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Gradient Estimation Revitalized Usman R. Alim 1 oller 1 Laurent Condat 2 Torsten M 1 Graphics, Usability, and Visualization (GrUVi) Lab. School of Computing Science Simon Fraser University 2 GREYC Lab. Image Team Caen, France gruvi graphics

  1. Gradient Estimation Revitalized Usman R. Alim 1 oller 1 Laurent Condat 2 Torsten M¨ 1 Graphics, Usability, and Visualization (GrUVi) Lab. School of Computing Science Simon Fraser University 2 GREYC Lab. Image Team Caen, France gruvi graphics + usability + visualization

  2. gruvi graphics + usability + visualization Motivation Good renderings need good gradients Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 2/22

  3. gruvi graphics + usability + visualization Motivation Good renderings need good gradients Finite Differencing Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 2/22

  4. gruvi graphics + usability + visualization Motivation Good renderings need good gradients Orthogonal Projection Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 2/22

  5. gruvi graphics + usability + visualization Related Work: Quantitative Analysis Quantitative Fourier analysis of scalar reconstruction schemes [Unser and Blu ’99] Linear interpolation revitalized [Blu et al. ’04] Extension to derivatives in 1D [Condat and M¨ oller ’09] Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 3/22

  6. gruvi graphics + usability + visualization Overview ( R 3 ) scalar reconstruction f app ( x ) f ( x ) f [ k ] p ϕ ( x ) sampling on L prefilter Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 4/22

  7. gruvi graphics + usability + visualization Overview ( R 3 ) scalar reconstruction f app ( x ) f ( x ) f [ k ] p ϕ ( x ) sampling on L prefilter Why prefilter? 1 Ensures approximation and original function agree at the lattice sites 2 Exploits the full approximation power of ϕ Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 4/22

  8. gruvi graphics + usability + visualization Overview ( R 3 ) scalar reconstruction f app ( x ) f ( x ) f [ k ] p ϕ ( x ) sampling on L prefilter dir. der. filter reconstruction f l 1 app d 1 ϕ ( x − l 1 / 2) f l 2 ( ∇ f ) app ( x ) � app i f l i app l i ϕ ( x − l 2 / 2) d 2 gradient estimation f l 3 app ϕ ( x − l 3 / 2) d 3 l i : Principal lattice directions Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 4/22

  9. gruvi graphics + usability + visualization Overview ( R 3 ) scalar reconstruction f app ( x ) f ( x ) f [ k ] p ϕ ( x ) sampling on L prefilter dir. der. filter reconstruction f l 1 app d 1 ϕ ( x − l 1 / 2) f l 2 ( ∇ f ) app ( x ) � app i f l i app l i ϕ ( x − l 2 / 2) d 2 gradient estimation f l 3 app ϕ ( x − l 3 / 2) d 3 l i : Principal lattice directions More general case considered in the paper Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 4/22

  10. gruvi graphics + usability + visualization Principal Directions 2D Cartesian: 2 Hexagonal: 3 Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 5/22

  11. gruvi graphics + usability + visualization Principal Directions 2D Cartesian: 2 Hexagonal: 3 CC: 3 BCC: 4 Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 5/22

  12. gruvi graphics + usability + visualization Formal Description Approximate derivatives in the principal directions l i Interested in a digital filter that approximates in the shift-invariant space V ( L h , ϕ i ) , i.e. � 1 ∂ l i f ( x ) ≈ f l i h ( f ∗ p ∗ d i )[ k ] ϕ i app ( x ) = h, k ( x ) k ∈ Z s ϕ i ( x ) := ϕ ( x − l i h, k ( x ) := ϕ i ( x 2 ) and ϕ i h − Lk ) The filter d i should be chosen so that � ∂ l f − f l app � L 2 = O ( h n ) where n is the approximation order of ϕ . Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 6/22

  13. gruvi graphics + usability + visualization Formal Description Approximate derivatives in the principal directions l i Interested in a digital filter that approximates in the shift-invariant space V ( L h , ϕ i ) , i.e. � 1 ∂ l i f ( x ) ≈ f l i h ( f ∗ p ∗ d i )[ k ] ϕ i app ( x ) = h, k ( x ) k ∈ Z s ϕ i ( x ) := ϕ ( x − l i h, k ( x ) := ϕ i ( x 2 ) and ϕ i h − Lk ) The filter d i should be chosen so that � ∂ l f − f l app � L 2 = O ( h n ) where n is the approximation order of ϕ . Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 6/22

  14. gruvi graphics + usability + visualization Formal Description Approximate derivatives in the principal directions l i Interested in a digital filter that approximates in the shift-invariant space V ( L h , ϕ i ) , i.e. � 1 ∂ l i f ( x ) ≈ f l i h ( f ∗ p ∗ d i )[ k ] ϕ i app ( x ) = h, k ( x ) k ∈ Z s ϕ i ( x ) := ϕ ( x − l i h, k ( x ) := ϕ i ( x 2 ) and ϕ i h − Lk ) The filter d i should be chosen so that � ∂ l f − f l app � L 2 = O ( h n ) where n is the approximation order of ϕ . Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 6/22

