Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � 0 A j := , (1) 2 j 0 j = 2 G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � � 1 � 0 k A j := , S k := . (1) 2 j 0 0 1 k = 0 G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � � 1 � 0 k A j := , S k := . (1) 2 j 0 0 1 k = 1 / 4 G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � � 1 � 0 k A j := , S k := . (1) 2 j 0 0 1 k = 1 / 2 G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � � 1 � 0 k A j := , S k := . (1) 2 j 0 0 1 The system SH ( ψ ) := { ψ j,k,m := 2 3 j/ 2 ψ ( S k A j · − m ) | j ∈ Z , k ∈ Z , m ∈ Z 2 } (2) is the discrete shearlet system defined by ψ . G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � � 1 � 0 k A j := , S k := . (1) 2 j 0 0 1 The system SH ( ψ ) := { ψ j,k,m := 2 3 j/ 2 ψ ( S k A j · − m ) | j ∈ Z , k ∈ Z , m ∈ Z 2 } (2) is the discrete shearlet system defined by ψ . Analogous continuous setting with parameters a ∈ R + , s ∈ R , t ∈ R 2 . G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets Comes into Play: What Are They? Definition (Discrete shearlet system [Kutyniok & Labate, 2006]) Let ψ ∈ L 2 ( R 2 ) . For j ∈ Z and k ∈ R , define the parabolic scaling matrix A j and the shear matrix S k by � 2 2 j � � 1 � 0 k A j := , S k := . (1) 2 j 0 0 1 The system SH ( ψ ) := { ψ j,k,m := 2 3 j/ 2 ψ ( S k A j · − m ) | j ∈ Z , k ∈ Z , m ∈ Z 2 } (2) is the discrete shearlet system defined by ψ . Analogous continuous setting with parameters a ∈ R + , s ∈ R , t ∈ R 2 . Many more evolved constructions (cone-adapted, bandlimited, compactly supported, . . . ). G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data , 1st ed. New York: Springer Verlag, 2012. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets and Wavefront Sets Resolution of Wavefront Sets (simplified from [Kutyniok & Labate, 2006], [Grohs, 2011]) ( t 0 , s 0 ) ∈ R 2 × [ − 1 , 1] : for ( t , s ) in neighborhood U of ( t 0 , s 0 ) : WF( f ) c = � � |SH ψ f ( a, s, t ) | = O ( a k ) as a − → 0 , ∀ k ∈ N , unif. over U where L 2 ( R 2 ) ∋ f �→ SH ψ f ( a, s, t ) = � f, ψ a,s, t � , ( a, s, t ) ∈ R + × R × R 2 denotes the continuous shearlet transform . x 2 s 0 t 0 x 1 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets and Wavefront Sets Resolution of Wavefront Sets (simplified from [Kutyniok & Labate, 2006], [Grohs, 2011]) ( t 0 , s 0 ) ∈ R 2 × [ − 1 , 1] : for ( t , s ) in neighborhood U of ( t 0 , s 0 ) : WF( f ) c = � � |SH ψ f ( a, s, t ) | = O ( a k ) as a − → 0 , ∀ k ∈ N , unif. over U where L 2 ( R 2 ) ∋ f �→ SH ψ f ( a, s, t ) = � f, ψ a,s, t � , ( a, s, t ) ∈ R + × R × R 2 denotes the continuous shearlet transform . x 2 f is smooth in t 0 and shearing direction s = ⇒ fast decay of shearlet coefficients s 0 = ⇒ sparsity! t 0 wavelets only characterize singular support of f ( i.e. , no directional information) x 1 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlets and Wavefront Sets Resolution of Wavefront Sets (simplified from [Kutyniok & Labate, 2006], [Grohs, 2011]) WF( f ) c = ( t 0 , s 0 ) ∈ R 2 × [ − 1 , 1] : for ( t , s ) in neighborhood U of ( t 0 , s 0 ) : � � |SH ψ f ( a, s, t ) | = O ( a k ) as a − → 0 , ∀ k ∈ N , unif. over U where L 2 ( R 2 ) ∋ f �→ SH ψ f ( a, s, t ) = � f, ψ a,s, t � , ( a, s, t ) ∈ R + × R × R 2 denotes the continuous shearlet transform . f is smooth in t 0 and shearing direction s θ ⇒ fast decay of shearlet coefficients = ⇒ sparsity! = wavelets only characterize singular support of f ( i.e. , no directional information) x 2 x 1 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
∗ -lets and Limited Angle Tomography I Forward problem carefully analyzed by Frikel (2013) for curvelets The index set of curvelets can be split into a part that is “visible” under R φ and a part that is “invisible”, i.e. , R φ ψ j,k, l = 0 (via Fourier slice theorem ). J. Frikel, Sparse regularization in limited angle tomography , Appl. Comp. Harm. Anal. 34 (1), 117–141, 2013. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
∗ -lets and Limited Angle Tomography I Forward problem carefully analyzed by Frikel (2013) for curvelets The index set of curvelets can be split into a part that is “visible” under R φ and a part that is “invisible”, i.e. , R φ ψ j,k, l = 0 (via Fourier slice theorem ). Holds for shearlets and other directional representation systems as well ξ 2 ξ 2 ξ 2 C v C h ξ 1 ξ 1 ξ 1 R C h W φ C v Invisible Visible Semi-visible Visible Wedge Semi-visible Semi-visible J. Frikel, Sparse regularization in limited angle tomography , Appl. Comp. Harm. Anal. 34 (1), 117–141, 2013. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
