Lissajous sampling and adaptive spectral filtering for the reduction of the Gibbs phenomenon in Magnetic Particle Imaging (MPI) S. De Marchi 1 , W. Erb 2 and F. Marchetti 1 1 University of Padova 2 University of Hawaii Bernried, 27 Feb. - 3 Mar. 2017 S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 1 / 57
S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 2 / 57
Starting points 1 Padua points lie on a Lissajous curve [Bos, De Marchi et al. JAT 2006] 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 2 Magnetic Particle Imaging: “The trajectory of the field-free point (FFP) describes a Lissajous curve” [Weizenecker et al., Phy. in Med. 2007] 3 Reconstruction of discontinuos and piecewise regular functions by trun. Fourier series � Gibbs phenomenon S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 3 / 57
Outline 1 Magnetic Particle Imaging 2 Lissajous curves 3 Fourier series and Gibbs phenomenon 4 Examples and parameter estimation 5 MPI applications S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 4 / 57
Magnetic Particle Imaging Magnetic Particle Imaging S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 5 / 57
Magnetic Particle Imaging Magnetic Particle Imaging (MPI) The MPI is an emerging technology in the (pre)clinical imaging [B. Gleich, J. Weizenecker (Philips Research, Hamburg) - Nature 2005]. Detection of a tracer consisting of super-paramagnetic (iron oxide) nanoparticles injected in the bloodstream ( � emissive tomography) 3D Field of View with high sensitivity, high resolution ( ∼ 0 . 4 mm) and high imaging speed ( ∼ 20 ms) The acquisition of the signal, which comes from the particles, is performed moving a field-free point (FFP) along trajectories: (Lissajous curves) No radiation, no iodine, no background noise (high contrast). 1000 times faster than PET; 100 times more sensitive than MRI. S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 6 / 57
Magnetic Particle Imaging MPI scanners topologies Figure: Left: two pairs of transmit coils and two pairs of receivers coils. Right: one sided from IMT Lübeck Two-two scanner: the design imposes size limitation on the object One side: the target no longer has to be small enough to fit inside the scanner. S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 7 / 57
Magnetic Particle Imaging MPI scanners Figure: Left: scanner for humans. Right: the “Bruker-Philips BioSpin MPI” for animals S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 8 / 57
Lissajous curves Lissajous curves S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 9 / 57
Lissajous curves Bowditch figures or Lissajous curves 1 Are planar parametric curves studied by Nathaniel Bowditch (1815) and Jules A. Lissajous (1857) of the form γ ( t ) = ( A x cos( ω x t + α x ) , A y sin( ω y t + α y )) . A x , A y are amplitudes, ω x , ω y are pulsations and α x , α y are phases. 2 Chebyshev polynomials ( T k or U k ) are Lissajous curves (cf. J. C. Merino 2003). In fact a parametrization of y = T n ( x ) , | x | ≤ 1 is � x = cos t � nt − π � y = − sin 0 ≤ t ≤ π 2 S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 10 / 57
Lissajous curves Bowditch figures or Lissajous curves 1 Are planar parametric curves studied by Nathaniel Bowditch (1815) and Jules A. Lissajous (1857) of the form γ ( t ) = ( A x cos( ω x t + α x ) , A y sin( ω y t + α y )) . A x , A y are amplitudes, ω x , ω y are pulsations and α x , α y are phases. 2 Chebyshev polynomials ( T k or U k ) are Lissajous curves (cf. J. C. Merino 2003). In fact a parametrization of y = T n ( x ) , | x | ≤ 1 is � x = cos t � nt − π � y = − sin 0 ≤ t ≤ π 2 S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 10 / 57
Lissajous curves Figure: Left: N. Bowditch (March 26, 1773 - March 16, 1838), American mathematician remembered for his work on ocean navigation. Right: J. Lissajous (March 4, 1822 - June 24, 1880), French physicist S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 11 / 57
