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- 1. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 1 Direct Linear Transformation & Computer Vision Models Chapter 7-A4
- 2. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 2 Photogrammetry Vs. Computer Vision • Conventional Photogrammetry is focusing on precise geometric information extraction from imagery. – Topographic mapping from space borne and airborne imagery – Metrological information extraction through close-range photogrammetry (terrestrial photogrammetry) • Object-to-camera distance is less than 100meter • Computer Vision (CV) is mainly concerned with automated image understanding: – Object recognition, – Navigation and obstacle avoidance, and – Object modeling
- 3. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 3 Airborne Photogrammetric Mapping
- 4. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 4 Airborne Photogrammetric Mapping
- 5. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 5 Close-Range Photogrammetric Mapping
- 6. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 6 CV: Object Recognition
- 7. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 7 CV: Navigation & Obstacle Avoidance
- 8. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 8 Photogrammetry Vs. Computer Vision • Photogrammetry is always concerned with precise geometric information extraction. – Photogrammetric mapping considers potential deviations from the assumed perspective projection. • For Computer Vision (CV): – Focus is always on automation. – Object recognition and navigation applications do not require precise derivation of geometric information. – Depending on the application, object modeling might require precise geometric information extraction. – CV usually assumes that the collinearity of the object point, perspective center, and corresponding image point is maintained, even for un-calibrated cameras.
- 9. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 9 Object-to-Image Coordinate Transformation in Photogrammetry Collinearity Equations
- 10. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 10 o a A oa = λ oA These vectors should be defined w.r.t. the same coordinate system. Collinearity Equations
- 11. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 11 Oi xc yc zc A XA YA ZA a + (xa, ya) R( , , ) ω φ κ XG YG ZG OG pp + c (Perspective Center) XO ZO YO Collinearity Equations
- 12. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 12 Collinearity Equations The vector connecting the perspective center to the image point w.r.t. the image coordinate system o a − − − − − = − − − = = c dist y y dist x x c y x dist y dist x r v y p a x p a p p y a x a c oa i 0
- 13. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 13 Collinearity Equations The vector connecting the perspective center to the object point w.r.t. the ground coordinate system o A − − − = − = = o A o A o A o o o A A A m oA o Z Z Y Y X X Z Y X Z Y X r V
- 14. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 14 Where: λ is a scale factor (+ve). Collinearity Equations 11 12 13 21 22 23 31 32 33 ( , , ) c c m i oa O m oA a p x A o a p y A o A o v r M V R r x x dist m m m X X y y dist m m m Y Y c m m m Z Z λ ω ϕ κ λ λ = = = − − − − − = − − − oA oa λ =
- 15. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 15 Collinearity Equations c m R M = y o A o A o A o A o A o A p a x o A o A o A o A o A o A p a dist Z Z m Y Y m X X m Z Z m Y Y m X X m c y y dist Z Z m Y Y m X X m Z Z m Y Y m X X m c x x + − + − + − − + − + − − = + − + − + − − + − + − − = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 33 32 31 23 22 21 33 32 31 13 12 11 m c R R = y o A o A o A o A o A o A p a x o A o A o A o A o A o A p a dist Z Z r Y Y r X X r Z Z r Y Y r X X r c y y dist Z Z r Y Y r X X r Z Z r Y Y r X X r c x x + − + − + − − + − + − − = + − + − + − − + − + − − = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 33 23 13 32 22 12 33 23 13 31 21 11
- 16. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 16 Object-to-Image Coordinate Transformation Direct Linear Transformation Computer Vision Model
