Application: Visualization with X-Slits Cameras 29/47 Synthesized sequence as if taken by a (non-central) camera inside the circle
Application: Visualization with X-Slits Cameras 30/47 Synthesized sequence as if taken by a (non-central) camera inside the circle
Main Results 31/47 Oblique camera
Hierarchy of cameras 32/47 central camera all other cameras ? all rays intersect at C − → some rays intersect ← − no rays intersect l X X c C l2 l1 • • •
Oblique camera 33/47 X Definition An oblique camera is a collection of lines such that every point in space is contained in exactly one line.
Oblique camera 33/47 X Definition An oblique camera is a collection of lines such that every point in space is contained in exactly one line. Observation Rays of an oblique camera do not intersect.
Oblique camera 33/47 X Definition An oblique camera is a collection of lines such that every point in space is contained in exactly one line. Observation Rays of an oblique camera do not intersect. ? Do oblique cameras exist
Oblique cameras exist 34/47 A set of lines generated by the linnear mapping σ line [ X σ ( X )] point X − y x x y y x − → span span z z w − z w w picture by Rolf Riesinger Lines are reguli of pairwise non-intersecting rotational hyperboloids s s X T X = 0 , s ∈ [0 , 1] s − 1 s − 1 Remark: OC are called spreads & wild spreads (not cospreads) exist! picture by Hans Havlicek
Stereo geometry of oblique cameras 35/47 l A set of lines generated by the mappings: point C 1 = red lines C 2 = blue lines − y − y x x x k y y x y x − → − w z z w z − z w w w z form two reguli of pairwise non-intersecting rotational hyperboloids + two lines l , k Remark: lines l ( s = 1) and k ( s = 0) are in both cameras C 1 and C 2 .
Linear OC ≡ generated by collineations over P 3 36/47 Oblique cameras can be generated by a linear mapping σ : P 3 → P 3 as { line [ X σ ( X )] | ∀ X ∈ P 3 } for some mappings Result: linear mappings that generate OC Q ∈ R 4 × 4 σ : Y = Q X , (X) σ change of coordinates: σ R 4 × 4 , ∃ S ∈ rank S = 4 , ∃ α ∈ R, α � = 0 (Y) σ α 1 X − 1 α Y σ S − 1 QS = α 1 − 1 α
Linear OC ≡ generated by collineations over P 3 36/47 Oblique cameras can be generated by a linear mapping σ : P 3 → P 3 as { line [ X σ ( X )] | ∀ X ∈ P 3 } for some mappings Result: linear mappings that generate OC Q ∈ R 4 × 4 σ : Y = Q X , (X) σ change of coordinates: σ R 4 × 4 , ∃ S ∈ rank S = 4 , ∃ α ∈ R, α � = 0 (Y) σ α 1 X − 1 α Y σ S − 1 QS = α 1 − 1 α Observation: Central, Pushbroom, X-Slits, and some Oblique cameras are linearly generated
Realization of oblique cameras 37/47 l s l3 r l1 T o l4 C l2 1. A subset of rays of an oblique camera can be realized 2. by rotating a catadioptric camera (telecentric optics + conical mirror) 3. used by Nayar and Karmarkar CVPR’2000 for mosaicing.
Realization of oblique cameras 38/47 l l s l3 e r l1 a b k c T o C d l4 l2 l1 l C l2 By storing two curves of pixels from each image, rays passing through the volume swept by the mirror are generated.
Summary 39/47 Non-central cameras and their stereo geometries were introduced. In particular, I have: 1. Provided a review of existing cameras and image acquisition techniques showing that non-central cameras are practically important and that there is a need to understand their stereo geometries. 2. Generalized the stereo geometry of central cameras to stereo geometries of non-central cameras consisting of quadratic stereo correspondence surfaces and shown that the generalization supports an efficient scene reconstruction. 3. Demonstrated that the stereo geometry with quadric stereo correspondence surfaces explains stereo geometry of symmetric concentric panoramas. 4. Found oblique cameras, the example of the most non-central cameras, and their stereo geometry.
Publications 40/47 [Paj99] T. Pajdla. Non-classical ray cameras. Research Report CTU-CMP-1999-11, November 1999. [HPa00] F. Huang and T. Pajdla. Epipolar geometry in concentric panoramas. Research Report CTU–CMP–2000–07, March 2000. [Paj01a] T. Pajdla. Characterization of epipolar geometries of non-classical cameras. Research Report CTU–CMP–2001–05, February 2001. [Paj01b] T. Pajdla. Epipolar geometry of some non-classical cameras. In B Likar, editor, Computer Vision Winter Workshop, pages 223–233, Ljubljana, Slovenia, February 2001. [Paj01c] T. Pajdla. Oblique cameras generated by collineations. Research Report CTU–CMP–2001–14, April 2001. [Paj01d] T. Pajdla. Rotational hyperboloids as a class of oblique cameras with double ruled quadric visibility closures. Research Report CTU–CMP–2001–10, March 2001. [Paj01e] T. Pajdla. Stereo with oblique cameras. In Bradski G.R and T.E Boult, editors, IEEE Workshop on Stereo and Multi-Baseline Vision, pages 85–91. IEEE Computer Society Press, December 2001. [Paj02a] T. Pajdla. Geometry of two-slit camera. Research Report CTU–CMP–2002–02, March 2002. [Paj02b] T. Pajdla. Stereo with oblique cameras. International Journal of Computer Vision, 47(1-3):161–170, May 2002.
The END 41/47 Anaglyph of a stereo panorama taken by an X-Slits (generalized pushbroom) camera. (D. Feldman, D. Weinshall, T. Pajdla) Stereo correspondence curves are horizontal lines.
42/47
Examples of stereo geometries 43/47 Examples of stereo geometries of central cameras, mosaics, and panoramas, were reviewed and it has been shown that some of them posses a generalization of epipolar geometry, e.g. the following concentric symmetric stereo panorama and the corresponding stereo Stereo correspondence lines in correspondence surfaces. a concentric symmetric stereo panorama (Courtesy of J. ˇ Sivic) while others, e.g. concentric non-symmetric stereo panoramas, do not.
Precision of reconstruction 44/47 A higher precision of reconstruction can be achieved with mosaics than with perspective images in many situations. For instance, 2.5 2 1.5 1 log 10 (r) 0.5 0 −0.5 The logarithm of the radius r = 1 /δ of the uncertainty circle for a stereo pair of −1 central cameras (blue) as well as for a 360 × −1.5 360 stereo mosaic (red) as a function of the 0 0.5 1 1.5 2 2.5 3 y distance from the respective camera. shows that 360 × 360 stereo mosaics have smaller reconstruction error, compared to a pair of standard cameras, in the plane of rotation of the mosaicing camera.
Generalization of epipolar planes 45/47 Definition 1. Let there hold for two sets U 1 , U 2 of lines in P 3 : C1. There are at least three distinct lines in both sets. C2. Every line from one set intersects all lines in the other set. Theorem 1. Lines in U 1 , U 2 are only in one of the following configurations a b c d e f g h i j k l
6. All stereo geometries of concentric stereo panoramas 46/47 Π C C C C C C O O O O O O (a-1) (b-1) (c-1) (d-1) (e-1) (f-1) (a-2) (b-2) (c-2) (d-2) (e-2) (f-2) (a-3) (b-3) (c-3) (d-3) (e-3) (f-3) All stereo geometries of concentric stereo panoramas. (*–1) Central camera orientation w.r.t. the circle of motion, (*–2) the resulting stereo correspondence surfaces, and (*–3) the corresponding stereo correspondence lines in images.
Pinhole
X C
Recommend
More recommend
