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CRF-F: D P F 
         S S E
            B L, D F  L...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
Intro               Transformation of Directed Model to CRF               Application             Experimental Results


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Intro       Transformation of Directed Model to CRF    Application             Experimental Results


 D  D...
Intro                Transformation of Directed Model to CRF           Application             Experimental Results


 D...
Intro                    Transformation of Directed Model to CRF    Application             Experimental Results


 D...
Intro                    Transformation of Directed Model to CRF    Application                Experimental Results


 D...
Intro        Transformation of Directed Model to CRF            Application        Experimental Results


 A P  ...
Intro                         Transformation of Directed Model to CRF            Application        Experimental Results

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Intro                         Transformation of Directed Model to CRF            Application        Experimental Results

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Intro                         Transformation of Directed Model to CRF            Application        Experimental Results

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O


  1   I: S E U D M

  2   T  D M  CRF
       ...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
Intro                    Transformation of Directed Model to CRF           Application     Experimental Results


 I: ...
Intro                    Transformation of Directed Model to CRF           Application     Experimental Results


 I: ...
Intro                    Transformation of Directed Model to CRF           Application      Experimental Results


 I:...
Intro                    Transformation of Directed Model to CRF           Application       Experimental Results


 I...
Intro                    Transformation of Directed Model to CRF            Application       Experimental Results


 I...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
Intro              Transformation of Directed Model to CRF              Application             Experimental Results


 CR...
Intro              Transformation of Directed Model to CRF              Application             Experimental Results


 CR...
Intro              Transformation of Directed Model to CRF              Application             Experimental Results


 CR...
Intro               Transformation of Directed Model to CRF         Application         Experimental Results


 T P...
Intro               Transformation of Directed Model to CRF         Application          Experimental Results


 T P...
Intro               Transformation of Directed Model to CRF         Application           Experimental Results


 T P...
Intro               Transformation of Directed Model to CRF             Application               Experimental Results


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Intro       Transformation of Directed Model to CRF                   Application   Experimental Results


 R: S...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
Intro              Transformation of Directed Model to CRF             Application    Experimental Results


 M ...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
Intro          Transformation of Directed Model to CRF          Application   Experimental Results


 U  CRF  ...
Intro          Transformation of Directed Model to CRF          Application   Experimental Results


 U  CRF  ...
Intro          Transformation of Directed Model to CRF           Application     Experimental Results


 U  CRF ...
Intro          Transformation of Directed Model to CRF           Application   Experimental Results


 U  CRF  ...
Intro          Transformation of Directed Model to CRF           Application      Experimental Results


 U  CRF ...
Intro          Transformation of Directed Model to CRF           Application    Experimental Results


 U  CRF ...
Intro          Transformation of Directed Model to CRF          Application     Experimental Results


 U  CRF ...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
Intro     Transformation of Directed Model to CRF     Application   Experimental Results


 D    ...
Intro     Transformation of Directed Model to CRF     Application    Experimental Results


 D    ...
Intro     Transformation of Directed Model to CRF     Application      Experimental Results


 D   ...
Intro     Transformation of Directed Model to CRF     Application      Experimental Results


 D   ...
Intro     Transformation of Directed Model to CRF     Application      Experimental Results


 D   ...
Intro       Transformation of Directed Model to CRF     Application      Experimental Results


 D   ...
Intro       Transformation of Directed Model to CRF               Application                 Experimental Results


 L...
O


  1   I: S E U D M

  2   T  D M  CRF
       ...
Intro     Transformation of Directed Model to CRF              Application       Experimental Results


 E R...
Intro     Transformation of Directed Model to CRF              Application         Experimental Results


 E R...
Intro           Transformation of Directed Model to CRF   Application   Experimental Results


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Intro           Transformation of Directed Model to CRF   Application   Experimental Results


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Intro           Transformation of Directed Model to CRF   Application   Experimental Results


 C



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Intro           Transformation of Directed Model to CRF   Application   Experimental Results


 C



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Intro           Transformation of Directed Model to CRF   Application   Experimental Results


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CRF-Filters: Discriminative Particle Filters for Sequential State Estimation

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Seminar talk on the paper by Limketkai, Fox and Liao.

