Optical Tomography Based on the Method of Rotated Reference Frames - PowerPoint PPT Presentation

[Applied Inverse Problems, Vienna, July 20, 2009] Optical Tomography Based on the Method of Rotated Reference Frames Manabu Machida (Dept. of Bioengineering, Univ. of Pennsylvania) In collaboration with George Y. Panasyuk (U. Penn), Zheng-Min Wang (U. Penn), Vadim A. Markel (U. Penn), and John C. Schotland (U. Penn)

Outline • Introduction • Green’s function • Inverse problem • Simulation • Experiment

Optical Tomography 3D slab geometry ( ) � d , z = L h s z ( ) � s , z = 0 h d 10 4 sources � 10 5 detectors e.g.,

Transport (RTE) Regime OCT Maxwell eqs •Multiple scattering •Low scattering Transport eq •Near sources •Near boundaries •Thin samples Diffusion approx •etc. DOT depth Needs to use the radiative transport equation (RTE)

OT by Radiative Transport Equation  O. Dorn, Inverse Problems (1998)  A. D. Klose and A. H. Hielscher, Med. Phys. (1999)  S. Wright, M. Schweiger, and S. R. Arridge, Meas. Sci. Technol. (2007)  T. Tarvainen, M. Vauhkonen, and S. R. Arridge, J. Quant. Spec. Rad. Trans. (2008)  P. Gonzalez-Rodriguez and A. D. Kim, Inverse Problems (2009) RTE in the 3D slab geometry ☛ No correlation among photons ☛ No time dependence ☛ Background is homogeneous

Outline • Introduction • Green’s function • Inverse problem • Simulation • Experiment

Green’s Function for the RTE ( ) � z ( ) = � � � � ( ) � ˆ ( ) � � s �� + µ t � µ s ˆ � I 0 � , z ,ˆ s � ˆ ˆ L s z � 0 d 2 � ( ) = ( ) ( ) I 0 � , z ,ˆ � � ˆ LI 0 � , z ,ˆ s ,ˆ � s A ˆ s s s ( ) ( ) I 0 � , z ,ˆ = G � , z ,ˆ s ; � 0 , z 0 ,ˆ : specific intensity s s 0 ( ) � = x , y µ t = µ a + µ s µ s , µ a [ : scattering and absorption] µ t and µ s = const ( ) s � ˆ � A ˆ : phase function s d 2 � ( ) A ˆ ( ) � g = s � ˆ � s ,ˆ � ˆ s s s : scattering asymmetry parameter

Boundary Conditions ( ) I 0 � , z b , � � � ˆ ( ) = R n ( ) , I 0 � , z b , � ˆ s , � ˆ n � ˆ s , � ˆ n � ˆ s < 0 ˆ ˆ s s s ( ) s = � ˆ s , � ˆ ˆ Snell and Fresnel s 2 + r � 2 r � x � x c p s ( ) = 1 n 1 � R n x 2 � ˆ ˆ n n ˆ x < x c s � 1 p = x � � x 0 s = x 0 � � x r r x + � x 0 x 0 + � x n 2 � 1 ( ) 1 � n 2 1 � x 2 x 0 = x c = n

Method of Rotated Reference Frames  V. A. Markel, Waves Random Media 14 (2004) L13  G. Panasyuk, J. C. Schotland, and V. A. Markel, J. Phys. A: Math. Gen. 39 (2006) 115  J. C. Schotland and V. A. Markel, Inv. Prob. Imag. 1 (2007) 181  M. M. , G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, submitted. Green’s function for the 3D slab geometry ( ) + µ t G � , z ,ˆ ( ) = s �� G � , z ,ˆ s ; � s ; � ˆ 0 ,0, ˆ 0 ,0, ˆ z z µ s d 2 � ( ) + � � � � ( ) � z ( ) ( ) � ˆ ( ) � s ,ˆ � G � , z ,ˆ � s ; � s � ˆ s A ˆ 0 ,0, ˆ s z z 0

