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How Useful Is Projective Geometry? Patrick Gros, Richard Hartley, Roger Mohr, Long Quan

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How Useful Is Projective Geometry? Patrick Gros, Richard Hartley, Roger Mohr, Long Quan How Useful is Projective Geometry? Patrick Gros, Richard Hartley, Roger Mohr, Long Quan To cite this version: Patrick Gros, Richard Hartley, Roger Mohr, Long Quan. How Useful is Projective Geometry?. Com- puter Vision and Image Understanding, Elsevier, 1997, 65 (3), pp.442–446. 10.1006/cviu.1996.0496. inria-00548360 HAL Id: inria-00548360 https://hal.inria.fr/inria-00548360 Submitted on 22 Dec 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. How useful is pro jective geometry 1 1 1 2 P Gros R Mohr L Quan R Hartley 1 LIFIA INRIA RhoneAlp es avenue F Viallet Grenoble Cedex France 2 GE CRD Schenectady NY USA In this resp onse we will weigh two dierent approaches to shap e recognition On the one hand we have the use of restricted camera mo dels as advo cated in the pap er of Pizlo et al to give a closer approximation to real calibrated cameras The alternative approach is to use a full pro jective camera mo del and take advantage of the machinery of pro jective geometry Ab out the p ersp ective pro jection Before discussing the usefulness of pro jective geometry we revisit p ersp ective pro jection A pin hole camera can b e mo deled as a linear mapping in homogeneous co ordinates from the D space onto a plane Usually this mapping is represented by the pro duct of a rigid motion in space with matrix D followed by a standard p ersp ective pro jection expressed as a matrix P and nally a rescaling in the image due to the camera parameters represented by a 0 matrix K So the nal pro jection is represented by a matrix P P KP D 0 s u u 0 R t C B B C v A A v 0 Such a transformation has dof degrees of freedom for K plus dof for D for the rotation R and for the translation t This makes a total of How many degrees of freedom has a plane to plane pro jection Let us rst consider the numb er of degrees of freedom of a pro jection b etween two planes This question is also discussed in the discussion pap er of Pizlo et al Without loss of generality we can assume that the p oints are selected from the x y plane The camera mapping induces a planetoplane pro jective mapping and therefore can at most have degrees of freedom so all the dof are not indep endent The pro jection matrix b etween two planes can b e related to the physical parameters of the imaging system and the co ordinate system of the ob ject plane as follows 0 A KT SD 33 cos sin t cos cos sin s u x u 0 B C B C B C B C sin cos t cos sin cos v A A A A y v 0 C sin 0 In this relation K is as b efore the matrix of intrinsic parameters accounting for dof D is a Euclidean displacement in the ob ject plane dof S is a scaling matrix dof and T represents the orientation of the ob ject plane with resp ect to the camera dof Note that S can b e absorb ed either by T or by D When it is absorb ed by T it can b e interpreted as the third parameter of the ob ject plane Otherwise it can b e interpreted as a global scaling factor in the ob ject plane Calibrated camera If the camera is calibrated the calibration matrix K is known and so there remain only dof Are they indep endent There exists an elegant pro of in this calibrated case Consider the image of a circle in space It is well known that a circle can b e pro jected as a general conic see Ap ollonios de Perga BC or more recent work on determining the orientation of a plane containing a circle This makes already ve dof since a conic is sp ecied by ve parameters Now if a p oint is distinguished on this conic by rotating the circle on itself we can bring the pro jection of this p oint anywhere on the conic This additional degree of freedom is the missing one which now leads us to a total numb er of degrees of freedom A more systematic algebraic pro of can b e provided as follows Consider three p oints in a plane and after rotating and translating this conguration in space through D pro ject them onto the image The co ordinates of these three p oints dene a vector of dimension It only remains to check if the manifold spanned by the parameters of the rigid motion has dimension Computing the determinant of the Jacobian shows easily that except for singular p oints the Jacobian has full rank and therefore the manifold is of dimension This computation can b e veried using Maple or Mathematica Therefore a p ersp ective pro jection of a plane onto the image plane of a calibrated camera has degrees of freedom Uncalibrated camera If now the camera is no longer assumed to b e calibrated obviously the numb er of dof cannot b e higher than So if we consider that two of the intrinsic parameters might change for instance the fo cal length and the asp ect ratio or the p osition of the camera principal p oint then using the manifold technique one can easily check that the numb er of dof is in each of these cases It might b e noticed however that real parameters are really limited in range this reduces the p ossible parameter space to a small region of the whole space but it should noticed that this region is still of dimension This p oint will b e discussed later on The p ersp ective pro jection from the full D space It is also interesting to compare the numb er of dof with that of the general pro jection of D non planar p oints The general mo del has dof as the general pro jective mapping from 3 2 the pro jective space P on the pro jective plane P However it may b e noted that for standard cameras the parameter s in K has value thus only dof remain So for the uncalibrated case the numb er of dof for a p ersp ective pro jection is lower than that of the full pro jective pro jection Now considering the case of a calibrated camera the numb er of dof comes to as in the case for a plane to plane pro jection notably fewer than the dof of the full pro jective pro jection What can b e extracted from these numb ers First notice that in the general case of uncalibrated cameras the pro jective mo del has a numb er of dof equal or close to the numb er of dof of the p ersp ective pro jection Consequently the pro jective mo del is a very go o d approximation to the calibrated camera mo del This fact added to simplicity and to the p ossibiliti es describ ed in the next section means that the pro jective mo del is a very useful mo del for analysis of the imaging pro cess However it is true that the manifold of shap es obtained via pro jective pro jection is larger than the one obtained by p ersp ective pro jection This is mainly due to the fact that camera parameters are limited in range for instance the ratio ranges usually b etween to u v s is close to Similarly the p ositions of shap e in space are also constrained for instance they do not cross the viewer eye and extreme congurations are not probable Such considerations have led to the intro duction of quasiinvariants These quasiinvariants may just b e considered as approximate values of invariants In practice they app ear to b e very useful for qualitative tasks like identication On the other hand if accuracy is needed then an exact mo del is necessary and exact invariants ie the one provided by the pro jective group are to b e used Comments on the approach of Pizlo et al In their pap er Pizlo et al work with calibrated cameras thus K is known The shape de ned in is implicitly constituted of similarity invariants mo dulo a scaled planar Euclidean transformation The concept of shap e is clear in but is quite confusing in the discussion pap er When similarity invariants are concerned SD is absorb ed and only T remains The determination of T is equivalent to that of the vanishing line of the ob ject plane If and were known we could compute the exact similarity invariants of the shap e But for general recognition purp ose we do not have a priori knowledge on the plane in which the ob ject lies so T remains unknown At this stage the computed invariants cannot b e indep endent of the unknown parameters of T With some assumptions on and an approximation of the similarity invariants might b e exp ected This is discussed in the previous pap er of Pizlo et al In this discussion pap er FCDP is dened by Pizlo et al to b e comp osed of TS which has dof together with the constraint that the ob ject is in front of the camera This constraint has b een previously exploited by and also by p eople working on computer graphics E FCDP is dened to b e comp osed of T S and D and not of the calibration matrix K Thus E FCDP has dof and is nothing but the transformation b etween the calibrated camera and the ob ject plane It can b e easily checked that T is a twoparameter transformation which fails to dene a trans formation group since the pro duct of two such transformations is no longer of the desired form Therefore neither FCDP nor E FCDP can b e a group as clearly indicated by the authors It means that neither FCDP nor E FCDP can b e a geometry in the mo dern view of the geometries Even worse it means that no FCDP or E FCDP invariant can b e dened other
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