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1 Introduction 2 the Quadric Reference Surface

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1 Introduction 2 the Quadric Reference Surface Intro duction The general problem of achieving corresp ondence or optical ow as it is known in the motion literature is to recover the D displacement eld b etween p oints across two images Typical applications for which full corresp ondence that is corresp ondence for all image p oints is initially required include the measurement of motion stereopsis structure from motion D reconstruction from p oint corresp ondences and recently visual recognition active vision and computer graphics animation In this pap er we fo cus on two problemsone theoretical and the other more practical On the practical side we address the problem of establishing the full p ointwise displacement eld b etween two views greylevel images of a general D ob ject We achieve this by rst considering a theoretical problem of establishing a quadric surface reference frame on the ob ject In other words given any two views of some unknown textured opaque quadric 3 surface in D pro jective space P is there a nite numb er of corresp onding p oints across the two views that uniquely determine all other corresp ondences coming from p oints on the quadric A constructive answer to this question readily suggests that we can asso ciate a virtual quadric surface with any D ob ject not necessarily itself a quadric and use it for describing shap e but more imp ortantly for achieving full corresp ondence b etween the two views On the conceptual level we prop ose combining geometric constraints captured from knowledge of a small numb er of corresp onding p oints manually given for example and pho tometric constraints captured by the instantaneous spatiotemp oral changes in image light intensities conventional optical ow The geometric constraints we prop ose are related to the virtual quadric surface mentioned ab ove These constraints lead to a transformation a nominal quadratic transformation that is applied to one of the views with the result of bringing b oth views closer together The remaining displacements residuals are recovered by using the spatial and temp oral derivatives of image light intensityeither by correlation of image patches or by optical ow techniques The Quadric Reference Surface 3 We consider ob ject space to b e the D pro jective space P and image space to b e the D 2 pro jective space P b oth over the eld C of complex numb ers Views are denoted by i 0 indexed by i The epip oles are denoted by v and v and we assume their lo cation 1 2 denotes equality up to a scale is known for metho ds see for example The symb ol GL stands for the group of n n matrices P GL is the group dened up to a scale and n n S P GL is the symmetric sp ecialization of P GL n n 2 3 Result Given two arbitrary views P of a quadric surface Q P C with 1 2 0 3 0 centers of pro jection at O O P and O O Q then ve corresp onding p oints across the two views uniquely determine all other corresp ondences 0 0 0 Proof Let x x x and x x x b e co ordinates of and resp ectively and z 0 1 2 1 2 0 0 1 2 z b e co ordinates of Q Let O then the quadric surface may b e given as 3 the lo cus z z z z and as the pro jection from O onto the plane 0 3 1 2 1 z In case where the centers of pro jection are on Q the line through O meets Q in 3 exactly one other p oint and thus the mapping Q is generically onetoone and so 1 2 has a rational inverse x x x x x x x x x x Because all quadric surfaces of 0 1 2 0 1 0 2 1 2 0 the same rank are pro jectively equivalent we can p erform a similar blowup from with 2 02 0 0 0 0 0 0 the result x x x x x x x The pro jective transformation A P GL b etween the two 4 0 0 1 0 2 1 2 representations of Q can then b e recovered from ve corresp onding p oints b etween the two images The result do es not hold when the centers of pro jection are not on the quadric surface 2 This is b ecause the mapping b etween Q and P is not onetoone a ray through the center of pro jection meets Q in two p oints and therefore a rational inverse do es not exist We are interested in establishing a more general result that applies when the centers of pro jection are not on the quadric surface One way to enforce a onetoone mapping is by making opacity assumptions dened b elow Denition Opacity Constraint Given an ob ject Q fP P g we assume there 1 n exists a plane through the camera center O that do es not intersect any of the chords P P i j ie Q is observed from only one side of the camera Secondly we assume that the surface is opaque which means that among all the surface p oints along a ray from O the closest p oint to O is the p oint that also pro jects to the second view The rst constraint 2 