: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction

gαn (ρ) =ρ∫

√R2+ρ2+2ρR cosα −R

Kn(ρ,u)√ρ−u Fn(u)du

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

y

gαn (ρ) =ρ∫

R−√

R2+ρ2−2ρR cosα

Kn(ρ,u)√ρ−u Fn(u)du

Object

α

ρ

R

φ

x axis

y axis

Tabish (IIIT-H) July 1, 2016 36 / 39

Page 119: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction

gαn (ρ) =ρ∫

√R2+ρ2+2ρR cosα −R

Kn(ρ,u)√ρ−u Fn(u)du

Object

x axis

y axis

O

φθ

ρ α

Rr

C(ρ, φ)

y

gαn (ρ) =ρ∫

R−√

R2+ρ2−2ρR cosα

Kn(ρ,u)√ρ−u Fn(u)du

Object

α

ρ

R

φ

x axis

y axis

Tabish (IIIT-H) July 1, 2016 36 / 39

Page 120: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction

Figure: Phantom with support outside the acquisition circle.

Tabish (IIIT-H) July 1, 2016 36 / 39

Page 121: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction

(a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76

(e) α = 21 (f) α = 31 (g) α = 46 (h) α = 76

Figure: Reconstructed images corresponding to different α before (row 1), and after artifactsuppression (row 2), for the support outside case.Tabish (IIIT-H) July 1, 2016 36 / 39

Page 122: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Experiments and Results: Artifact Reduction

(a) α = 21 (b) α = 31 (c) α = 46 (d) α = 76

(e) α = 21 (f) α = 31 (g) α = 46 (h) α = 76

Figure: Reconstructed images corresponding to different α before (row 1), and after artifactsuppression (row 2), for the support outside case.Tabish (IIIT-H) July 1, 2016 36 / 39

Page 123: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work.

3 We Proposed a method of numerical inversion of circular arc Radontransform, a limited view generalization of circular Radon transform.

3 We also proposed a strategy to reduce the artifacts which arise in theimage due to limited view.

3 Provide a rigorous mathematical justification of the artifacts.

3 Derive a closed form solution of the Volterra integral equation arisingin the transform.

Tabish (IIIT-H) July 1, 2016 37 / 39

Page 124: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work.

3 We Proposed a method of numerical inversion of circular arc Radontransform, a limited view generalization of circular Radon transform.

3 We also proposed a strategy to reduce the artifacts which arise in theimage due to limited view.

3 Provide a rigorous mathematical justification of the artifacts.

3 Derive a closed form solution of the Volterra integral equation arisingin the transform.

Tabish (IIIT-H) July 1, 2016 37 / 39

Page 125: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work.

3 We Proposed a method of numerical inversion of circular arc Radontransform, a limited view generalization of circular Radon transform.

3 We also proposed a strategy to reduce the artifacts which arise in theimage due to limited view.

3 Provide a rigorous mathematical justification of the artifacts.

3 Derive a closed form solution of the Volterra integral equation arisingin the transform.

Tabish (IIIT-H) July 1, 2016 37 / 39

Page 126: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Summary & Future Work.

3 We Proposed a method of numerical inversion of circular arc Radontransform, a limited view generalization of circular Radon transform.

3 We also proposed a strategy to reduce the artifacts which arise in theimage due to limited view.

3 Provide a rigorous mathematical justification of the artifacts.

3 Derive a closed form solution of the Volterra integral equation arisingin the transform.

Tabish (IIIT-H) July 1, 2016 37 / 39

Page 127: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Related Publications

PET Image Reconstruction And Denoising On Hexagonal Lattices.Syed T. A. and Sivaswamy J.International Conference on Image Processing(ICIP) 2015, Quebeccity.

Numerical inversion of circular arc Radon transformSyed T. A., Krishnan V. P. and Sivaswamy J.(Under review).

Tabish (IIIT-H) July 1, 2016 38 / 39

Page 128: Tomographic Image Reconstruction in Noisy and Limited ......We propose a 2 step reconstruction process onto Hexagonal lattice:!Step 1: Noisy Reconstruction using Filtered Back Projection.!Step

Thank You

Tabish (IIIT-H) July 1, 2016 39 / 39


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