∗ -lets and Limited Angle Tomography II Obtain � � f = � f, ψ j,k, l � ψ j,k, l + � f, ψ j,k, l � ψ j,k, l ( j,k, l ) ∈I vis ( j,k, l ) ∈I inv = f vis + f inv . Use in: � � z � 1 , w + 1 � SH T 2 � R φ SH T ψ ( z ) − y � 2 argmin ψ 2 z Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
∗ -lets and Limited Angle Tomography II Obtain � � f = � f, ψ j,k, l � ψ j,k, l + � f, ψ j,k, l � ψ j,k, l ( j,k, l ) ∈I vis ( j,k, l ) ∈I inv = f vis + f inv . Use in: � � z � 1 , w + 1 � SH T 2 � R φ SH T ψ ( z ) − y � 2 argmin ψ 2 z ∈I vis dimension reduction Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
∗ -lets and Limited Angle Tomography II Obtain � � f = � f, ψ j,k, l � ψ j,k, l + � f, ψ j,k, l � ψ j,k, l ( j,k, l ) ∈I vis ( j,k, l ) ∈I inv = f vis + f inv . Use in: � � z � 1 , w + 1 � SH T 2 � R φ SH T ψ ( z ) − y � 2 argmin ψ 2 z ∈I vis dimension reduction drawbacks: no positivity constraint, synthesis formulation Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Idea Facts: parts of the WF are available only “here and there”. shearlets are proven to resolve the WF Idea : use shearlet coefficients to fill in the missing parts of the wavefront set. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Idea Facts: parts of the WF are available only “here and there”. shearlets are proven to resolve the WF Idea : use shearlet coefficients to fill in the missing parts of the wavefront set. ℓ 1 -minimization reconstruction “candy-wrap” Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Idea Facts: parts of the WF are available only “here and there”. shearlets are proven to resolve the WF Idea : use shearlet coefficients to fill in the missing parts of the wavefront set. ℓ 1 -minimization shearlet cube: visibile + invisible reconstruction “candy-wrap” “candy-wrap” Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Idea Facts: parts of the WF are available only “here and there”. shearlets are proven to resolve the WF Idea : use shearlet coefficients to fill in the missing parts of the wavefront set. ℓ 1 -minimization “inpainted” shearlet cube: visibile + invisible reconstruction reconstruction “candy-wrap” “candy-wrap” fill in missing coeff. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
The Idea Facts: parts of the WF are available only “here and there”. shearlets are proven to resolve the WF Idea : use shearlet coefficients to fill in the missing parts of the wavefront set. ℓ 1 -minimization DL? “inpainted” shearlet cube: visibile + invisible reconstruction reconstruction “candy-wrap” “candy-wrap” fill in missing coeff. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Shearlet Cube Numerically, the shearlet transform does the following: f ∈ R n × n �− → F ∈ R n × n × L SH ψ f ∈ R n × n F ∈ R n × n × L Each subband F (: , : , i ) corresponds to the inner products � f , ψ j,k, · � , F is referred to as the shearlet coefficients cube of f . Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
“Candy–wrap” Structure Due to the sorting, shearlets coefficients follow a specific structure in each scale: SH ψ f ∈ R n × n highest scale Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
“Candy–wrap” Structure Due to the sorting, shearlets coefficients follow a specific structure in each scale: SH ψ f ∈ R n × n highest scale Invisibility of limited angle tomography “creates” holes in it: (a) full-angle CT (b) limited angle CT Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
“Candy–wrap” Prior? First Idea Handcraft a prior that promotes this specific “candy–wrap” structure of the shearlet coefficients: enforce continuity of WF. Too complicated? Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
“Candy–wrap” Prior? First Idea Handcraft a prior that promotes this specific “candy–wrap” structure of the shearlet coefficients: enforce continuity of WF. Too complicated? “messy” shearlet coefficients easy rule for the human eye, hard to grasp mathematically (a) shearlet cube for Shepp-Logan (b) “cleaned” shearlet cube Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