therefore is a camera opacity assumption and the second constraint is a surface opacity assumption together we call them the opacity constraint 3 With an appropriate reparameterization of P we can obtain the following result 2 3 Theorem Given two arbitrary views P of an opaque quadric surface Q P 1 2 then nine corresponding points across the two views uniquely determine al l other correspon dences The following auxiliary prop ositions are used as part of the pro of 0 0 0 0 be four p p p Lemma Relative Ane Parameterization Let p p p p and p o 1 2 3 3 2 1 o corresponding points coming from four noncoplanar points in space Let A be a col lineation 0 0 2 0 v Final ly let v be scaled of P determined by the equations Ap p j and Av j j 0 0 3 0 such that p Ap v Then for any point P P projecting onto p and p we have o o 0 0 p Ap k v The coecient k k p is independent of ie is invariant to the choice of the second 2 view and the coordinates of P are x y k The lemma its pro of and its theoretical and practical implications are discussed in detail in The scalar k is called a relative ane invariant and can b e computed with the aid of a second arbitrary view For future reference let stand for the plane passing through 2 P P P in space 1 2 3 Pro of of Theorem From Lemma any p oint P can b e represented by the co ordinates x y k and k can b e computed from Equation Since Q is a quadric surface then there > exists H S P GL such that P HP for all p oints P of the quadric Because H 4 is symmetric and determined up to a scale it contains only nine indep endent parameters Therefore given nine corresp onding image p oints we can solve for H as a solution of a 0 linear system each corresp onding pair p p provides one linear equation in H of the form > x y k H x y k Given that we have solved for H the mapping due to the quadric Q can 1 2 0 b e determined uniquely ie for every p we can nd the corresp onding p as 1 2 > follows The equation P HP gives rise to a second order equation in k of the form 2 ak bpk cp where the co ecient a is constant dep ends only on H and the co ecients b c dep end also on the lo cation of p Therefore we have two solutions for k br 0 1 2 where and by Equation two solutions for p The two solutions for k are k k 2a p 2 r b ac The nding shown in the next auxiliary lemma is that if the surface Q is opaque then the sign of r is xed for all p Therefore the sign of r for p that leads to 1 o a p ositive ro ot recall that k determines the sign of r for all other p o 1 p 2 Lemma Given the opacity constraint the sign of the term r b ac is xed for al l points p 1 Proof Let P b e a p oint on the quadric pro jecting onto p in the rst image and let the 1 2 1 2 OP intersect the quadric at p oints P P and let k k b e the ro ots of the quadratic ray 2 equation ak bpk cp The opacity assumption is that the intersection closer to O 0 is the p oint pro jecting onto p and p 0 Recall that P is a p oint on the quadric in this case used for setting the scale of v in o Equation ie k Therefore all p oints that are on the same side of as P have o o p ositive k asso ciated with them and vice versa There are two cases to b e considered either P is b etween O and ie O P or is b etween O and P ie O P that o o o o 1 2 is O and P are on opp osite sides of In the rst case if k k then the nonnegative ro ot o 1 2 is closer to O ie k maxk k If b oth ro ots are negative the one closer to zero is closer 1 2 to O again k maxk k Finally if b oth ro ots are p ositive then the larger ro ot is closer 1 2 to O Similarly in the second case we have k mink k for all combinations Because P can satisfy either of these two cases the opacity assumption gives rise to a consistency o requirement in picking the right ro ot either the maximum ro ot should b e uniformly chosen for all p oints or the minimum ro ot In the next section we will show that Theorem can b e used to surround an arbitrary D surface by a virtual quadric ie to create quadric reference surfaces which in turn can b e used to facilitate the corresp ondence problem b etween two views of a general ob ject The remainder of this section takes Theorem further to quantify certain useful relationships b etween the centers of two cameras and the family of quadrics that pass through arbitrary congurations of eight p oints whose pro jections on the two views are known 3 2 Theorem Given a quadric surface Q P projected onto views P with centers 1 2 0 3 of projection O O P there exists a parameterization of the image planes that 1 2 yields a representation H S P GL of Q such that h when O Q and the sum of the 4 44 0 elements of H vanishes when O Q Proof The reparameterization describ ed here was originally intro duced in as part of 3 the pro of of Lemma We rst assign
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