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A Computational Model of View Degeneracy and Its Application to Active Focal Length Control Please do not remove this page A Computational Model of View Degeneracy and its Application to Active Focal Length Control Wilkes, David; Dickinson, Sven J.; Tsotsos, John K. https://scholarship.libraries.rutgers.edu/discovery/delivery/01RUT_INST:ResearchRepository/12643458520004646?l#13643535820004646 Wilkes, D., Dickinson, S. J., & Tsotsos, J. K. (1997). A Computational Model of View Degeneracy and its Application to Active Focal Length Control. Rutgers University. https://doi.org/10.7282/T3R214W9 This work is protected by copyright. You are free to use this resource, with proper attribution, for research and educational purposes. Other uses, such as reproduction or publication, may require the permission of the copyright holder. Downloaded On 2021/10/04 10:05:38 -0400 A Computational Mo del of View Degeneracy and its Application to Active Focal Length Control David Wilkes Department of Computer Science University of Toronto Sven J Dickinson Rutgers University Center for Cognitive Science RuCCS and Department of Computer Science Rutgers University John K Tsotsos Department of Computer Science University of Toronto Abstract We quantify the observation by Kender and Freudenstein that degenerate views o ccupy a signicant fraction of the viewing sphere surrounding an ob ject This demonstrates that systems for recognition must explicitly account for the p ossibility of view degeneracy We show that view degeneracy cannot b e detected from a single camera viewp oint As a result systems designed to recognize ob jects from a single arbitrary viewp oint must b e able to function in spite of p ossible undetected degeneracies or else op erate with imaging parameters that cause acceptably low probabilities of degeneracy To address this need we give a prescription for active control of fo cal length that allows a principled tradeo b etween the camera eld of view and probability of view degeneracy Introduction Image segmentation is the task of transforming a signallevel image description into a symbolic description consisting of sets of segments or features of some sort Examples of features in common use are homogeneous regions and also discontinuities describ ed by line segments or curves Such features may b e dicult to b oth identify and interpret due to the accidental alignment of spatially distinct scene parts b elonging to a single ob ject or multiple ob jects as seen from the camera viewp oint This alignment is often referred to as view degeneracy This sup erp osition of features can lead to erroneous part counts or bad parameterizations of a mo del For example if two ob ject edges are abutted and collinear one longer edge is seen in their place Given arbitrarily high resolution such alignments would not b e a problem b ecause they would o ccur only for a vanishingly small fraction of the p ossible viewp oints Unfortunately cameras and featureextracting op erators have nite resolution We shall see in this pap er that this enlarges the set of viewp oints that are problematic We b egin by dening view degeneracy precisely and showing how our denition covers many interesting classes of degeneracy We develop a mo del of degeneracy under p ersp ective pro jection It is parameterized by the distance to the ob ject to b e recognized the separation of features on the ob ject the minimum separation of the corresp onding image features in order for them to b e detectible as distinct features and camera fo cal length The mo del gives the probability of b eing in a degenerate view as a function of the mo del parameters The evaluation of the mo del at realistic parameterizations indicates that degenerate views can have signicant probabiliti es We will show that there is a signicant additional problem for recognition systems op erating from a single arbitrary viewp oint Namely view degeneracy is not detectible from a single viewp oint As a result singleviewp oint systems must b e able to p erform unambiguous ob ject identication in spite of p ossible degeneracies The alternative is to minimize the probability of degeneracy We will show that the probability of degeneracy is very sensitive to camera fo cal length This will lead nally to a prescription for fo cal length control that allows a tradeo to b e achieved b etween the width of the eld of view and probability of view degeneracy Denitions of degeneracy Kender and Freudenstein presented an analysis of degenerate views that p ointed out prob lems with existing denitions of the concept The denitions that they prop ose in place of earlier problematic ones incorp orate the notion that what constitutes a degenerate view is dep endent on what heuristics the system uses to invert the pro jection from three dimensions to two One of