Challenges in Improved Sensitivity of Quantification of PET Data for Alzheimer s Disease Studies

Transcription

1 Challenges in Improved Sensitivity of Quantification of PET Data for Alzheimer s Disease Studies Rosemary Renaut, Hongbin Guo, Kewei Chen and Wolfgang Stefan Supported by Arizona Alzheimer s Research Center and NIH Arizona State University March 30, 2007 Georgia State University Colloquium Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 1 / 50

2 Outline 1 PET Images 2 FDG-PET Quantification The Standard Model Practical Difficulties - obtaining the input function Possible Solution: Arterial ROI TAC Review 3 Parameter Estimation Modeling the input function The New Input Estimating Parameters of the Model Results 4 Image Restoration Regularized Optimization Numerical Results 5 Conclusions and Future Work Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 2 / 50

3 Outline 1 PET Images 2 FDG-PET Quantification The Standard Model Practical Difficulties - obtaining the input function Possible Solution: Arterial ROI TAC Review 3 Parameter Estimation Modeling the input function The New Input Estimating Parameters of the Model Results 4 Image Restoration Regularized Optimization Numerical Results 5 Conclusions and Future Work Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 2 / 50

4 Outline 1 PET Images 2 FDG-PET Quantification The Standard Model Practical Difficulties - obtaining the input function Possible Solution: Arterial ROI TAC Review 3 Parameter Estimation Modeling the input function The New Input Estimating Parameters of the Model Results 4 Image Restoration Regularized Optimization Numerical Results 5 Conclusions and Future Work Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 2 / 50

5 Outline 1 PET Images 2 FDG-PET Quantification The Standard Model Practical Difficulties - obtaining the input function Possible Solution: Arterial ROI TAC Review 3 Parameter Estimation Modeling the input function The New Input Estimating Parameters of the Model Results 4 Image Restoration Regularized Optimization Numerical Results 5 Conclusions and Future Work Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 2 / 50

6 Outline 1 PET Images 2 FDG-PET Quantification The Standard Model Practical Difficulties - obtaining the input function Possible Solution: Arterial ROI TAC Review 3 Parameter Estimation Modeling the input function The New Input Estimating Parameters of the Model Results 4 Image Restoration Regularized Optimization Numerical Results 5 Conclusions and Future Work Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 2 / 50

7 Example of typical PET scan Typical PET Images show High noise content (non Gaussian) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 3 / 50

8 Example of typical PET scan Typical PET Images show High noise content (non Gaussian) High blurring Partial volume Effects Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 3 / 50

9 Example of typical PET scan Typical PET Images show High noise content (non Gaussian) High blurring Partial volume Effects Reconstruction artifacts Reconstruction using filtered backprojection Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 3 / 50

10 Example of Dynamic PET series Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 4 / 50

11 Example of Dynamic PET series Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 4 / 50

12 Example of Dynamic PET series Dynamic data Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 4 / 50

13 Example of Dynamic PET series Dynamic data Very poor initial scans Noise levels change across scans Time interval increases with scan Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 4 / 50

14 Example of Dynamic PET series Dynamic data Very poor initial scans Noise levels change across scans Time interval increases with scan Movement of patient Physiological Movement Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 4 / 50

15 Goals/Methods of the Study Improved Sensitivity for identifying features in images: Identify anomalies in single images. Identify changes over time: Longitudinal Studies of AD Assessing disease state- AD or MCI (mild cognitive impairment) Assess impact of drug treatment Noninvasive assistance in AD studies. Solve Ordinary Differential System of Equations. Use basic Inverse Problems, Optimization and Statistics. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 5 / 50

16 Goals/Methods of the Study Improved Sensitivity for identifying features in images: Identify anomalies in single images. Identify changes over time: Longitudinal Studies of AD Assessing disease state- AD or MCI (mild cognitive impairment) Assess impact of drug treatment Noninvasive assistance in AD studies. Solve Ordinary Differential System of Equations. Use basic Inverse Problems, Optimization and Statistics. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 5 / 50

17 Goals/Methods of the Study Improved Sensitivity for identifying features in images: Identify anomalies in single images. Identify changes over time: Longitudinal Studies of AD Assessing disease state- AD or MCI (mild cognitive impairment) Assess impact of drug treatment Noninvasive assistance in AD studies. Solve Ordinary Differential System of Equations. Use basic Inverse Problems, Optimization and Statistics. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 5 / 50

18 Goals/Methods of the Study Improved Sensitivity for identifying features in images: Identify anomalies in single images. Identify changes over time: Longitudinal Studies of AD Assessing disease state- AD or MCI (mild cognitive impairment) Assess impact of drug treatment Noninvasive assistance in AD studies. Solve Ordinary Differential System of Equations. Use basic Inverse Problems, Optimization and Statistics. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 5 / 50

19 Goals/Methods of the Study Improved Sensitivity for identifying features in images: Identify anomalies in single images. Identify changes over time: Longitudinal Studies of AD Assessing disease state- AD or MCI (mild cognitive impairment) Assess impact of drug treatment Noninvasive assistance in AD studies. Solve Ordinary Differential System of Equations. Use basic Inverse Problems, Optimization and Statistics. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 5 / 50

20 PET Imaging System Input, Response and Output Input Tracer Density in plasma u(t) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 6 / 50

21 PET Imaging System Input, Response and Output Brain Tissue: Impulse Response Function (IRF) h(t, K 1, k 2, k 3, k 4 ) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 6 / 50

22 PET Imaging System Input, Response and Output Output: Tissue time activity curve y(t) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 6 / 50