Structure of the Green’s Function ( ) = G � , z ,ˆ s ; � 0 ,0, ˆ z MRRF d 2 q ( ) � ( ) � , z ,ˆ ( ) � ( ) � , z ,ˆ ( ) I q µ ( ) I q µ � ( ) + F ( ) � � � + + � � 2 F s s � � ( ) q µ q µ 0 0 2 � µ Boundary Conditions s � ˆ z > 0 ˆ ( ) � ˆ ( ) ( ) = � � � � ( ) + R n ˆ ( ) G � ,0, � � � ˆ G � ,0,ˆ s � ˆ s � ˆ s , � ˆ s z z 0 s s � ˆ z < 0 ˆ ( ) ( ) = R n � ˆ ( ) G � , L , � � � ˆ G � , L ,ˆ s � ˆ s , � ˆ s z s

( ) � , z ,ˆ ( ) � , z ,ˆ ( ) , I q µ ( ) + � Basis Modes I q µ s s ( ) � , z ,ˆ ( ) � , z ,ˆ ) � µ s d 2 � ( ) � , z ,ˆ � ( ) + µ t I q µ ( ( ) = 0 ( ) � ± ± ± s �� I q µ s ,ˆ � ˆ s A ˆ s s s I q µ s Plane-wave decomposition ( ) z � � exp i q � � � Q µ q � � Spherical harmonics in a rotated reference frame ( ) = ( ) l ( ) � ( ) D � s ; ˆ � ˆ k , � ˆ Y lm ˆ s l Y lm ˆ ˆ k Y l s k ,0 � m m m m = � l � ( ) ˆ k = � i q � � ± Q µ q z ◆ Numerical diagonalization of a tri-diagonal matrix. � ◆ Dependences on , , and are analytical. z ˆ s

Green’s Function � d 2 q l ( ) = � � ( ) ( ) e ( ) � � im � q k lm q , z ;n � i q � � � � 0 G � , z ,ˆ s ; � 0 ,0, ˆ Y lm ˆ z 2 e s ( ) 2 � l = 0 m = � l ( ) l � n M ( ) � ( ) = ( ) � � i � q � n M l k lm q , z ;n d mM � � � l Mn ) � 1 l + m + M f Mn ( ) + e ( ) ( ) ( ) z f Mn ( ) q ;n ( ) L � z ( ( ) q ;n � � � Q Mn q + � Q Mn q � � e � � q 2 + � µ ( ) = � 2 Q Mn q ( ) � l = µ a + µ s 1 � g l [Henyey-Greenstein phase function]

Green’s Function � d 2 q l ( ) = � � ( ) ( ) e ( ) � � im � q k lm q , z ;n � i q � � � � 0 G � , z ,ˆ s ; � 0 ,0, ˆ Y lm ˆ z 2 e s ( ) 2 � l = 0 m = � l spherical harmonics eigenvector Wigner’s d-function ( ) l � n M ( ) � ( ) = ( ) b.c. � � i � q � n M l k lm q , z ;n d mM � � � l Mn ) � 1 l + m + M f Mn ( ) + e ( ) ( ) ( ) z f Mn ( ) q ;n ( ) L � z ( ( ) q ;n � � � Q Mn q + � Q Mn q � � e � � q 2 + � µ ( ) = � 2 eigenvalue Q Mn q ( ) � l = µ a + µ s 1 � g l [Henyey-Greenstein phase function]

Outline • Introduction • Green’s function • Inverse problem • Simulation • Experiment

Structure of Our Method Absorbing inhomogeneity  Markel and Schotland, µ a ( � , z ) = µ a + � µ a ( � , z ) Phys. Rev. E (2004)  Schotland and Markel, Inv. Prob. Imag. (2007) Forward Problem ( ) � µ a I 0 � s ; � d , L ,ˆ Background s ( ) � µ a � , z ( ) I � s ; � d , L ,ˆ Signal s ( ) � µ a � , z ( ) ( ) G � , z ,ˆ � I 0 � I � � z ; � , z ,ˆ s ; � d , L , ˆ s ,0, ˆ G s z Inverse Problem ( ) � µ a � , z

Radiative Transport Equation ( ) � µ s ˆ ( ) = � � , z ,ˆ ( ) � � s �� + µ t + � µ a � , z � I � , z ,ˆ ˆ L s s � ( ) = � � , z ,ˆ ( ) � � s �� + µ t � µ s ˆ � I 0 � , z ,ˆ ˆ L s s � d 2 � ( ) = ( ) ( ) I � , z ,ˆ � � ˆ LI � , z ,ˆ s ,ˆ � s A ˆ s s s ( ) � z ( ) � � � � � ( ) � ˆ ( ) � � , z ,ˆ s � ˆ source: s z s ( ) ( ) I � I 0 � Only and in the normal d , L , ˆ d , L , ˆ z z direction is detected through the aperture.

