Invariance of l and the Conic Dual to Circular Points C

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1 Invariance of l and the Conic Dual to Circular Points C [ ] A t l = (0, 0, 1) is preserved under H = v iff H is an affinity: w [ ] l H l H A l l v 0 [ t 0 v! = = w w] 0 0 v = C = diag(1, 1, 0) is preserved under H iff H is a similarity the converse works as well [ ] [ ] [ ] [ ] A t I 0 C HC H A v AA Av v w 0 0 t = w v A v v = 0, A = sr v Q: Are there any points x invariant to similarity H S? x = λh S x, H S = A: Yes: I = H S I, J = H S J where I = (1, i, 0), J = (1, i, 0) they are called circular points the converse works as well» s R t 0 1 Every circle intersects l at I, J: circle points: x x2 2 x2 3 r2 = 0, l points: x 3 = 0 then IJ + JI = diag(1, 1, 0) = C indeed, C is invariant under H S: H S I = I, H S J = J H S (IJ + JI )H S = (IJ + JI )H S = (IJ + JI ) indeed, C l = 0 since l I J: (IJ + JI )(I J) = 0 3D Computer Vision, R. Šára, CMP (p. 120/157) rev. November 23, 2010

2 Summary and Interpretation of our Findings C consists of complex points I, J (hence it is dual to them), l = I J C l = 0 C 4 DOF (pointwise) (3 3, symmetric, homogeneous, rank 2) l, C are invariant entities attached to each plane in P 3 l does not see the difference between affinities C does not see the difference between similarities C transforms with non-similarity only, it sees non-similarities cos θ is a gauge for measuring angles in plane ρ via its image in image plane π and (15) If C is in canonical form C = diag(1, 1, 0), we can measure angles directly using the coordinates of the entities, otherwise we have to use (15) and C is all that is needed to know about ρ to be able to measure angles A homography that brings C to its canonical form brings us to metric frame this is a basis for autocalibration 3D Computer Vision, R. Šára, CMP (p. 121/157) rev. November 23, 2010

3 Canonical, Metric, and Non-Metric Entities canonical camera P = [I, 0] similarity PH s rotates, translates, scales the metric frame = metric camera we can measure proper angles, length ratios non-similarity PH brings the camera to non-metric frame we cannot measure angles directly homography KP is a change of basis in image plane undoing the effect of H: bringing the camera to metric frame undoing the effect of K: camera autocalibration besides cameras, we will have some other entities whose canonical representations are canonical form name invariant to l = (0, 0, 1) line at infinity affinity in P 2 π = (0, 0, 0, 1) plane at infinity affinity in P 3 C = diag(1, 1, 0) conic dual to circular points similarity in P2 Q = diag(1, 1, 1, 0) absolute dual quadric similarity in P3 Ω = diag(1, 1, 1) absolute conic similarity in P 3 ω = diag(1, 1, 1) image of the absolute conic (IAC) those invariant under similarities: canonical = metric non-similarities bring them to non-metric frame 3D Computer Vision, R. Šára, CMP (p. 122/157) rev. November 23, 2010

4 Quadrics and their Image Projections point quadric x Q x = 0, x P 3 tangent plane quadric ρ Q ρ = 0 camera P projects the outline of quadric Q to dual conic by C = PQ P. É its rim Ò Lines n tangent to C satisfy n C n = 0 corresponding optical planes are ρ = P n Then ρ Q ρ = n PQ P n = n C n = 0 and this is true for all tangent lines we say C is the dual of the image of the rim quadric Q 3D Computer Vision, R. Šára, CMP (p. 123/157) rev. November 23, 2010

5 Measuring Angles between Planes in P 3 by Q the canonical (metric) form of absolute dual quadric Q is Q = diag(1, 1, 1, 0) Q invariant under collineation H iff H is a similarity the plane at infinity π is the null vector of Q the angle between planes π 1 = (p 1, d 1 ), π 2 = (p 2, d 2 ) is given by p i normals cos Θ = π 1 Q π 2 (π )( ) 1 Q π 1 π 2 Q π 2 a homography that brings Q to its canonical form brings us to metric frame projection of canonical Q to general camera = DIAC (dual of IAC): PQ P = [ Q q ] [ ] Q diag(1, 1, 1, 0) = QQ = KK = ω q projection of general Q to general camera looks the same in the same projective frame P = PH 1, Q = HQ H, P Q P = PQ P 3D Computer Vision, R. Šára, CMP (p. 124/157) rev. November 23, 2010