Lissajous curves Two dimensional general definition [Erb et al. DRNA2015] Definition � u 1 cos( n 2 t − κ 1 π/ (2 n 1 )) � γ n κ , u ( t ) = , t ∈ [0 , 2 π ] , u 2 cos( n 1 t − κ 2 π/ (2 n 2 )) with n = ( n 1 , n 2 ) ∈ N 2 , κ = ( κ 1 , κ 2 ) ∈ R 2 and u = {− 1 , 1 } 2 . n 1 , n 2 are called frequencies (like for the pendulum) and u reflection parameter. Proposition There exist t ′ ∈ R , η ∈ [0 , 2) and u ′ ∈ {− 1 , 1 } 2 s.t. γ n κ , u ( t − t ′ ) := γ n (0 ,η ) , u ′ ( t ) , t ∈ [0 , 2 π ] (1) Obs: if κ ∈ Z 2 , the value of η can be always chosen as { 0 , 1 } . S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 12 / 57
Lissajous curves Two dimensional general definition [Erb et al. DRNA2015] Definition � u 1 cos( n 2 t − κ 1 π/ (2 n 1 )) � γ n κ , u ( t ) = , t ∈ [0 , 2 π ] , u 2 cos( n 1 t − κ 2 π/ (2 n 2 )) with n = ( n 1 , n 2 ) ∈ N 2 , κ = ( κ 1 , κ 2 ) ∈ R 2 and u = {− 1 , 1 } 2 . n 1 , n 2 are called frequencies (like for the pendulum) and u reflection parameter. Proposition There exist t ′ ∈ R , η ∈ [0 , 2) and u ′ ∈ {− 1 , 1 } 2 s.t. γ n κ , u ( t − t ′ ) := γ n (0 ,η ) , u ′ ( t ) , t ∈ [0 , 2 π ] (1) Obs: if κ ∈ Z 2 , the value of η can be always chosen as { 0 , 1 } . S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 12 / 57
Lissajous curves Lissajous curves (cont’) Lissajous curves on the square Let n = ( n 1 , n 2 ) with n 1 , n 2 ∈ N relatively primes. Then we can ǫ : [0 , 2 π ] → [ − 1 , 1] 2 consider the parametric curves γ n � � cos( n 2 t ) γ n ǫ ( t ) := γ n (0 ,ǫ − 1) , 1 ( t ) = (2) cos( n 1 t + ( ǫ − 1) π/ (2 n 2 )) with ǫ ∈ { 1 , 2 } and fixed reflection parameter 1 = (1 , 1) . Special cases ǫ = 1 (i.e. η = 0 in (1) and I = [0 , π ] ), γ n 1 ( t ) is called a degenerate curve [Erb AMC2016] ǫ = 2 the curve is called non-degenerate [Erb et al. NM2016]. S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 13 / 57
Lissajous curves Lissajous curves (cont’) Lissajous curves on the square Let n = ( n 1 , n 2 ) with n 1 , n 2 ∈ N relatively primes. Then we can ǫ : [0 , 2 π ] → [ − 1 , 1] 2 consider the parametric curves γ n � � cos( n 2 t ) γ n ǫ ( t ) := γ n (0 ,ǫ − 1) , 1 ( t ) = (2) cos( n 1 t + ( ǫ − 1) π/ (2 n 2 )) with ǫ ∈ { 1 , 2 } and fixed reflection parameter 1 = (1 , 1) . Special cases ǫ = 1 (i.e. η = 0 in (1) and I = [0 , π ] ), γ n 1 ( t ) is called a degenerate curve [Erb AMC2016] ǫ = 2 the curve is called non-degenerate [Erb et al. NM2016]. S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 13 / 57
Lissajous curves The Lissajous (node) points Lissajous nodes Let γ n ǫ be a Lissajous curve with ǫ ∈ { 1 , 2 } and let πk t ǫ n k = , k = 0 , . . . , 2 ǫn 1 n 2 − 1 . ǫn 1 n 2 The set LS n ǫ = { γ n ǫ ( t ǫ n k ) : k = 0 , . . . , 2 ǫn 1 n 2 − 1 } is the set of Lissajous node points related to γ n ǫ . We need also to introduce the set of indices � i j � Γ ǫ n = ( i, j ) ∈ N 2 0 : + < 1 ∪ { (0 , ǫn 2 ) } . ǫn 1 ǫn 2 S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 14 / 57
Lissajous curves Examples Figure: plots of γ (5 , 6) and γ (5 , 6) 1 2 # LS (5 , 6) 5 = # PD 5 , # LS (5 , 6) = 21 = dim Π 2 = 71 < dim Π 2 11 = 78 1 2 ǫ = #Γ ǫ n = ( ǫn 1 + 1)( ǫn 2 + 1) − ( ǫ − 1) # LS n (3) 2 S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 15 / 57
Lissajous curves Padua and Morrow-Patterson points Padua points correspond to the degenerate Lissajous curve (cf. (2) ) γ ( n,n +1) 0 , u or γ ( n +1 ,n ) , n ∈ N . 0 , u Up to reflection u , they are given by the curves � cos ( n + 1) t � � � cos nt γ n or γ n 1 ( t ) = 1 ( t ) = . cos( n + 1) t cos nt The Morrow-Patterson come from γ ( n +2 ,n +3) which are the self-intersection 1 points of the Padua’s curve γ ( n,n +1) . 1 S. De Marchi (short) Lissajous sampling and adaptive spectral filtering for 27/2-3/3 the reduction of the Gibbs 16 / 57
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