23 PET Imaging System Input, Response and Output Convolution Solution of Differential Equation y(t)=u h Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 6 / 50

24 FDG-PET Compartmental Modelling Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 7 / 50

25 The Differential System of Equations ẏ 1 = K 1 u(t) (k 2 + k 3 )y 1 (t) + k 4 y 2 (t) ẏ 2 = k 3 y 1 (t) k 4 y 2 (t). (1) K 1 and k 2 FDG transport rate k 3 and k 4 phosphorylation and dephosphorylation rate. K LCMRglc: 1 k 3 C p k 2 +k 3 LC = K Cp LC. y 1 (t) is the FDG in tissue y 2 (t) the phosphorylated FDG in tissue. u(t) is the FDG input in plasma. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 8 / 50

26 The Differential System of Equations ẏ 1 = K 1 u(t) (k 2 + k 3 )y 1 (t) + k 4 y 2 (t) ẏ 2 = k 3 y 1 (t) k 4 y 2 (t). (1) K 1 and k 2 FDG transport rate k 3 and k 4 phosphorylation and dephosphorylation rate. K LCMRglc: 1 k 3 C p k 2 +k 3 LC = K Cp LC. y 1 (t) is the FDG in tissue y 2 (t) the phosphorylated FDG in tissue. u(t) is the FDG input in plasma. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 8 / 50

27 The Differential System of Equations ẏ 1 = K 1 u(t) (k 2 + k 3 )y 1 (t) + k 4 y 2 (t) ẏ 2 = k 3 y 1 (t) k 4 y 2 (t). (1) K 1 and k 2 FDG transport rate k 3 and k 4 phosphorylation and dephosphorylation rate. K LCMRglc: 1 k 3 C p k 2 +k 3 LC = K Cp LC. y 1 (t) is the FDG in tissue y 2 (t) the phosphorylated FDG in tissue. u(t) is the FDG input in plasma. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 8 / 50

28 The Differential System of Equations ẏ 1 = K 1 u(t) (k 2 + k 3 )y 1 (t) + k 4 y 2 (t) ẏ 2 = k 3 y 1 (t) k 4 y 2 (t). (1) K 1 and k 2 FDG transport rate k 3 and k 4 phosphorylation and dephosphorylation rate. K LCMRglc: 1 k 3 C p k 2 +k 3 LC = K Cp LC. y 1 (t) is the FDG in tissue y 2 (t) the phosphorylated FDG in tissue. u(t) is the FDG input in plasma. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 8 / 50

29 The Differential System of Equations ẏ 1 = K 1 u(t) (k 2 + k 3 )y 1 (t) + k 4 y 2 (t) ẏ 2 = k 3 y 1 (t) k 4 y 2 (t). (1) K 1 and k 2 FDG transport rate k 3 and k 4 phosphorylation and dephosphorylation rate. K LCMRglc: 1 k 3 C p k 2 +k 3 LC = K Cp LC. y 1 (t) is the FDG in tissue y 2 (t) the phosphorylated FDG in tissue. u(t) is the FDG input in plasma. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 8 / 50

30 Standard Nonlinear Estimation Method Method Given u(t) and y(t) = y 1 (t) + y 2 (t), estimate K 1, k 2, k 3, k 4 and K. Let k 4 = 0, the solution of the system is easily found and we have y(t) = u(t) h(t, K 1, k 2, k 3 ) ( K1 k 3 = u(t) + K ) 1k 2 e (k 2+k 3 )t. k 2 + k 3 k 2 + k 3 This gives the nonlinear least squares estimation min K 1,k 2,k 3 y(t) u(t) h(t, K 1, k 2, k 3 ) W. Here W indicates that this is in a weighted norm. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 9 / 50

31 Obtain Input Function by Blood Sampling Gold Standard: arterial sampling causes discomfort. There are potential risks, e.g.,arterial thrombosis, arterial sclerosis, and ischemia to the extremity. Arterialized venous sampling method, the limb is heated to avoid discomfort,(phelps et al, 1979). Still requires frequent blood sampling. Population based input function, (Takikawa et al, 1993 and Erberl et al, 1997). Not individual specific. Challenge Reduce/avoid blood sampling. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 10 / 50

32 Obtain Input Function by Blood Sampling Gold Standard: arterial sampling causes discomfort. There are potential risks, e.g.,arterial thrombosis, arterial sclerosis, and ischemia to the extremity. Arterialized venous sampling method, the limb is heated to avoid discomfort,(phelps et al, 1979). Still requires frequent blood sampling. Population based input function, (Takikawa et al, 1993 and Erberl et al, 1997). Not individual specific. Challenge Reduce/avoid blood sampling. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 10 / 50

33 Obtain Input Function by Blood Sampling Gold Standard: arterial sampling causes discomfort. There are potential risks, e.g.,arterial thrombosis, arterial sclerosis, and ischemia to the extremity. Arterialized venous sampling method, the limb is heated to avoid discomfort,(phelps et al, 1979). Still requires frequent blood sampling. Population based input function, (Takikawa et al, 1993 and Erberl et al, 1997). Not individual specific. Challenge Reduce/avoid blood sampling. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 10 / 50

34 Obtain Input Function by Blood Sampling Gold Standard: arterial sampling causes discomfort. There are potential risks, e.g.,arterial thrombosis, arterial sclerosis, and ischemia to the extremity. Arterialized venous sampling method, the limb is heated to avoid discomfort,(phelps et al, 1979). Still requires frequent blood sampling. Population based input function, (Takikawa et al, 1993 and Erberl et al, 1997). Not individual specific. Challenge Reduce/avoid blood sampling. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 10 / 50

