Determinantal sets, singularities and application to optimal control in medical imagery Bernard Bonnard 1 , 3 Jean-Charles Faugère 2 Alain Jacquemard 1 , 2 Mohab Safey El Din 2 Thibaut Verron 2 1 Institut de Mathématiques de Bourgogne, Dijon, France, UMR CNRS 5584 2 Sorbonne Universités, UPMC Univ Paris 06, CNRS, Inria Paris, PolSys team 3 Inria Sophia-Antipolis, McTao project ISSAC’16, 20 July 2016 1
Physical problem (N)MRI = (Nuclear) Magnetic Resonance Imagery 1. Apply a magnetic field to a body 2. Measure the radio waves emitted in reaction Goal = optimize the contrast = distinguish two biological matters from this measure ? � Bad contrast (no enhancement) Good contrast (enhanced) 2
Physical problem (N)MRI = (Nuclear) Magnetic Resonance Imagery 1. Apply a magnetic field to a body 2. Measure the radio waves emitted in reaction Goal = optimize the contrast = distinguish two biological matters from this measure ? � Bad contrast (no enhancement) Good contrast (enhanced) Known methods: ◮ inject contrast agents to the patient: potentially toxic ◮ make the field variable to exploit differences in relaxation times = ⇒ requires finding optimal settings depending on the relaxation parameters 2
Physical problem (N)MRI = (Nuclear) Magnetic Resonance Imagery 1. Apply a magnetic field to a body 2. Measure the radio waves emitted in reaction Goal = optimize the contrast = distinguish two biological matters from this measure ? � Bad contrast (no enhancement) Good contrast (enhanced) Examples of relaxation parameters: ◮ Water: γ = Γ = 0 . 01Hz ◮ Cerebrospinal fluid: γ = 0 . 02Hz, Γ = 0 . 10Hz ◮ Fat: γ = 0 . 15Hz, Γ = 0 . 31Hz 2
Numerical approach and computational problem The Bloch equations Saturation method � Find a path u so that after some time T : ˙ y i = − Γ i y i − uz i ◮ matter 1 saturated: y 1 ( T ) = z 1 ( T ) = 0 ˙ z i = − γ i ( 1 − z i )+ uy i ◮ matter 2 “maximized”: | ( y 2 ( T ) , z 2 ( T )) | maximal ( i = 1 , 2) Glaser’s team, 2012 : method from Optimal Control Theory 3
Numerical approach and computational problem The Bloch equations Saturation method � Find a path u so that after some time T : ˙ y i = − Γ i y i − uz i ◮ matter 1 saturated: y 1 ( T ) = z 1 ( T ) = 0 ˙ z i = − γ i ( 1 − z i )+ uy i ◮ matter 2 “maximized”: | ( y 2 ( T ) , z 2 ( T )) | maximal ( i = 1 , 2) Glaser’s team, 2012 : method from Optimal Control Theory Problem: analyze the behavior of the control through algebraic invariants ◮ Example: singular feedback control: u = D ′ D ( D , D ′ polynomials in y , z , γ , Γ ) ◮ Geometry of { D = 0 } ? ◮ Study of the singular points of { D = 0 } for each value of γ 1 , Γ 1 , γ 2 , Γ 2 ◮ Examples with water: [Bonnard, Chyba, Jacquemard, Marriott, 2013] ◮ Water/Fat : 1 point ◮ Water/Cerebrospinal fluid : 1 point 3
Numerical approach and computational problem The Bloch equations Saturation method � Find a path u so that after some time T : ˙ y i = − Γ i y i − uz i ◮ matter 1 saturated: y 1 ( T ) = z 1 ( T ) = 0 ˙ z i = − γ i ( 1 − z i )+ uy i ◮ matter 2 “maximized”: | ( y 2 ( T ) , z 2 ( T )) | maximal ( i = 1 , 2) Glaser’s team, 2012 : method from Optimal Control Theory Problem: analyze the behavior of the control through algebraic invariants ◮ Example: singular feedback control: u = D ′ D ( D , D ′ polynomials in y , z , γ , Γ ) ◮ Geometry of { D = 0 } ? ◮ Study of the singular points of { D = 0 } for each value of γ 1 , Γ 1 , γ 2 , Γ 2 ◮ Examples with water: [Bonnard, Chyba, Jacquemard, Marriott, 2013] ◮ Water/Fat : 1 point ◮ Water/Cerebrospinal fluid : 1 point Questions ◮ Is there always 1 singular point for pairs involving water? ◮ If not, how many possible families of parameters can we separate? 3
Statement of the semi-algebraic problem The D invariant: equations of a determinantal system − Γ 1 y 1 − z 1 − 1 − Γ 1 +( γ 1 − Γ 1 ) z 1 ( 2 γ 1 − 2 Γ 1 ) y 1 − γ 1 z 1 y 1 ( γ 1 − Γ 1 ) y 1 2 Γ 1 − γ 1 − ( 2 γ 1 − 2 Γ 1 ) z 1 ◮ M := − Γ 2 y 2 − z 2 − 1 − Γ 2 +( γ 2 − Γ 2 ) z 2 ( 2 γ 2 − 2 Γ 2 ) y 2 − γ 2 z 2 y 2 ( γ 2 − Γ 2 ) y 2 2 Γ 2 − γ 2 − ( 2 γ 2 − 2 Γ 2 ) z 2 ◮ D := determinant ( M ) � D = ∂ D = ∂ D = ∂ D = ∂ D � ◮ V := = 0 ∂ y 1 ∂ z 1 ∂ y 2 ∂ z 2 The Bloch ball: inequalities y 12 +( z 1 + 1 ) 2 ≤ 1 � � ◮ B := y 22 +( z 2 + 1 ) 2 ≤ 1 Goal Classification of the real fibers of the projection of V ∩ B onto the parameter space 4
State of the art and contributions State of the art: ◮ General tool: Cylindrical Algebraic Decomposition [Collins, 1975] ◮ Specific tools for roots classification [Yang, Hou, Xia, 2001] [Lazard, Rouillier, 2007] 5