- 17. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 17 DLT & Computer Vision Models • The DLT and computer vision models encompass: – Collinearity Equations, – Non-orthogonality (α) between the axes of the image/camera coordinate system, and – Two scale factors (Sx, Sy) along the axes of the image coordinate system. • DLT & CV models can directly deal with pixel coordinates. • We will start with modifying the rotation matrix to consider the impact of the non-orthogonality (α). – Primary rotation 𝜔 @ the 𝑋-axis of the ground coord. system – Secondary rotation 𝜑 @ the 𝑌 -axis – Tertiary rotation 𝜅 & (𝜅 + 𝛼) @ the𝑍 -axis
- 18. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 18 Primary Rotation (ω) X Z Y & Xω Yω Zω ω
- 19. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 19 Primary Rotation (ω) = − = ω ω ω ω ω ω ω ω ω ω ω z y x R z y x z y x z y x cos sin 0 sin cos 0 0 0 1
- 20. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 20 Secondary Rotation (φ) φ Yω Zω Xω Xωφ Zωφ & Yωφ
- 21. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 21 Secondary Rotation (φ) = − = φ ω φ ω φ ω φ ω ω ω φ ω φ ω φ ω ω ω ω φ φ φ φ z y x R z y x z y x z y x cos 0 sin 0 1 0 sin 0 cos
- 22. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 22 Tertiary Rotation (κ) Xωφ Zωφ Yωφ Yωφκ Xωφκ & Zωφκ κ κ κ
- 23. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 23 Tertiary Rotation (κ) = − = κ φ ω κ φ ω κ φ ω κ φ ω φ ω φ ω κ φ ω κ φ ω κ φ ω φ ω φ ω φ ω κ κ κ κ z y x R z y x z y x z y x 1 0 0 0 cos sin 0 sin cos
- 24. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 24 Rotation in Space = κ φ ω κ φ ω κ φ ω κ φ ω z y x R R R z y x // to the ground coordinate system // to the image coordinate system
- 25. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 25 Rotation in Space φ ω κ φ ω κ ω κ φ ω κ ω φ ω κ φ ω κ ω κ φ ω κ ω φ κ φ κ φ κ φ ω cos cos sin sin cos cos sin cos sin cos sin sin cos sin sin sin sin cos cos cos sin sin sin cos sin sin cos cos cos : 33 32 31 23 22 21 13 12 11 33 32 31 23 22 21 13 12 11 = + = − = − = − = + = = − = = = = r r r r r r r r r where r r r r r r r r r R R R R
- 26. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 26 Xωφ Zωφ Yωφ Yωφκ Xωφκ & Zωφκ κ κ Consideration of the Non-Orthogonality (α) + + − = κ φ ω κ φ ω κ φ ω φ ω φ ω φ ω α κ κ α κ κ Z Y X Z Y X 1 0 0 0 ) cos( sin 0 ) sin( cos
- 27. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 27 + + − = κ φ ω κ φ ω κ φ ω φ ω φ ω φ ω α κ κ α κ κ Z Y X Z Y X 1 0 0 0 ) cos( sin 0 ) sin( cos − − − = κ φ ω κ φ ω κ φ ω φ ω φ ω φ ω κ α κ κ κ α κ κ Z Y X Z Y X 1 0 0 0 sin cos sin 0 cos sin cos Consideration of the Non-Orthogonality (α) κ α κ α κ α κ α κ κ α κ α κ α κ α κ sin cos sin sin cos cos ) cos( cos sin sin cos cos sin ) sin( − = − = + + = + = + Assuming small non-orthogonality angle (α)
- 28. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 28 Consideration of the Non-Orthogonality (α) − − = − − − 1 0 0 0 1 0 0 1 1 0 0 0 cos sin 0 sin cos 1 0 0 0 sin cos sin 0 cos sin cos α κ κ κ κ κ α κ κ κ α κ κ − = − = κ φ ω κ φ ω κ φ ω κ φ ω κ φ ω κ φ ω κ φ ω α α Z Y X R Z Y X R R R Z Y X 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1
- 29. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 29 Consideration of the Non-Orthogonality (α) = = Z Y X R Z Y X z y x T 1 0 0 0 1 0 0 1 α κ φ ω κ φ ω κ φ ω // to the image coordinate system // to the ground coordinate system − = κ φ ω κ φ ω κ φ ω α Z Y X R Z Y X 1 0 0 0 1 0 0 1 Note: 1 −𝛼 0 0 1 0 0 0 1 = 1 𝛼 0 0 1 0 0 0 1
- 30. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 30 Consideration of the Non-Orthogonality (α) • Collinearity Equations while considering the non- orthogonality (α) between the axes of the image coordinate system. − − − = − − − O O O T p p Z Z Y Y X X R c y y x x 1 0 0 0 1 0 0 1 α λ
- 31. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 31 − − − = − − − O O O T y p x p Z Z Y Y X X R c s y y s x x 1 0 0 0 1 0 0 1 / ) ( / ) ( α λ − − − − = − − − − O O O T y p x p Z Z Y Y X X R c cs y y cs x x 1 0 0 0 1 0 0 1 / 1 ) /( ) ( ) /( ) ( α λ • Divide both sides by (-c). Consideration of the Scale Factors • Collinearity Equations while considering the non- orthogonality (α) between the axes of the image coordinate system & different scale factors.