35 Using the Region of Interest (ROI): Carotid Arterial ROI TAC and Blood Samples Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 11 / 50

36 Recent Representative Methods Without blood samples Image-derived input function: no blood samples. Blood ROIs are identified by aligning MR and PET, and average blood TAC, v(t), is used. (Litton, 1997, Liptrot et al, 2004, Wahl et al, 1999). Simultaneous Estimation of Input and Output(SIME): u(t) = (A 1 t A 2 A 3 )e λ 1t + A 2 e λ 2t + A 3 e λ 3t Let P m = {k 1,, k 4 } for m th tissue TACS, and solve (6 input parameters) (Feng et al,1997). min M A i,λ j,p m m=1 n=1 N { } 2 w mn ym (t n ) (h m (t, P m ) u(t, A i, λ j )) n. Difficulties: former requires MR images and registration, latter assumes y not contaminated by partial volume or spillover. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 12 / 50

37 Recent Representative Methods Without blood samples Image-derived input function: no blood samples. Blood ROIs are identified by aligning MR and PET, and average blood TAC, v(t), is used. (Litton, 1997, Liptrot et al, 2004, Wahl et al, 1999). Simultaneous Estimation of Input and Output(SIME): u(t) = (A 1 t A 2 A 3 )e λ 1t + A 2 e λ 2t + A 3 e λ 3t Let P m = {k 1,, k 4 } for m th tissue TACS, and solve (6 input parameters) (Feng et al,1997). min M A i,λ j,p m m=1 n=1 N { } 2 w mn ym (t n ) (h m (t, P m ) u(t, A i, λ j )) n. Difficulties: former requires MR images and registration, latter assumes y not contaminated by partial volume or spillover. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 12 / 50

38 Recent Representative Methods Without blood samples Image-derived input function: no blood samples. Blood ROIs are identified by aligning MR and PET, and average blood TAC, v(t), is used. (Litton, 1997, Liptrot et al, 2004, Wahl et al, 1999). Simultaneous Estimation of Input and Output(SIME): u(t) = (A 1 t A 2 A 3 )e λ 1t + A 2 e λ 2t + A 3 e λ 3t Let P m = {k 1,, k 4 } for m th tissue TACS, and solve (6 input parameters) (Feng et al,1997). min M A i,λ j,p m m=1 n=1 N { } 2 w mn ym (t n ) (h m (t, P m ) u(t, A i, λ j )) n. Difficulties: former requires MR images and registration, latter assumes y not contaminated by partial volume or spillover. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 12 / 50

39 Using a Limited Number of Blood Samples SIME based method, (Sanabria-Bohorquez et al, 2003), Let u(t) = sf F (t) v(t) + A te Bt F(t) = a e bt + 1 a where F(t) models the [ 11 C]-FMZ fraction in plasma, b is decay, and sf and a are determined using three late blood samples. Then A and B are calculated by SIME with information from tissue TACs. Linear relationship based method: (Chen et al, 1998, 2007), assume v(t) = α u(t) + β y(t). Solve for α and β by using three late venous samples to correct partial volume and spillover effects. Proposed method (Guo et al). Model late portion by A e λ(t τ)δ and early portion by correcting the partial volume of v(t). Do not use the later portion of v(t). Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 13 / 50

40 Real Data FDG-PET data from 951/31 ECAT for 18 healthy subjects. Images are reconstructed by filtered back projection. Each reconstructed data set includes 31 slices with 3.375mm separation and each slice has voxels with resolution of approximately 9.5 mm full width at half maximum (FWHM). The scaning time durations in minutes for the frames are 0.2, , , 0.2, 0.5, 2 1, 2 1.5, 3.5, 2 5, 10 and 30. Sequential arterial blood samples are drawn every 5 seconds for the first minute, every 10 seconds for the second minute, every 30 seconds for the next 2 minutes, the 5, 6, 8, 10, 12, 15, 20, 25, 30, 40, 50 and 60 minutes. We represent the blood samples by u bs (t j ), j = 1, 2,, 34. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 14 / 50

41 Compare CA-ROI TAC with Blood Samples Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 15 / 50

42 Proposed Model for the input function Formulation u e (t, θ, λ, δ) = { θ v(t) t [0, τ] (Window 1) θ v(τ) e λ(t τ)δ t [τ, T ] (Window 2) 1 τ separates W 1 and W 2. On W 1, the curve is high quality, spillover can be ignored. On W 2 use the analytic formula. τ is a parameter that has to be determined so as to trade off between the best characteristics of the two windows. 2 θ v(τ) assures continuity at τ. 3 Blood samples at t = 10, 20 and 60 min. are used in fitting to obtain the model s parameters. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 16 / 50

43 Rationale Compare the proposed formulation with two standard models u Phelps = A 1 e λ1(t τ) + A 2 e λ2(t τ) + A 3 e λ3(t τ), u Feng = (A 1 (t τ 0 ) A 2 A 3 )e λ 1(t τ 0 ) + A 2 e λ 2(t τ 0 ) + A 3 e λ 3(t τ 0 ) Fit blood samples on W 2 using Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 17 / 50

44 Rationale Compare the proposed formulation with two standard models u Phelps = A 1 e λ1(t τ) + A 2 e λ2(t τ) + A 3 e λ3(t τ), u Feng = (A 1 (t τ 0 ) A 2 A 3 )e λ 1(t τ 0 ) + A 2 e λ 2(t τ 0 ) + A 3 e λ 3(t τ 0 ) Fit blood samples on W 2 using Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 17 / 50

