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Reconstruction (of micro-objects) based on focus-sets using blind deconvolution   Reconstruction (of micro-objects) based ...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                         MINIMAN-project   ...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                          people: SHU staff ...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                           people: UKA staff...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution  overview: environment• Leica DM RXA micro...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                          overview: objecti...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                basics: discrete fourier tr...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution             basics: 1D discrete fourier tr...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution             basics: 1D discrete fourier tr...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution           sparse matrices and vectors: ima...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution         sparse matrices and vectors: convo...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution       sparse matrices and vectors: circula...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution           sparse matrices and vectors: fou...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution       sparse matrices and vectors: convolu...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution         sparse matrices and vectors: convo...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                    EM-algorithm: descripti...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution          EM-algorithm: overview of represe...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                      EM-Algorithm: derivat...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                   EM-Algorithm: derivation...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                   EM-Algorithm: derivation...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                      EM-Algorithm: derivat...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                      EM-Algorithm: derivat...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                     EM-Algorithm: implemen...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                       results: deblur pict...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                      results: statistic an...
Reconstruction (of micro-objects) based on focus-sets using blind deconvolution                                result: com...
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Reconstruction (of micro-objects) based on focus-sets using blind deconvolution (2001)

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Reconstruction (of micro-objects) based on focus-sets using blind deconvolution (2001)

  1. 1. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution Reconstruction (of micro-objects) based on focus-sets using blind deconvolution Jan Wedekind November 19th, 2001 Jan.Wedekind@stud.uni-karlsruhe.de http://www.uni-karlsruhe.de/~unoh University of Karlsruhe (TH) -1- Sheffield Hallam University
  2. 2. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution MINIMAN-project Esprit Project No. 33915 Miniaturised Robot for Micro Manipulation http://www.miniman-project.com/ Miniman III University of Karlsruhe (TH) -2- Sheffield Hallam University
  3. 3. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution people: SHU staff Sheffield Hallam University microsystems & machine vision lab http://www.shu.ac.uk/mmvl/Prof. J. Travis B. Amavasai F. Caparrelli no picture A. Selvan University of Karlsruhe (TH) -3- Sheffield Hallam University
  4. 4. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution people: UKA staffUniversit¨t Karlsruhe (TH) aInstitute f¨ r Prozeßrechentechnik, uAutomation und Robotikhttp://wwwipr.ira.uka.de/microrobots/ J. Wedekind University of Karlsruhe (TH) -4- Sheffield Hallam University
  5. 5. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution overview: environment• Leica DM RXA microscope – 2 channel illumination – motorized z-table (piezo-driven 0.1 µm) – filter-module• motorized xy-table• Dual Pentium III with 1GHz processors – Linux OS – C++ and KDE/QT• CCD camera 768 × 576 Leica DM RXA microscope µm ⇒ resolution up to 0.74 pixel University of Karlsruhe (TH) -5- Sheffield Hallam University