the denitions incorp orating this systemdep endence is the negation of the denition of a general view point A general viewp oint is such that there is some p ositive epsilon for which a camera movement of epsilon in any direction can b e taken without eect on the resulting semantic analysis of the image This denition was noted by Kender and Freudenstein to b e incomplete in the sense that there exist general viewp oints on certain ob jects that are unlikely to allow a system to instantiate a prop er mo del for the ob ject relative to other views of the same ob ject eg the view of the base of a pyramid They argue for somehow quantifying degeneracy according to the amount of additional work exp ected to b e required to unambiguously instantiate a mo del corresp onding to the ob ject b eing viewed We are interested in a denition that is indep endent of the particular recognition system in use and are willing to trade away the completeness of the denition to achieve this Figure illustrates a denition of degeneracy that covers a large subset of the cases app earing in the literature The denition has to do with collinearity of the front no dal p oint of the lens and a pair of p oints on or dened by the ob ject We shall consider a view to b e degenerate if at least one of the following two conditions holds a zerodimensional p ointlike ob ject feature is collinear with another zerodimensional ob ject feature and the front no dal p oint of the lens a zerodimensional ob ject feature is collinear with object point feature object point feature front nodal point object point features object point feature front nodal point Figure Our denition of view degeneracy The two cases are the collinearity of the front no dal p oint of the lens with two ob ject p oint features top or with one ob ject p oint feature and a p oint on one line b ottom or line segment some p oint on either a an innite line or b a nite line segment dened by two zerodimensional ob ject features and the front no dal p oint Each of the sp ecic imaging degeneracies enumerated by Kender and Freudenstein for p olyhedral ob jects b elongs to one of these types Below we discuss each of their example degeneracies in turn Vertices imaged in the plane of scene edges This is encompassed by case b ab ove and is depicted in gure The lower right scene edge on the upp er ob ject is the line segment of case b The vertex on the lower ob ject is the zerodimensional ob ject feature Parallel scene lines imaged in their own plane We depict this situation in gure This is case a with an innite line The p ortion of the viewing sphere in which the degeneracy o ccurs is that for which the front no dal p oint of the lens is in the plane dened by an end p oint of the edge on the front ob ject and the other scene line extended to have innite length Coincident scene lines imaged in their own plane This generalizes to coplanar lines imaged in their own plane We depict this situation in gure This is equivalent to case b using an innite line An endp oint of the edge on the righthand ob ject may b e used as the zerodimensional feature The edge on the lefthand ob ject may b e extended to dene the innite line Figure An example of view degeneracy in which a vertex is imaged in the plane of a scene edge Figure An example of view degeneracy in which parallel scene lines are imaged in their own plane Figure An example of view degeneracy in which coplanar scene lines are imaged in their own plane tetrahedron Figure An example of view degeneracy in which p erfect symmetry gives preference to a at D interpretation of the lines Perfect symmetry Kender and Freudenstein include p erfect symmetry in the enumeration of types of degeneracy b ecause they say that it leads to a tendency to interpret the symmetric structure as at An example is shown in gure In the case of a radial symmetry this is equivalent to case with the two zerodimensional p oints chosen to b e any two distinct p oints on the axis of symmetry In the case of symmetry ab out a plane this is equivalent to case a with b oth features chosen arbitrarily to lie in the plane of symmetry Note that in the case of symmetry the features dening the degeneracy are more likely to b e abstract p oints on the ob ject eg the centroid of an ob ject face rather than p oints extractible directly by lowlevel vision The use of abstract ob ject p oints rather than obvious feature p oints has no impact on the estimation of probabilities of degeneracy that follows What matters is that there is some means to enumerate the types of degeneracy that are imp ortant to the approach This enumeration is discussed later in the pap er Dep endence on eective resolution If a computer vision system had innite ability to resolve ob ject features that were close to one another then all of the degeneracies discussed
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