45 Rationale Compare the proposed formulation with two standard models u Phelps = A 1 e λ1(t τ) + A 2 e λ2(t τ) + A 3 e λ3(t τ), u Feng = (A 1 (t τ 0 ) A 2 A 3 )e λ 1(t τ 0 ) + A 2 e λ 2(t τ 0 ) + A 3 e λ 3(t τ 0 ) Fit blood samples on W 2 using all available nodes t i τ Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 17 / 50

46 Rationale Compare the proposed formulation with two standard models u Phelps = A 1 e λ1(t τ) + A 2 e λ2(t τ) + A 3 e λ3(t τ), u Feng = (A 1 (t τ 0 ) A 2 A 3 )e λ 1(t τ 0 ) + A 2 e λ 2(t τ 0 ) + A 3 e λ 3(t τ 0 ) Fit blood samples on W 2 using just using nodes at τ and 10, 20, and 60 min.. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 17 / 50

47 Extrapolation for Different Subjects for t < 10 Fit the proposed formula by blood samples at t = 10, 20 and 60 min. to estimate parameters θ, λ and δ. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 18 / 50

48 Extrapolation for Different Subjects for t < 10 Fit the proposed formula by blood samples at t = 10, 20 and 60 min. to estimate parameters θ, λ and δ. θ = Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 18 / 50

49 Extrapolation for Different Subjects for t < 10 Fit the proposed formula by blood samples at t = 10, 20 and 60 min. to estimate parameters θ, λ and δ. θ = Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 18 / 50

50 Interpolation and Picking θ Fit at t = 10, 20 and 60 min. and [τ, θ v(τ)]. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 19 / 50

51 Interpolation and Picking θ Fit at t = 10, 20 and 60 min. and [τ, θ v(τ)]. θ = 6 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 19 / 50

52 Interpolation and Picking θ Fit at t = 10, 20 and 60 min. and [τ, θ v(τ)]. θ = 2.8 θ is the only independent parameter, once a good θ is obtained, the whole curve is recovered. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 19 / 50

53 Overview of fitting: The choice of θ is important θ must be chosen to fit the measured data Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 20 / 50

54 Overview of fitting: The choice of θ is important θ must be chosen to fit the measured data Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 20 / 50

55 Overview of fitting: The choice of θ is important θ must be chosen to fit the measured data Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 20 / 50

56 Overview of fitting: The choice of θ is important θ must be chosen to fit the measured data Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 20 / 50

57 θ is the Only Independent Parameter Method For a given value of θ, parameters λ, δ are determined by [λ(θ), δ(θ)] = argmin λ,δ 3 i=1 [ θv(τ)e λ(t i τ) δ u bs (t i )] 2. where (t i, u bs (t i )), i = 1, 2, 3 are venous blood samples. θ is the inverse of the recovery coefficient, and measures the ability to recover the actual blood data from the brain measurement. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 21 / 50

58 θ is the Only Independent Parameter Method For a given value of θ, parameters λ, δ are determined by [λ(θ), δ(θ)] = argmin λ,δ 3 i=1 [ θv(τ)e λ(t i τ) δ u bs (t i )] 2. where (t i, u bs (t i )), i = 1, 2, 3 are venous blood samples. θ is the inverse of the recovery coefficient, and measures the ability to recover the actual blood data from the brain measurement. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 21 / 50

59 Nonlinear Method Method (Here using two TTAC curves obtained by clustering) min θ,p 1,P 2,α i subject to 2 i=1 j=1 n [ w ij yi (t j ) α i (h i u e ) j (1 α i ) u e (t j, θ) ] 2, K (i) 1 0.3, k (i) , 0.01 k (i) 3 0.2, 0.9 α i 1, 1.2 θ 4 1 Weight w ij = t j /y i (t j ) 2 Parameter α i corrects spillover from blood to tissue 3 Tissue clusters y i (t) are generated by a fast PET data clustering algorithm, (Guo et al, 2003). Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 22 / 50

60 Nonlinear Method Method (Here using two TTAC curves obtained by clustering) min θ,p 1,P 2,α i subject to 2 i=1 j=1 n [ w ij yi (t j ) α i (h i u e ) j (1 α i ) u e (t j, θ) ] 2, K (i) 1 0.3, k (i) , 0.01 k (i) 3 0.2, 0.9 α i 1, 1.2 θ 4 1 Weight w ij = t j /y i (t j ) 2 Parameter α i corrects spillover from blood to tissue 3 Tissue clusters y i (t) are generated by a fast PET data clustering algorithm, (Guo et al, 2003). Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 22 / 50

61 Nonlinear Method Method (Here using two TTAC curves obtained by clustering) min θ,p 1,P 2,α i subject to 2 i=1 j=1 n [ w ij yi (t j ) α i (h i u e ) j (1 α i ) u e (t j, θ) ] 2, K (i) 1 0.3, k (i) , 0.01 k (i) 3 0.2, 0.9 α i 1, 1.2 θ 4 1 Weight w ij = t j /y i (t j ) 2 Parameter α i corrects spillover from blood to tissue 3 Tissue clusters y i (t) are generated by a fast PET data clustering algorithm, (Guo et al, 2003). Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 22 / 50

62 Nonlinear Method Method (Here using two TTAC curves obtained by clustering) min θ,p 1,P 2,α i subject to 2 i=1 j=1 n [ w ij yi (t j ) α i (h i u e ) j (1 α i ) u e (t j, θ) ] 2, K (i) 1 0.3, k (i) , 0.01 k (i) 3 0.2, 0.9 α i 1, 1.2 θ 4 1 Weight w ij = t j /y i (t j ) 2 Parameter α i corrects spillover from blood to tissue 3 Tissue clusters y i (t) are generated by a fast PET data clustering algorithm, (Guo et al, 2003). Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 22 / 50