  6. 6. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution overview: objective• reconstruct from focus-set: – surface – luminosity and coloring• identify model-parameters and quality of assembly University of Karlsruhe (TH) -6- Sheffield Hallam University
  7. 7. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution basics: discrete fourier transform• 1D DFT N −1 2πxki − definition Fk := fx e N where i2 = −1 x=0 N −1 2πxki 1 + ⇔ fx = Fk e N N k=0• 2D DFT N −1 2πxki 2πyli − − analogous Fkl := fxy e N e N , i2 = −1 x,y=0 N −1 2πxki 2πyli 1 + + ⇔ fxy = Fkl e N e N N2 k,l=0 University of Karlsruhe (TH) -7- Sheffield Hallam University
  8. 8. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution basics: 1D discrete fourier transform image domain * = x x x frequency domain ⊗ = ω x ω ∞ ∞ ∀x ∈ R : f (x) = f (x) δ(x − T n) dx −∞ n=−∞ ∞ 2π 2π ⇔ ∀ω ∈ C : F (ω) = (F ⊗ λω. δ(ω − n ) )(ω) T n=−∞ T University of Karlsruhe (TH) -8- Sheffield Hallam University
  9. 9. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution basics: 1D discrete fourier transform image domain * = x x x frequency domain 1 sinc(x) 0.8 = 0.6 ω ⊗ 0.4 ω 0.2 0 -8 -6 -4 -2 0 2 4 6 8 w t ∀x ∈ R : f (x) = f (x) rect( ) 2T sin(ωT ) ⇔ ∀ω ∈ C : F (ω) = (F ⊗ λω. 2T )(ω) ωT University of Karlsruhe (TH) -9- Sheffield Hallam University
  10. 10. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution sparse matrices and vectors: images   x00 ··· x0(N −1) . .   ..conservative: x =  . .  ∈ RN ×N   . . .   x(N −1)0 · · · x(N −1)(N −1)   x00     x0   x01      . .   x1       .  vector repr.: x = =  ∈ RN N  .     .   x10   .    .  xN −1   . .     x(N −1)(N −1)University of Karlsruhe (TH) - 10 - Sheffield Hallam University
  11. 11. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution sparse matrices and vectors: convolution (i) matrix repr. of convolution with a 1D-LSI-filters   h0 0 ··· 0   .     .. .  g0  h1 h0 . . f   .  0  .. ..   g1   .   . . 0   f1     . = ∗ .    .   . .    hK−1 h0   .         .   ..  gK+N −1   0 . h1   fN −1  . . . .. . ∈R(K+N −1) . . . ∈RN ∈R(K+N −1)×N• vectors of different size ⇒ not feasible• matrix is clumsy and baffles mathematical approaches University of Karlsruhe (TH) - 11 - Sheffield Hallam University
  12. 12. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution sparse matrices and vectors: circulant matrices   a(0) a(N − 1) · · · a(1)  .. .. . .  . .    a(1) .  A=  .  . ..    . . a(N − 1)  a(N − 1) ··· a(1) a(0) N −1 2πkjiwithout − • eigenvalues of A: λ(k) = a(j) e Nproof: j=0 • A = FQA F −1 with – QA = diag(λ1 , λ2 , . . . , λN ) and – fourier-kernel F −1 University of Karlsruhe (TH) - 12 - Sheffield Hallam University
  13. 13. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution sparse matrices and vectors: fourier kernel 2πukl idefinition of fourier-kernel: F −1 := e N   0 0 0 ···   0 1 2 · · ·   with U :=   and i2 = −1 0 2 4  · · · . . . ..  . . . . . . .note: • X = DFT{x} = X = F −1 x • x = DFT−1 {X} = x = FX where F = 1 N (F −1 )∗ • F =F University of Karlsruhe (TH) - 13 - Sheffield Hallam University
  14. 14. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution sparse matrices and vectors: convolution (ii) approximated 1D convolution       g0 h0 hN −1 ··· h1 f   0  .   .. ..      g1   h1   . . .   f1  .  . ≈ . ∗   .       .   . .. .   .   . . hN −1   .     gN −1 hN −1 ··· h1 h0 fN −1 =:g∈RN =:H∈RN ×N =:f ∈RN g = HfUniversity of Karlsruhe (TH) - 14 - Sheffield Hallam University
  15. 15. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution sparse matrices and vectors: convolution (iii) N −1 gk = H(k−l) mod N fl l=0       g0 H0 HN −1 ··· H1 f   0  .   .. ..      g1   H1   . . .   f1  .  . ≈ . ∗   .       .   . .. .   .   . . HN −1   .     gN −1 HN −1 ··· H1 H0 fN −1 =:g∈RN 2 =:H∈RN 2 ×N 2 =:f ∈RN 2⇒ g = H f is 2D convolution!⇒ H is (block) circulant University of Karlsruhe (TH) - 15 - Sheffield Hallam University
  16. 16. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-algorithm: description • “incomplete” data-set y • many-to-one transform given: y = g(z1 , z2 , . . . , zM ) = g(z) • pdf pz (z; θ) and cond. pdf p(z|y; θ) 1. expectation: Determine expected log-likelihood of complete data U (θ, θp ) :=Ez {ln pz (z; θ)|y; θp } EM-algorithm: = ln pz (z; θ) p(z|y; θp )dz 2. maximisation: Maximize θp+1 = argmax U (θ, θp ) θ 3. goto step 1 until θ convergesUniversity of Karlsruhe (TH) - 16 - Sheffield Hallam University
