Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion An ADMM algorithm for constrained material decomposition in spectral CT May, 23 th , 2018 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 1 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Spectral CT Spectral CT records energy-dependant data Log 10 of data 6 Bin 1 50 5 100 4 3 150 100 200 300 400 500 600 6 50 Bin 2 5 100 4 3 150 100 200 300 400 500 600 6 50 Bin 3 5 100 4 150 3 100 200 300 400 500 600 5.5 5 Bin 4 50 4.5 4 100 3.5 150 3 100 200 300 400 500 600 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 2 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Spectral CT Spectral CT records energy-dependant data Log 10 of data 6 Bin 1 50 5 100 4 Attenuation coefficients depend on energy 3 150 100 200 300 400 500 600 Soft tissues 10 2 Bones 6 Gadolinium 50 Bin 2 5 100 4 10 1 Attenuation 3 150 100 200 300 400 500 600 6 10 0 50 Bin 3 5 100 4 20 30 40 50 60 70 80 90 100 110 120 150 3 Energy 100 200 300 400 500 600 5.5 5 Bin 4 50 4.5 4 100 3.5 150 3 100 200 300 400 500 600 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 2 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Spectral CT Spectral CT records energy-dependant data Log 10 of data Allow decomposing materials 6 Bin 1 50 5 Material maps 100 4 Soft tissues 3 30 150 50 100 200 300 400 500 600 20 100 10 6 150 50 0 Bin 2 5 100 200 300 400 500 600 100 4 10 3 150 Bones 8 50 100 200 300 400 500 600 6 100 4 6 2 150 0 50 Bin 3 5 100 200 300 400 500 600 100 4 Gadolinium 0.2 150 3 50 0.15 100 200 300 400 500 600 0.1 100 5.5 0.05 150 5 0 Bin 4 50 100 200 300 400 500 600 4.5 4 100 3.5 150 3 100 200 300 400 500 600 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 2 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Spectral CT Spectral CT records energy-dependant data Log 10 of data Allow decomposing materials 6 Bin 1 50 5 Material maps 100 4 Soft tissues 3 30 150 50 100 200 300 400 500 600 20 100 10 6 150 50 0 Bin 2 5 100 200 300 400 500 600 100 4 Solve the non-convex − − − − − − − − − − − − − − − → 10 3 150 Bones 8 nonlinear problem s = F ( a ) 50 100 200 300 400 500 600 6 100 4 6 2 150 0 50 Bin 3 5 100 200 300 400 500 600 100 4 Gadolinium 0.2 150 3 50 0.15 100 200 300 400 500 600 0.1 100 5.5 0.05 150 5 0 Bin 4 50 100 200 300 400 500 600 4.5 4 100 3.5 150 3 100 200 300 400 500 600 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 2 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Introduction Previous work A Gauss-Newton algorithm was developed [1] + Fast convergence + Better decomposition error than first order algorithms + Spatially regularized – Can have negative values Gauss-Newton Soft tissues Bones Gadolinium 6 0.2 30 20 20 20 0.15 4 20 0.1 40 40 40 2 0.05 60 10 60 60 0 0 80 80 80 0 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 3 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Introduction Previous work A Gauss-Newton algorithm was developed [1] + Fast convergence + Better decomposition error than first order algorithms + Spatially regularized – Can have negative values Gauss-Newton Soft tissues Bones Gadolinium 6 0.2 30 20 20 20 0.15 4 20 0.1 40 40 40 2 0.05 60 10 60 60 0 0 80 80 80 0 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 Constrained algorithms Positivity constraints were added for materials decomposition Long and Fessler 2014 [2] Noh and Fessler 2009 [3] Sidky and Pan 2014 [4] Ding and Long 2017 [5] Lake Como Summer School Tom Hohweiller CREATIS Laboratory 3 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Introduction Previous work A Gauss-Newton algorithm was developed [1] + Fast convergence + Better decomposition error than first order algorithms + Spatially regularized – Can have negative values Gauss-Newton Soft tissues Bones Gadolinium 6 0.2 30 