  15. gruvi graphics + usability + visualization Revitalization scalar reconstruction f ( x ) f [ k ] f app ( x ) p ϕ ( x ) sampling on L prefilter dir. der. filter reconstruction f l 1 app ϕ ( x − l 1 / 2) d 1 f l 2 ( ∇ f ) app ( x ) � app app l i i f l i d 2 ϕ ( x − l 2 / 2) gradient estimation f l 3 app d 3 ϕ ( x − l 3 / 2) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 7/22

  16. gruvi graphics + usability + visualization Shift Example (Cubic B-spline) 0.6 0.5 0.4 0.3 0.2 0.1 � 3 � 2 � 1 1 2 3 ϕ ( x ) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 8/22

  17. gruvi graphics + usability + visualization Shift Example (Cubic B-spline) 0.6 0.5 0.4 0.3 0.2 0.1 � 3 � 2 � 1 1 2 3 ϕ ( x − k ) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 8/22

  18. gruvi graphics + usability + visualization Shift Example (Cubic B-spline) 0.6 0.5 0.4 0.3 0.2 0.1 � 3 � 2 � 1 1 2 3 ϕ 1 ( x ) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 8/22

  19. gruvi graphics + usability + visualization Shift Example (Cubic B-spline) 0.6 0.5 0.4 0.3 0.2 0.1 � 3 � 2 � 1 1 2 3 ϕ 1 ( x − k ) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 8/22

  20. gruvi graphics + usability + visualization Why shift? (1D) 1.0 0.7 0.6 0.5 0.5 0.4 � 3 � 2 � 1 1 2 3 0.3 0.2 � 0.5 0.1 � 1.0 � 2 � 1 1 2 β 2 ( x ) - Quadratic B-spline 2 ( x ) = β 1 ( x + 1 2 ) − β 1 ( x − 1 β ′ 2 ) (blue) V ( Z , β 1 ( x )) can’t recover the exact derivative (purple) V ( Z , β 1 ( x − 1 / 2)) can! (blue) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 9/22

  21. gruvi graphics + usability + visualization Why shift? (1D) 1.0 0.7 0.6 0.5 0.5 0.4 � 3 � 2 � 1 1 2 3 0.3 0.2 � 0.5 0.1 � 1.0 � 2 � 1 1 2 β 2 ( x ) - Quadratic B-spline 2 ( x ) = β 1 ( x + 1 2 ) − β 1 ( x − 1 β ′ 2 ) (blue) V ( Z , β 1 ( x )) can’t recover the exact derivative (purple) V ( Z , β 1 ( x − 1 / 2)) can! (blue) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 9/22

  22. gruvi graphics + usability + visualization How to shift in higher dimensions? Directional derivative of a 4th order hexagonal box spline is a linear combination of two lower order shifted box splines Shifts are in the direction of the derivative 1 Choose s linearly independent principal directions 2 Shift the symmetric box spline along those principal directions Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 10/22

  23. gruvi graphics + usability + visualization How to shift in higher dimensions? Directional derivative of a 4th order hexagonal box spline is a linear combination of two lower order shifted box splines Shifts are in the direction of the derivative 1 Choose s linearly independent principal directions 2 Shift the symmetric box spline along those principal directions Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 10/22

  24. gruvi graphics + usability + visualization Error Analysis How to predict the directional derivative error, given an approximation space V ( L , ψ ) and a filter r Error Kernel � � 2 � � � � � � 2 � � ψ ( ω ) � � R ( ω ) � − � ⋆ � + � ˚ E l ( ω ) := 1 − A ψ ( ω ) � ψ ( ω ) � � � j l T ω � A ψ ( ω ) � �� � � �� � E min ( ω ) E l res ( ω ) � l : A principal lattice direction R ↔ r : Filter applied to samples � A ψ : Autocorrelation sequence Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 11/22

  25. gruvi graphics + usability + visualization Filter Design scalar reconstruction f app ( x ) f ( x ) f [ k ] p ϕ ( x ) sampling on L prefilter dir. der. filter reconstruction f l 1 app d 1 ϕ ( x − l 1 / 2) f l 2 ( ∇ f ) app ( x ) � app i f l i app l i ϕ ( x − l 2 / 2) d 2 gradient estimation f l 3 app ϕ ( x − l 3 / 2) d 3 Combined directional derivative filter r i = ( p ∗ d i ) Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 12/22

  26. gruvi graphics + usability + visualization Asymptotic Optimality � � 2 � � � � � � 2 � � ψ ( ω ) � � R ( ω ) � − � ⋆ � ˚ + � E l ( ω ) := 1 − A ψ ( ω ) ψ ( ω ) � � � j l T ω � � A ψ ( ω ) � �� � � �� � E min ( ω ) E l res ( ω ) For a minimum error approximation: E l res ( ω ) = 0 , not realizable! Choose r so that E min ( ω ) ∼ E l res ( ω ) (as h → 0 ) Plug-in our basis function ϕ i and combined filter r i = p ∗ d i Optimality Criterion 2 l i T ω ) + O ( | ω | n +1 ) d i ↔ � D i = j l i T ω exp( j No dependence on ϕ Alim et al. , GrUVi, GREYC Gradient Estimation Revitalized 13/22

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