  17. 17. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-algorithm: overview of representationsmeaning conversative sparse matrices real fourier real fourier N ×N N ×N N 2 ,N 2 ×N 2 N 2 ,N 2 ×N 2sets R C R Cimage x X x XPSF h ∆ D QDcov. image Cx Sx Λx Qx 2 2cov. noise σv I (σv ) Λv Qvconvolution x⊗h X ∆ Dx QD XDFT - DFT{x} - F −1 xinv. DFT DFT−1 {X} - FX - University of Karlsruhe (TH) - 17 - Sheffield Hallam University
  18. 18. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-Algorithm: derivation (i)• incomplete/complete data-set: y ∈ RN / (x , y ) ∈ R2N   x• many-to-one transform: y =: g( ) ∀y y• pdf of z is zero-mean normal distr.: pz (z, θ = {D, Λx , Λv }) =  −1     1x Λ Λx D −    x x   2 y   DΛx DΛx D + Λv   y 1 e =:C Λx Λx D(2π)2N 2 DΛx DΛx D + Λv University of Karlsruhe (TH) - 18 - Sheffield Hallam University
  19. 19. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-Algorithm: derivation (ii)• conditional pdf p(z|y, θ) = p(x|y, θ) is (conditional) normal distr.: – with mean Mx|y = Λx D (DΛx D + Λv )−1 y and – covariance Sx|y = Λx − Λx D (DΛx D + Λv )−1 DΛx derivation of   2 1 1 x −1   ln pz (z; θ) = −N ln(2π) − ln |C| − (x , y ) C : 2 2 yΛx Λx D = det(Λx (DΛx D + Λv ) − DΛx Λx D )DΛx DΛx D + Λv = det(Λx Λv ) = |Λx ||Λv |, because Λx and Λv are symmetric and positive definite. University of Karlsruhe (TH) - 19 - Sheffield Hallam University
  20. 20. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-Algorithm: derivation (iii) 2 N2 • observe that |Λv | = (σv ) . • |Λx | = |FQx F −1 | = |Qx | = Sx (k, l) kl   A A12using A =  11 = A21 A22  (A11 − A12 A−1 A21 )−1 22 A−1 A12 (A21 A−1 A12 − A22 )−1 11 11  (A21 A−1 A12 − A22 )−1 A21 A−1 11 11 (A22 − A21 A−1 A12 )−1 11  −1 Λ Λx D −1we get C =  x  = DΛx DΛx D + Λv University of Karlsruhe (TH) - 20 - Sheffield Hallam University
  21. 21. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-Algorithm: derivation (iv)   (I − D (DΛx D + Λv )−1 DΛx )−1 Λ−1 x −D Λ−1 vC −1 =  . −Λ−1 D v Λ−1 vWith u Av= u∗ FQA F −1 v = (F ∗ u)∗ QA F −1 v 1 −1 ∗ −1 1 ∗ = ( 2 F u) QA F v = 2 U QA V we obtain N N N2 1 1 2ln pz (z; θ) =−N ln(2π) − 2 ln(σv ) − ln Skl − 2 Y ∗ Q−1 Y v 2 2 2N kl 1 1 + 2 Re{Y ∗ QD Q−1 X} v − Qx (QD Q∗ Q−1 + Q−1 ) D v x N 2 kl 1 − 2 X ∗ (QD Q∗ Q−1 + Q−1 )X D v x 2N University of Karlsruhe (TH) - 21 - Sheffield Hallam University
  22. 22. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-Algorithm: derivation (v)Replacing the remaining diagonal matrices with DFT-arrays yields: N2 1 1 2ln pz (z; θ) =−N ln(2π) − 2 ln(σv ) − ln Skl − 2 Y∗ Y(σv )−1 2 2 2 2N kl 1 ∗ 1 + 2 Re{Y ∆(σv )−1 X} 2 − Sx (∆∆∗ (σv )−1 + Sx ) 2 −1 N 2 1 − 2 X∗ (∆∆∗ (σv )−1 + Sx )X 2 −1 2N The last steps are: • substitute X and X∗ ... X with their expected values. ⇒ U (θ, θk ) δU (θ, θk ) • determine argmax by setting derivative to zero. θ δθ University of Karlsruhe (TH) - 22 - Sheffield Hallam University
  23. 23. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution EM-Algorithm: implementationThe resulting iteration step: (p+1) (p) 1 • Sx (k, l) = Sx|y (k, l) + 2 |Mx|y (k, l)|2 N ∗ 1 Y (k, l)Mx|y (k, l) • ∆(p+1) (k, l) = N2 (p) 1 Sx|y (k, l) + 2 |Mx|y (k, l)|2 N 2 1 (p+1) 2 (p) 1 • σv = kl |∆ (k, l)| Sx|y (k, l) + |Mx|y (k, l)|2 + N2 N 2 1 |Y (k, l)|2 − 2Re Y ∗ (k, l)∆(p+1) (k, l)Mx|y (k, l) N2 University of Karlsruhe (TH) - 23 - Sheffield Hallam University
  24. 24. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution results: deblur picture conditional mean after 10. iterationUniversity of Karlsruhe (TH) - 24 - Sheffield Hallam University
  25. 25. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution results: statistic and psf log-DFTUniversity of Karlsruhe (TH) - 25 - Sheffield Hallam University
  26. 26. Reconstruction (of micro-objects) based on focus-sets using blind deconvolution result: comparison Jan Wedekindaddress: . . . . . . . . . . . . . . . . . . . . . . . . . . Scheffelstr. 65, D-76135 Karlsruheemail: . . . . . . . . . . . . . . . . . . . . . . . . . . jan.wedekind@stud.uni-karlsruhe.dewww: . . . . . . . . . . . . . . . . . . . . . . . http://www.uni-karlsruhe.de/~unohpgp: . . . . . . . . . . . . . . . . . EE FA AF 15 D8 ED 11 4A 5A 76 35 5F 2D 20 C4 E8 University of Karlsruhe (TH) - 26 - Sheffield Hallam University

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