Publié dans : Technologie, Économie & finance
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CRF-Filters: Discriminative Particle Filters for Sequential State Estimation

  1. 1. CRF-F: D P F  S S E B L, D F  L L Hannes Schulz University of Freiburg, ACS Feb 2008
  2. 2. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  3. 3. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  4. 4. Intro Transformation of Directed Model to CRF Application Experimental Results C: S E C D M A  S E ut−2 ut−1 xt−2 xt−1 xt ... ... n 1 2 n zt−1 1 zt 2 zt zt zt−1 zt−1 P (xt |u1:t −1 , z1:t ) = ηP (zt |xt ) P (xt |ut −1 , xt −1 )P (xt −1 |u1:t −2 , z1:t −1 ) dxt −1
  5. 5. Intro Transformation of Directed Model to CRF Application Experimental Results D  D M P p (zt |xt ) = n i =1 p (zti |xt ) p (xt +1 |xt , u)
  6. 6. Intro Transformation of Directed Model to CRF Application Experimental Results D  D M P p (zt |xt ) = n i =1 p (zti |xt ) p (xt +1 |xt , u) i P (zt |xt) ˆi zt zmax zrand
  7. 7. Intro Transformation of Directed Model to CRF Application Experimental Results D  D M P p (zt |xt ) = n i =1 p (zti |xt ) p (xt +1 |xt , u) i P (zt |xt) ˆi zt zmax zrand
  8. 8. Intro Transformation of Directed Model to CRF Application Experimental Results D  D M P p (zt |xt ) = n i =1 p (zti |xt ) p (xt +1 |xt , u) i P (zt |xt) ˆi zt zmax δrot2 zrand xt δtrans δrot1 xt−1 u = (δrot1 , δrot2 , δtrans ) executed with gaussian noise
  9. 9. Intro Transformation of Directed Model to CRF Application Experimental Results A P   D A p (zti |xt ) are not cond. independent zt xt
  10. 10. Intro Transformation of Directed Model to CRF Application Experimental Results A P   D A ut−2 ut−1 xt−2 xt−1 xt p (zti |xt ) are not cond. independent Sensor models can only be ... ... n 1 2 n zt−1 1 2 zt zt−1 zt−1 zt zt generated seperatly for each beam i P (zt |xt) ˆi zt zmax zrand
  11. 11. Intro Transformation of Directed Model to CRF Application Experimental Results A P   D A ut−2 ut−1 xt−2 xt−1 xt p (zti |xt ) are not cond. independent Sensor models can only be ... ... n 1 2 n zt−1 1 2 zt zt−1 zt−1 zt zt generated seperatly for each beam Assumption that measurements are independent: “Works i P (zt |xt) ˆi zt zmax surprisingly well”. . . if. . . zrand
  12. 12. Intro Transformation of Directed Model to CRF Application Experimental Results A P   D A ut−2 ut−1 xt−2 xt−1 xt p (zti |xt ) are not cond. independent Sensor models can only be ... ... n 1 2 n zt−1 1 2 zt zt−1 zt−1 zt zt generated seperatly for each beam Assumption that measurements are independent: “Works i P (zt |xt) ˆi zt zmax surprisingly well”. . . if. . . increasing uncertainty (tweaking) using every 10th measurement zrand ...
  13. 13. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  14. 14. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  15. 15. Intro Transformation of Directed Model to CRF Application Experimental Results I: CRF Undirected graphical models ut−2 ut−1 xt−2 xt−1 xt zt−1 zt
  16. 16. Intro Transformation of Directed Model to CRF Application Experimental Results I: CRF Undirected graphical models Every (possible) dependency ut−2 ut−1 represented by edge xt−2 xt−1 xt zt−1 zt
  17. 17. Intro Transformation of Directed Model to CRF Application Experimental Results I: CRF Undirected graphical models Every (possible) dependency ut−2 ut−1 represented by edge Distribution defined over products xt−2 xt−1 xt of functions over cliques zt−1 zt