- 32. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 32 Consideration of the Scale Factors − − − ′ = − − − − O O O T y p x p Z Z Y Y X X R c y y c x x 1 0 0 0 1 0 0 1 1 ) /( ) ( ) /( ) ( α λ − − − ′ = − − − − O O O T p p y x Z Z Y Y X X R y y x x c c 1 0 0 0 1 0 0 1 1 ) ( ) ( 1 0 0 0 / 1 0 0 0 / 1 α λ − − − − − ′ = − − O O O T y x p p Z Z Y Y X X R c c y y x x 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 ) ( ) ( α λ • csx→ cx , csy→ cy & -λ/c → λ`.
- 33. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 33 DLT & Computer Vision Models − − − − − − ′ = − − O O O T y x x p p Z Z Y Y X X R c c c y y x x 1 0 0 0 0 0 1 ) ( ) ( α λ − − − − − − ′ = − − O O O y x x p p Z Z Y Y X X r r r r r r r r r c c c y y x x 33 23 13 32 22 12 31 21 11 1 0 0 0 0 0 1 ) ( ) ( α λ 𝒙 − 𝒙𝒑 𝒚 − 𝒚𝒑 𝟏 = 𝟏 𝟎 −𝒙𝒑 𝟎 𝟏 −𝒚𝒑 𝟎 𝟎 𝟏 𝒙 𝒚 𝟏 𝟏 𝟎 −𝒙𝒑 𝟎 𝟏 −𝒚𝒑 𝟎 𝟎 𝟏 𝟏 = 𝟏 𝟎 𝒙𝒑 𝟎 𝟏 𝒚𝒑 𝟎 𝟎 𝟏 &
- 34. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 34 DLT & Computer Vision Models 𝒙 − 𝒙𝒑 𝒚 − 𝒚𝒑 𝟏 = 𝛌′ −𝒄𝒙 −𝜶𝒄𝒙 𝟎 𝟎 −𝒄𝒚 𝟎 𝟎 𝟎 𝟏 𝑹𝑻 𝑿 − 𝑿𝑶 𝒀 − 𝒀𝑶 𝒁 − 𝒁𝑶 𝒙 𝒚 𝟏 = 𝛌′ 𝟏 𝟎 𝒙𝒑 𝟎 𝟏 𝒚𝒑 𝟎 𝟎 𝟏 −𝒄𝒙 −𝜶𝒄𝒙 𝟎 𝟎 −𝒄𝒚 𝟎 𝟎 𝟎 𝟏 𝑹𝑻 𝑿 − 𝑿𝑶 𝒀 − 𝒀𝑶 𝒁 − 𝒁𝑶 𝒙 𝒚 𝟏 = 𝛌′ −𝒄𝒙 −𝜶𝒄𝒙 𝒙𝒑 𝟎 −𝒄𝒚 𝒚𝒑 𝟎 𝟎 𝟏 𝑹𝑻 𝑿 − 𝑿𝑶 𝒀 − 𝒀𝑶 𝒁 − 𝒁𝑶 𝒙 𝒚 𝟏 = 𝛌′ −𝒄𝒙 −𝜶𝒄𝒙 𝒙𝒑 𝟎 −𝒄𝒚 𝒚𝒑 𝟎 𝟎 𝟏 𝑹𝑻 −𝑹𝑻𝑿𝑶 𝑿 𝒀 𝒁 𝟏 𝑿𝑶 = 𝑿𝑶 𝒀𝑶 𝒁𝑶 𝑻
- 35. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 35 DLT & Computer Vision Models 𝒙 𝒚 𝟏 = 𝛌′ −𝒄𝒙 −𝜶𝒄𝒙 𝒙𝒑 𝟎 −𝒄𝒚 𝒚𝒑 𝟎 𝟎 𝟏 𝑹𝑻 −𝑹𝑻𝑿𝒐 𝑿 𝒀 𝒁 𝟏 𝒙 𝒚 𝟏 = 𝛌 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏 𝑲 = −𝒄𝒙 −𝜶𝒄𝒙 𝒙𝒑 𝟎 −𝒄𝒚 𝒚𝒑 𝟎 𝟎 𝟏 Where:
- 36. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 36 DLT & Computer Vision Models [ ] [ ] 3 3 ' 1 1 0 { } 0 0 1 { } T O x x p y p T O X x Y y K R I X Z c c x K c y Calibration Matrix R I X Exterior Orientation Matrix λ α = − − − = − ≡ − ≡ 𝑬𝒙𝒕𝒆𝒓𝒊𝒐𝒓 𝑶𝒓𝒊𝒆𝒏𝒕𝒂𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙 = 𝑹𝑻 𝟏 𝟎 𝟎 −𝑿𝑶 𝟎 𝟏 𝟎 −𝒀𝑶 𝟎 𝟎 𝟏 −𝒁𝑶
- 37. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 37 DLT & Computer Vision Models • The Direct Linear Transformation (DLT), which has been developed by the photogrammetric community, is an alternative to the collinearity equations that allows for direct transformation between machine/pixel coordinates and corresponding ground coordinates. – 𝑥 = & 𝑦 = • The DLT can be also represented by the following form: – 𝑥 𝑦 1 = 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝑋 𝑌 𝑍 1
- 38. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 38 DLT & Computer Vision Models 1 2 3 5 6 7 9 10 11 ' 0 0 0 1 x x p T y p L L L c c x D L L L c y R L L L α λ − − = = − 4 8 12 ' 0 0 0 1 x x p O T y p O O L c c x X L c y R Y L Z α λ − − =− − DLT: Direct Linear Transformation 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 = 𝛌 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐
- 39. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 39 DLT & CV Models: Pixel Coordinates • The DLT & CV models can also consider the direct transformation from pixel to ground coordinates. 𝒙 𝒚 𝟏 = 𝒖 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ − 𝒗 × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟏 u v
- 40. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 40 DLT & CV Models: Pixel Coordinates 𝒙 𝒚 𝟏 = 𝒖 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ − 𝒗 × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟏 𝒙 𝒚 𝟏 = 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 −𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 𝟎 𝟏 𝒖 𝒗 𝟏 𝒙 𝒚 𝟏 = 𝛌 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 −𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 𝟎 𝟏 𝒖 𝒗 𝟏 = 𝛌 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏
- 41. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 41 DLT & CV Models: Pixel Coordinates 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 −𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 𝟎 𝟏 𝒖 𝒗 𝟏 = 𝛌 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏 𝒖 𝒗 𝟏 = 𝛌 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 −𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 𝟎 𝟏 𝟏 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 − 𝒏𝒄 𝟐 ⁄ × 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 −𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝒏𝒓 𝟐 ⁄ × 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 𝟎 𝟎 𝟏 𝟏 = 𝟏 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝟎 𝒏𝒄 𝟐 ⁄ 𝟎 − 𝟏 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝒏𝒓 𝟐 ⁄ 𝟎 𝟎 𝟏 𝒖 𝒗 𝟏 = 𝛌 𝟏 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝟎 𝒏𝒄 𝟐 ⁄ 𝟎 − 𝟏 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝒏𝒓 𝟐 ⁄ 𝟎 𝟎 𝟏 𝑲𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏
- 42. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 42 DLT & CV Models: Pixel Coordinates • Modified Calibration Matrix: 𝐾 = 𝟏 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝟎 𝒏𝒄 𝟐 ⁄ 𝟎 − 𝟏 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝒏𝒓 𝟐 ⁄ 𝟎 𝟎 𝟏 −𝒄𝒙 −𝜶𝒄𝒙 𝒙𝒑 𝟎 −𝒄𝒚 𝒚𝒑 𝟎 𝟎 𝟏 𝐾 = −𝒄𝒙 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ −𝜶𝒄𝒙 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ 𝒙𝒑 𝒙_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ + 𝒏𝒄 𝟐 ⁄ 0 𝒄 𝒚_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ −𝑦𝒑 𝑦_𝒑𝒊𝒙_𝒔𝒊𝒛𝒆 ⁄ + 𝒏 𝟐 ⁄ 0 0 1 𝒖 𝒗 𝟏 = 𝛌 𝑲 𝑹𝑻 𝑰𝟑 −𝑿𝒐 𝑿 𝒀 𝒁 𝟏
- 43. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 43 DLT & CV Models: Pixel Coordinates • For DLT when working with pixel coordinates, we have the following model. – 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 = 𝛌 𝑲 𝑹𝑻 𝑰𝟑 −𝑿𝒐 • 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 = 𝛌 𝑲 𝑹𝑻 • 𝐿 𝐿 𝐿 = −𝛌 𝑲 𝑹𝑻 𝑿𝒐 𝑌𝒐 𝑍𝒐
- 44. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 44 Modern Photogrammetry & Computer Vision • Modern Photogrammetry and Computer Vision are converging fields. Art and science of tool development for automatic generation of spatial and descriptive information from multi-sensory data and/or systems