“Candy–wrap” Prior? First Idea Handcraft a prior that promotes this specific “candy–wrap” structure of the shearlet coefficients: enforce continuity of WF. Too complicated? “messy” shearlet coefficients easy rule for the human eye, hard to grasp mathematically (a) shearlet cube for Shepp-Logan (b) “cleaned” shearlet cube Idea Train a deep neural network to fill in the gaps of the WF. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility f gt FBP Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility ℓ 1 shearlet solution f ∗ f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility � � SH T f gt SH ψ ( f FBP ) I vis + SH ψ ( f gt ) I inv ψ Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility � � SH T SH ψ ( f ∗ ) I vis + SH ψ ( f gt ) I inv f gt ψ Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility � � SH T SH ψ ( f ∗ ) I vis + SH ψ ( f gt ) I inv f gt ψ SH ψ ( f ∗ ) I vis close to SH ψ ( f gt ) I vis ⇒ learning the invisible coefficients should be sufficient Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Numerical Simulation: Verify the Concept of (In-)Visibility f gt spoiler: � � SH ψ ( f ∗ ) I vis + F N N SH T ψ I inv SH ψ ( f ∗ ) I vis close to SH ψ ( f gt ) I vis ⇒ learning the invisible coefficients should be sufficient Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Some Literature on DL for Inverse Problems on FBP: Kang et al. (2017): contourlets of FBP + U-net, 2nd place Mayo low-dose challenge & many more works from this group! Zhang et al. (2016): 2-layer network on FBP Jin et al. (2017): U-Net on FBP incorporating forward model via optimization scheme: Hammernik et al. (2017): learning weights for FBP, then filtering Meinhardt et al. (2017): learning proximal operators Adler et al. (2017): learned primal dual The Closest Method: Gu & Ye (2017) “Based on the observation that the artifacts from limited angles have some directional property and are globally distributed, we propose a novel multi-scale wavelet domain residual learning architecture, which compensates for the artifacts.” Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Some Observations Concerning “denoising” of the FBP (or its coefficients) with DL: missing theory, unclear what the NN really does: entire image is processed which features are modified? lack of a clear interpretation (?) NN needs to learn a lot of streaking artifacts (+ noise) Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Some Observations Concerning “denoising” of the FBP (or its coefficients) with DL: missing theory, unclear what the NN really does: entire image is processed which features are modified? lack of a clear interpretation (?) NN needs to learn a lot of streaking artifacts (+ noise) We can do better (in limited angle CT)! only the invisible boundaries need to be learned shearlets help to access them the coefficients follow the “candy–wrap” structure Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Our Approach Step 1, “recover the visible”: best available classical solution (little artifacts, denoised) � SH ψ ( f ) � 1 ,w + 1 f ∗ := argmin 2 � R φ f − y � 2 2 f ≥ 0 Allows to access WF via sparsity prior on shearlets: for ( j, k, l ) ∈ I inv : SH ψ ( f ∗ ) ( j,k, l ) ≈ 0 for ( j, k, l ) ∈ I vis : SH ψ ( f ∗ ) ( j,k, l ) reliable and near perfect Step 2, “learn the invisible”: supervised learning of invisible coefficients � � ! NN θ : SH ψ ( f ∗ ) I inv ≈ SH ψ ( f gt ) I inv F Step 3, “combine”: f LtI = SH T ψ (SH ψ ( f ∗ ) I vis + F ) Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Our Approach y SH T f LtI SH( f ∗ ) I vis + F Step 1 Step 3 nonlin. rec. of vis. coeff. combine both parts (ADMM) Σ Step 2 PhantomNet NN θ learn the invisible F visible coeff. I vis SH( f ∗ ) invisible coeff. I inv learned coeff. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Our Approach – Step 2: CNN PhantomNet Convolutional Neural Network that minimizes over the empirical risk: N 1 � �NN θ (SH( f ∗ j )) − SH( f j ) I inv � 2 min w , 2 . N θ j =1 ( 512 × 512 , 128 ) ( 512 × 512 , 64) ( 512 × 512 , 64 ) ( 512 × 512 , 59 ) ( 256 × 256 , 256 ) ( 256 × 256 , 128 ) ( 256 × 256 , 128 ) ( 512 × 512 , 128 ) ( 128 × 128 , 512 ) ( 128 × 128 , 256 ) ( 128 × 128 , 256 ) ( 256 × 256 , 256 ) ( 64 × 64 , 512 ) ( 64 × 64 , 512 ) ( 128 × 128 , 512 ) Transition Up Copy and Concatenate Convolution Trimmed-DenseBlock Transition Down Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Learning the Invisible Model based & data driven: only learn what needs to be learned! Possible advantages: faithfulness by learning only what is not visible in the data better performance due to better input NN does not process entire image less blurring by U-net fewer unwanted artifacts better generalization T.A. Bubba, G. Kutyniok, M. Lassas, M. M¨ arz, W. Samek, S. Siltanen and V. Srinivasan, Learning the Invisible: a hybrid deep learning-shearlets framework for limited angle computed tomography , Inverse Problems 2019 (in press). Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Learning the Invisible Model based & data driven: only learn what needs to be learned! Possible advantages: faithfulness by learning only what is not visible in the data better performance due to better input NN does not process entire image less blurring by U-net fewer unwanted artifacts better generalization Disadvantage: speed: dominated by ℓ 1 -minimization T.A. Bubba, G. Kutyniok, M. Lassas, M. M¨ arz, W. Samek, S. Siltanen and V. Srinivasan, Learning the Invisible: a hybrid deep learning-shearlets framework for limited angle computed tomography , Inverse Problems 2019 (in press). Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Setup Experimental Scenarios: Mayo Clinic 5 : human abdomen scans provided by the Mayo Clinic for the AAPM Low-Dose CT Grand Challenge. 10 patients ( 2378 slices of size 512 × 512 with thickness 3 mm) 9 patients for training ( 2134 slices) and 1 patient for testing ( 244 slices) Mayo- 60 ◦ : missing wedge of 60 ◦ Mayo- 75 ◦ : missing wedge of 30 ◦ Lotus Root: real data measured with the µ CT in Helsinki to check generalization properties of our method (training is on Mayo- 60 ◦ ) Lotus- 60 ◦ : missing wedge of 60 ◦ Lotus- 75 ◦ : missing wedge of 30 ◦ Operators: R φ : Astra toolbox (fanbeam geometry fits lotus root’s acquisition setup) SH ψ : α -shearlet transform toolbox (bandlimited shearlets with 5 scales, i.e. , 59 subbands of size 512 × 512 ) 5 We would like to thank Dr. Cynthia McCollough, the Mayo Clinic, the American Association of Physicists in Medicine (AAPM), and grant EB01705 and EB01785 from the National Institute of Biomedical Imaging and Bioengineering for providing the Low-Dose CT Grand Challenge data set. Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt f FBP : RE = 0.50, HaarPSI=0.35 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt f TV : RE = 0.21, HaarPSI=0.41 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f ∗ : RE = 0.19, HaarPSI=0.43 f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt f [Gu & Ye, 2017] : RE = 0.22, HaarPSI=0.40 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt NN θ ( f FBP ) : RE = 0.16, HaarPSI=0.53 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt NN θ (SH( f FBP )) : RE = 0.16, HaarPSI=0.58 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 60 ◦ f gt f LtI : RE = 0.09, HaarPSI=0.76 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt f FBP : RE = 0.30, HaarPSI=0.46 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt f TV : RE = 0.10, HaarPSI=0.63 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f ∗ : RE = 0.09, HaarPSI=0.64 f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt f [Gu & Ye, 2017] : RE = 0.21, HaarPSI=0.43 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt NN θ ( f FBP ) : RE = 0.09, HaarPSI=0.82 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt NN θ (SH( f FBP )) : RE = 0.06, HaarPSI=0.82 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Mayo- 75 ◦ f gt f LtI : RE = 0.03, HaarPSI=0.92 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f gt f FBP : RE = 0.49, HaarPSI=0.49 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f gt f TV : RE = 0.21, HaarPSI=0.60 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f ∗ : RE = 0.19, HaarPSI=0.61 f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f gt f [Gu & Ye, 2017] : RE = 0.42, HaarPSI=0.56 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f gt NN θ (SH( f FBP )) : RE = 0.27, HaarPSI=0.63 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 60 ◦ f gt f LtI : RE = 0.15, HaarPSI=0.74 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 75 ◦ f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 75 ◦ f gt f FBP : RE = 0.30, HaarPSI=0.36 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 75 ◦ f gt f TV : RE = 0.12, HaarPSI=0.80 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 75 ◦ f ∗ : RE = 0.10, HaarPSI=0.79 f gt Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
Lotus- 75 ◦ f gt f [Gu & Ye, 2017] : RE = 0.25, HaarPSI=0.65 Tatiana Bubba The Mathematics of X-ray Tomography Very Finnish IP2019
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