Born Approximation ( ) I 0 � , z ,ˆ s ( ) � I 0 � ( ) I � d , L , ˆ d , L , ˆ z z ( ) � µ a � , z ( ) I � , z ,ˆ ( ) � � d 2 � d z d 2 s G � z ; � , z ,ˆ d , L , ˆ s s Data Function: ( ) � I 0 � ( ) � I � ( ) � � s , � d , L , ˆ d , L , ˆ z z d ( ) � µ a � , z ( ) ( ) G � , z ,ˆ � � � d 2 � d z d 2 s G � z ; � , z ,ˆ s ; � d , L , ˆ s ,0, ˆ s z

Fourier Transform (1) ( ) = ( ) � � ( ) � � � i q s � � s + q d � � d � q s , q d s , � e d � s � d ( ) ( ) = � � d 2 � e i q � � � µ a � , z � µ a q , z ( ) = G � , z ,ˆ s ; � 0 ,0, ˆ z � d 2 q l � � ( ) ( ) e ( ) � � im � q k lm q , z ;n � i q � � � � 0 Y lm ˆ 2 e s ( ) 2 � l = 0 m = � l

Fourier Transform (2) 1 ( ) � ( ) � ( ) � � � q s , q d d z � q s , � q d , z � µ a q s + q d , z 2 h d 2 h s ( ) k lm q s , z ;n ( ) = ( ) k lm q d , L � z ;n ( ) � � im � q s + � q d � q s , � q d , z e lm q s = q q d = q 2 + p , 2 � p � � � q 2 + p , q 2 � ( ) � h s � q , p 2 � p 2 h d � � � � � � ) � � p + q 2 , p � q ( � � K q , p , z 2 , z � �

Fourier Transform (3) ( ) = � � s , � d ( ) � µ a � , z ( ) ( ) G � , z ,ˆ � d 2 � d z d 2 s G � z ; � , z ,ˆ s ; � d , L , ˆ s ,0, ˆ s z ( ) = ( ) � ( ) � � q , p � µ a q , z d z K q , p , z K � � = ˆ � µ a

Singular Value Decomposition ( ) K † = ˆ � � 1 � µ a = ˆ K † � ˆ K ˆ ˆ K * K * ( ) = ( ) K * q , � ( ) � M p � p q d z K q , p , z p , z Ill-posed � ( ) v j ( ) ( ) v j ( ) ( ) q ( ) q 2 q � = E j p p M p � p q Regularizer � p ( ) � E j , � � � � ( ) = ( ) v j ( ) � K * q , � ( ) ( ) * � � � µ a q , z p p p , z v j q , p 2 E j � j p p d 2 q ( ) = ( ) � � 2 e � i q � � � µ a � , z � µ a q , z ( ) 2 �

Outline • Introduction • Green’s function • Inverse problem • Simulation • Experiment

x - y planes Bars at depth z [RTE] [DA] L = 6 � * , h s = 0.2 � * , h d = 0.1 � * , � * = 1 mm ( ) = 10 cm -1 , µ a = 0.05 cm -1 , µ s 1 � g g = 0.9,

x - y planes x z Balls at depth z [RTE] [DA] L = 6 � * , h s = 0.2 � * , h d = 0.1 � * , � * = 1 mm ( ) = 10 cm -1 , µ a = 0.05 cm -1 , µ s 1 � g g = 0.9,

Outline • Introduction • Green’s function • Inverse problem • Simulation • Experiment

Real World (1) 3mm Width of slab: L =10mm 3mm CCD Laser