State of the art and contributions State of the art: ◮ General tool: Cylindrical Algebraic Decomposition [Collins, 1975] ◮ Specific tools for roots classification [Yang, Hou, Xia, 2001] [Lazard, Rouillier, 2007] Problem ◮ None of these algorithms can solve the problem efficiently: ◮ 1050 s in the case of water ( γ 1 = Γ 1 = 1 → 2 parameters) ◮ > 24h in the general case (3 parameters) ◮ Can we exploit the determinantal structure to go further? 5
State of the art and contributions State of the art: Main results ◮ General tool: Cylindrical Algebraic ◮ Dedicated strategy for real roots Decomposition classification for determinantal systems [Collins, 1975] ◮ Can use existing tools for elimination ◮ Specific tools for roots classification ◮ Main refinements: [Yang, Hou, Xia, 2001] ◮ Rank stratification [Lazard, Rouillier, 2007] ◮ Incidence varieties Problem ◮ None of these algorithms can solve the problem efficiently: ◮ 1050 s in the case of water ( γ 1 = Γ 1 = 1 → 2 parameters) ◮ > 24h in the general case (3 parameters) ◮ Can we exploit the determinantal structure to go further? 5
State of the art and contributions State of the art: Main results ◮ General tool: Cylindrical Algebraic ◮ Dedicated strategy for real roots Decomposition classification for determinantal systems [Collins, 1975] ◮ Can use existing tools for elimination ◮ Specific tools for roots classification ◮ Main refinements: [Yang, Hou, Xia, 2001] ◮ Rank stratification [Lazard, Rouillier, 2007] ◮ Incidence varieties Problem ◮ Faster than general algorithms: ◮ None of these algorithms can solve the ◮ 10 s in the case of water problem efficiently: ◮ 4 h in the general case ◮ 1050 s in the case of water ◮ Results for the application ( γ 1 = Γ 1 = 1 → 2 parameters) ◮ Full classification ◮ > 24h in the general case ◮ Answers to the experimental (3 parameters) questions for water: there can be ◮ Can we exploit the determinantal 1, 2 or 3 singularities structure to go further? 5
Classification strategy X B V π G 3 4 2 2 G 6
Classification strategy X meeting the border B V π G 3 4 2 2 G In our case, the only points where the number of roots may change are: ◮ projections of points where V meets the border of the semi-algebraic domain 6
Classification strategy X meeting the border B V π G critical 3 4 2 2 G In our case, the only points where the number of roots may change are: ◮ projections of points where V meets the border of the semi-algebraic domain ◮ critical values of π G restricted to V 6
Classification strategy X meeting the border B V π G singular critical 3 4 2 2 G In our case, the only points where the number of roots may change are: ◮ projections of points where V meets the border of the semi-algebraic domain ◮ critical values of π G restricted to V ◮ projections of singular points of V 6
Classification strategy X meeting the border B V π G singular critical 3 4 2 2 G In our case, the only points where the number of roots may change are: ◮ projections of points where V meets the border of the semi-algebraic domain ◮ critical values of π G restricted to V ◮ projections of singular points of V We want to compute P ∈ Q [ G ] with P � = 0 and P vanishing at all these points 6
How to compute these points Goal: compute P ∈ Q [ γ , Γ] such that V ( P ) ⊃ π (( V ∩ ∂ B ) ∪ Sing ( V ) ∪ Crit ( π , V )) with ◮ π : projection onto the parameters γ i , Γ i � � D = ∂ D = ∂ D = ∂ D = ∂ D ◮ V = = 0 ∂ y 1 ∂ z 1 ∂ y 2 ∂ z 2 � 2 +( z 1 + 1 ) 2 ≤ 1 , y 2 2 +( z 2 + 1 ) 2 ≤ 1 � ◮ B = y 1 Intersection with the border Compute: 2 +( z 1 + 1 ) 2 = 1 } ◮ V ∩{ y 1 2 +( z 2 + 1 ) 2 = 1 } ◮ V ∩{ y 2 and their image through π (polynomial elimination) 7
How to compute these points Goal: compute P ∈ Q [ γ , Γ] such that V ( P ) ⊃ π (( V ∩ ∂ B ) ∪ Sing ( V ) ∪ Crit ( π , V )) with ◮ π : projection onto the parameters γ i , Γ i � � D = ∂ D = ∂ D = ∂ D = ∂ D ◮ V = = 0 ∂ y 1 ∂ z 1 ∂ y 2 ∂ z 2 � 2 +( z 1 + 1 ) 2 ≤ 1 , y 2 2 +( z 2 + 1 ) 2 ≤ 1 � ◮ B = y 1 Intersection with the border Critical and singular points Compute: ( y , z , γ , Γ) ∈ Sing ( V ) ∪ Crit ( π , V ) 2 +( z 1 + 1 ) 2 = 1 } ◮ V ∩{ y 1 ⇐ ⇒ Jac ( F , ( y , z )) has rank < d 2 +( z 2 + 1 ) 2 = 1 } ◮ V ∩{ y 2 and their image through π (polynomial elimination) Requirements ◮ F generates the ideal of V = ⇒ radical ◮ V is equidimensional with codimension d 7
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