63 Representative Results Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 23 / 50

64 Representative Results Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 23 / 50

65 Representative Results Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 23 / 50

66 Representative Results Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 23 / 50

67 Tabulated Results: Calculation compared to Patlak straight line fit Table: Linear regression: K calculated by the Patlak method for 50 clusters of each subject, comparing blood samples u bs and estimated input function u e. Subject u bs mean K u e mean K Correlation Slope Intercept ± standard deviation ± standard deviation Coefficient Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 24 / 50

68 Tabulated Results: Calculation compared to Patlak straight line fit Table: Linear regression: K calculated by the Patlak method for 50 clusters of each subject, comparing blood samples u bs and estimated input function u e. Subject u bs mean K u e mean K Correlation Slope Intercept ± standard deviation ± standard deviation Coefficient e 02 ± 7.4e e 02 ± 7.4e e e 04 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 24 / 50

69 Tabulated Results: Calculation compared to Patlak straight line fit Table: Linear regression: K calculated by the Patlak method for 50 clusters of each subject, comparing blood samples u bs and estimated input function u e. Subject u bs mean K u e mean K Correlation Slope Intercept ± standard deviation ± standard deviation Coefficient e 02 ± 7.4e e 02 ± 7.4e e e e 02 ± 4.9e e 02 ± 5.0e e e 03 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 24 / 50

70 Tabulated Results: Calculation compared to Patlak straight line fit Table: Linear regression: K calculated by the Patlak method for 50 clusters of each subject, comparing blood samples u bs and estimated input function u e. Subject u bs mean K u e mean K Correlation Slope Intercept ± standard deviation ± standard deviation Coefficient e 02 ± 7.4e e 02 ± 7.4e e e e 02 ± 4.9e e 02 ± 5.0e e e e 02 ± 5.6e e 02 ± 5.7e e e e 02 ± 3.7e e 02 ± 3.7e e e e 02 ± 5.9e e 02 ± 5.8e e e e 02 ± 6.3e e 02 ± 6.1e e e e 02 ± 5.0e e 02 ± 4.9e e e e 02 ± 5.0e e 02 ± 5.1e e e e 02 ± 5.1e e 02 ± 5.3e e e e 02 ± 4.7e e 02 ± 4.8e e e e 02 ± 7.8e e 02 ± 7.8e e e e 02 ± 4.2e e 02 ± 4.6e e e e 02 ± 6.4e e 02 ± 6.4e e e e 02 ± 1.1e e 02 ± 1.0e e e e 02 ± 6.7e e 02 ± 6.7e e e e 02 ± 5.3e e 02 ± 5.7e e e e 02 ± 4.9e e 02 ± 4.9e e e e 02 ± 4.6e e 02 ± 4.8e e e 04 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 24 / 50

71 Regression for K Over All Subjects Linear regression for K calculated with blood samples u bs and estimated input function u e using the Patlak method. Correlation , slope.9878 and intercept 7.1e 04. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 25 / 50

72 Regression for Individual Rate Constants Parameters for one subject. The correlation coefficients are , , and Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 26 / 50

73 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

74 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

75 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

76 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

77 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

78 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

79 Observations: Very Simple Problem requiring basic mathematics Mathematical Techniques used 1 Solution of simple ODEs. 2 Nonlinear fit - two options, one a simple nonlinear fit. 3 Optimization. 4 Statistics: Clustering and Regression. 5 Image analysis 6 Matlab programming. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 27 / 50

80 More About PET images Recall blurring of standard PET images: Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 28 / 50

81 More About PET images Recall blurring of standard PET images: Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 28 / 50

82 More About PET images Recall blurring of standard PET images: Assume spatially invariant blur: g=f*h+n given g and h with unknown n. g is the recorded image, f the unknown real image, h the point spread function(psf) and n unknown noise. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 28 / 50

83 More About PET images Recall blurring of standard PET images: Assume spatially invariant blur: g=f*h+n given g and h with unknown n. g is the recorded image, f the unknown real image, h the point spread function(psf) and n unknown noise. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 28 / 50

84 More About PET images Recall blurring of standard PET images: Assume spatially invariant blur: g=f*h+n given g and h with unknown n. g is the recorded image, f the unknown real image, h the point spread function(psf) and n unknown noise. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 28 / 50

85 Inverse Problem Goal Given g find f Problem is ill-posed. We cannot invert to find f. Regularization is needed. ˆf = arg min { g f h λr(f )} f R(f ) is a regularization term which picks a best image according to desired properties. Difficulty images cannot be improved prior to kinetic estimation, because values change. Benefit May show anatomical change without quantification. Important to identify edges. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 29 / 50

86 Inverse Problem Goal Given g find f Problem is ill-posed. We cannot invert to find f. Regularization is needed. ˆf = arg min { g f h λr(f )} f R(f ) is a regularization term which picks a best image according to desired properties. Difficulty images cannot be improved prior to kinetic estimation, because values change. Benefit May show anatomical change without quantification. Important to identify edges. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 29 / 50

87 Inverse Problem Goal Given g find f Problem is ill-posed. We cannot invert to find f. Regularization is needed. ˆf = arg min { g f h λr(f )} f R(f ) is a regularization term which picks a best image according to desired properties. Difficulty images cannot be improved prior to kinetic estimation, because values change. Benefit May show anatomical change without quantification. Important to identify edges. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 29 / 50