  18. 18. Intro Transformation of Directed Model to CRF Application Experimental Results I: CRF Undirected graphical models Every (possible) dependency ut−2 ut−1 represented by edge Distribution defined over products xt−2 xt−1 xt of functions over cliques zt−1 zt Functions are called clique potentials
  19. 19. Intro Transformation of Directed Model to CRF Application Experimental Results I: CRF Undirected graphical models Every (possible) dependency ut−2 ut−1 represented by edge Distribution defined over products xt−2 xt−1 xt of functions over cliques zt−1 zt Functions are called clique potentials Clique potentials represent compatibility of their variables
  20. 20. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  21. 21. Intro Transformation of Directed Model to CRF Application Experimental Results CRF-M  S E ut−2 ut−1 xt−2 xt−1 xt zt−1 zt T 1 p (x0:T |z1:T , u0:T −1 ) = ϕp (xt , xt −1 , ut −1 )ϕm (xt , zt ) Z (z1:T , u1:T −1 ) t =1
  22. 22. Intro Transformation of Directed Model to CRF Application Experimental Results CRF-M  S E ut−2 ut−1 xt−2 xt−1 xt zt−1 zt T 1 p (x0:T |z1:T , u0:T −1 ) = ϕp (xt , xt −1 , ut −1 )ϕm (xt , zt ) Z (z1:T , u1:T −1 ) t =1 Z (·): all trajectories ϕp (·)ϕm (·)
  23. 23. Intro Transformation of Directed Model to CRF Application Experimental Results CRF-M  S E ut−2 ut−1 xt−2 xt−1 xt zt−1 zt T 1 p (x0:T |z1:T , u0:T −1 ) = ϕp (xt , xt −1 , ut −1 )ϕm (xt , zt ) Z (z1:T , u1:T −1 ) t =1 Z (·): all trajectories ϕp (·)ϕm (·) How to define ϕp (·) and ϕm (·)?
  24. 24. Intro Transformation of Directed Model to CRF Application Experimental Results T P P φp ut −1 = (δrot1 , δtrans , δrot2 ) odometry ut −1 = (δrot1 , δtrans , δrot2 ) derived odometry ˆ ˆ ˆ ˆ δrot2 2 Before: Gaussian noise N uti −1 , σi xt δtrans δrot1 xt−1
  25. 25. Intro Transformation of Directed Model to CRF Application Experimental Results T P P φp ut −1 = (δrot1 , δtrans , δrot2 ) odometry ut −1 = (δrot1 , δtrans , δrot2 ) derived odometry ˆ ˆ ˆ ˆ δrot2 2 Before: Gaussian noise N uti −1 , σi xt  (δrot1 − δrot1 )2 ˆ      δtrans   fp (xt , xt −1 , ut −1 ) =  (δtrans − δtrans )2       ˆ      3 features   (δrot2 − δrot2 )2 ˆ     δrot1 xt−1
  26. 26. Intro Transformation of Directed Model to CRF Application Experimental Results T P P φp ut −1 = (δrot1 , δtrans , δrot2 ) odometry ut −1 = (δrot1 , δtrans , δrot2 ) derived odometry ˆ ˆ ˆ ˆ δrot2 2 Before: Gaussian noise N uti −1 , σi xt  (δrot1 − δrot1 )2 ˆ      δtrans   fp (xt , xt −1 , ut −1 ) =  (δtrans − δtrans )2       ˆ      3 features   (δrot2 − δrot2 )2 ˆ     δrot1 φp (xt , xt −1 , ut −1 ) = exp wp , fp (xt , xt −1 , ut −1 ) xt−1
  27. 27. Intro Transformation of Directed Model to CRF Application Experimental Results T P P φp ut −1 = (δrot1 , δtrans , δrot2 ) odometry ut −1 = (δrot1 , δtrans , δrot2 ) derived odometry ˆ ˆ ˆ ˆ δrot2 2 Before: Gaussian noise N uti −1 , σi xt  (δrot1 − δrot1 )2 ˆ      δtrans   fp (xt , xt −1 , ut −1 ) =  (δtrans − δtrans )2       ˆ      3 features   (δrot2 − δrot2 )2 ˆ     δrot1 φp (xt , xt −1 , ut −1 ) = exp wp , fp (xt , xt −1 , ut −1 ) xt−1 1 (a − a )2 ˆ N a, = exp − σ2 2σ2 Gaussian noise N uti −1 , 1 −2wpi if wp < 0 i
  28. 28. Intro Transformation of Directed Model to CRF Application Experimental Results R: S M   N¨ B A  i P (zt |xt) ˆi zt zmax zrand n p (zt |xt ) = p (zti |xt ) i =1
  29. 29. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ  
  30. 30. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ   mti ∈ {1, 0} measured zmax
  31. 31. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ   mti ∈ {1, 0} measured zmax mti ∈ {1, 0} expected zmax ˆ