6 Autocalibration from DIAC ω suitable when we cannot observe π directly Idea of calibration: find H to bring Q to canonical form: see also Slide 95 P i Q P i This brings us to a metric frame up to similarity. = P i H 1 HQ }{{ H H } P i canonical 1. Q projects to image as [H&Z, Sec. 19.3] P i Q P i = λ i ω i, λ i 0 (16) eq. (16) maps a known constraint on ω i on a constraint on Q via the known P i. we know that eg. some entries in ω i = K i K i vanish because eg. of zero skew in K i 2. Q computed from (16) can be decomposed to as on Slide 119 Q = H I 3 H, where I 3 = diag(1, 1, 1, 0) this provides a common update homography H: P i P i H 1, X HX which gives structure and motion `X, {P i } up to similarity H s since H s Q H s Q alternatively: K i is obtained by Choleski decomposition from ω i P i Q P i all cameras must be in a common projective frame, using, eg. the camera system reconstruction process for uncalibrated cameras 3D Computer Vision, R. Šára, CMP (p. 125/157) rev. November 23, 2010

7 Example: The Case for ZSUA + Known Principal Point let K i = f i f i 0, then ω i = K i K i = f i fi 2 0, i = 1,... k λ i ω i = P i Q P i def = M i = [ m i jk ] λ 1 i Q are unknown 8 + k constraints needed symmetric 4 4 homogeneous mtx of rank 2 Under our choice of ω we have m i 12 = 0, m i 13 = 0, m i 23 = 0, m i 11 m i 22 = 0. m i kl are functions of unknown parameters Q k = 1 camera: 4 linear constraints on Q are stacked to k = 2 cameras: 8 constraints an additional constraint needed: D q = 0, q R 10, q = 1 we can then use a method similar to the 7-point algorithm 1. det Q = 0 quartic constraint 2. when known that f 1 = f 2 m2 11 m 1 11 = m2 22 m 1 22 quadratic constraint k 3 cameras: enough linear constraints we find the closest singular matrix to Q use autocalibration from DIAC with caution; needs good data 3D Computer Vision, R. Šára, CMP (p. 126/157) rev. November 23, 2010

8 Stereovision mostly covered by [1] Šára, R. How To Teach Stereoscopic Vision. Proc. ELMAR 2010 referenced as [SP] Stereovision = dense correspondence problem dense 3D model

9 What Do You See? 3D Computer Vision, R. Šára, CMP (p. 127/157) rev. November 23, 2010

10 What Do You See? (Why?) 3D Computer Vision, R. Šára, CMP (p. 128/157) rev. November 23, 2010

11 What Do You See? Centrum för teknikstudier at Malmö Högskola, Sweden 3D Computer Vision, R. Šára, CMP (p. 129/157) rev. November 23, 2010

12 How Difficult Is Stereo? when we do not recognize the scene and cannot use high-level constraints the problem seems difficult (right, less so in center) most stereo matching algorithms do not require scene understanding prior to matching the success of a model-free stereo matching algorithm is unlikely: WTA Matching: for every pixel in the left image find the most similar pixel in the right image left image disparity map disparity map from WTA 3D Computer Vision, R. Šára, CMP (p. 130/157) rev. November 23, 2010

13 No Continuity Model Structural Ambiguity in Stereo left image right image A 2 B 2 C 3 C 3 A 1 B 1 A B C A B C interpretation 1 interpretation 2 3D Computer Vision, R. Šára, CMP (p. 131/157) rev. November 23, 2010

14 But What Kind of Continuity Model Applies Here? 3D Computer Vision, R. Šára, CMP (p. 132/157) rev. November 23, 2010

15 Repetition: How Many Scenes Correspond to a Stereopair? Consider the fence and the fortress worlds...? 3D Computer Vision, R. Šára, CMP (p. 133/157) rev. November 23, 2010

16 occluded Ö ½ surface pt. Understanding the Basic Occlusion Types Ö Ö ¾ ¾ ½ transparent Ð half occlusion mutual occlusion surface point at the intersection of rays l and r 1 occludes a world point at the intersection (l, r 3 ) and implies the world point (l, r 2 ) is transparent (l, r 3 ) and (l, r 2 ) are ruled-out by (l, r 1 ) in half-occlusion, every world point such as X 1 or X 2 is ruled out by a binocularly visible surface point in mutual occlusion this is no longer the case half occluded mutually occluded 3D Computer Vision, R. Šára, CMP (p. 134/157) rev. November 23, 2010

17 The End

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20 half occluded mutually occluded