88 Inverse Problem Goal Given g find f Problem is ill-posed. We cannot invert to find f. Regularization is needed. ˆf = arg min { g f h λr(f )} f R(f ) is a regularization term which picks a best image according to desired properties. Difficulty images cannot be improved prior to kinetic estimation, because values change. Benefit May show anatomical change without quantification. Important to identify edges. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 29 / 50

89 Inverse Problem Goal Given g find f Problem is ill-posed. We cannot invert to find f. Regularization is needed. ˆf = arg min { g f h λr(f )} f R(f ) is a regularization term which picks a best image according to desired properties. Difficulty images cannot be improved prior to kinetic estimation, because values change. Benefit May show anatomical change without quantification. Important to identify edges. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 29 / 50

90 Inverse Problem Goal Given g find f Problem is ill-posed. We cannot invert to find f. Regularization is needed. ˆf = arg min { g f h λr(f )} f R(f ) is a regularization term which picks a best image according to desired properties. Difficulty images cannot be improved prior to kinetic estimation, because values change. Benefit May show anatomical change without quantification. Important to identify edges. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 29 / 50

91 Regularization Methods (short overview) Common methods are Tikhonov (TK). R(f ) = TK(f ) = f (x) 2 dx. Total Variation (TV) R(f ) = TV(f ) = Ω Ω f (x) dx. Sparse deconvolution (L 1 ) (not relevant for PET images) R(f ) = f 1 = f (x) dx. Ω Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 30 / 50

92 Regularization Methods (short overview) Common methods are Tikhonov (TK). R(f ) = TK(f ) = f (x) 2 dx. Total Variation (TV) R(f ) = TV(f ) = Ω Ω f (x) dx. Sparse deconvolution (L 1 ) (not relevant for PET images) R(f ) = f 1 = f (x) dx. Ω Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 30 / 50

93 Regularization Methods (short overview) Common methods are Tikhonov (TK). R(f ) = TK(f ) = f (x) 2 dx. Total Variation (TV) R(f ) = TV(f ) = Ω Ω f (x) dx. Sparse deconvolution (L 1 ) (not relevant for PET images) R(f ) = f 1 = f (x) dx. Ω Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 30 / 50

94 Regularization Review ˆf = arg min { g f h λr(f )} f λ Governs the trade off between the fit to the data and the smoothness of the reconstruction and can be picked by the L-curve approach, (see Hansen, Inverse Problems) TV yields a piece wise constant reconstruction and preserves the edges of the image. TK yields a smooth reconstruction. L1 yields spike trains Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 31 / 50

95 Regularization Review ˆf = arg min { g f h λr(f )} f λ Governs the trade off between the fit to the data and the smoothness of the reconstruction and can be picked by the L-curve approach, (see Hansen, Inverse Problems) TV yields a piece wise constant reconstruction and preserves the edges of the image. TK yields a smooth reconstruction. L1 yields spike trains Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 31 / 50

96 Regularization Review ˆf = arg min { g f h λr(f )} f λ Governs the trade off between the fit to the data and the smoothness of the reconstruction and can be picked by the L-curve approach, (see Hansen, Inverse Problems) TV yields a piece wise constant reconstruction and preserves the edges of the image. TK yields a smooth reconstruction. L1 yields spike trains Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 31 / 50

97 Regularization Review ˆf = arg min { g f h λr(f )} f λ Governs the trade off between the fit to the data and the smoothness of the reconstruction and can be picked by the L-curve approach, (see Hansen, Inverse Problems) TV yields a piece wise constant reconstruction and preserves the edges of the image. TK yields a smooth reconstruction. L1 yields spike trains Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 31 / 50

98 Optimization Objective function of the data fit term is convex TK is a linear least squares (LS) problem ˆf = arg min f { g Hf λ f 2 2 )} The TV objective function is non differentiable J(f ) = g Hf λ f 1 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 32 / 50

99 Optimization Objective function of the data fit term is convex TK is a linear least squares (LS) problem ˆf = arg min f { g Hf λ f 2 2 )} The TV objective function is non differentiable J(f ) = g Hf λ f 1 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 32 / 50

100 Optimization Objective function of the data fit term is convex TK is a linear least squares (LS) problem ˆf = arg min f { g Hf λ f 2 2 )} The TV objective function is non differentiable J(f ) = g Hf λ f 1 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 32 / 50

101 Differentiability of TV - 1D (tensor product in 2D) R(f ) = i f i+1 f i For a small β define R β = i (f i+1 f i ) 2 + β choose β in 10 5 to 10 9 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 33 / 50

102 Differentiability of TV - 1D (tensor product in 2D) R(f ) = i f i+1 f i For a small β define R β = i (f i+1 f i ) 2 + β choose β in 10 5 to 10 9 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 33 / 50

103 Numerical optimization scheme To find the minimum we use a limited memory BFGS (see Vogel, Computational Methods for Inverse Problems and Nocedal) A quasi Newton Method where the estimated Hessian in each step is updated by a rank 2 update. Only a limited number of update vectors are kept, e.g. 10. Evaluation of the OF and its gradient is cheap (some FFTs and sparse matrix-vector multiplications) Problems are usually large and many iterations are needed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 34 / 50

104 Numerical optimization scheme To find the minimum we use a limited memory BFGS (see Vogel, Computational Methods for Inverse Problems and Nocedal) A quasi Newton Method where the estimated Hessian in each step is updated by a rank 2 update. Only a limited number of update vectors are kept, e.g. 10. Evaluation of the OF and its gradient is cheap (some FFTs and sparse matrix-vector multiplications) Problems are usually large and many iterations are needed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 34 / 50