  32. 32. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ   mti ∈ {1, 0} measured zmax mti ∈ {1, 0} expected zmax ˆ cti ∈ {1, 0} zti − zti < 20cm ˆ
  33. 33. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ   mti ∈ {1, 0} measured zmax mti ∈ {1, 0} expected zmax ˆ cti ∈ {1, 0} zti − zti < 20cm ˆ
  34. 34. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ   mti ∈ {1, 0} measured zmax mti ∈ {1, 0} expected zmax ˆ cti ∈ {1, 0} zti − zti < 20cm ˆ
  35. 35. Intro Transformation of Directed Model to CRF Application Experimental Results M P φm i P (zt |xt) ˆi zt zmax n     φm (xt , zt ) = exp  wm , fm (zt , xt )  i         i =0 zrand (¬mti ∧ ¬mti )cti (zti − zti )2 ˆ ˆ         i i i        ˆ (¬mt ∧ ¬mt )¬ct      fm (zt , xt ) =  i   (¬mti ∧ mti ) ˆ            ( mti ∧ ¬mti )        ˆ        ( mti ∧ mti ) ˆ   mti ∈ {1, 0} measured zmax mti ∈ {1, 0} expected zmax ˆ cti ∈ {1, 0} zti − zti < 20cm ˆ
  36. 36. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  37. 37. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  38. 38. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp Same as sampling from N uti −1 , ˆ 1 i −2wp Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  39. 39. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp u Same as sampling from N uti −1 , ˆ 1 i −2wp Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  40. 40. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp moved Same as sampling from N uti −1 , ˆ 1 −2wpi particles Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  41. 41. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp added Same as sampling from N uti −1 , noise 1 ˆ −2wpi Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  42. 42. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp ...sense... Same as sampling from N uti −1 , ˆ 1 −2wpi Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  43. 43. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp weights Same as sampling from N uti −1 , ˆ 1 −2wpi Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  44. 44. Intro Transformation of Directed Model to CRF Application Experimental Results U  CRF    P F At each time step t: Prediction Move particles according to gaussian noise determined by wp resample Same as sampling from N uti −1 , ˆ 1 i −2wp Correction Particle at xt gets weight φm (xt , zt ) Resample (includes normalization)
  45. 45. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  46. 46. Intro Transformation of Directed Model to CRF Application Experimental Results D     wp  wm Drive around in test area
  47. 47. Intro Transformation of Directed Model to CRF Application Experimental Results D     wp  wm Drive around in test area Use high-quality scanmatcher to generate “ground truth” trajectory x∗
  48. 48. Intro Transformation of Directed Model to CRF Application Experimental Results D     wp  wm Drive around in test area Use high-quality scanmatcher to generate “ground truth” trajectory x∗ ˆ Using arbitrary weights, generate trajectory x with CRF-filter
  49. 49. Intro Transformation of Directed Model to CRF Application Experimental Results D     wp  wm Drive around in test area Use high-quality scanmatcher to generate “ground truth” trajectory x∗ ˆ Using arbitrary weights, generate trajectory x with CRF-filter Use difference of summed features as weight update(−) : wk = wk −1 + α ( f (x∗ , u, z) − f (x, u, z)) ˆ