- 45. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 45 DLT → IOPs & EOPs Approach # 1
- 46. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 46 DLT → IOP & EOP 1 2 3 5 6 7 9 10 11 0 0 0 1 x x p T y p L L L c c x D L L L c y R L L L α λ − − = = − 4 8 12 0 0 0 1 x x p O T y p O O L c c x X L c y R Y L Z α λ − − =− − 𝐿 𝐿 𝐿 = − 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝑋 𝑌𝒐 𝑍𝒐
- 47. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 47 DLT → IOP & EOP No Sign Ambiguity − = O O O Z Y X L L L L L L L L L L L L 11 10 9 7 6 5 3 2 1 12 8 4 • Given: • Then: − = − 12 8 4 1 11 10 9 7 6 5 3 2 1 L L L L L L L L L L L L Z Y X O O O
- 48. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 48 DLT → IOP & EOP − − − − − − = 1 0 0 0 1 0 0 0 2 p p y x x p y p x x T y x c c c y c x c c D D α α λ = = = 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 ) ( ) ( L L L L L L L L L L L L L L L L L L KK R K R K D D T T T T T λ λ λ 2 2 11 2 10 2 9 3 3 ) ( λ = + + = × L L L D D T } { 2 11 2 10 2 9 Ambiguity Sign L L L + + ± = λ Then:
- 49. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 49 DLT → IOP & EOP p T x L L L L L L D D 2 3 11 2 10 1 9 1 3 ) ( λ = + + = × ) ( ) ( 2 11 2 10 2 9 3 11 2 10 1 9 L L L L L L L L L xp + + + + = No Sign Ambiguity Then: = = = 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 ) ( ) ( L L L L L L L L L L L L L L L L L L KK R K R K D D T T T T T λ λ λ
- 50. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 50 p T y L L L L L L D D 2 7 11 6 10 5 9 2 3 ) ( λ = + + = × DLT → IOP & EOP ) ( ) ( 2 11 2 10 2 9 7 11 6 10 5 9 L L L L L L L L L yp + + + + = No Sign Ambiguity Then: = = = 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 ) ( ) ( L L L L L L L L L L L L L L L L L L KK R K R K D D T T T T T λ λ λ
- 51. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 51 ) ( ) ( 2 2 2 2 7 2 6 2 5 2 2 y p T c y L L L D D + = + + = × λ DLT → IOP & EOP = = = 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 ) ( ) ( L L L L L L L L L L L L L L L L L L KK R K R K D D T T T T T λ λ λ 5 . 0 2 2 11 2 10 2 9 2 7 2 6 2 5 ) ( − + + + + = p y y L L L L L L c No Sign Ambiguity Then:
- 52. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 52 DLT → IOP & EOP ) ( ) ( 2 7 3 6 2 5 1 2 1 p p y x T y x c c L L L L L L D D + = + + = × α λ = = = 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 ) ( ) ( L L L L L L L L L L L L L L L L L L KK R K R K D D T T T T T λ λ λ − + + + + = p p y x y x L L L L L L L L L c c ) ( / 1 2 11 2 10 2 9 7 3 6 2 5 1 α No Sign Ambiguity Then:
- 53. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 53 ) ( ) ( 2 2 2 2 2 2 3 2 2 2 1 1 1 p x x T x c c L L L D D + + = + + = × α λ DLT → IOP & EOP = = = 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 ) ( ) ( L L L L L L L L L L L L L L L L L L KK R K R K D D T T T T T λ λ λ 5 . 0 2 2 2 2 11 2 10 2 9 2 3 2 2 2 1 ) ( − − + + + + = p x x x c L L L L L L c α No Sign Ambiguity Then:
- 54. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 54 DLT → IOP & EOP • Given: • Then: φ λ λ sin 13 9 = = r L Sign Ambiguity − − − = 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 1 0 0 0 r r r r r r r r r y c x c c L L L L L L L L L p y p x x α λ
- 55. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 55 Collinearity Equations • Objective: Resolve the sign ambiguity in λ • Since the scale factor is always +ve • Assuming that the origin (0, 0, 0) is visible in the imagery ve Z Z r Y Y r X X r O O O − − + − + − ) ( ) ( ) ( 33 23 13 ve Z r Y r X r O O O − − − − 33 23 13 − + − + − − + − + − − + − + − = − − − ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 33 23 13 32 22 12 31 21 11 O O O O O O O O O p p Z Z r Y Y r X X r Z Z r Y Y r X X r Z Z r Y Y r X X r S c y y x x
- 56. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 56 DLT → IOP & EOP • By choosing L12 = 1. 12 13 23 33 13 23 33 13 23 33 ( ) 1 ( ) 1 ( ) O O O O O O O O O L r X r Y r Z r X r Y r Z r X r Y r Z λ λ λ =− + + = − − − = − − − λ is Negative 2 11 2 10 2 9 L L L + + − = λ − − − − = O O O T p y p x x Z Y X R y c x c c L L L 1 0 0 0 12 8 4 α λ
- 57. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 57 DLT → IOP & EOP • No sign Ambiguity 2 11 2 10 2 9 9 13 9 sin sin L L L L r L + + − = = = φ φ λ λ
- 58. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 58 DLT → IOP & EOP φ ω λ λ φ ω λ λ cos cos cos sin 33 11 23 10 = = − = = r L r L 11 10 tan L L − = ω No Sign Ambiguity − − − = 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 1 0 0 0 r r r r r r r r r y c x c c L L L L L L L L L p y p x x α λ
- 59. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 59 DLT → IOP & EOP − − − = − − − = − 1 0 0 0 1 0 0 0 1 11 10 9 7 6 5 3 2 1 33 32 31 23 22 21 13 12 11 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 p y p x x p y p x x y c x c c L L L L L L L L L r r r r r r r r r r r r r r r r r r y c x c c L L L L L L L L L α λ α λ • Retrieve κ • Note: There is an ambiguity in 𝜅 determination (±𝜅 cannot be distinguished). φ κ cos cos 11 r =
- 60. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 60 DLT → IOP & EOP Approach # 2: Matrix Factorization
- 61. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 61 DLT → IOP (Factorization # 1) • Conceptual basis: Direct derivation of the calibration matrix • Cholesky Decomposition of DDT→ λK (Calibration Matrix)? Wrong − − − − − − = = = = 1 0 0 0 1 0 0 0 ) ( ) ( 2 11 7 3 10 6 2 9 5 1 11 10 9 7 6 5 3 2 1 2 p p y x x p y p x x T T T T T T y x c c c y c x c c D D L L L L L L L L L L L L L L L L L L KK R K R K D D α α λ λ λ λ
- 62. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 62 1 1 ) ( − − = = N M M M N CHO T T M M T D D N = T K λ K λ T T T K K M M N M M N 2 1 1 1 λ = = = − − − 1 1 )] } ({ [ − − = T DD CHO K λ T K λ K λ DLT → IOP (Factorization # 2)