105 Numerical optimization scheme To find the minimum we use a limited memory BFGS (see Vogel, Computational Methods for Inverse Problems and Nocedal) A quasi Newton Method where the estimated Hessian in each step is updated by a rank 2 update. Only a limited number of update vectors are kept, e.g. 10. Evaluation of the OF and its gradient is cheap (some FFTs and sparse matrix-vector multiplications) Problems are usually large and many iterations are needed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 34 / 50

106 Numerical optimization scheme To find the minimum we use a limited memory BFGS (see Vogel, Computational Methods for Inverse Problems and Nocedal) A quasi Newton Method where the estimated Hessian in each step is updated by a rank 2 update. Only a limited number of update vectors are kept, e.g. 10. Evaluation of the OF and its gradient is cheap (some FFTs and sparse matrix-vector multiplications) Problems are usually large and many iterations are needed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 34 / 50

107 Numerical optimization scheme To find the minimum we use a limited memory BFGS (see Vogel, Computational Methods for Inverse Problems and Nocedal) A quasi Newton Method where the estimated Hessian in each step is updated by a rank 2 update. Only a limited number of update vectors are kept, e.g. 10. Evaluation of the OF and its gradient is cheap (some FFTs and sparse matrix-vector multiplications) Problems are usually large and many iterations are needed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 34 / 50

108 Point Spread Function The PSF is usually unknown or only estimated Estimates exist for PET scanners from phantom scans Actually PSF is spatially variant and also depends on the scanned object Hence even if provided PSF is always only an estimate For the PET scans presented here, a 6mm half width Gaussian was assumed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 35 / 50

109 Point Spread Function The PSF is usually unknown or only estimated Estimates exist for PET scanners from phantom scans Actually PSF is spatially variant and also depends on the scanned object Hence even if provided PSF is always only an estimate For the PET scans presented here, a 6mm half width Gaussian was assumed. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 35 / 50

110 Simulated PET On the left simulated PET from blurred segmented MRI scan using Gaussian PSF and noise added. On the right deblurred PET with TV and known PSF. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 36 / 50

111 Simulated PET On the left simulated PET from blurred segmented MRI scan using Gaussian PSF and noise added. On the right deblurred PET with TV and known PSF. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 36 / 50

112 Recover real PET image Reconstruction done using Filtered Back Projection PSF estimated by a Gaussian TV regularization Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 37 / 50

113 Observations Image improvement is possible even with a rough estimation of the PSF (non-blind deconvolution) Total Variation regularization (piecewise constant solution) is appropriate: intensity levels depend on the tissue type. Improvement requires better approximation of the PSF Total Least Squares Increased Artifacts and noise. (More post processing can improve this) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 38 / 50

114 Observations Image improvement is possible even with a rough estimation of the PSF (non-blind deconvolution) Total Variation regularization (piecewise constant solution) is appropriate: intensity levels depend on the tissue type. Improvement requires better approximation of the PSF Total Least Squares Increased Artifacts and noise. (More post processing can improve this) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 38 / 50

115 Observations Image improvement is possible even with a rough estimation of the PSF (non-blind deconvolution) Total Variation regularization (piecewise constant solution) is appropriate: intensity levels depend on the tissue type. Improvement requires better approximation of the PSF Total Least Squares Increased Artifacts and noise. (More post processing can improve this) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 38 / 50

116 Observations Image improvement is possible even with a rough estimation of the PSF (non-blind deconvolution) Total Variation regularization (piecewise constant solution) is appropriate: intensity levels depend on the tissue type. Improvement requires better approximation of the PSF Total Least Squares Increased Artifacts and noise. (More post processing can improve this) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 38 / 50

117 Total least squares (TLS) Rewrite convolution as matrix vector product: g = Hf + n H is a Toeplitz matrix of Point Spread Function TLS assumes error in H and g i.e. g = (H + E)f + n Total least squares solution f TLS solves min E n 2 F subject to g = (H + E)f + n f TLS can be found from SVD of [H, g] (Golub et al) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 39 / 50

118 Total least squares (TLS) Rewrite convolution as matrix vector product: g = Hf + n H is a Toeplitz matrix of Point Spread Function TLS assumes error in H and g i.e. g = (H + E)f + n Total least squares solution f TLS solves min E n 2 F subject to g = (H + E)f + n f TLS can be found from SVD of [H, g] (Golub et al) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 39 / 50

119 Total least squares (TLS) Rewrite convolution as matrix vector product: g = Hf + n H is a Toeplitz matrix of Point Spread Function TLS assumes error in H and g i.e. g = (H + E)f + n Total least squares solution f TLS solves min E n 2 F subject to g = (H + E)f + n f TLS can be found from SVD of [H, g] (Golub et al) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 39 / 50

120 Total least squares (TLS) Rewrite convolution as matrix vector product: g = Hf + n H is a Toeplitz matrix of Point Spread Function TLS assumes error in H and g i.e. g = (H + E)f + n Total least squares solution f TLS solves min E n 2 F subject to g = (H + E)f + n f TLS can be found from SVD of [H, g] (Golub et al) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 39 / 50

121 Rayleigh Quotient Formulation The TLS solution minimizes Rayleigh Quotient: Include regularization: min f Hf g f 2 2 min f Hf g f λr(f ) p=2 e.g. Golub et al (1999), Renaut et al (2005) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 40 / 50

122 Rayleigh Quotient Formulation The TLS solution minimizes Rayleigh Quotient: Include regularization: min f Hf g f 2 2 min f Hf g f λr(f ) p=2 e.g. Golub et al (1999), Renaut et al (2005) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 40 / 50