  50. 50. Intro Transformation of Directed Model to CRF Application Experimental Results D     wp  wm Drive around in test area Use high-quality scanmatcher to generate “ground truth” trajectory x∗ ˆ Using arbitrary weights, generate trajectory x with CRF-filter Use difference of summed features as weight update(−) : wk = wk −1 + α ( f (x∗ , u, z) − f (x, u, z)) ˆ Decrease α if new Filter cannot track
  51. 51. Intro Transformation of Directed Model to CRF Application Experimental Results D     wp  wm Drive around in test area Use high-quality scanmatcher to generate “ground truth” trajectory x∗ ˆ Using arbitrary weights, generate trajectory x with CRF-filter Use difference of summed features as weight update(−) : wk = wk −1 + α ( f (x∗ , u, z) − f (x, u, z)) ˆ Decrease α if new Filter cannot track loop Adapts weights to task, sensor dependencies/environment, sensor noise, particle filter parameters
  52. 52. Intro Transformation of Directed Model to CRF Application Experimental Results L A Averaged Perceptron Algorithm (Collins 2002) for tagging w k = w k −1 + α f (x∗ , u, z) − f (x, u, z) ˆ Proven to converge even in presence of errors in training data Intuition of learning algorithm: If PF works correctly, then f (xn , un−1 , zn ) = ∗ f (xn , un−1 , zn ) ˆ f i occurs less often in x∗ than in x → decrease influence of f i ˆ on particle filter by decreasing w i
  53. 53. O 1 I: S E U D M 2 T  D M  CRF Short Introduction to CRF CRF-Model for State Estimation 3 A CRF-Filter Algorithm Learning the Parameters 4 E R
  54. 54. Intro Transformation of Directed Model to CRF Application Experimental Results E R Properties of the learned weights Norm of weight vector decreases with number of laser beams in z believes the features/measurements less equivalent to initially introduced “tweaking”?!
  55. 55. Intro Transformation of Directed Model to CRF Application Experimental Results E R Properties of the learned weights Norm of weight vector decreases with number of laser beams in z believes the features/measurements less equivalent to initially introduced “tweaking”?! Two specialized CRF-filters compared to generative particle filter trained using expectation maximization Tracking Global Error Localization Accuracy Generative 7.52 cm 30% CRF-Filter 7.07 cm 96%
  56. 56. Intro Transformation of Directed Model to CRF Application Experimental Results C 1 A CRF is an alternative, undirected graphical model
  57. 57. Intro Transformation of Directed Model to CRF Application Experimental Results C 1 A CRF is an alternative, undirected graphical model 2 CRF-Filters use a continuous CRF for recursive state estimation
  58. 58. Intro Transformation of Directed Model to CRF Application Experimental Results C 1 A CRF is an alternative, undirected graphical model 2 CRF-Filters use a continuous CRF for recursive state estimation 3 . . . can be trained to maximize filter performance depending on the task
  59. 59. Intro Transformation of Directed Model to CRF Application Experimental Results C 1 A CRF is an alternative, undirected graphical model 2 CRF-Filters use a continuous CRF for recursive state estimation 3 . . . can be trained to maximize filter performance depending on the task 4 . . . can deal with correlated measurements
  60. 60. Intro Transformation of Directed Model to CRF Application Experimental Results C 1 A CRF is an alternative, undirected graphical model 2 CRF-Filters use a continuous CRF for recursive state estimation 3 . . . can be trained to maximize filter performance depending on the task 4 . . . can deal with correlated measurements 5 . . . do not explicitly account for dependencies between sensor data

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