20 20 20 0.15 4 20 0.1 40 40 40 2 0.05 60 10 60 60 0 0 80 80 80 0 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 Constrained algorithms Positivity constraints were added for materials decomposition Long and Fessler 2014 [2] Noh and Fessler 2009 [3] First oder algorithms in projection and image domain Sidky and Pan 2014 [4] Ding and Long 2017 [5] Lake Como Summer School Tom Hohweiller CREATIS Laboratory 3 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Introduction Previous work A Gauss-Newton algorithm was developed [1] + Fast convergence + Better decomposition error than first order algorithms + Spatially regularized – Can have negative values Gauss-Newton Soft tissues Bones Gadolinium 6 0.2 30 20 20 20 0.15 4 20 0.1 40 40 40 2 0.05 60 10 60 60 0 0 80 80 80 0 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 Constrained algorithms Positivity constraints were added for materials decomposition Long and Fessler 2014 [2] Noh and Fessler 2009 [3] First oder algorithms in projection and image domain Sidky and Pan 2014 [4] Ding and Long 2017 [5] } Second order algorithm in image domain Lake Como Summer School Tom Hohweiller CREATIS Laboratory 3 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Proposed method Positivity / Inequality We propose to enforce the positivity of the materials maps Lake Como Summer School Tom Hohweiller CREATIS Laboratory 4 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Proposed method Positivity / Inequality We propose to enforce the positivity of the materials maps Equality Quantity of injected marker is known Should be retrieved in the decompositions Lake Como Summer School Tom Hohweiller CREATIS Laboratory 4 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Proposed method Positivity / Inequality We propose to enforce the positivity of the materials maps Equality Quantity of injected marker is known Should be retrieved in the decompositions Algorithm Use of an alternating direction method of multipliers (ADMM) : update of a with a second order algorithm Lake Como Summer School Tom Hohweiller CREATIS Laboratory 4 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Forward problem Denoting s ∈ R IP the data vector and a ∈ R MP the materials maps, we have : s = [ s 1 , 1 , . . . , s I , 1 , . . . , s I , P ] T a = [ a 1 , 1 , ..., a m , 1 , ..., a M , P ] T Lake Como Summer School Tom Hohweiller CREATIS Laboratory 5 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Forward problem Denoting s ∈ R IP the data vector and a ∈ R MP the materials maps, we have : s = [ s 1 , 1 , . . . , s I , 1 , . . . , s I , P ] T a = [ a 1 , 1 , ..., a m , 1 , ..., a M , P ] T Using the standard spectral CT forward model is : � M � � � n 0 s i , p = i ( E ) exp − a m , p τ m ( E ) d E R m = 1 with n 0 i ( E ) the effective spectrum, τ m ( E ) a function representing the material attenuation and a m , p the projected mass of material m on pixel p . Lake Como Summer School Tom Hohweiller CREATIS Laboratory 5 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion Inverse problem We propose to solve : � a ≥ 0 min C ( a , s ) s.t (1) � a p a m , p = c m with c m is the quantity of m th material. We chose C ( a , s ) = D ( a , s ) + α R R ( a ) where D ( a , s ) = || s − F ( a ) || 2 W R ( a ) = || ∆ a soft || 2 2 + ||∇ a bone || 1 + ||∇ a Gd || 1 Lake Como Summer School Tom Hohweiller CREATIS Laboratory 6 / 15
Introduction Material decomposition Constrained algorithm Numerical experiments Results Conclusion ADMM The Lagrangian function is minimized by an ADMM algorithm : L ( a , b ,α α α I , α E , s ) = D ( a , s ) + α R R ( a ) + H E ( a , α E ) + G E ( a ) + H I ( a , b ,α α α I ) + G I ( a , b ) + 1 ( b ) Lake Como Summer School Tom Hohweiller CREATIS Laboratory 7 / 15
Recommend
More recommend