123 Rayleigh Quotient Formulation The TLS solution minimizes Rayleigh Quotient: Include regularization: min f Hf g f 2 2 min f Hf g f λr(f ) p=2 e.g. Golub et al (1999), Renaut et al (2005) Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 40 / 50

124 Scaled TLS: different noise levels Theory (Paige and Strakos, Numerische Mathematik) min E n 2 F subject to g = (H + E)f + n γ Minimum is obtained as the minimum singular value of [H, γg] For flexibility use Rayleigh quotient formulation min f Hf g γ 2 f 2 2 γ = 0 is the LS problem γ = 1 is the standard TLS problem γ accounts for different noise levels in H and g. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 41 / 50

125 Scaled TLS: different noise levels Theory (Paige and Strakos, Numerische Mathematik) min E n 2 F subject to g = (H + E)f + n γ Minimum is obtained as the minimum singular value of [H, γg] For flexibility use Rayleigh quotient formulation min f Hf g γ 2 f 2 2 γ = 0 is the LS problem γ = 1 is the standard TLS problem γ accounts for different noise levels in H and g. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 41 / 50

126 Scaled TLS: different noise levels Theory (Paige and Strakos, Numerische Mathematik) min E n 2 F subject to g = (H + E)f + n γ Minimum is obtained as the minimum singular value of [H, γg] For flexibility use Rayleigh quotient formulation min f Hf g γ 2 f 2 2 γ = 0 is the LS problem γ = 1 is the standard TLS problem γ accounts for different noise levels in H and g. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 41 / 50

127 Scaled TLS: different noise levels Theory (Paige and Strakos, Numerische Mathematik) min E n 2 F subject to g = (H + E)f + n γ Minimum is obtained as the minimum singular value of [H, γg] For flexibility use Rayleigh quotient formulation min f Hf g γ 2 f 2 2 γ = 0 is the LS problem γ = 1 is the standard TLS problem γ accounts for different noise levels in H and g. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 41 / 50

128 Scaled Total Least Squares with Regularization Regularize scaled RQ: Hf g 2 2 min f 1 + γ 2 f 2 + λr(f ) 2 Permits careful investigation of effect of noise levels in H and g. Which is greater, the error in the PSF or the error in the measured data? Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 42 / 50

129 Scaled Total Least Squares with Regularization Regularize scaled RQ: Hf g 2 2 min f 1 + γ 2 f 2 + λr(f ) 2 Permits careful investigation of effect of noise levels in H and g. Which is greater, the error in the PSF or the error in the measured data? Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 42 / 50

130 Scaled Total Least Squares with Regularization Regularize scaled RQ: Hf g 2 2 min f 1 + γ 2 f 2 + λr(f ) 2 Permits careful investigation of effect of noise levels in H and g. Which is greater, the error in the PSF or the error in the measured data? Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 42 / 50

131 Test Problem Noisy Shepp Logan Phantom Shepp Logan Phantom Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 43 / 50

132 Test Problem Noisy Shepp Logan Phantom x 2 e 2σ 2 Blur with Gaussian h(x) = 1 σ 2π width) Take forward Radon transform with 45 angles Add Poisson Noise to sinogram Transform back, with filtered back projection with σ = 1.5 (6mm half Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 43 / 50

133 Deconvolving the Shepp-Logan Phantom Gauss PSF with σ = 2 and TV regularization γ 2 = 0 γ 2 = 7.3e 9 γ 2 = 3.7e 7 Scaling shows how to improve impact of badly chosen PSF. Scaling Hf = g = 1 (Notice for given image g ) For scaled problem γ 2 is 1, 50, resp. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 44 / 50

134 Deconvolving the Shepp-Logan Phantom Gauss PSF with σ = 2 and TV regularization γ 2 = 0 γ 2 = 7.3e 9 γ 2 = 3.7e 7 Scaling shows how to improve impact of badly chosen PSF. Scaling Hf = g = 1 (Notice for given image g ) For scaled problem γ 2 is 1, 50, resp. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 44 / 50

135 Deconvolving the Shepp-Logan Phantom Gauss PSF with σ = 2 and TV regularization γ 2 = 0 γ 2 = 7.3e 9 γ 2 = 3.7e 7 Scaling shows how to improve impact of badly chosen PSF. Scaling Hf = g = 1 (Notice for given image g ) For scaled problem γ 2 is 1, 50, resp. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 44 / 50

136 Deconvolving the Shepp-Logan Phantom Gauss PSF with σ = 2 and TV regularization γ 2 = 0 γ 2 = 7.3e 9 γ 2 = 3.7e 7 Scaling shows how to improve impact of badly chosen PSF. Scaling Hf = g = 1 (Notice for given image g ) For scaled problem γ 2 is 1, 50, resp. Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 44 / 50

137 Real PET data Real PET data Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 45 / 50

138 Numerical Results Scaled RTVTLS Use a PSF with 6mm half width Gaussian and then restore γ = 0 Least Squares Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 46 / 50

139 Numerical Results Scaled RTVTLS Use a PSF with 6mm half width Gaussian and then restore γ = 3.7e 7 Total Least Squares Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 46 / 50

140 Numerical Results Scaled RTVTLS Use a PSF with 6mm half width Gaussian and then restore γ = 1.5e 5 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 46 / 50

141 Numerical Results Scaled RTVTLS Use a PSF with 6mm half width Gaussian and then restore γ = 1e 4 Renaut, Guo, Chen and Stefan (ASU) Information from PET Images GSU 46 / 50