- 63. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 63 − − − = − − − = − 1 0 0 0 1 0 0 0 1 11 10 9 7 6 5 3 2 1 33 32 31 23 22 21 13 12 11 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 p y p x x p y p x x y c x c c L L L L L L L L L r r r r r r r r r r r r r r r r r r y c x c c L L L L L L L L L α λ α λ • Using the rotation matrix R, one can derive the individual rotation angles ω, φ and κ. DLT → Rotation Angles
- 64. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 64 Analysis
- 65. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 65 Perspective Center • (XO, YO , ZO) is the intersection point of three different planes whose surface normals are (L1, L2, L3), (L5, L6, L7) and (L9, L10, L11), respectively. 𝐿 𝐿 𝐿 = − 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝑋 𝑌 𝑍 𝐿 𝑋 + 𝐿 𝑌 +𝐿 𝑍 = −𝐿 𝐿 𝑋 + 𝐿 𝑌 +𝐿 𝑍 = −𝐿 𝐿 𝑋 + 𝐿 𝑌 +𝐿 𝑍 = −𝐿
- 66. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 66 Perspective Center − − − = 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 1 0 0 0 r r r r r r r r r y c x c c L L L L L L L L L p y p x x α λ • Assuming: – xp ≈ 0.0 and yp ≈ 0.0 – -αcx ≈ 0.0 − − − − − − = 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 r r r r c r c r c r c r c r c L L L L L L L L L y y y x x x λ • The three surfaces are orthogonal to each other. – This would lead to better intersection.
- 67. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 67 − − − = 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 1 0 0 0 r r r r r r r r r y c x c c L L L L L L L L L p y p x x α λ • Assuming: – xp ≠ 0.0 and yp ≠ 0.0 – -αcx ≈ 0.0 + − + − + − + − + − + − = 33 23 13 33 32 23 22 13 12 33 31 23 21 13 11 11 10 9 7 6 5 3 2 1 r r r r y r c r y r c r y r c r x r c r x r c r x r c L L L L L L L L L p y p y p y p x p x p x λ • As xp and yp increase, the surface normals become almost parallel. – This would lead to weak intersection. Perspective Center
- 68. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 68 • The rows of D are not correlated: – They are orthogonal to each other. • L-1 is well defined. − − − = − 1 0 0 0 1 11 10 9 7 6 5 3 2 1 33 32 31 23 22 21 13 12 11 p y p x x y c x c c L L L L L L L L L r r r r r r r r r α λ • Assuming: – xp ≈ 0.0 and yp ≈ 0.0 – -αcx ≈ 0.0 − − − − − − = 33 23 13 32 22 12 31 21 11 11 10 9 7 6 5 3 2 1 r r r r c r c r c r c r c r c L L L L L L L L L y y y x x x λ Rotation Angles
- 69. CE 59700: Digital Photogrammetric Systems Ayman F. Habib 69 − − − = − 1 0 0 0 1 11 10 9 7 6 5 3 2 1 33 32 31 23 22 21 13 12 11 p y p x x y c x c c L L L L L L L L L r r r r r r r r r α λ • Assuming: – xp ≠ 0.0 and yp ≠ 0.0 – -αcx ≈ 0.0 + − + − + − + − + − + − = 33 23 13 33 32 23 22 13 12 33 31 23 21 13 11 11 10 9 7 6 5 3 2 1 r r r r y r c r y r c r y r c r x r c r x r c r x r c L L L L L L L L L p y p y p y p x p x p x λ • The rows of D tend to be highly correlated. • L-1 is not well defined. Rotation Angles